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0.4939.11
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\section{Proof of Theorem~\ref{THM1}: Wright-Fisher model}\label{sec2}
Recall the description in the introduction of the Wright-Fisher model with neutral mutation in a haploid population of constant size~$N$.
The process is driven by offspring vector having distribution~$\mathrm{MN}(N;1/N, \ldots, 1/N)$, and the
mutation structure is general with~$K$ types. The process is a time-homogeneous Markov chain~$\tX(0), \tX(1), \ldots$, where~$\tX(n)$ is a~$(K-1)$ dimensional vector that represents the counts of the first~$K-1$ alleles in the population, so~$\tX(n)/N \in \-\Delta_K$.
Since~$(\tX(n))_{n\geq0}$ is a Markov chain on a finite state space, it has a stationary distribution, and we apply Theorem~\ref{THM3} to prove the
bound on the approximation of this stationary distribution by the Dirichlet distribution given by Theorem~\ref{THM1}.
To define a stationary pair~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$, let~$\tX$ be distributed as a stationary distribution of the chain of Theorem~\ref{THM1}
and let~$\tX'$ be a step in the chain from~$\tX$.
Set~$\textnormal{\textbf{W}}=\tX/N$ and~$\textnormal{\textbf{W}}'=\tX'/N$.
It is not difficult to see that
the distribution of~$\tX'$ given~$\tX$ is the first~$K-1$ coordinates of a multinomial with~$N$ trials with success probabilities given by the vector
$\tq(\tX)$, where
\ben{\label{42}
q_j(\tX)
=\sum_{k=1}^Kp_{ k j}\frac{X_ k}{N} = \sum_{k=1}^{K-1} p_{ k j}\frac{X_ k}{N} + p_{Kj}\bbbklr{1-\sum_{k=1}^{K-1}\frac{X_k}{N}}.
}
Hence
\be{
\mathrm{I}E[W_j'|\textnormal{\textbf{W}}]
= p_{jj}W_j + \sum_{\substack{k=1\\k\neq j}}^{K-1}p_{ k j} W_k + p_{Kj} - \sum_{k=1}^{K-1}p_{Kj} W_k,
}
so that
\bes{
\mathrm{I}E[W_j'-W_j|\textnormal{\textbf{W}}]
& = -(1-p_{jj})W_j + \sum_{\substack{k=1\\k\neq j}}^{K-1}p_{ k j} W_k + p_{Kj} - \sum_{k=1}^{K-1}p_{Kj} W_k \\
& =\frac{1}{2N}(a_j -sW_j) + R_j(\textnormal{\textbf{W}}),
}
where
\besn{\label{43}
R_j(\textnormal{\textbf{W}})
& = -\frac{a_j}{2N}+ \bbklr{\frac{s}{2N}-(1-p_{jj})}W_j+\sum_{\substack{k=1\\k\neq j}}^{K-1}p_{ k j} W_k + p_{Kj} - \sum_{k=1}^{K-1}p_{Kj} W_k \\
& = \bbklr{p_{Kj}-\frac{a_j}{2N}}(1-W_j)+\bbbklr{\,\sum_{\substack{k=1\\k\neq j}}^K\bbklr{\frac{a_k}{2N}-p_{jk}}}W_j+\sum_{\substack{k=1\\k\neq j}}^{K-1}(p_{ k j}-p_{Kj}) W_k.
}
Thus we are in the setting of Theorem~\ref{THM3} with~$\ta$ as above,~$\Lambda=(2N)^{-1}\times \mathrm{I}d$, and~$\tR$ given by~\epsqref{43}.
Applying the theorem is a relatively straightforward but somewhat tedious calculation involving conditioning and multinomial moment formulas. We need the quantities
\bg{
T_j=p_{K j}+\sum_{ \substack{k=1\\k \neq j}}^{K-1} (p_{ k j}- p_{K j}) W_{ k },\\
\sigma_j= p_{Kj} + \sum_{\substack{k=1\\k\neq j}}^K p_{jk},\qquad \tau_j = p_{Kj} + \sum_{\substack{k=1\\k\neq j}}^K\abs{p_{kj}-p_{Kj}}, \qquad 1\leq j\leq K-1.
}
Note that we can write
\ben{\label{44}
q_j:=q_j(\tX)=W_j(1-\sigma_j)+T_j.
}
We also record the following multinomial moment formula lemma. Let~$(n)_{ k \deltaownarrow}=n(n-1)\cdots(n- k +1)$ denote the falling factorial.
\begin{lemma}\label{lem10}
For~$(\tX, \tX')$ defined above,~$i,j,k\in\{1,\ldots, K-1\}$ all distinct and non-negative integers~$ k _i, k _j, k _k$,
\be{
\mathrm{I}E\left[ \left(X_i'\right)_{ k _i\deltaownarrow}\left(X_j'\right)_{ k _j\deltaownarrow}\left(X_k'\right)_{ k _k\deltaownarrow} \big| \tX \right]
=\left( N \right)_{( k _i+ k _j+ k _k) \deltaownarrow} q_i(\tX)^{ k _i} q_j(\tX)^{ k _j} q_k(\tX)^{ k _k}.
}
\epsnd{lemma}
\begin{lemma}\label{lem11}
For~$(\textnormal{\textbf{W}},\textnormal{\textbf{W}}')$ defined above,
\bes{
\mathrm{I}E\left[(W_j'-W_j)^2\big| \textnormal{\textbf{W}} \right]&=W_j^2\left[-\frac{1}{N}+\frac{2\sigma_j}{N} +\sigma_j^2\left(1-\frac{1}{N}\right)\right] \\
&\qquad + W_j\left[\frac{1}{N} - \frac{2T_j}{N}-\frac{\sigma_j}{N}-2T_j\sigma_j\left(1-\frac{1}{N}\right)\right]
+T_j\left[T_j+\frac{1-T_j}{N}\right].
}
\epsnd{lemma}
\begin{proof}
We first expand
\be{
\mathrm{I}E\left[(W_j'-W_j)^2\big| \textnormal{\textbf{W}} \right]=\frac{1}{N^2}\mathrm{I}E\left[ X_j'\left(X_j'-1\right) \big| \textnormal{\textbf{W}}\right]
-\left(2W_j-\frac{1}{N}\right) \mathrm{I}E[W_j'|\textnormal{\textbf{W}}] + W_j^2.
}
Using Lemma~\ref{lem10} and the expression for~$\tq$ given at~\epsqref{44} we find
\be{
\frac{1}{N^2}\mathrm{I}E\left[ X_j'\left(X_j'-1\right) \big| \textnormal{\textbf{W}}\right]=\frac{N-1}{N} \left(W_j \left(1-\sigma_j\right)+T_j\right)^2,
}
and
\be{
\mathrm{I}E[W_j'|\textnormal{\textbf{W}}]=W_j \left(1-\sigma_j\right)+T_j.
}
Combining these last three displays and simplifying yields the result.
\epsnd{proof}
\begin{lemma}\label{lem12}
For~$(\textnormal{\textbf{W}},\textnormal{\textbf{W}}')$ defined above, and~$i\not=j$,
\bes{
\mathrm{I}E\left[(W_i'-W_i)(W_j'-W_j) \big| \textnormal{\textbf{W}} \right]&=W_iW_j\left[-\frac{1}{N}+\frac{\sigma_i+\sigma_j}{N}+\sigma_i\sigma_j\left(1-\frac{1}{N}\right)\right] +T_iT_j\left(1-\frac{1}{N}\right) \\
&\qquad
-W_iT_j\left (\sigma_i+\frac{1-\sigma_i}{N}\right) -W_jT_i\left (\sigma_j+\frac{1-\sigma_j}{N}\right).
}
\epsnd{lemma}
\begin{proof}
We first expand
\be{
\mathrm{I}E\left[(W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}\right]=\mathrm{I}E [W_i'W_j' | \textnormal{\textbf{W}}] -W_i\mathrm{I}E[W_j'|\textnormal{\textbf{W}}]-W_j\mathrm{I}E[W_i'|\textnormal{\textbf{W}}]+W_i W_j.
}
Using Lemma~\ref{lem10} and the expression for~$\tq$ given at~\epsqref{44} we find
\be{
\mathrm{I}E [W_i'W_j' | \textnormal{\textbf{W}}]=\frac{N-1}{N} \left(W_i \left(1-\sigma_i\right)+T_i\right) \left(W_j \left(1-\sigma_j\right)+T_j\right),
}
and
\be{
W_i\mathrm{I}E[W_j'|\textnormal{\textbf{W}}]=W_i\left(W_j \left(1-\sigma_j\right)+T_j\right).
}
Combining these last three displays and simplifying yields the result.
\epsnd{proof}
\begin{lemma}\label{lem13}
For~$(\textnormal{\textbf{W}},\textnormal{\textbf{W}}')$ defined above, and~$\lambda=(2N)^{-1}~$,
\ba{
\sum_{i,j=1}^{K-1} \mathrm{I}E &\left| W_i(\deltaelta_{i j}-W_j)-\frac{1}{2 \lambda}\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}] \right|\leq N\sum_{i,j=1}^{K-1}(\sigma_i+\tau_i) \left(\sigma_j+\tau_j+\frac{2}{N}\right).
}
\epsnd{lemma}
\begin{proof}
The lemma follows in a straightforward way from Lemmas~\ref{lem11} and~\ref{lem12}, the triangle inequality, that~$0\leq W_i\leq 1$,
and~$\abs{T_j}\leq \tau_j$.
\epsnd{proof}
\begin{lemma}\label{lem14}
For~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$ defined above and~$\lambda=(2N)^{-1}~$,
\ba{
\frac{1}{\lambda}\sum_{i,j,k=1}^{K-1}&\mathrm{I}E \abs{(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)}\\
&\leq \frac{2}{N^{1/2}} \left(\sum_{i=1}^{K-1} \left[\sqrt2+\sqrt N (\tau_i + \sigma_i) \right]\right)^{2} \left(\sum_{i=1}^{K-1} \left[1+\sqrt N (\tau_i + \sigma_i)\right]\right).
}
\epsnd{lemma}
\begin{proof}
Conditional on~$\tX$,~$\tX'$ is distributed as the first~$(K-1)$ entries of a multinomial distribution with~$N$ trials
and success probabilities given by the vector at~\epsqref{44}:
\be{
q_ k :=q_ k (\tX)=\left(W_ k \left(1-\sigma_ k \right)+T_ k \right).
}
Decompose
\ba{
X_i' - X_i&=X_i' -\mathrm{I}E[X_i'| X_i] + \mathrm{I}E[X_i'| X_i] - X_i \\
&= [X_i' - (X_i(1-\sigma_i) + NT_i)] + [NT_i - \sigma_iX_i]\\
&=: E_i + G_i.
}
Using H\"older's inequality followed by Minkowski's inequality, we find
\ban{
\sum_{i,j,k}^{K-1} &\mathrm{I}E \abs{(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)} = \frac{1}{N^3}\sum_{i,j,k}^{K-1}\mathrm{I}E \abs{(E_i + G_i)(E_j + G_j)(E_k + G_k)} \notag \\
&\leq \frac{1}{N^3}\sum_{i,j,k}^{K-1}\left[\mathrm{I}E (E_i+G_i)^4 \mathrm{I}E (E_j+G_j)^4 \right]^{1/4}\left[\mathrm{I}E (E_k+G_k)^2\right]^{1/2} \notag \\
&\leq \frac{1}{N^3}\left(\sum_{i=1}^{K-1}\left[(\mathrm{I}E E_i^4)^{1/4}+(\mathrm{I}E G_i^4)^{1/4} \right]\right)^2\sum_{k=1}^{K-1}\left[(\mathrm{I}E E_k^2)^{1/2}+(\mathrm{I}E G_i^2)^{1/2}\right]. \label{45}
}
Now noting that for~$Y\sim\mathop{\mathrm{Bi}}n(n,p)$,
\bes{
\mathrm{I}E(Y-np)^4 &= 3(np(1-p))^2 + np(1-p)(1-6p(1-p)) \leq 3(np(1-p))^2 + np(1-p),
}
which, along with the variance formula for the binomial distribution, yields
\ba{
\mathrm{I}E E_i^4 &= 3(X_iq_i(1-q_i))^2 + X_iq_i(1-q_i) \leq 4N^2, \\
\mathrm{I}E E_i^2 &= X_i q_i (1-q_i) \leq N.
}
Plugging these bounds along with~$\abs{G_i} = \abs{NT_i - \sigma_i X_i} \leq N\tau_i + N\sigma_i$ into~\epsqref{45} yields the result.
\epsnd{proof}
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0.4939.12
|
\begin{lemma}\label{lem13}
For~$(\textnormal{\textbf{W}},\textnormal{\textbf{W}}')$ defined above, and~$\lambda=(2N)^{-1}~$,
\ba{
\sum_{i,j=1}^{K-1} \mathrm{I}E &\left| W_i(\deltaelta_{i j}-W_j)-\frac{1}{2 \lambda}\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}] \right|\leq N\sum_{i,j=1}^{K-1}(\sigma_i+\tau_i) \left(\sigma_j+\tau_j+\frac{2}{N}\right).
}
\epsnd{lemma}
\begin{proof}
The lemma follows in a straightforward way from Lemmas~\ref{lem11} and~\ref{lem12}, the triangle inequality, that~$0\leq W_i\leq 1$,
and~$\abs{T_j}\leq \tau_j$.
\epsnd{proof}
\begin{lemma}\label{lem14}
For~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$ defined above and~$\lambda=(2N)^{-1}~$,
\ba{
\frac{1}{\lambda}\sum_{i,j,k=1}^{K-1}&\mathrm{I}E \abs{(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)}\\
&\leq \frac{2}{N^{1/2}} \left(\sum_{i=1}^{K-1} \left[\sqrt2+\sqrt N (\tau_i + \sigma_i) \right]\right)^{2} \left(\sum_{i=1}^{K-1} \left[1+\sqrt N (\tau_i + \sigma_i)\right]\right).
}
\epsnd{lemma}
\begin{proof}
Conditional on~$\tX$,~$\tX'$ is distributed as the first~$(K-1)$ entries of a multinomial distribution with~$N$ trials
and success probabilities given by the vector at~\epsqref{44}:
\be{
q_ k :=q_ k (\tX)=\left(W_ k \left(1-\sigma_ k \right)+T_ k \right).
}
Decompose
\ba{
X_i' - X_i&=X_i' -\mathrm{I}E[X_i'| X_i] + \mathrm{I}E[X_i'| X_i] - X_i \\
&= [X_i' - (X_i(1-\sigma_i) + NT_i)] + [NT_i - \sigma_iX_i]\\
&=: E_i + G_i.
}
Using H\"older's inequality followed by Minkowski's inequality, we find
\ban{
\sum_{i,j,k}^{K-1} &\mathrm{I}E \abs{(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)} = \frac{1}{N^3}\sum_{i,j,k}^{K-1}\mathrm{I}E \abs{(E_i + G_i)(E_j + G_j)(E_k + G_k)} \notag \\
&\leq \frac{1}{N^3}\sum_{i,j,k}^{K-1}\left[\mathrm{I}E (E_i+G_i)^4 \mathrm{I}E (E_j+G_j)^4 \right]^{1/4}\left[\mathrm{I}E (E_k+G_k)^2\right]^{1/2} \notag \\
&\leq \frac{1}{N^3}\left(\sum_{i=1}^{K-1}\left[(\mathrm{I}E E_i^4)^{1/4}+(\mathrm{I}E G_i^4)^{1/4} \right]\right)^2\sum_{k=1}^{K-1}\left[(\mathrm{I}E E_k^2)^{1/2}+(\mathrm{I}E G_i^2)^{1/2}\right]. \label{45}
}
Now noting that for~$Y\sim\mathop{\mathrm{Bi}}n(n,p)$,
\bes{
\mathrm{I}E(Y-np)^4 &= 3(np(1-p))^2 + np(1-p)(1-6p(1-p)) \leq 3(np(1-p))^2 + np(1-p),
}
which, along with the variance formula for the binomial distribution, yields
\ba{
\mathrm{I}E E_i^4 &= 3(X_iq_i(1-q_i))^2 + X_iq_i(1-q_i) \leq 4N^2, \\
\mathrm{I}E E_i^2 &= X_i q_i (1-q_i) \leq N.
}
Plugging these bounds along with~$\abs{G_i} = \abs{NT_i - \sigma_i X_i} \leq N\tau_i + N\sigma_i$ into~\epsqref{45} yields the result.
\epsnd{proof}
\begin{proof}[Proof of Theorem~\ref{THM1}] We apply Theorem~\ref{THM3} with~$\Lambda=(2N)^{-1}\times \mathrm{I}d$. Using the bounds in Lemmas~\ref{lem13} for~$A_2$ and~\ref{lem14} for~$A_1$ along with a straightforward bound on~$\abs{R_j}$ ($R_j$ given at~\epsqref{43}) for~$A_1$, we obtain
\ba{
A_1 &\leq 2N\sum_{j=1}^{K-1} \left[ | p_{K j} - \frac{a_j}{2N}| +\sum_{\substack{k=1\\k\not=j}}^{K}\abs{p_{j k}-\frac{a_k}{2N}}+\sum_{\substack{k=1\\k\not=j}}^{K-1} \abs{p_{k j}- p_{Kj}}\right],\\
A_2&\leq N\sum_{i,j=1}^{K-1}(\sigma_i+\tau_i) \bbklr{\sigma_j+\tau_j+\frac{2}{N}}, \\
A_3&\leq \frac{2}{N^{1/2}}\left(\,\sum_{i=1}^{K-1} \bklr{\sqrt{2}+\sqrt{N}(\sigma_i+ \tau_i)}\right)^{2}\left(\,\sum_{i=1}^{K-1} \bklr{1+\sqrt{N}(\sigma_i+ \tau_i)}\right),
}
The final bound in Theorem~\ref{THM1} is now obtained through straightforward manipulations and applying some standard analytic inequalities, in particular
that~$\abs{x+y}^p\leq 2^{p-1}(\abs{x}^p+\abs{y}^p)$ for~$p\geq1$, and that
\be{
\sum_{i=1}^{K-1}(\sigma_i+\tau_i)\leq 2\sum_{i=1}^{K-1}p_{K i}+
\sum_{i=1}^{K-1}\sum_{\substack{ j\neq i}}^{K}p_{i j}+\sum_{i=1}^{K-1}\sum_{\substack{ j\neq i}}^{K-1}p_{ij}
+(K-2)\sum_{i=1}^{K-1} p_{K i}
\leq K \sum_{i=1}^K\sum_{\substack{j=1\\j\neq i}}^K p_{ij}
=K\mu.
}
\epsnd{proof}
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0.4939.13
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\section{Proof of Theorem~\ref{THM2}: Cannings model}\label{sec3}
Recall the description in the introduction of the Cannings exchangeable
model with neutral PIM mutation in a haploid population of constant size~$N$.
The process is driven by a generic exchangeable offspring vector
$\tV$ with mutation structure such that~$p_{ij} = \pi_j$ for~$1\leq i \neq j\leq K$ and~$p_{ii}=1-\sum_{j\not=i} \pi_j$. To distinguish from the~$\pi_i$, we write~$p_i:=1-p_{ii}$ for the chance that an individual with parent of type~$i$ is not of type~$i$. As in the previous section, we apply Theorem~\ref{THM3}, and to define a stationary pair~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$, let~$\tX$ be distributed as a stationary distribution of the chain
and let~$\tX'$ be a step in the chain from~$\tX$.
Set~$\textnormal{\textbf{W}}=\tX/N$ and~$\textnormal{\textbf{W}}'=\tX'/N$.
We first compute~$\mathrm{I}E[X_i'-X_i|\tX]$.
The first thing to note is that we can decompose the number of individuals of
type~$i$ in the~$\tX'$-generation into those that have parent of type~$i$ and those that do not. In particular, if we denote by~$M_i$ the number of offspring in~$\tV$ that originate from a parent of type~$i$ in the~$\tX$-generation and write~$\textnormal{\textbf{M}}=(M_1,\ldots, M_K)$, then
\ben{
\mathscr{L}(X_i' | \textnormal{\textbf{M}})=\mathscr{L}(Y_1(\textnormal{\textbf{M}})+Y_2(\textnormal{\textbf{M}})), \label{46}
}
where~$Y_1(\textnormal{\textbf{M}})\sim\mathop{\mathrm{Bi}}n(N-M_i, \pi_i)$,~$Y_2(\textnormal{\textbf{M}})\sim \mathop{\mathrm{Bi}}n(M_i,1- p_i)$,
and these two variables are independent given~$\textnormal{\textbf{M}}$. From here we easily have
\be{
\mathrm{I}E(X_i'|\tX,\textnormal{\textbf{M}}) = \pi_i(N-M_i) + (1-p_i) M_i.
}
Now noting the exchangeability of~$\tV$ implies~$\mathrm{I}E V_j = 1$, and hence~$\mathrm{I}E (M_i|\tX) = X_i$, take the expectation with respect to~$\textnormal{\textbf{M}}$ to find
\bes{
\mathrm{I}E(X_i'|\tX) &= \pi_i(N-X_i) + (1-p_i)X_i\\
&= \pi_i N + (1-\sigma)X_i,
}
where~$\sigma = \sum_{i=1}^K \pi_i$. If we now set~$\textnormal{\textbf{W}} = \tX/N$ and~$\textnormal{\textbf{W}}' = \tX' / N$ we find that
\be{
\mathrm{I}E[\textnormal{\textbf{W}}' - \textnormal{\textbf{W}} | \textnormal{\textbf{W}}] = \boldsymbol{\pi}-\sigma\textnormal{\textbf{W}}.
}
Recalling our definition of~$\alpha$ from~\epsqref{3} and letting
\be{
\ta=\frac{2(N-1)}{\alpha} \boldsymbol{\pi},
}
we are in the setting of Theorem~\ref{THM3} with
$\Lambda=\frac{\alpha}{2(N-1)}\times \mathrm{I}d$ and~$\tR=0$.
As in Section~\ref{sec2}, applying the theorem is a relatively straightforward but
tedious calculation involving conditioning and computing various moment formulas.
For the latter we record the following lemma.
\begin{lemma}\label{47}
If\/~$\tV$ is a Cannings exchangeable offspring vector, if
$\alpha$,~$\beta$, and~$\gamma$ are the moments defined at~\epsqref{3}, and
$\deltaelta:=\mathrm{I}E \klg{V_1(V_1-1)(V_1-2)(V_1-3)}$,
then
\ban{
\mathrm{I}E V_1^2 &= 1+\alpha, \label{48}\\
\mathrm{I}E V_1V_2 &=1 - \alpha\frac{1}{N-1},\label{49}\\
\mathrm{I}E V_1^3 &= 1+3\alpha+\beta, \label{50}\\
\mathrm{I}E V_1 V_2 V_3&=1-\alpha\frac{3}{N-1}+\beta\frac{2}{(N-1)(N-2)}, \label{51}\\
\mathrm{I}E V_1^2 V_2&= 1+\alpha \frac{N-3}{N-1}-\beta\frac{1}{N-1},\label{52} \\
\mathrm{I}E V_1^2 V_2^2 &= 1+ \alpha \frac{2N-5}{N-1} -\beta \frac{2}{N-1} + \gamma,\label{53}\\
\mathrm{I}E V_1^4&= 1+7\alpha + 6 \beta + \deltaelta \label{54}, \\
\begin{split}\label{55}
\mathrm{I}E V_1V_2V_3V_4&=1-\alpha\frac{6}{N-1}+\beta\frac{8}{(N-1)(N-2)} \\
&\qquad+\gamma\frac{3}{(N-2)(N-3)}-\deltaelta\frac{3}{(N-1)(N-2)(N-3)},
\epsnd{split}\\
\mathrm{I}E V_1^2 V_2 V_3&= 1+\alpha\frac{N-6}{N-1}-\beta \frac{2N-8}{(N-1)(N-2)}-\gamma\frac{1}{N-2}+\deltaelta\frac{1}{(N-1)(N-2)}, \label{56} \\
\mathrm{I}E V_1^3 V_2 &= 1+\alpha \frac{3N-7}{N-1} +\beta \frac{N-6}{N-1}-\deltaelta \frac{1}{N-1}. \label{57}
}
\epsnd{lemma}
\begin{proof}
Since~$\sum_{i=1}^N V_1 =N$ and the~$V_i$'s are exchangeable, we have that~$\mathrm{I}E V_1=1$.
Thus~$\alpha=\mathrm{I}E V_1 (V_1-1)=\mathrm{I}E V_1^2 -1$ which is~\epsqref{48}. Note that similarly,
\be{
N=\mathrm{I}E V_1 (V_1+\cdots+V_N) = \mathrm{I}E V_1^2 + (N-1) \mathrm{I}E V_1 V_2 = \alpha+1 +(N-1)\mathrm{I}E V_1V_2,
}
and rearranging gives~\epsqref{49}.
For~\epsqref{50}, we have that
\be{
\mathrm{I}E V_1^3= \mathrm{I}E V_1(V_1-1)(V_1-2)+3\mathrm{I}E V_1^2 -2\mathrm{I}E V_1=\beta+3(\alpha+1)-2,
}
and further,
\ba{
N^2&=\mathrm{I}E V_1(V_1+\cdots + V_N)^2 = \mathrm{I}E V_1^3 + 3(N-1)\mathrm{I}E V_1^2 V_2 + (N-1)(N-2) \mathrm{I}E V_1 V_2 V_3,\\
(\alpha+1)N&=\mathrm{I}E V_1^2(V_1+\cdots+V_N)=\mathrm{I}E V_1^3 + (N-1) \mathrm{I}E V_1^2 V_2.
}
Solving these two equations yields the expressions for~\epsqref{51} and~\epsqref{52}.
Moving forward similarly, we have
\ba{
\mathrm{I}E V_1^2V_2^2&= \mathrm{I}E V_1(V_1-1)V_2(V_2-1)+2 \mathrm{I}E V_1^2V_2-\mathrm{I}E V_1 V_2, \\
\mathrm{I}E V_1^4&=\mathrm{I}E V_1(V_1-1)(V_1-2)(V_1-3)+6\mathrm{I}E V_1 (V_1-1)(V_1-2)+7\mathrm{I}E V_1(V_1-1)+\mathrm{I}E V_1,
}
and using previous expressions gives~\epsqref{53} and~\epsqref{54}. Along the same lines, we have
\ba{
\mathrm{I}E(V_1V_2V_3(V_1 + \cdots + V_N)) &= N \mathrm{I}E(V_1V_2V_3) = 3\mathrm{I}E(V_1^2V_2V_3) + (N-3) \mathrm{I}E(V_1V_2V_3V_4)\\
\mathrm{I}E(V_1^2V_2(V_1 + \cdots + V_N)) &= N\mathrm{I}E(V_1^2V_2) = \mathrm{I}E(V_1^3V_2) + \mathrm{I}E(V_1^2V_2^2) + (N-2)\mathrm{I}E(V_1^2V_2V_3)\\
\mathrm{I}E(V_1^3(V_1 + \cdots + V_N)) &= N \mathrm{I}E(V_1^3) = \mathrm{I}E(V_1^4) + (N-1)\mathrm{I}E(V_1^3V_2).
}
Plugging in values for known quantities in these three equations and solving yields~\epsqref{55},~\epsqref{56}, and~\epsqref{57}.
\epsnd{proof}
We first work on the~$A_2$ term from Theorem~\ref{THM3} which only requires
two moments.
\begin{lemma}\label{58}
For~$\tV,\tX, \textnormal{\textbf{M}}$ defined above,~$\alpha$ defined at~\epsqref{3}, and~$1\leq i\not=j\leq (K-1),$
\bes{
\mathrm{I}E(M_i|\tX) &= X_i,\\
\mathrm{I}E(M_i^2|\tX) &= X_i^2\left(1-\frac{\alpha}{N-1}\right) + X_i\frac{\alpha N}{N-1},\\
\mathrm{I}E(M_iM_j|\tX)& = X_iX_j\left(1 - \frac{\alpha}{N-1}\right).
}
\epsnd{lemma}
\begin{proof}
Using exchangeability, without loss of generality,
\bes{
\mathrm{I}E(M_i|\tX) &= \mathrm{I}E[V_1 + \cdots + V_{X_i}|\tX] = X_i, \\
\mathrm{I}E(M_i^2|\tX) &= \mathrm{I}E[( V_1 + \cdots + V_{X_i})^2|\tX] = X_i \mathrm{I}E(V_1^2) + X_i(X_i-1)\mathrm{I}E(V_1V_2),\\
\mathrm{I}E(M_iM_j|\tX) &= \mathrm{I}E[ (V_1 + \cdots + V_{X_i})(V_{X_i+1} + \cdots + V_{X_i+X_j})|\tX]= X_i X_j \mathrm{I}E(V_1V_2),
}
The lemma now follows by using the formulas
for the moments of the~$V_i$ in Lemma~\ref{47}.
\epsnd{proof}
| 3,141 | 53,110 |
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0.4939.14
|
For~\epsqref{50}, we have that
\be{
\mathrm{I}E V_1^3= \mathrm{I}E V_1(V_1-1)(V_1-2)+3\mathrm{I}E V_1^2 -2\mathrm{I}E V_1=\beta+3(\alpha+1)-2,
}
and further,
\ba{
N^2&=\mathrm{I}E V_1(V_1+\cdots + V_N)^2 = \mathrm{I}E V_1^3 + 3(N-1)\mathrm{I}E V_1^2 V_2 + (N-1)(N-2) \mathrm{I}E V_1 V_2 V_3,\\
(\alpha+1)N&=\mathrm{I}E V_1^2(V_1+\cdots+V_N)=\mathrm{I}E V_1^3 + (N-1) \mathrm{I}E V_1^2 V_2.
}
Solving these two equations yields the expressions for~\epsqref{51} and~\epsqref{52}.
Moving forward similarly, we have
\ba{
\mathrm{I}E V_1^2V_2^2&= \mathrm{I}E V_1(V_1-1)V_2(V_2-1)+2 \mathrm{I}E V_1^2V_2-\mathrm{I}E V_1 V_2, \\
\mathrm{I}E V_1^4&=\mathrm{I}E V_1(V_1-1)(V_1-2)(V_1-3)+6\mathrm{I}E V_1 (V_1-1)(V_1-2)+7\mathrm{I}E V_1(V_1-1)+\mathrm{I}E V_1,
}
and using previous expressions gives~\epsqref{53} and~\epsqref{54}. Along the same lines, we have
\ba{
\mathrm{I}E(V_1V_2V_3(V_1 + \cdots + V_N)) &= N \mathrm{I}E(V_1V_2V_3) = 3\mathrm{I}E(V_1^2V_2V_3) + (N-3) \mathrm{I}E(V_1V_2V_3V_4)\\
\mathrm{I}E(V_1^2V_2(V_1 + \cdots + V_N)) &= N\mathrm{I}E(V_1^2V_2) = \mathrm{I}E(V_1^3V_2) + \mathrm{I}E(V_1^2V_2^2) + (N-2)\mathrm{I}E(V_1^2V_2V_3)\\
\mathrm{I}E(V_1^3(V_1 + \cdots + V_N)) &= N \mathrm{I}E(V_1^3) = \mathrm{I}E(V_1^4) + (N-1)\mathrm{I}E(V_1^3V_2).
}
Plugging in values for known quantities in these three equations and solving yields~\epsqref{55},~\epsqref{56}, and~\epsqref{57}.
\epsnd{proof}
We first work on the~$A_2$ term from Theorem~\ref{THM3} which only requires
two moments.
\begin{lemma}\label{58}
For~$\tV,\tX, \textnormal{\textbf{M}}$ defined above,~$\alpha$ defined at~\epsqref{3}, and~$1\leq i\not=j\leq (K-1),$
\bes{
\mathrm{I}E(M_i|\tX) &= X_i,\\
\mathrm{I}E(M_i^2|\tX) &= X_i^2\left(1-\frac{\alpha}{N-1}\right) + X_i\frac{\alpha N}{N-1},\\
\mathrm{I}E(M_iM_j|\tX)& = X_iX_j\left(1 - \frac{\alpha}{N-1}\right).
}
\epsnd{lemma}
\begin{proof}
Using exchangeability, without loss of generality,
\bes{
\mathrm{I}E(M_i|\tX) &= \mathrm{I}E[V_1 + \cdots + V_{X_i}|\tX] = X_i, \\
\mathrm{I}E(M_i^2|\tX) &= \mathrm{I}E[( V_1 + \cdots + V_{X_i})^2|\tX] = X_i \mathrm{I}E(V_1^2) + X_i(X_i-1)\mathrm{I}E(V_1V_2),\\
\mathrm{I}E(M_iM_j|\tX) &= \mathrm{I}E[ (V_1 + \cdots + V_{X_i})(V_{X_i+1} + \cdots + V_{X_i+X_j})|\tX]= X_i X_j \mathrm{I}E(V_1V_2),
}
The lemma now follows by using the formulas
for the moments of the~$V_i$ in Lemma~\ref{47}.
\epsnd{proof}
\begin{lemma}\label{59}
For~$1\leq i\leq (K-1)$,~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}'), \pi_i, p_i, \sigma$ defined above, and~$\alpha$
defined at~\epsqref{3},
\bes{
\mathrm{I}E[ (W_i' - W_i)^2 | \textnormal{\textbf{W}}] &= W_i^2\left[\frac{-\alpha}{N-1} - \alpha \left(\frac{\sigma^2 - 2\sigma}{N-1}\right) + \sigma^2\right]\\
&\ \ \ + W_i\left[ \frac{\alpha}{N-1} - \alpha \left(\frac{2\sigma -\sigma^2}{N-1}\right) + \frac{p_i(1-p_i) - \pi_i (1-\pi_i)}{N} - 2\pi_i\sigma\right]\\
&\ \ \ + \pi_i(1-\pi_i)/N + \pi_i^2.
}
\epsnd{lemma}
\begin{proof}
Using the decomposition of~\epsqref{46},
\bes{
\mathrm{I}E[ (X_i' - X_i)^2 | \tX, \textnormal{\textbf{M}}] &= (N-M_i)\pi_i(1-\pi_i) + (N-M_i)^2\pi_i^2 + M_i(1-p_i)p_i + M_i^2(1-p_i)^2\\
&\ \ \ + 2(N-M_i)\pi_iM_i(1-p_i)- 2X_i(\pi_i N + (1-\sigma)M_i) + X_i^2\\
&= M_i^2(1-\sigma)^2\\
&\ \ \ + M_i[p_i(1-p_i) -\pi_i(1-\pi_i) - 2N\pi_i^2 + 2N \pi_i(1-p_i)-2X_i(1-\sigma)]\\
&\ \ \ + N\pi_i(1-\pi_i) + N^2 \pi_i^2 - 2N \pi_iX_i + X_i^2. \\
}
Now taking expectation with respect to~$M_i$ using Lemma~\ref{58},
\bes{
\mathrm{I}E[ (X_i' - X_i)^2 | \tX]
&= \left[X_i^2\left(1-\frac{\alpha}{N-1}\right) + X_i\frac{\alpha N}{N-1}\right](1-\sigma)^2\\
&\qquad + X_i[ p_i(1-p_i) -\pi_i(1-\pi_i) - 2N\pi_i^2 + 2N\pi_i(1-p_i)-2X_i(1-\sigma)]\\
&\qquad + N\pi_i(1-\pi_i) + N^2 \pi_i^2 - 2N \pi_iX_i + X_i^2 \\
&= X_i^2\left[1 + \left(1-\frac{\alpha}{N-1}\right)(1-\sigma)^2 - 2(1-\sigma)\right]\\
&\ \ \ + X_i\left[ \frac{\alpha N}{N-1}(1-\sigma)^2- 2N\pi_ip_i + p_i(1-p_i) - \pi_i(1-\pi_i) -2N\pi_i^2 \right]\\
&\ \ \ + N\pi_i(1-\pi_i) + N^2 \pi_i^2.
}
Dividing this last expression by~$N^2$ and rearranging gives the lemma.
\epsnd{proof}
\begin{lemma}\label{60}
For~$1\leq i\not=j \leq (K-1)$,~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}'), \pi_i, \pi_j, p_i, p_j,\sigma$ defined above, and~$\alpha$
defined at~\epsqref{3},
\bes{
\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j) | \textnormal{\textbf{W}} ] &= W_iW_j\left[ -\frac{\alpha}{N-1} + \frac{\alpha\sigma(2-\sigma)}{N-1} + \sigma^2\right]\\
&\qquad+\left(W_i\pi_j+W_j\pi_i\right)\left[ -\frac{1}{N} -\frac{N-1}{N}\sigma \right]+\frac{N-1}{N}\pi_i\pi_j.
}
\epsnd{lemma}
\begin{proof}
Given~$\textnormal{\textbf{M}}$, we can write~$(X_i', X_j', N-X_i'-X_j')$
as the sum of three independent multinomial random
variables corresponding to the counts of types~$i$,~$j$ and neither~$i$ or~$j$
in~$\tX'$ coming from
individuals in the previous~$\tX$-generation having types~$i$,~$j$, and neither~$i$ or~$j$.
Then the parameters of these multinomials are
$M_i, (1-p_i, \pi_j, 1-p_i-\pi_j)$;
$M_j, (\pi_i,1-p_j, \pi_j, 1-p_j-\pi_i)$; and
$N-M_i-M_j, (\pi_i, \pi_j, 1-\pi_i-\pi_j)$.
From this description and multinomial moment formulas (e.g., Lemma~\ref{lem10}),
it's straightforward to find that
\bes{
\mathrm{I}E[X_i'X_j'| \tX, \textnormal{\textbf{M}}] &= M_i(M_i-1)(1-p_i)\pi_j + M_j(M_j-1)\pi_i(1-p_j) \\
&\qquad + (N-M_i-M_j)(N-M_i-M_j-1)\pi_i\pi_j\\
&\qquad+M_i(1-p_i)M_j(1-p_j) + M_i(1-p_i)(N-M_i-M_j) \pi_j \\
&\qquad+ M_j\pi_iM_i\pi_j+ M_j\pi_i(N-M_i-M_j)\pi_j \\
&\qquad+ (N-M_i-M_j)\pi_iM_i\pi_j + (N-M_i-M_j)\pi_i M_j (1-p_j)\\
&=M_i M_j (1-\sigma)^2+(M_i \pi_j+M_j \pi_i)(1-\sigma)(N-1)+N(N-1)\pi_i \pi_j.
}
Also note that
\be{
\mathrm{I}E[X_i'X_j | \tX, \textnormal{\textbf{M}}] = X_j[(N-M_i)\pi_i + (1-p_i)M_i],
}
so that these last two displays and Lemma~\ref{58}
imply
\bes{
\mathrm{I}E[(X_i'-X_i)(X_j'-X_j)|\tX] &= X_iX_j\left[ (1-\sigma)^2\left(1-\frac{\alpha}{N-1}\right) - 2(1-\sigma) + 1\right]\\
&\qquad+ ( X_i \pi_j+X_j\pi_i)[-(N-1) \sigma -1]+ N(N-1) \pi_i\pi_j.
}
Dividing this last expression by~$N^2$ and rearranging gives the lemma.
\epsnd{proof}
The next lemma summarizes the bound on the~$A_2$ term of Theorem~\ref{THM3}
for this example.
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0.4939.15
|
The next lemma summarizes the bound on the~$A_2$ term of Theorem~\ref{THM3}
for this example.
\begin{lemma}\label{lem15}
For~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$ and~$\sigma$ defined above and~$\alpha$ defined at~\epsqref{3},
if~$\lambda = \alpha/(2(N-1))$, then
\bes{
\frac{1}{\lambda} \sum_{i,j=1}^{K-1} \mathrm{I}E& \left|\lambda W_i(\deltaelta_{i j}-W_j)-\frac{1}{2}\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}] \right|\\
&\leq \sigma^2\left[ (K-1)^2 + \frac{N-1}{\alpha}\left( K^2 +1\right)\right] + \sigma \left[ 2(K-1)^2 + \frac{3K-5}{\alpha} \right].
}
\epsnd{lemma}
\begin{proof}
Using Lemmas~\ref{59} and~\ref{60},
\bes{
\frac{1}{\lambda} &\sum_{i,j=1}^{K-1} \mathrm{I}E \left|\lambda W_i(\deltaelta_{i j}-W_j)-\frac{1}{2}\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}] \right|\\
&\leq \frac{N-1}{\alpha}\sum_{i=1}^{K-1} \left(\alpha \left(\frac{\sigma^2 + 2\sigma}{N-1}\right) + \sigma^2 + \frac{p_i(1-p_i) + \pi_i (1-\pi_i)}{N} + 2\pi_i\sigma+ \pi_i^2\right) \\
&\ \ \ + \frac{N-1}{\alpha} \sum_{i \neq j}^{K-1} \left( \frac{\alpha}{N-1}(2\sigma + \sigma^2) + \frac{\pi_i + \pi_j}{N} + \sigma^2 +\sigma\pi_i + \sigma\pi_j + \pi_i \pi_j\right)\\
&\leq (K-1)(\sigma^2 + 2\sigma) + \frac{N-1}{\alpha}[(K-1)\sigma^2 + (K-1)\sigma/N + 3\sigma^2 ]\\
&\quad + (K-1)(K-2)(\sigma^2 + 2\sigma) + \frac{N-1}{\alpha}\left[\frac{ 2(K-2)\sigma}{N} + \sigma^2((K-1)(K-2)+2(K-2)+1) \right],\\
&\leq \sigma^2\left[ (K-1)^2 + \frac{N-1}{\alpha}\left( K^2 +1\right)\right] + \sigma \left[ 2(K-1)^2 + \frac{3K-5}{\alpha} \right].\qed
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0.4939.16
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here
}
\epsnd{proof}
To compute the~$A_3$ term of Theorem~\ref{THM3}, we
need higher moment information.
\begin{lemma}\label{61}
For~$\tV,\tX, \textnormal{\textbf{M}}$ defined above
and~$1\leq i\leq (K-1),$
\bes{
\mathrm{I}E(M_i^3|\tX)
&= X_i^3 [\mathrm{I}E(V_1V_2V_3)] + X_i^2[ 3\mathrm{I}E(V_1^2V_2) - 3\mathrm{I}E(V_1V_2V_3)]\\
& \qquad + X_i[ \mathrm{I}E(V_1^3) - 3\mathrm{I}E(V_1^2V_2) + 2\mathrm{I}E(V_1V_2V_3)]\\
\mathrm{I}E(M_i^4|\tX)
&= X_i^4 [\mathrm{I}E(V_1V_2V_3V_4)] + X_i^3[ 6\mathrm{I}E(V_1^2V_2V_3) - 6\mathrm{I}E(V_1V_2V_3V_4)]\\
&\qquad + X_i^2 [ 4\mathrm{I}E(V_1^3V_2) + 3\mathrm{I}E(V_1^2V_2^2) - 18\mathrm{I}E(V_1^2V_2V_3) + 11\mathrm{I}E(V_1V_2V_3V_4)]\\
&\qquad + X_i[ \mathrm{I}E(V_1^4) - 4\mathrm{I}E(V_1^3V_2) - 3\mathrm{I}E(V_1^2V_2^2) + 12\mathrm{I}E(V_1^2V_2V_3) - 6 \mathrm{I}E(V_1V_2V_3V_4)].
}
\epsnd{lemma}
\begin{proof}
Similar to the proof of Lemma~\ref{47}, exchangeability implies
\bes{
\mathrm{I}E(M_i^3|\tX) &= \mathrm{I}E[(V_1 + \cdots + V_{X_i})^3|\tX],\\
&= X_i \mathrm{I}E(V_1^3) + 3X_i(X_i-1)\mathrm{I}E(V_1^2V_2) + X_i(X_i-1)(X_i-2) \mathrm{I}E(V_1V_2V_3),\\
\mathrm{I}E(M_i^4|\tX) &= \mathrm{I}E[(V_1 + \cdots + V_{X_i})^4|\tX],\\
&=X_i \mathrm{I}E(V_1)^4 + 4X_i(X_i-1)\mathrm{I}E(V_1^3V_2) + 3X_i(X_i-1)\mathrm{I}E(V_1^2V_2^2),\\
&\qquad + 6X_i(X_i-1)(X_i-2) \mathrm{I}E(V_1^2V_2V_3) + X_i(X_i-1)(X_i-2)(X_i-3) \mathrm{I}E(V_1V_2V_3V_4).
}
The lemma now follows by rearranging these equations.
\epsnd{proof}
\begin{lemma}\label{lem16}
For~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$ and~$\sigma$ defined above and~$\alpha,\beta,\gamma, \deltaelta$ defined at~\epsqref{3}
and~$N>1$,
if~$\lambda = \alpha/(2(N-1))$, then
\ba{
&\frac{1}{\lambda}\sum_{i,j,k=1}^{K-1}\mathrm{I}E |(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)|\\
&\quad \leq 2(K-1)^3\left(\left( \frac{3\sigma^2}{N\alpha} + \frac{\sigma}{N^2\alpha} \right)^{1/4} + \left( \frac{\rho}{N^3\alpha}\right)^{1/4} + \left(\frac{N\sigma^4}{\alpha}\right)^{1/4} \right)^2 \left(\sqrt{\frac\sigma\alpha}+ 1 + \sqrt{\frac{N\sigma^2}{\alpha}}\right).
}
where
\be{
\rho :=
\frac{ N^2\beta}{2(N-1)}+ \frac{(3 N^4) \gamma }{(N-2)(N-3)}+ \frac{(4N^4+3N^2)\deltaelta}{(N-1)(N-2)(N-3)}.
}
\epsnd{lemma}
\begin{proof}
Decompose
\ba{
X_i' - X_i &= [X_i' - (M_i(1-p_i) + (N-M_i)\pi_i)] + [(M_i-X_i)(1-\sigma)] + [N\pi_i - X_i \sigma]\\
&=:E_i + F_i + G_i.
}
Using H\"older's inequality followed by Minkowski's inequality, we find
\ban{
\sum_{i,j,k}^{K-1}& \mathrm{I}E \abs{(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)} \notag\\
&\qquad= \frac{1}{N^3}\sum_{i,j,k}^{K-1}\mathrm{I}E \abs{(E_i +F_i+ G_i)(E_j +F_j+ G_j)(E_k +F_k+ G_k)}\notag \\
&\qquad \leq \frac{1}{N^3}\sum_{i,j,k}^{K-1}\left[\mathrm{I}E (E_i+F_i+G_i)^4 \mathrm{I}E (E_j+F_j+G_j)^4 \right]^{1/4}\left[\mathrm{I}E (E_k+F_k+G_k)^2\right]^{1/2}\notag\\
\begin{split}\label{62}
& \qquad \leq \frac{1}{N^3}\left(\sum_{i=1}^{K-1}\left[(\mathrm{I}E E_i^4)^{1/4}+(\mathrm{I}E F_i^4)^{1/4}+(\mathrm{I}E G_i^4)^{1/4} \right]\right)^2 \\
&\qquad\qquad\qquad\qquad \times \sum_{k=1}^{K-1}\left[(\mathrm{I}E E_k^2)^{1/2}+(\mathrm{I}E F_k^2)^{1/2}+(\mathrm{I}E G_k^2)^{1/2}\right].
\epsnd{split}
}
Recall the decomposition~\epsqref{46} of~$\mathscr{L}(X_i'|\textnormal{\textbf{M}})=\mathscr{L}(Y_1(\textnormal{\textbf{M}})+Y_2(\textnormal{\textbf{M}}))$ as a sum of
conditionally (on~$\textnormal{\textbf{M}}$)
independent binomials
and note that if~$Y\sim\mathop{\mathrm{Bi}}n(n,p)$ then
\bes{
\mathrm{I}E(Y-np)^4 &= 3(np(1-p))^2 + np(1-p)(1-6p(1-p)) \leq 3(np(1-p))^2 + np(1-p),
}
so that
\ba{
\mathrm{I}E[E_i^4|\textnormal{\textbf{M}}]&= \mathrm{I}E[(Y_1(\textnormal{\textbf{M}}) - \mathrm{I}E [Y_1(\textnormal{\textbf{M}})|\textnormal{\textbf{M}}] + Y_2(\textnormal{\textbf{M}})- \mathrm{I}E [Y_2(\textnormal{\textbf{M}})|\textnormal{\textbf{M}}])^4| \textnormal{\textbf{M}}] \\
&\leq 3 (M_ip_i(1-p_i))^2 + M_ip_i(1-p_i) + 6 M_i(1-p_i)p_i(N-M_i)\pi_i(1-\pi_i)\\
& \ \ \ + 3((N-M_i)\pi_i(1-\pi_i))^2 + (N-M_i)\pi_i(1-\pi_i)\\
&\leq 3(N(p_i(1-p_i) + \pi_i(1-\pi_i)))^2 + N(p_i(1-p_i) + \pi_i(1-\pi_i))\\
&\leq 3(N \sigma)^2 + N\sigma.
}
Using a similar argument for the second moment, we thus have for all~$1\leq i \leq K-1$,
\ben{\label{63}
\mathrm{I}E E_i^2\leq N \sigma, \hspace{1cm} \mathrm{I}E E_i^4\leq 3(N\sigma)^2 + N\sigma.
}
Now note that~$|G_i| \leq (N-X_i) \pi_i + X_i(\sigma-\pi_i) \leq N\sigma$, so that for all~$1\leq i \leq K-1$,
\ben{\label{64}
\mathrm{I}E G_i^2\leq (N\sigma)^2, \hspace{1cm} \mathrm{I}E G_i^4\leq (N\sigma)^4.
}
For the~$F_i=M_i-X_i$ moments, first note that Lemma~\ref{58} implies
\ben{\label{65}
\mathrm{I}E[F_i^2|\tX]=\mathrm{I}E[(M_i - X_i)^2|\tX]
=\frac{\alpha X_i (N-X_i)}{N-1} \leq \frac{\alpha N^2}{N-1}.
}
Furthermore, using Lemmas~\ref{47},~\ref{58}, and~\ref{61},
\ba{
&\mathrm{I}E[(M_i - X_i)^4| \tX] = \mathrm{I}E(M_i^4 | \tX) - 4X_i \mathrm{I}E(M_i^3|\tX) + 6X_i^2 \mathrm{I}E(M_i^2|\tX) - 4X_i^3 \mathrm{I}E(M_i|X_i) + X_i^4\\
&\quad= X_i^4\left\{ \frac{3\gamma}{(N-2)(N-3)} + \frac{(-3)\deltaelta}{(N-1)(N-2)(N-3)}\right\}\\
&\qquad + X_i^3\left\{ \frac{-6 N\gamma}{(N-2)(N-3)} + \frac{6N \deltaelta}{(N-1)(N-2)(N-3)}\right\}\\
&\qquad + X_i^2 \left\{ \frac{-\alpha}{N-1} + \frac{(-2N+4)\beta}{(N-1)(N-2)} + \frac{(3N^2 + 3N -3)\gamma}{(N-2)(N-3)} + \frac{(-4N^2 + 2N - 3)\deltaelta}{(N-1)(N-2)(N-3)} \right\}\\
&\qquad + X_i \left\{ \frac{(-5N + 6)\alpha}{N-1} + \frac{(2N^2 -4N)\beta}{(N-1)(N-2)}+ \frac{(-3N^2+3N)\gamma}{(N-2)(N-3)} + \frac{(N^3 - 2N^2 + 3N)\deltaelta}{(N-1)(N-2)(N-3)}\right\}.
}
Now using that~$0\leq X_i \leq N$ (and assuming~$N > 1$),
we have
\ba{
&\alpha X_i (-X_i -5N+6) \leq 0, & &\beta X_i((-2N+4)X_i+(2N^2-4N))\leq \beta N^2(N-2)/2, \\
& 3\gamma X_i^3(X_i-2N)\leq 0, & &3\gamma X_i((N^2 + N -1)X_i-N^2+N)\leq 3\gamma N^4, \\
& 3 \deltaelta X_i^3(-X_i+2N)\leq 3\deltaelta N^4, & & \deltaelta[X_i^2(-4N^2 + 2N - 3)+X_i(N^3 - 2N^2 + 3N)]\leq \deltaelta(N^4+3N^2).
}
Combining these inequalities with the previous display, we have
\ben{\label{66}
\mathrm{I}E F_i^4=\mathrm{I}E[(M_i - X_i)^4] \leq \frac{ N^2\beta}{2(N-1)}+ \frac{(3 N^4) \gamma }{(N-2)(N-3)}+ \frac{(4N^4+3N^2)\deltaelta}{(N-1)(N-2)(N-3)}= \rho.
}
Now using the inequalities~\epsqref{63},~\epsqref{64},~\epsqref{65}, and~\epsqref{66} in~\epsqref{62} yields
the lemma.
\epsnd{proof}
\begin{proof}[Proof of Theorem~\ref{THM2}] We apply Theorem~\ref{THM3} with~$\Lambda = \frac{\alpha}{2(N-1)}\times \mathrm{I}d$.
From Lemmas~\ref{lem15} for~$A_2$
and~\ref{lem16} for~$A_3$ we obtain
\ba{
A_2&\leq \sigma^2\left[ (K-1)^2 + \frac{N-1}{\alpha}\left( K^2 +1\right)\right] + \sigma \left[ 2(K-1)^2 + \frac{3K-5}{\alpha} \right],\\
A_3 &\leq 2(K-1)^3\left(\left( \frac{3\sigma^2}{N\alpha} + \frac{\sigma}{N^2\alpha} \right)^{1/4} + \left( \frac{\rho}{N^3\alpha}\right)^{1/4} + \left(\frac{N\sigma^4}{\alpha}\right)^{1/4} \right)^2 \left(\sqrt{\frac\sigma\alpha}+ 1 + \sqrt{\frac{N\sigma^2}{\alpha}}\right),
}
where
\be{
\rho :=
\frac{ N^2\beta}{2(N-1)}+ \frac{(3 N^4) \gamma }{(N-2)(N-3)}+ \frac{(4N^4+3N^2)\deltaelta}{(N-1)(N-2)(N-3)}.
}
The final bound in Theorem~\ref{THM2} is now obtained through straightforward manipulations and applying some standard analytic inequalities,
in particular,~$\sigma=\epsta(\alpha/N)$ and~$\deltaelta\leq (N-3)\beta$.
\epsnd{proof}
\section*{Acknowledgments}
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\begin{proof}[Proof of Theorem~\ref{THM2}] We apply Theorem~\ref{THM3} with~$\Lambda = \frac{\alpha}{2(N-1)}\times \mathrm{I}d$.
From Lemmas~\ref{lem15} for~$A_2$
and~\ref{lem16} for~$A_3$ we obtain
\ba{
A_2&\leq \sigma^2\left[ (K-1)^2 + \frac{N-1}{\alpha}\left( K^2 +1\right)\right] + \sigma \left[ 2(K-1)^2 + \frac{3K-5}{\alpha} \right],\\
A_3 &\leq 2(K-1)^3\left(\left( \frac{3\sigma^2}{N\alpha} + \frac{\sigma}{N^2\alpha} \right)^{1/4} + \left( \frac{\rho}{N^3\alpha}\right)^{1/4} + \left(\frac{N\sigma^4}{\alpha}\right)^{1/4} \right)^2 \left(\sqrt{\frac\sigma\alpha}+ 1 + \sqrt{\frac{N\sigma^2}{\alpha}}\right),
}
where
\be{
\rho :=
\frac{ N^2\beta}{2(N-1)}+ \frac{(3 N^4) \gamma }{(N-2)(N-3)}+ \frac{(4N^4+3N^2)\deltaelta}{(N-1)(N-2)(N-3)}.
}
The final bound in Theorem~\ref{THM2} is now obtained through straightforward manipulations and applying some standard analytic inequalities,
in particular,~$\sigma=\epsta(\alpha/N)$ and~$\deltaelta\leq (N-3)\beta$.
\epsnd{proof}
\section*{Acknowledgments}
We thank the anonymous referee for helpful comments and for pointing out an omission in an earlier version of the manuscript (proof of existence of partial derivatives of the solution to the Stein equation).
NR received support from ARC grant DP150101459; AR received support from NUS Research Grant R-155-000-124-112. This work was done
partially while the authors were visiting the Institute for Mathematical Sciences, National University of Singapore in 2015. The visit was supported by the Institute. HG would also like to thank the School of Mathematics at the University of Melbourne for their hospitality while some of this work was done.
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\epsnd{document}
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{{\mathfrak{m}}athfrak{b}}egin{document}
{\mathfrak{m}}aketitle
{{\mathfrak{m}}athfrak{b}}egin{abstract}
We study the birational geometry of irregular varieties and the singularities of Theta divisors of PPAV's in positive characteristic by applying recent generic vanishing results of Hacon and Patakfalvi. In particular, we prove that irreducible Theta divisors of principally polarized abelian varieties are strongly F-regular, which extends an old result of Ein and Lazarsfeld to fields of positive characteristic. In order to prove this, we formulate a positive characteristic analogue of another result of Ein and Lazarsfeld, to the effect that the Albanese image of a smooth projective variety of maximal Albanese dimension with vanishing holomorphic Euler characteristic is fibered by abelian subvarieties.
{\epsilon}nd{abstract}
\tableofcontents
\section{Introduction}
The purpose of this paper is to apply recent generic vanishing results in positive characteristic due to Hacon and Patakfalvi ~{{\mathfrak{m}}athfrak{c}}ite{hp13} to the study of the birational geometry of irregular varieties and the singularities of Theta divisors of principally polarized abelian varieties. Over fields of characteristic zero, seminal work of Ein and Lazarsfeld ~{{\mathfrak{m}}athfrak{c}}ite{el97} applied generic vanishing techniques over the complex numbers to settle a number of questions concerning the geometry of irregular varieties. One of their main results states that irreducible Theta divisors on principally polarized abelian varieties have mild singularities:
{{\mathfrak{m}}athfrak{b}}egin{thm}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 1]{el97})
Let $A$ be an abelian variety and let $\Theta\subset A$ be a principal polarization (i.e. an ample divisor such that $h^0(A,{\mathcal{O}}_A(\Theta))=1$). If $\Theta$ is irreducible, then it is normal and has rational singularities. \label{el97-main-theorem}
{\epsilon}nd{thm} {\mathfrak{m}}edskip
The conclusion of the theorem is captured by the adjoint ideal of $\Theta$: given any log resolution ${\mathfrak{m}}u:A' {\rightarrow} A$ of the pair $(A,\Theta)$ and writing ${\mathfrak{m}}u^{{{\mathfrak{m}}athfrak{a}}st}\Theta=\Theta'+F$ with $\Theta'$ smooth and $F$ ${\mathfrak{m}}u$-exceptional, one may define ${{\mathfrak{m}}athfrak{a}}dj(A,\Theta)={\mathfrak{m}}u_{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{O}}_{A'}(K_{A'/A}-F)$. Standard arguments show that ${{\mathfrak{m}}athfrak{a}}dj(A,\Theta)={\mathcal{O}}_A$ is equivalent to $\Theta$ being normal and having rational singularities (see section 9.3.E in ~{{\mathfrak{m}}athfrak{c}}ite{laz04}). Bearing this in mind, Ein and Lazarsfeld's argument breaks into the following steps: let $X{\rightarrow} \Theta$ be a resolution of singularities.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item If $\Theta$ is irreducible, then $X$ is of general type. This relies on a classical argument due to Ueno (see ~{{\mathfrak{m}}athfrak{c}}ite{mor00}, Theorem 3.7), characterizing the Itaka fibration and the Kodaira dimension of an irreducible subvariety of an abelian variety.
\item If $X$ is of general type, then ${{\mathfrak{m}}athfrak{c}}hi({\omega}_X)>0$. More concretely, if $X$ is a smooth projective variety of maximal Albanese dimension and ${{\mathfrak{m}}athfrak{c}}hi(X,\omega_X)=0$, then the image of the Albanese map is fibred by tori. In particular, this shows that if $X$ is birational onto its image under the Albanese map, then $X$ is not of general type.
\item Generic vanishing theorems and Nadel vanishing yield ${{\mathfrak{m}}athfrak{a}}dj(\Theta)={\mathcal{O}}_A {\Lambda}ongleftrightarrow {{\mathfrak{m}}athfrak{c}}hi(X,\omega_X)>0$.
{\epsilon}nd{enumerate} {\mathfrak{m}}edskip
Therefore if $\Theta$ is irreducible, its adjoint ideal must be trivial, and by the characterization described above, it must be normal an have rational singularities. Work of Abramovich ~{{\mathfrak{m}}athfrak{c}}ite{abr95} shows that the statement in (i) remains valid in positive characteristic once an appropriate notion of Kodaira dimension is defined for possibly singular varieties. In this paper we provide positive characteristic analogues of items (ii) and (iii). {\mathfrak{m}}edskip
The only known results in this direction are due to Hacon ~{{\mathfrak{m}}athfrak{c}}ite{hac11}, where he proved that for principally polarized abelian varieties $(A,\Theta)$ over algebraically closed fields of positive characteristic, the pair $(A,\Theta)$ is a limit of strongly F-regular pairs. More precisely:
{{\mathfrak{m}}athfrak{b}}egin{thm}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 1.1]{hac11})
Let $(A,\Theta)$ be a principally polarized abelian variety over an algebraically closed field of characteristic $p>0$. If $D\in |m\Theta|$, then $\left(A,\frac{1-{\epsilon}psilon}{m}D\right)$ is strongly F-regular for any rational number $0<{\epsilon}psilon<1$
{\epsilon}nd{thm} {\mathfrak{m}}edskip
We summarize briefly our main results. The arguments employed in the proofs bear a strong resemblance to their characteristic zero analogues, albeit plenty of technicalities arise. Not only is resolution of singularities unavailable in general, but also generic vanishing for canonical sheaves is known to fail in positive characteristic (c.f. ~{{\mathfrak{m}}athfrak{c}}ite{hk12}). Nevertheless, recent work of Hacon and Patakfalvi ~{{\mathfrak{m}}athfrak{c}}ite{hp13} provides strong generic vanishing statements for objects arising from Cartier modules (see section 2.4 for precise statements): given a coherent Cartier module ${\mathcal{O}}mega_0\in {{\mathfrak{m}}athbb C}oh(A)$, the traces of the Frobenius iterates yield an inverse system $${{\mathfrak{m}}athfrak{c}}dots {\rightarrow} F_{{{\mathfrak{m}}athfrak{a}}st}^e{{\mathcal{O}}mega}_0 {\rightarrow} F_{{{\mathfrak{m}}athfrak{a}}st}^{e-1}{{\mathcal{O}}mega}_0 {\rightarrow} {{\mathfrak{m}}athfrak{c}}dots$$ and denoting by ${{\mathcal{O}}mega}=\varprojlim F_{{{\mathfrak{m}}athfrak{a}}st}^e{{\mathcal{O}}mega}_0$ its inverse limit, there exists a closed subset $Z\subset \hat{A}$ such that $H^i(A,{{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})=0$ for every $i>0$ and every ${{\mathfrak{m}}athfrak{a}}lpha\in Z$ such that $p^e{{\mathfrak{m}}athfrak{a}}lpha\notin Z$ for all $e>>0$ (c.f. Corollary 3.3.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}). This grounds on the following Theorem, which is the main result in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}.
{{\mathfrak{m}}athfrak{b}}egin{thm}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 3.1.1 and Lemma 3.1.2]{hp13})
Let $k$ be an algebraically closed field of characteristic $p>0$ and
$A$ be an abelian variety over $k$. Let $\{{\mathcal{O}}mega_e\}$ be Cartier module on $A$. If for any sufficiently
ample line bundle $L$ on $\hat{A}$ and any $e\gg 0$,
$H^i(A,{\mathcal{O}}mega_e\otimes \hat{L}^\vee)=0$ for all $i>0$, then the
complex\footnote{Here $S_{A,\hat{A}}$ denotes the Fourier-Mukai functor with kernel given by the Poincar{\'e} bundle of $A\times \hat{A}$} $${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}} RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$$ is a
quasi-coherent sheaf in degree 0, i.e.,
${\Lambda}ambda={\mathfrak{m}}athcal{H}^0({\Lambda}ambda)$.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
This result is generalized further in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}, where a notion of M-regularity in positive characteristic is also introduced. Concretely, one has the following:
{{\mathfrak{m}}athfrak{b}}egin{thm}[c.f. Theorem 4.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}]
Let $A$ be an abelian variety and
$\{{\mathcal{O}}mega_e\}$ be a GV-inverse system of coherent sheaves on $A$
such that $\{{\mathcal{O}}mega_e\}$ is M-regular, in the sense that ${\mathfrak{m}}athcal{H}^0({\Lambda}ambda)$ is torsion-free. Then for any scheme-theoretic point $P\in A$, if ${\textrm{dim }} P\geqslant i$, then $P$ is not in the support of $$Im (R^i\hat{S}({\mathcal{O}}mega) {\rightarrow} R^i\hat{S}({\mathcal{O}}mega_e))$$ for any $e$.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
Our main technical result is a partial converse to the previous theorem: the presence of torsion in ${\mathcal{H}}^0({\Lambda})$ induces the following non-vanishing statement:
{{\mathfrak{m}}athfrak{b}}egin{thm}[c.f. Theorem 3.1]
Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on a
$g$-dimensional abelian variety satisfying the Mittag-Leffler
condition and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let
${\Lambda}ambda_e=RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$.
Suppose that $\{{{\mathcal{O}}mega}_e\}$ is a GV-inverse system, in the sense that ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$. If ${\mathcal{H}}^0({\Lambda})$ is has a torsion point $P$ of dimension $g-k$, then the maps $$\varprojlim \left(R^kS_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {\rightarrow} R^kS_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)$$ are non-zero for every $e>>0$.
{\epsilon}nd{thm}
Using this, we can derive a fibration statement similar to that of Ein and Lazarsfeld:
{{\mathfrak{m}}athfrak{b}}egin{thm}[c.f. Theorem 4.2]
Let $X$ be a smooth projective variety of maximal Albanese dimension and denote by $a:X{\rightarrow} A$ the Albanese map. Let $g={\textrm{dim }} A$. Consider the inverse system $\{{{\mathcal{O}}mega}_e:=F_{{{\mathfrak{m}}athfrak{a}}st}^eS^0a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X\}_e$ and denote ${{\mathcal{O}}mega}=\varprojlim {{\mathcal{O}}mega}_e$. Define ${\Lambda}_e=RS_{A,\hat{A}}D_A({{\mathcal{O}}mega}_e)$ and assume that the sheaf ${\mathcal{H}}^0({\Lambda})=\varinjlim {\mathcal{H}}^0({\Lambda}_e)$ has torsion. Then the image of the Albanese map is fibered by abelian subvarieties of $\hat{A}$.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
An identical argument to the one we employ to prove the previous theorem also yields the following result \label{Main-theorem} describing the singularities of Theta divisors in positive characteristic.
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $A$ be an ordinary abelian variety over an algebraically closed field of positive characteristic and let $\Theta$ be an irreducible Theta divisor. Then $\Theta$ is strongly F-regular. \label{Main-theorem}
{\epsilon}nd{thm} {\mathfrak{m}}edskip
Over fields of positive characteristic, work of Smith and Hara (c.f. {{\mathfrak{m}}athfrak{c}}ite{smi97}, {{\mathfrak{m}}athfrak{c}}ite{har98}) shows that F-rationality is the positive characteristic counterpart to rational singularities, and the former is implied by strong F-regularity, so in this sense Theorem \ref{Main-theorem} is stronger than one might expect. {\mathfrak{m}}edskip
This paper is structured as follows. We start by recording all the background results we need in section 2: in 2.1 we recall the main definitions and some useful properties of the Fourier-Mukai transform in the context of abelian varieties, in 2.2 we record the relevant definitions of F-singularities; in 2.3 we record results of Pink and Roessler characterizing subvarieties of abelian varieties in positive characteristic; in 2.4 we outline a few useful facts concerning inverse systems that will ease the exposition of the proofs and in 2.5 we collect the generic vanishing statements in positive characteristic that will be needed in the sequel. Sections 3 and 4 constitute the technical core of the paper: section 3 contains the proof of the non-vanishing statement in the presence of torsion of ${\mathcal{H}}^0({\Lambda})$ and in section 4 we generalize Ein and Lazarsfeld's fibration statement. Finally in section 5 we present the proof of Theorem \ref{Main-theorem} on the singularities of Theta divisors. We start with the case of simple abelian varieties in section 5.1, which is a simple computation following easily from the results in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that does not require the arguments from sections 3 and 4. The general case is more involved and is presented section 5.2.
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\section{Preliminaries}
\subsection{Derived categories and Fourier-Mukai transforms}
Let $A$ be a g-dimensional abelian variety, denote by $\hat{A}={{\mathfrak{m}}athbb P}ic^0(A)$ its dual and let ${\mathbb{F}}F\in {{\mathfrak{m}}athbb C}oh(A)$. Let ${{\mathfrak{m}}athbb P}P\in {{\mathfrak{m}}athbb P}ic(A\times \hat{A})$ be the Poincar{\'e} bundle and denote and consider the usual Fourier-Mukai functors: $$RS_{A,\hat{A}}^{{{\mathfrak{m}}athbb P}P}:D(A) {\rightarrow} D(\hat{A}), {{\mathfrak{m}}athfrak{q}}quad RS_{A,\hat{A}}^{{{\mathfrak{m}}athbb P}P}({{\mathfrak{m}}athfrak{b}}ullet)=Rp_{\hat{A}{{\mathfrak{m}}athfrak{a}}st}(p_A^{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athfrak{b}}ullet)\otimes {{\mathfrak{m}}athbb P}P)$$ even though we will most often omit ${{\mathfrak{m}}athbb P}P$ from the notation and simply write $RS_{A,\hat{A}}({{\mathfrak{m}}athfrak{b}}ullet)$. {\mathfrak{m}}edskip
We start by stating Mukai's inversion theorem in the derived category of quasi-coherent sheaves:
{{\mathfrak{m}}athfrak{b}}egin{thm}[~{{\mathfrak{m}}athfrak{c}}ite{muk81}]
If $[-g]$ denotes the rightwise shift by $g$ places and $-1_A$ is the inverse on $A$, the following equalities hold on $D_{qc}(A)$ and $D_{qc}(\hat{A})$ $$RS_{\hat{A},A} {{\mathfrak{m}}athfrak{c}}irc RS_{A,\hat{A}}= (-1_A)^{{{\mathfrak{m}}athfrak{a}}st}[-g], {{\mathfrak{m}}athfrak{q}}quad RS_{A,\hat{A}} {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{A},A}= (-1_{\hat{A}})^{{{\mathfrak{m}}athfrak{a}}st}[-g]$$
{\epsilon}nd{thm} {\mathfrak{m}}edskip
We will also be using the following two results:
{{\mathfrak{m}}athfrak{b}}egin{lemma}[~{{\mathfrak{m}}athfrak{c}}ite{muk81}, Proposition 3.8]
The Fourier-Mukai transform commutes with the dualizing functor in $D_{qc}(\hat{A})$ up to inversions and shifts, namely $$D_A {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{A},A} \simeq \left((-1_{A})^{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{A},A} {{\mathfrak{m}}athfrak{c}}irc D_{\hat{A}}\right)[g]$$
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f, ~{{\mathfrak{m}}athfrak{c}}ite[Lemma 3.4]{muk81})
Let ${{\mathfrak{m}}athfrak{p}}hi:A{\rightarrow} B$ be an isogeny between abelian varieties and denote by $\hat{{{\mathfrak{m}}athfrak{p}}hi}:\hat{B} {\rightarrow} \hat{A}$ the dual isogeny. Then the following equalities hold on $D_{qc}(B)$ and $D_{qc}(A)$ respectively. $${{\mathfrak{m}}athfrak{p}}hi^{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{B},B} \simeq RS_{\hat{A},A} {{\mathfrak{m}}athfrak{c}}irc \hat{{{\mathfrak{m}}athfrak{p}}hi}_{{{\mathfrak{m}}athfrak{a}}st}, {{\mathfrak{m}}athfrak{q}}quad {{\mathfrak{m}}athfrak{p}}hi_{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{A},A} \simeq RS_{\hat{B},B} {{\mathfrak{m}}athfrak{c}}irc \hat{{{\mathfrak{m}}athfrak{p}}hi}^{{{\mathfrak{m}}athfrak{a}}st}$$ In particular, this holds for the (e-th iterate) Frobenius map $F^e$ and its dual isogeny, namely the Verschiebung map $V^e=\hat{F}^e$. \label{Mukai-vs-isogenies}
{\epsilon}nd{lemma}
We will also be using the following simple remark.
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Exercise 5.12]{huy06})
Let ${{\mathfrak{m}}athfrak{p}}i:B{\rightarrow} A$ be a morphism between abelian varieties and let ${{\mathfrak{m}}athbb P}P$ be a locally free sheaf on $A\times \hat{A}$. Denote ${{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{p}}i}=({{\mathfrak{m}}athfrak{p}}i \times 1_{\hat{A}})^{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athbb P}P)$. Then $$S_{{{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{p}}i}} \simeq S_{{{\mathfrak{m}}athbb P}P} {{\mathfrak{m}}athfrak{c}}irc {{\mathfrak{m}}athfrak{p}}i_{{{\mathfrak{m}}athfrak{a}}st}$$ \label{Mukai-vs-push-forward}
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
We next record the notions of homotopy limits and colimits in the derived category. Given a direct system of objects ${{\mathfrak{m}}athbb C}C_i\in D(A)$ $${{\mathfrak{m}}athbb C}C_1 {\rightarrow} {{\mathfrak{m}}athbb C}C_2 {\rightarrow} \ldots$$ its homotopy colimit ${\textrm{hocolim}_{\rightarrow}} {{\mathfrak{m}}athbb C}C_i$ is defined by the triangle $$\oplus {{\mathfrak{m}}athbb C}C_i {\longrightarrow} \oplus {{\mathfrak{m}}athbb C}C_i {\longrightarrow} {\textrm{hocolim}_{\rightarrow}} {{\mathfrak{m}}athbb C}C_i \stackrel{[+1]}{{\longrightarrow}}$$ where the first map is the homomorphism given by $id-shift$ where $shift:\oplus {{\mathfrak{m}}athbb C}C_i {\rightarrow} \oplus {{\mathfrak{m}}athbb C}C_i$ is given on ${{\mathfrak{m}}athbb C}C_i$ by the composition ${{\mathfrak{m}}athbb C}C_i {\rightarrow} {{\mathfrak{m}}athbb C}C_{i+1} \hookrightarrow \oplus {{\mathfrak{m}}athbb C}C_j$. {\mathfrak{m}}edskip
Given an inverse system of objects ${{\mathfrak{m}}athbb C}C_i \in D_{qc}(X)$ $${{\mathfrak{m}}athbb C}C_1 \longleftarrow {{\mathfrak{m}}athbb C}C_2 \longleftarrow {{\mathfrak{m}}athfrak{c}}dots$$ its homotopy limit ${\textrm{holim}_{\leftarrow}} {{\mathfrak{m}}athbb C}C_i$ is given by the triangle $${\textrm{holim}_{\leftarrow}} {{\mathfrak{m}}athbb C}C_i {\longrightarrow} {{\mathfrak{m}}athfrak{p}}rod {{\mathfrak{m}}athbb C}C_i {\longrightarrow} {{\mathfrak{m}}athfrak{p}}rod {{\mathfrak{m}}athbb C}C_i \stackrel{+1}{{\longrightarrow}}$$ where the map between products is ${{\mathfrak{m}}athfrak{p}}rod(id-shift)$ and where by product we mean product of chain complexes as opposed to the product inside $D_{qc}(X)$. {\mathfrak{m}}edskip
Note that if ${{\mathfrak{m}}athbb C}C_i$ are coherent sheaves, then ${\textrm{hocolim}_{\rightarrow}} {{\mathfrak{m}}athbb C}C_i = \varinjlim {{\mathfrak{m}}athbb C}C_i$. {\mathfrak{m}}edskip
If $X$ is an n-dimensional variety over a field $k$ and ${\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}$ denotes its dualizing complex, so that ${\mathcal{H}}^{-{\textrm{dim }} X}({\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}) \simeq {\omega}_X$, we define the dualizing functor $D_X$ on $D_{qc}(X)$ as $D_X({\mathbb{F}}F)= R{\mathcal{H}} om({\mathbb{F}}F,{\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet})$. In this context, Grothendieck duality reads as follows:
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $f:X{\rightarrow} Y$ be a proper morphism of quasi-projective varieties over a field $k$. Then for any complex ${\mathbb{F}}F\in D_{qc}(X)$ we have an isomorphism $$Rf_{{{\mathfrak{m}}athfrak{a}}st}D_X({\mathbb{F}}F) \simeq D_Y Rf_{{{\mathfrak{m}}athfrak{a}}st}({\mathbb{F}}F)$$ Assuming that $X$ and $Y$ are smooth, then we equivalently have that for any ${\mathbb{F}}F \in D_{qc}(X)$ and ${\mathcal{E}}\in D_{qc}(Y)$, if ${\omega}_f={\omega}_X\otimes f^{{{\mathfrak{m}}athfrak{a}}st}{\omega}_Y$ denotes the relative dualizing sheaf, there is a functorial isomorphism $$Rf_{{{\mathfrak{m}}athfrak{a}}st} R{\mathcal{H}} om({\mathbb{F}}F,Lf^{{{\mathfrak{m}}athfrak{a}}st}({\mathcal{E}}) \otimes {\omega}_f[{\textrm{dim }} X-{\textrm{dim }} Y]) \simeq R{\mathcal{H}} om (Rf_{{{\mathfrak{m}}athfrak{a}}st}{\mathbb{F}}F,{\mathcal{E}})$$ \label{grothendieck-duality}
{\epsilon}nd{thm}
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\subsection{F-singularities and linear subvarieties of abelian subvarieties}
In this section we recall the basic notions from the theory of F-singularities following {{\mathfrak{m}}athfrak{c}}ite{sch12} and ~{{\mathfrak{m}}athfrak{c}}ite{bst12}. Let $X$ be a separated scheme of finite type over an F-finite perfect field of characteristic $p>0$. A variety is a connected reduced equidimensional scheme over $k$. We denote the canonical sheaf of $X$ by ${\omega}_X={\mathcal{H}}^{-{\textrm{dim }} X}({\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet})$, where ${\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}={\epsilon}ta^{{{\mathfrak{m}}athfrak{a}}st}k$ is the dualizing complex of $X$ and ${\epsilon}ta:X {\rightarrow} k$ is the structural map. If $X$ is normal, a \textit{canonical divisor} on $X$ is any divisor $K_X$ such that ${\omega}_X \simeq {\mathcal{O}}_X(K_X)$. {\mathfrak{m}}edskip
By a pair $(X,{\Delta})$ we mean the combined information of a normal integral scheme $X$ and an effective ${{\mathfrak{m}}athbb Q}$-divisor ${\Delta}$. Denote by $F^e:X {\rightarrow} X$ the e-th iterated absolute Frobenius, where the source has structure map ${\epsilon}ta{{\mathfrak{m}}athfrak{c}}irc F^e:X{\rightarrow} k$. Since $(F^e)^!{\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet} = (F^e)^!{\epsilon}ta^!k={\epsilon}ta^! (F^e)^!k={\epsilon}ta^!k={\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}$. In general for a finite morphism $f:X{\rightarrow} Y$, a coherent sheaf ${\mathbb{F}}F$ on $X$ and a quasi-coherent sheaf ${\mathcal{G}}$ on $Y$, we have the duality ${\mathcal{H}} om(f_{{{\mathfrak{m}}athfrak{a}}st}{\mathbb{F}}F,{\mathcal{G}}) \simeq f_{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{H}} om({\mathbb{F}}F, f^!{\mathcal{G}})$, so the identity ${\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet} {\rightarrow} {\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet} \simeq (F^e)^!{\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}$ yields a trace map $F_{{{\mathfrak{m}}athfrak{a}}st}^e{\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet} {\rightarrow} {\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}$ and taking cohomology we obtain ${{\mathfrak{m}}athbb P}hi^e:F_{{{\mathfrak{m}}athfrak{a}}st}^e{\omega}_X {\rightarrow} {\omega}_X$. Given a variety $X$, the \textit{parameter test submodule} $\tau({\omega}_X)$ of $X$ is the unique smallest ${\mathcal{O}}_X$-submodule $M\subseteq {\omega}_X$, non-zero on any component of $X$, such that ${{\mathfrak{m}}athbb P}hi^1(F_{{{\mathfrak{m}}athfrak{a}}st}M) \subseteq M$. {\mathfrak{m}}edskip
Assume that $(X,{\Delta}elta)$ is a pair such that $K_X+{\Delta}$ is ${{\mathfrak{m}}athbb Q}$-Cartier with index not divisible by $p$. Choose $e>0$ such that $(p^e-1)(K_X+{\Delta})$ is Cartier and define the line bundle ${\mathfrak{m}}athcal{L}_{e,{\Delta}}={\mathcal{O}}_X((1-p^e)(K_X+{\Delta}))$. By ~{{\mathfrak{m}}athfrak{c}}ite{sch09}, there is a canonically determined map ${{\mathfrak{m}}athfrak{p}}hi_{e,{\Delta}elta}:F_{{{\mathfrak{m}}athfrak{a}}st}^e{\mathfrak{m}}athcal{L}_{e,{\Delta}} {\rightarrow} {\mathcal{O}}_X$. We define the \textit{test ideal} $\tau(X,{\Delta})$ of the pair $(X,{\Delta})$ to be the smallest non-zero ideal $J\subseteq {\mathcal{O}}_X$ such that $${{\mathfrak{m}}athfrak{p}}hi_{e,{\Delta}}(F_{{{\mathfrak{m}}athfrak{a}}st}^e(J{{\mathfrak{m}}athfrak{c}}dot {\mathfrak{m}}athcal{L}_{e,{\Delta}})) \subseteq J.$$ Similarly one defines the \textit{non-F-pure ideal} $\sigma(X,{\Delta})$ of $(X,{\Delta})$ to be the the largest such ideal $J\subseteq {\mathcal{O}}_X$. {\mathfrak{m}}edskip
Ever since Hochster and Huneke introduced test ideals and tight closure theory in ~{{\mathfrak{m}}athfrak{c}}ite{hh90}, deep connections have been established between the classes of singularities defined in terms of Frobenius splittings and those arising within the minimal model program. For instance, a normal domain $(R,{\mathfrak{m}})$ of characteristic $p>0$ is said to be F-pure if the inclusion induced by the Frobenius $R\hookrightarrow F_{{{\mathfrak{m}}athfrak{a}}st}^eR{\epsilon}quiv R^{1/p^e}$ splits for every $e$. Similarly, a pair $(R,{\Delta})$ is said to be F-pure if the inclusion $R\hookrightarrow R^{1/p^e} \hookrightarrow R\left(\lceil(p^e-1){\Delta}\rceil\right)^{1/p^e}$ splits for every $e$ and it was shown in ~{{\mathfrak{m}}athfrak{c}}ite{hw02} that F-pure pairs are the analogues of log canonical pairs in characteristic zero, in the sense that if $(X,{\Delta})$ is a log canonical pair, then its reduction mod $p$ $(X_p,{\Delta}_p)$ is F-pure for all $p>>0$. {\mathfrak{m}}edskip
In this paper we shall be concerned with the two classes of F-singularities that we define next. We will be recording the original definition in terms of Frobenius splittings and we will then state their description in terms of test ideals that will be used in the sequel.
{{\mathfrak{m}}athfrak{b}}egin{defi}
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item A pair $(X={{\mathfrak{m}}athbb S}pec R,{\Delta})$ is \textit{strongly F-regular} if for every non-zero element $c\in R$, there exists $e$ such that the map $R\hookrightarrow R^{1/p^e} \hookrightarrow R((p^e-1){\Delta})^{1/p^e}$ that sends $1{\mathfrak{m}}apsto c^{1/p^e} {\mathfrak{m}}apsto c^{1/p^e}$ splits as an $R$-module homomorphism.
\item A reduced connected variety $X$ is F-rational if it is Cohen-Macaulay and there is no non-zero submodule $M\subsetneq {\omega}_R$ such that the Grothendieck trace map ${{\mathfrak{m}}athbb P}hi_X:F_{{{\mathfrak{m}}athfrak{a}}st}^e{\omega}_X {\rightarrow} {\omega}_X$ satisfies ${{\mathfrak{m}}athbb P}hi(F_{{{\mathfrak{m}}athfrak{a}}st}^e M)\subseteq M$.
{\epsilon}nd{enumerate}
{\epsilon}nd{defi} {\mathfrak{m}}edskip
Strongly F-regular pairs are the analog of log terminal pairs in characteristic zero (c.f. ~{{\mathfrak{m}}athfrak{c}}ite{hw02}) and F-rational varieties are the analogue of varieties with rational singularities (c.f. ~{{\mathfrak{m}}athfrak{c}}ite{smi97}). The notion of strong F-regularity is also captured by the test ideal, as the following well-known result shows.
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Proposition 2.4]{hw02})
A pair $(X,{\Delta})$ is strongly F-regular if, and only if, $\tau(X,{\Delta})={\mathcal{O}}_X$.
{\epsilon}nd{lemma}
Assume that $(X,{\Delta})$ is a pair, where $X$ is a normal proper variety over an algebraically closed field of characteristic $p>0$ and ${\Delta}\geq 0$ is a ${{\mathfrak{m}}athbb Q}$-divisor such that $K_X+{\Delta}$ is ${{\mathfrak{m}}athbb Q}$-Cartier with index not divisible by $p$. The map ${{\mathfrak{m}}athfrak{p}}hi_{{\Delta}elta}^e:F_{{{\mathfrak{m}}athfrak{a}}st}^e{\mathfrak{m}}athcal{L}_{e,{\Delta}} {\rightarrow} {\mathcal{O}}_X$ defined in ~{{\mathfrak{m}}athfrak{c}}ite{sch09} restricts to surjective maps $$F_{{{\mathfrak{m}}athfrak{a}}st}^e(\sigma(X,{\Delta}) \otimes {\mathfrak{m}}athcal{L}_{e,{\Delta}}) {\longrightarrow} \sigma(X,D), {{\mathfrak{m}}athfrak{q}}quad F_{{{\mathfrak{m}}athfrak{a}}st}^e(\tau(X,{\Delta}) \otimes {\mathfrak{m}}athcal{L}_{e,{\Delta}}) {\longrightarrow} \tau(X,D).$$
The power of vanishing theorems in characteristic zero relies on the fact that they allow us to lift global sections of adjoint bundles. The full space of global sections is not so well behaved in positive characteristic, so one instead focuses on a subspace of it that is stable under the Frobenius action. {\mathfrak{m}}edskip
If $M$ is any Cartier divisor, one thus defines the subspace $S^0(X, \tau(X,{\Delta}) \otimes {\mathcal{O}}_X(M))$ as
{{\mathfrak{m}}athfrak{b}}egin{multline*} S^0(X, \tau(X,{\Delta}) \otimes {\mathcal{O}}_X(M)) \\ := {{\mathfrak{m}}athfrak{b}}igcap_{n\geq0} Im\left( H^0(X,F_{{{\mathfrak{m}}athfrak{a}}st}^{ne}\tau(X,{\Delta}) \otimes {\mathfrak{m}}athcal{L}_{ne,{\Delta}}(p^{ne}M)) {\longrightarrow} H^0(X,\tau(X,{\Delta}) \otimes {\mathcal{O}}_X(M)) \right) \\ \subseteq H^0(X,{\mathcal{O}}_X(M)) {\epsilon}nd{multline*}
Among the many applications of these subspaces, for instance, they can be used to prove global generation statements: concretely, suppose that $X$ is a $d$-dimensional variety of characteristic $p>0$ and that ${\Delta}$ is a ${{\mathfrak{m}}athbb Q}$-divisor such that $K_X+{\Delta}$ is ${{\mathfrak{m}}athbb Q}$-Cartier with index not divisible by $p$. It was shown in ~{{\mathfrak{m}}athfrak{c}}ite{sch09} that if $L$ and $M$ are Cartier divisors such that $L-K_X-{\Delta}$ is ample and $M$ is ample and globally generated, then the sheaf $\tau(X,{\Delta})\otimes {\mathcal{O}}_X(L+nM)$ is globally generated for all $n\geq d$ by $S^0(X, \tau(X,{\Delta}) \otimes {\mathcal{O}}_X(L+nM))$. {\mathfrak{m}}edskip
\subsection{The Frobenius morphism on Abelian varieties}
Throughout this paper, $A$ will denote an abelian variety of dimension $g$ over a field $k$ and $\hat{A}={{\mathfrak{m}}athbb P}ic^0(A)$ will denote the dual abelian variety
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Proposition 2.13]{hp13})
For a g-dimensional abelian variety $A$ over a field $k$, the following conditions are equivalent.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item There are $p^g$ p-torsion points.
\item The Frobenius action $H^g(A,{\mathcal{O}}_A) {\rightarrow} H^g(A,{\mathcal{O}}_A)$ is bijective
\item The Frobenius action $H^i(A,{\mathcal{O}}_A) {\rightarrow} H^i(A,{\mathcal{O}}_A)$ is bijective for all $0\leq i \leq g$
\item $S^0(A,{\omega}_A) = H^0(A, {\omega}_A)$
{\epsilon}nd{enumerate}
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
If any of these equivalent conditions is satisfied we say that $A$ is \textit{ordinary}. Given an isogeny $\varphi:A {\rightarrow} B$ between abelian varieties of dimension $g$, $A$ is ordinary if and only if $B$ is ordinary. Given a surjective morphism $\varphi:A {\rightarrow} B$ of abelian varieties, if $A$ is ordinary then so is $B$ (see Lemmas 2.14 and 2.14 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}). {\mathfrak{m}}edskip
We finally record a characterization of linear subvarieties of abelian varieties following ~{{\mathfrak{m}}athfrak{c}}ite{pr03}. Let $A$ be an abelian variety endowed with an isogeny $\varphi:A {\rightarrow} A$. We say that $A$ is pure of positive weight if there exist integers $r,s>0$ such that $\varphi^s=F_{p^r}$ for some model of $A$ over ${\mathbb{F}}_{p^r}$. If $A$ is defined over a finite field, we say $A$ is \textit{supersingular} if and only if it is pure of positive weight for the isogeny given by multiplication by $p$; in general, we say that $A$ is supersingular if it is isogenous to a supersingular variety defined over a finite field. We say that $A$ \textit{has no supersingular factors} is there exist no non-trivial homomorphism to an abelian variety which is pure of positive weight for the isogeny given by multiplication by $p$. One sees that $A$ has no supersingular factors if there does not exist a non-trivial homomorphism to a supersingular abelian variety. In particular, if $A$ is an ordinary abelian variety, it follows from the observations in the previous paragraph that $A$ has no supersingular factors (see Lemma 2.16 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}). {\mathfrak{m}}edskip
The following result of Pink and Roessler characterizing linear subvarieties of abelian varieties will be crucial in our proof:
{{\mathfrak{m}}athfrak{b}}egin{thm}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 2.2]{pr03})
Let $A$ be an abelian variety over a field $K$ of characteristic $p>0$ and let $X\subset A$ be a reduced closed subscheme $p(X) \subset X$, where $p$ denotes the isogeny given by multiplication by $p$. If $A$ has no supersingular factors, then all the maximal dimensional irreducible components of $X$ are completely linear (namely, torsion translates of subabelian varieties).
{\epsilon}nd{thm} {\mathfrak{m}}edskip
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\subsection{Generalities on inverse systems and spectral sequences}
\textbf{Mittag-Leffler inverse systems}. We start by recording a few results that will be used in the sequel, most of which are taken directly from ~{{\mathfrak{m}}athfrak{c}}ite{har78}. Recall that a sheaf is countably quasi-coherent if it is quasi-coherent and locally countably generated. Also recall that an inverse system of coherent sheaves $\{{\mathcal{O}}mega_e\}$ is said to satisfy the Mittag-Leffler condition if for any $e\geq 0$, the image of ${\mathcal{O}}mega_{e'}{\rightarrow} {\mathcal{O}}mega_{e}$ stabilizes for $e'$ sufficiently large. The inverse limit functor is always left exact in the sense that if $$\xymatrix{0 {{\mathfrak{m}}athfrak{a}}r[r] & {\mathfrak{m}}athcal{F}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d] & {\mathfrak{m}}athcal{G}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d] & {\mathfrak{m}}athcal{H}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d] & 0 \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & {\mathfrak{m}}athcal{F}_{e-1} {{\mathfrak{m}}athfrak{a}}r[r] & {\mathfrak{m}}athcal{G}_{e-1} {{\mathfrak{m}}athfrak{a}}r[r] & {\mathfrak{m}}athcal{H}_{e-1} {{\mathfrak{m}}athfrak{a}}r[r] & 0}$$ is an exact sequence of inverse systems, then $$0 {\rightarrow} \varprojlim{\mathfrak{m}}athcal{F}_e {\rightarrow} \varprojlim{\mathfrak{m}}athcal{G}_e {\rightarrow} \varprojlim{\mathfrak{m}}athcal{H}_e$$ is exact in the category of quasi-coherent sheaves. By a theorem of Roos (c.f. Proposition I.4.1 in ~{{\mathfrak{m}}athfrak{c}}ite{har78}), the right derived functors $R^i\varprojlim$ are $0$ for $i\geqslant 2$. Hence, we have a long exact sequence $$0 {\rightarrow} \varprojlim{\mathfrak{m}}athcal{F}_e {\rightarrow} \varprojlim{\mathfrak{m}}athcal{G}_e {\rightarrow} \varprojlim{\mathfrak{m}}athcal{H}_e {\rightarrow} R^1\varprojlim{\mathfrak{m}}athcal{F}_e {\rightarrow} R^1\varprojlim{\mathfrak{m}}athcal{G}_e {\rightarrow} R^1\varprojlim{\mathfrak{m}}athcal{H}_e {\rightarrow} 0.$$
We start by recording a characterization of the Mittag-Leffler condition in terms of the first right-derived inverse limit.
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Proposition I.4.9]{har78})
Let $\{{{\mathcal{O}}mega}_e\}_e$ be an inverse system of countably quasi-coherent sheaves on a scheme $X$ of finite type. Then the following conditions are equivalent:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item $\{{{\mathcal{O}}mega}_e\}_e$ is satisfies the Mittag-Leffler condition.
\item $R^1\varprojlim {{\mathcal{O}}mega}_e=0$
\item $R^1\varprojlim {{\mathcal{O}}mega}_e$ is countably quasi-coherent.
{\epsilon}nd{enumerate}
\label{ML-charact}
{\epsilon}nd{lemma}{\mathfrak{m}}edskip
The following is basic result about the cohomology of an inverse system of sheaves:
{{\mathfrak{m}}athfrak{b}}egin{prop}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem I.4.5]{har78})
Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on a variety $X$. Let $T$ be a functor on $D(X)$ which commutes with arbitrary direct products. Suppose that $\{{\mathcal{O}}mega_e\}$ satisfies the Mittag-Leffler condition. Then for each $i$, there is an exact sequence $$0{\rightarrow} R^1\varprojlim R^{i-1}T({\mathcal{O}}mega_e){\rightarrow} R^iT(\varprojlim {\mathcal{O}}mega_e){\rightarrow} \varprojlim R^iT({\mathcal{O}}mega_e) {\rightarrow} 0.$$ In particular, if for some $i$, $\{R^{i-1}T({\mathcal{O}}mega_e)\}$ satisfies the Mittag-Leffler condition, then $R^iT(\varprojlim {\mathcal{O}}mega_e){{\mathfrak{m}}athfrak{c}}ong \varprojlim R^iT({\mathcal{O}}mega_e)$ (by Lemma \ref{ML-charact}). \label{inverse-limits-commute-functor}
{\epsilon}nd{prop} {\mathfrak{m}}edskip
We will be applying this theorem to the push-forward $f_{{{\mathfrak{m}}athfrak{a}}st}$ under a proper morphism of schemes. We finally record a standard statement about the commutation of inverse limits and tensor products. Recall that a sheaf is countably quasi-coherent if it is quasi-coherent and locally countably generated. Then one has the following:
{{\mathfrak{m}}athfrak{b}}egin{lemma}(see ~{{\mathfrak{m}}athfrak{c}}ite[Proposition 4.10]{har78})
Let $\{{\mathbb{F}}F_e\}_e$ be an inverse system of countably quasi-coherent sheaves on a scheme $X$ of finite type and let $E$ be a flat ${\mathcal{O}}_X$-module. Consider the natural map $${{\mathfrak{m}}athfrak{a}}lpha:(\varprojlim {\mathbb{F}}F_e)\otimes E \rightarrow \varprojlim ({\mathbb{F}}F_e \otimes E)$$ If $\varprojlim {\mathbb{F}}F_e$ is countably quasi-coherent then ${{\mathfrak{m}}athfrak{a}}lpha$ is injective and if furthermore $R^1\varprojlim {{\mathcal{O}}mega}_e$ is countably quasi-coherent, then ${{\mathfrak{m}}athfrak{a}}lpha$ is surjective. In particular, if $\{{\mathbb{F}}F_e\}$ is an inverse system of coherent sheaves satisfying the Mittag-Leffler condition on a scheme $X$ with generic point $w$, then there is an isomorphism $$\left(\varprojlim_e {\mathbb{F}}F_e \right) \otimes k(w) {\longrightarrow} \left(\varprojlim_e {\mathbb{F}}F_e \otimes k(w) \right)$$
\label{inverse-limit-tensor-product-commute}
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
\textbf{Inverse systems of convergent spectral sequences}. The following observation is taken from ~{{\mathfrak{m}}athfrak{c}}ite{car08}. Let $\{E(n)\}$ be an inverse system of spectral sequences with morphisms of spectral sequences $E(n) {\rightarrow} E(n-1)$ and consider the tri-graded abelian groups $E_{p,q}^r=\varprojlim_n E_{p,q}^r(n)$, with differentials given by the inverse limits of the differentials in the $E(n)$. Concretely, if $d^r(n)$ is the r-th differential in $E(n)$ and $x(n)\in E_{p,q}^r$, then the r-th differential $d^r$ in the limit sequence $E_{p,q}^r$ is given by $d^r(x(n))=d^r(n)(x(n))$. The resulting object is a spectral sequence provided that $H(E_{p,q}^r,d^r)=E_{p,q}^{r+1}$, which is in turn equivalent to showing that $$H(\varprojlim_n E_{p,q}^r(n),d^r)=\varprojlim_n H(E_{p,q}^r(n),d^r)$$ and this is precisely the statement of Proposition \ref{inverse-limits-commute-functor} above (for the functor of global sections). {\mathfrak{m}}edskip
Note that, in particular, if the terms $E_{p,q}^r$ are all finite-dimensional vector spaces, the hypotheses of Proposition \ref{inverse-limits-commute-functor} hold and all the $\varprojlim^{(1)}$ terms are 0. Besides, given that the inverse limit of the spectral sequences is again a spectral sequence and provided that every spectral sequence in the inverse system is bounded and convergent, one observes that the limit spectral sequence is also convergent: for fixed $p,q$, there is a fixed $N$ such that $E_{p,q}^{\infty}(n)=E_{p,q}^N(n)$.{\mathfrak{m}}edskip
\textbf{Morphisms between spectral sequences}. We recall the definition of a spectral sequence from EGA III [$0_{III}$, 1.1]. Let ${\mathfrak{m}}athscr{C}$ be an abelian category. A \textbf{(biregular) spectral sequence} $E$ on ${\mathfrak{m}}athscr{C}$ consists of the following ingredients:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item A family of objects $\{E^{p,q}_{r}\}$ in ${\mathfrak{m}}athscr{C}$,
where $p,q,r\in {\mathfrak{m}}athbb{Z}$ and $r\geqslant 2$, such that for any
fixed pair $(p,q)$, $E^{p,q}_r$ stabilizes when $r$ is sufficiently
large. We denote the stable objects by $E^{p,q}_\infty$.
\item A family of morphisms $d^{p,q}_r:E^{p,q}_r{\rightarrow}
E^{p+r,q-r+1}_r$ satisfying $$d^{p+r,q-r+1}_r{{\mathfrak{m}}athfrak{c}}irc d^{p,q}_r=0.$$
\item A family of isomorphisms
${{\mathfrak{m}}athfrak{a}}lpha^{p,q}_r:{\kappa}er(d^{p,q}_r)/Im(d^{p-r,q+r-1}_r)\stackrel{\rightarrow}{\sim}
E^{p,q}_{r+1}$.
\item A family of objects $\{E^n\}$ in ${\mathfrak{m}}athscr{C}$. For every
$E_n$, there is a bounded decreasing filtration $\{F^pE^n\}$ in the
sense that there is some $p$ such that $F^pE^n=E^n$ and there is
some $p$ such that $F^pE^n=0$.
\item A family of isomorphisms
${{\mathfrak{m}}athfrak{b}}eta^{p,q}:E^{p,q}_\infty\stackrel{\rightarrow}{\sim}F^pE^{p+q}/F^{p+1}E^{p+q}$.
{\epsilon}nd{enumerate}
We say that the spectral sequence $\{E^{p,q}_r\}$ converges to $\{E^n\}$
and write $$E^{p,q}_2{{\mathfrak{m}}athbb R}ightarrow E^{p+q}.$$
A morphism ${{\mathfrak{m}}athfrak{p}}hi:E{\rightarrow} H$ between two spectral sequences on
${\mathfrak{m}}athscr{C}$ is a family of morphisms ${{\mathfrak{m}}athfrak{p}}hi^{p,q}_r:E^{p,q}_r{\rightarrow}
H^{p,q}_r$ and ${{\mathfrak{m}}athfrak{p}}hi^n:E^n{\rightarrow} H^n$ such that ${{\mathfrak{m}}athfrak{p}}hi$ is compatible
with $d$, ${{\mathfrak{m}}athfrak{a}}lpha$, the filtration and ${{\mathfrak{m}}athfrak{b}}eta$. The following result is useful in order to obtain information about the limiting map ${{\mathfrak{m}}athfrak{p}}hi^n$ from the maps ${{\mathfrak{m}}athfrak{p}}hi_2^{p,q}$.
{{\mathfrak{m}}athfrak{b}}egin{lemma}[see Lemma 2.15 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}]
Let $$\xymatrix{ E_2^{i,j} {{\mathfrak{m}}athfrak{a}}r@2{->}[r] {{\mathfrak{m}}athfrak{a}}r^{{{\mathfrak{m}}athfrak{p}}hi_2^{i,j}}[d] & E^{i+j} {{\mathfrak{m}}athfrak{a}}r^{{{\mathfrak{m}}athfrak{p}}hi^{i+j}}[d] \\
H_2^{i,j} {{\mathfrak{m}}athfrak{a}}r@2{->}[r] & H^{i+j}}$$ be two spectral sequences with
commutative maps. Let $l$ and $a$ be integers. Suppose that
$E_2^{i,l-i}=0$ for $i<a$, $H_2^{i,l-i}=0$ for $i>a$ and
${{\mathfrak{m}}athfrak{p}}hi_2^{a,l-a}=0$. Then ${{\mathfrak{m}}athfrak{p}}hi^l=0$. \label{spec-seq-zero-map}
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
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\subsection{Generic vanishing in positive characteristic}
A smooth projective variety $X$ over an algebraically closed field is said to have maximal Albanese dimension if it admits a generically finite morphism to an abelian variety $X{\rightarrow} A$. Over fields of characteristic zero, the main tool that is employed when studying properties of varieties of maximal Albanese dimension is the generic vanishing theorem of Green and Lazarsfeld (~{{\mathfrak{m}}athfrak{c}}ite{gl87}, ~{{\mathfrak{m}}athfrak{c}}ite{gl91}). Even though it is shown in ~{{\mathfrak{m}}athfrak{c}}ite{hk12} that the obvious generalization of this result to fields of positive characteristic if false, recent work of Hacon and Patakfalvi ~{{\mathfrak{m}}athfrak{c}}ite{hp13} provides a weaker generic vanishing statement in positive characteristic which albeit necessarily weaker, is strong enough prove positive characteristic versions of Kawamata's celebrated characterization of abelian varieties. In this subsection we collect the results of ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that we shall be using throughout. {\mathfrak{m}}edskip
The following is the main theorem in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}:
{{\mathfrak{m}}athfrak{b}}egin{thm}( c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 3.1, Lemma 3.2]{hp13})
Let $A$ be an abelian variety defined over an algebraically closed field of positive characteristic and let ${\mathcal{O}}mega_{e+1} {\rightarrow} {{\mathcal{O}}mega}_e$ be an inverse system of coherent sheaves on $A$.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item If for any sufficiently ample line bundle $L\in {{\mathfrak{m}}athbb P}ic(\hat{A})$ and for any $e>>0$ we have $H^i(A,{{\mathcal{O}}mega}_e \otimes RS_{\hat{A},A}(L)^{\vee})=0$ for every $i>0$, then the complex ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}} RS_{A,\hat{A}}(D_A{{\mathcal{O}}mega}_e)$ (which in general is concentrated in degrees $[-g,\ldots,0]$), is actually a quasi-coherent sheaf concentrated in degree 0, namely ${\Lambda}ambda={\mathcal{H}}^0({\Lambda}ambda)$. Besides ${{\mathcal{O}}mega}=\varprojlim {{\mathcal{O}}mega}_e = \left((-1_A)^{{{\mathfrak{m}}athfrak{a}}st}D_A RS_{\hat{A},A}({\Lambda}ambda)\right)[g]$.
\item The condition in (i) is satisfied for coherent Cartier modules: if $F_{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_0 {\rightarrow} {{\mathcal{O}}mega}_0$ is a coherent Cartier module and we denote ${{\mathcal{O}}mega}_e=F_{{{\mathfrak{m}}athfrak{a}}st}^e{{\mathcal{O}}mega}_0$, then for any ample line bundle $L\in {{\mathfrak{m}}athbb P}ic(\hat{A})$ we have $$H^i(A,{{\mathcal{O}}mega}_e \otimes RS_{\hat{A},A}(L)^{\vee} \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0, {{\mathfrak{m}}athfrak{q}}uad \forall e>>0, {{\mathfrak{m}}athfrak{q}}uad \forall i>0, {{\mathfrak{m}}athfrak{q}}uad \forall {{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$$
{\epsilon}nd{enumerate} \label{generic-vanishing-char-p}
{\epsilon}nd{thm} {\mathfrak{m}}edskip
From the above result and the cohomology and base change theorem one derives the following corollary:
{{\mathfrak{m}}athfrak{b}}egin{corol}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Corollaries 3.5 and 3.6]{hp13})
With the same notations as above we have the following:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item For every ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$ we have ${\Lambda}ambda \otimes k({{\mathfrak{m}}athfrak{a}}lpha) \simeq \varinjlim H^0(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}$, and for every integer $e\geq0$, ${\mathcal{H}}^0({\Lambda}ambda_e) \otimes k({{\mathfrak{m}}athfrak{a}}lpha) \simeq H^0(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}$.
\item There exists a proper closed subset $Z\subset \hat{A}$ such that if $i>0$ and $p^ey\notin Z$ for all $e>>0$, then $\varinjlim H^i(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}=0$. Furthermore, if $W^i=\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}, {{\mathfrak{m}}athfrak{q}}uad \varprojlim H^i(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee} \neq 0\}$, then $$W^i \subset Z'=\overline{{{\mathfrak{m}}athfrak{b}}igcup_{e\geq 0} \left([p_{\hat{A}}^e]^{-1}(Z)\right)_{red}}$$ where $pZ' \subset Z'$ If besides $\hat{A}$ has no supersingular factors, then the top dimensional components of $Z'$ are a finite union of torsion translates of subtori of $A$.
{\epsilon}nd{enumerate} \label{GV-corollary1}
{\epsilon}nd{corol} {\mathfrak{m}}edskip
We quote two more results from ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that will provide a simple proof of a special case of our main theorem:
{{\mathfrak{m}}athfrak{b}}egin{prop}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Proposition 3.17]{hp13})
Let $A$ be an ordinary abelian variety and consider the same notations as above. Then each maximal dimensional irreducible component of the set $Z$ of points ${{\mathfrak{m}}athfrak{a}}lpha \in \hat{A}$ such that the image of the natural map $${\mathcal{H}}^0({\Lambda}ambda_0) \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} {\longrightarrow} {\mathcal{H}}^0({\Lambda}ambda) \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \simeq {\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha}$$ is non-zero, is a torsion translate of an abelian subvariety of $\hat{A}$ and ${\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \neq 0$ if and only if ${{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^e \in Z$. \label{GV-corollary2}
{\epsilon}nd{prop} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{prop}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Lemma 3.9, Corollary 3.10]{hp13})
Let ${{\mathcal{O}}mega}_0$ be a coherent sheaf on an abelian variety $A$ and assume that $F_{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_0 {\rightarrow} {{\mathcal{O}}mega}_0$ is surjective. Then ${{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega} = {{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega}_0$, so that ${{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega}$ is a closed subvariety. Let $\hat{B} \subset \hat{A}$ be an abelian subvariety such that $$V^0({{\mathcal{O}}mega}_0)=\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}: h^0({{\mathcal{O}}mega}_0 \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}) \neq 0\}$$ is contained in finitely many translates of $\hat{B}$. Then $t_x^{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega} \simeq {{\mathcal{O}}mega}$ for every $x\in {\omega}idehat{\hat{A}/\hat{B}}$. In particular, ${{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega}$ is fibered by the projection $A{\rightarrow} B$, namely ${{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega}$ is a union of fibers of $A{\rightarrow} B$. \label{GV-corollary3}
{\epsilon}nd{prop} {\mathfrak{m}}edskip
Note that, in particular, Proposition \ref{GV-corollary3} applies to the subvariety $$Z=\left\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}: {{\mathfrak{m}}athfrak{q}}uad Im \left({\mathcal{H}}^0({\Lambda}ambda_0)\otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} {\longrightarrow} {\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \right) \neq 0\right\}$$ from Proposition \ref{GV-corollary2}. {\mathfrak{m}}edskip
Finally, grounding on the results in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}, part of Pareschi and Popa's generic vanishing theory was extended to positive characteristic in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}. The main result in that paper is the following:
{{\mathfrak{m}}athfrak{b}}egin{thm} Let $A$ be an abelian variety. Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on $A$ satisfying the Mittag-Leffler condition and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let ${\Lambda}ambda_e=R\hat{S}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$. The following are equivalent:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item[(1)] For any ample line bundle $L$ on $\hat{A}$, $H^i(A,{\mathcal{O}}mega\otimes \hat{L}^\vee)=0$ for any $i>0$.
\item[(1')] For any fixed positive integer $e$ and any $i>0$, the homomorphism $$H^i(A, {\mathcal{O}}mega\otimes \hat{L}^\vee) {\rightarrow} H^i(A,{\mathcal{O}}mega_e\otimes \hat{L}^\vee)$$ is 0 for any sufficiently ample line bundle $L$.
\item[(2)] ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$.
{\epsilon}nd{enumerate}
These conditions imply the following:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item[(3)] For any scheme-theoretical point $P\in A$, if ${\textrm{dim }} P>g-i$, then $P$ is not in the support of the image of $$\varprojlim R^i\hat{S}({\mathcal{O}}mega_e) {\rightarrow} R^i\hat{S}({\mathcal{O}}mega_e)$$ for any $e$.
{\epsilon}nd{enumerate}
If $\{R^i\hat{S}({\mathcal{O}}mega_e)\}$ satisfies the Mittag-Leffler condition for any $i\geq 0$, then (3) also implies (1), (1') and (2) and, moreover, the support of the image of the map in (3) is a closed subset.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
We also record the following variant of the implication $(2){{\mathfrak{m}}athbb R}ightarrow (3)$ in the previous theorem.
{{\mathfrak{m}}athfrak{b}}egin{prop}
Let ${{\mathfrak{m}}athfrak{p}}i:A {{\mathfrak{m}}athfrak{p}}roj W$ be a projection between abelian varieties with generic fiber dimension $f$ and with ${\textrm{dim }} W=k$. Let $\{{{\mathcal{O}}mega}_e\}_e$ be a Cartier module on $A$ and let $S_{A,\hat{W}}$ be the Fourier-Mukai functor with kernel $\left({{\mathfrak{m}}athfrak{p}}i \times 1_{\hat{W}}\right)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$. Denote ${\Lambda}_e = RS_{A,\hat{W}} (D_A ({{\mathcal{O}}mega}_e))$. If $P\in \hat{W}$ is a scheme-theoretic point with ${\textrm{dim }}(P)>k+f-{\epsilon}ll$, then $P$ is not in the support of the image of the map $$\varprojlim_e R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) {\longrightarrow} R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)$$ Moreover, if the inverse system $\{R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the Mittag-Leffler condition, then the support of the image of the above map is closed and its codimension is $\geq {\epsilon}ll-f$. \label{GV-k}
{\epsilon}nd{prop}
{{\mathfrak{m}}athfrak{b}}egin{pf}
The proof is identical (modulo shifts) to that of Theorem 4.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}, but we include it for the sake of completeness. {\mathfrak{m}}edskip
We need to show that if $P\in \hat{W}$ is a scheme-theoretic point with ${\textrm{dim }}(P)>k+f-{\epsilon}ll$, then $$\left(R^{{\epsilon}ll}S_{A,\hat{W}}(\varprojlim {{\mathcal{O}}mega}_e)\right)_P \stackrel{0}{{\longrightarrow}} \left(R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_P$$
Note in the first place that for any ${\epsilon}ll$, we have the following isomorphisms
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Note that, in particular, Proposition \ref{GV-corollary3} applies to the subvariety $$Z=\left\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}: {{\mathfrak{m}}athfrak{q}}uad Im \left({\mathcal{H}}^0({\Lambda}ambda_0)\otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} {\longrightarrow} {\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \right) \neq 0\right\}$$ from Proposition \ref{GV-corollary2}. {\mathfrak{m}}edskip
Finally, grounding on the results in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}, part of Pareschi and Popa's generic vanishing theory was extended to positive characteristic in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}. The main result in that paper is the following:
{{\mathfrak{m}}athfrak{b}}egin{thm} Let $A$ be an abelian variety. Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on $A$ satisfying the Mittag-Leffler condition and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let ${\Lambda}ambda_e=R\hat{S}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$. The following are equivalent:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item[(1)] For any ample line bundle $L$ on $\hat{A}$, $H^i(A,{\mathcal{O}}mega\otimes \hat{L}^\vee)=0$ for any $i>0$.
\item[(1')] For any fixed positive integer $e$ and any $i>0$, the homomorphism $$H^i(A, {\mathcal{O}}mega\otimes \hat{L}^\vee) {\rightarrow} H^i(A,{\mathcal{O}}mega_e\otimes \hat{L}^\vee)$$ is 0 for any sufficiently ample line bundle $L$.
\item[(2)] ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$.
{\epsilon}nd{enumerate}
These conditions imply the following:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item[(3)] For any scheme-theoretical point $P\in A$, if ${\textrm{dim }} P>g-i$, then $P$ is not in the support of the image of $$\varprojlim R^i\hat{S}({\mathcal{O}}mega_e) {\rightarrow} R^i\hat{S}({\mathcal{O}}mega_e)$$ for any $e$.
{\epsilon}nd{enumerate}
If $\{R^i\hat{S}({\mathcal{O}}mega_e)\}$ satisfies the Mittag-Leffler condition for any $i\geq 0$, then (3) also implies (1), (1') and (2) and, moreover, the support of the image of the map in (3) is a closed subset.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
We also record the following variant of the implication $(2){{\mathfrak{m}}athbb R}ightarrow (3)$ in the previous theorem.
{{\mathfrak{m}}athfrak{b}}egin{prop}
Let ${{\mathfrak{m}}athfrak{p}}i:A {{\mathfrak{m}}athfrak{p}}roj W$ be a projection between abelian varieties with generic fiber dimension $f$ and with ${\textrm{dim }} W=k$. Let $\{{{\mathcal{O}}mega}_e\}_e$ be a Cartier module on $A$ and let $S_{A,\hat{W}}$ be the Fourier-Mukai functor with kernel $\left({{\mathfrak{m}}athfrak{p}}i \times 1_{\hat{W}}\right)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$. Denote ${\Lambda}_e = RS_{A,\hat{W}} (D_A ({{\mathcal{O}}mega}_e))$. If $P\in \hat{W}$ is a scheme-theoretic point with ${\textrm{dim }}(P)>k+f-{\epsilon}ll$, then $P$ is not in the support of the image of the map $$\varprojlim_e R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) {\longrightarrow} R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)$$ Moreover, if the inverse system $\{R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the Mittag-Leffler condition, then the support of the image of the above map is closed and its codimension is $\geq {\epsilon}ll-f$. \label{GV-k}
{\epsilon}nd{prop}
{{\mathfrak{m}}athfrak{b}}egin{pf}
The proof is identical (modulo shifts) to that of Theorem 4.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}, but we include it for the sake of completeness. {\mathfrak{m}}edskip
We need to show that if $P\in \hat{W}$ is a scheme-theoretic point with ${\textrm{dim }}(P)>k+f-{\epsilon}ll$, then $$\left(R^{{\epsilon}ll}S_{A,\hat{W}}(\varprojlim {{\mathcal{O}}mega}_e)\right)_P \stackrel{0}{{\longrightarrow}} \left(R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_P$$
Note in the first place that for any ${\epsilon}ll$, we have the following isomorphisms
{{\mathfrak{m}}athfrak{b}}egin{eqnarray} {\mathcal{E}} xt^p({\Lambda}_e,{\mathfrak{m}}athcal{O}_{\hat{W}}) &\simeq&
{\mathfrak{m}}athcal{H}^{p-k}(D_{\hat{W}}({\Lambda}_e)) \simeq
{\mathfrak{m}}athcal{H}^{p-k}(D_{\hat{W}}(RS_{A,\hat{W}}(D_A({{\mathcal{O}}mega}_e)))) \nonumber \\
&\stackrel{[{{\mathfrak{m}}athfrak{a}}st]}{\simeq}& {\mathfrak{m}}athcal{H}^{p}(
R\tilde{S}_{A,\hat{W}}(D_A(D_A({{\mathcal{O}}mega}_e)))) \simeq
{\mathfrak{m}}athcal{H}^{p}(R\tilde{S}_{A,\hat{W}}({{\mathcal{O}}mega}_e)) \label{comp1} {\epsilon}nd{eqnarray}
and
{{\mathfrak{m}}athfrak{b}}egin{eqnarray} {\mathcal{E}} xt^p({\Lambda},{\mathfrak{m}}athcal{O}_{\hat{W}}) &\simeq&
{\mathfrak{m}}athcal{H}^{p-g}(D_{\hat{W}}({\Lambda})) \simeq
{\mathfrak{m}}athcal{H}^{p-g}(D_{\hat{W}}({\textrm{hocolim}_{\rightarrow}}_e RS_{A,\hat{W}}(D_A({{\mathcal{O}}mega}_e)))) \nonumber \\
&\simeq& {\mathfrak{m}}athcal{H}^{p-g}({\textrm{holim}_{\leftarrow}}_e D_{\hat{W}}(RS_{A,\hat{W}}(D_A({{\mathcal{O}}mega}_e)))) \nonumber \\ &\simeq&
{\mathfrak{m}}athcal{H}^p({\textrm{holim}_{\leftarrow}}_e (-1_{\hat{W}})^* RS_{A,\hat{W}}(\tilde{{{\mathcal{O}}mega}}_e)) \nonumber \\
&\simeq&
(-1_{\hat{W}})^*{\mathcal{H}}^p\left({\textrm{holim}_{\leftarrow}}_e RS_{A,\hat{W}}(\tilde{{{\mathcal{O}}mega}}_e)\right) \label{comp2} {\epsilon}nd{eqnarray}
where in $[{{\mathfrak{m}}athfrak{a}}st]$ we used Lemma 2.2 in ~{{\mathfrak{m}}athfrak{c}}ite{pp11}. Here, if $S$ is the Fourier-Mukai functor with kernel ${{\mathfrak{m}}athbb P}P$, $\tilde{S}$ denotes the Fourier-Mukai functor with kernel ${{\mathfrak{m}}athbb P}P^{\vee}$; the codimension computation is unaffected by this, so we omit the tildes in the remainder of the proof. {\mathfrak{m}}edskip
From the following factorization of the map $R^{{\epsilon}ll}S_{A,\hat{W}}(\varprojlim {{\mathcal{O}}mega}_e) {\rightarrow} R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)$
$$R^{{\epsilon}ll}S_{A,\hat{W}}(\varprojlim_e {{\mathcal{O}}mega}_e) {\longrightarrow} {\mathcal{H}}^{{\epsilon}ll}\left( {\textrm{holim}_{\leftarrow}} RS_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) {\longrightarrow} {\mathcal{H}}^{{\epsilon}ll}\left( RS_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right)$$
we see that it suffices to show that the map $${\mathcal{E}} xt^{{\epsilon}ll}({\Lambda},{\mathcal{O}}_{\hat{W}})_P {\longrightarrow} {\mathcal{E}} xt^{{\epsilon}ll}({\Lambda}_e,{\mathcal{O}}_{\hat{W}})_P$$ is zero for ${\textrm{dim }}(P)>k+f-{\epsilon}ll$. In order to see this, we may proceed as in the proof of Theorem 4.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}, computing the above map via the commutative diagram $$\xymatrix{ {\mathcal{E}} xt^i({\mathcal{H}}^j({\Lambda}),{\mathcal{O}}_{\hat{W}})_P {{\mathfrak{m}}athfrak{a}}r[d]_{{{\mathfrak{m}}athfrak{p}}hi^{i,j}} {{\mathfrak{m}}athfrak{a}}r@{=>}[r] & {\mathcal{E}} xt^{{\epsilon}ll}({\Lambda},{\mathcal{O}}_{\hat{W}})_P {{\mathfrak{m}}athfrak{a}}r[d]^{{{\mathfrak{m}}athfrak{p}}hi^{{\epsilon}ll}} \\ {\mathcal{E}} xt^i({\mathcal{H}}^j({\Lambda}_e),{\mathcal{O}}_{\hat{W}})_P {{\mathfrak{m}}athfrak{a}}r@{=>}[r] & {\mathcal{E}} xt^{{\epsilon}ll}({\Lambda}_e,{\mathcal{O}}_{\hat{W}})_P}$$
with $i-j={\epsilon}ll$. We seek to apply Lemma \ref{spec-seq-zero-map} with $a={\epsilon}ll-1-f$. If $i>a$, then $i \geq {\epsilon}ll-f > [k+f-{\textrm{dim }}(P)] -f = k-{\textrm{dim }}(P)$ and hence ${\mathcal{E}} xt^i({\mathcal{H}}^j({\Lambda}_e),{\mathcal{O}}_{\hat{W}})_P=0$ and if $i\leq a$, then $j = i-{\epsilon}ll \leq ({\epsilon}ll -1 -f) -{\epsilon}ll = -1-f$, so that ${\mathcal{H}}^j({\Lambda})=0$ (c.f. proof of Theorem 3.1.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}). Lemma \ref{spec-seq-zero-map} then implies that ${{\mathfrak{m}}athfrak{p}}hi^{{\epsilon}ll}=0$ as claimed.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
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\section{Main technical result}
Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on a $g$-dimensional abelian variety satisfying the Mittag-Leffler condition and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let ${\Lambda}ambda_e=RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$. Suppose that $\{{{\mathcal{O}}mega}_e\}$ is a GV-inverse system, in the sense that ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$. It was shown in Theorem 4.2 of ~{{\mathfrak{m}}athfrak{c}}ite{wz14} that if ${\mathcal{H}}^0({\Lambda})$ is torsion-free, then the maps $$\left(\varprojlim R^kS_{A,\hat{A}}({{\mathcal{O}}mega}_e)\right)_P {\longrightarrow} R^kS_{A,\hat{A}}({{\mathcal{O}}mega}_e)_P$$ are zero for any point $P\in \hat{A}$ such that ${\textrm{dim }}(P) \geq g-k$. We next show a partial converse to this statement. {\mathfrak{m}}edskip
In the sequel, we will say that ${\mathcal{H}}^0({\Lambda})$ has torsion if it is not torsion-free. More concretely, we will say that ${\mathcal{H}}^0({\Lambda})$ has torsion at a point $P$ if there exists a section $s \in {\mathcal{O}}_{\hat{A}}$ such that the multiplication map ${\mathcal{H}}^0({\Lambda})_P \stackrel{\times s_P}{{\longrightarrow}} {\mathcal{H}}^0({\Lambda})_P$ is not injective. {\mathfrak{m}}edskip
Before stating our main result we need to introduce some notation. Consider the following commutative diagram {{\mathfrak{m}}athfrak{b}}egin{equation} \xymatrix{0 {{\mathfrak{m}}athfrak{a}}r[r] & {\Lambda}^t {{\mathfrak{m}}athfrak{a}}r[r] & {\Lambda} {{\mathfrak{m}}athfrak{a}}r[r] & {\mathbb{F}}F={\Lambda}/{\Lambda}^t {{\mathfrak{m}}athfrak{a}}r[r] & 0 \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & \tilde{{\Lambda}}_e^t {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & \tilde{{\Lambda}}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & {\mathbb{F}}F_e {{\mathfrak{m}}athfrak{a}}r[u] {{\mathfrak{m}}athfrak{a}}r[r] & 0} \label{main-thm-notation} {\epsilon}nd{equation} where ${\Lambda}^t$ denotes the torsion subsheaf of ${\Lambda}$, $\tilde{{\Lambda}}_e=\textrm{Im}({\Lambda}_e {\rightarrow} {\Lambda})$, ${\mathbb{F}}F_e=\textrm{Im}(\tilde{{\Lambda}}_e {\rightarrow} {\mathbb{F}}F)$ and $\tilde{{\Lambda}}_e^t={\kappa}er\left( \tilde{{\Lambda}}_e {\rightarrow} {\mathbb{F}}F\right)$. It is easy to see that the second row is exact and that $\tilde{{\Lambda}}_e^t$ is the torsion subsheaf of $\tilde{{\Lambda}}_e$. It is also clear by construction that ${\Lambda} = \varinjlim_e \tilde{{\Lambda}}_e$ (c.f. Theorem \ref{generic-vanishing-char-p}). {\mathfrak{m}}edskip
Since the direct limit is exact, by its universal property there is also a commutative diagram $$\xymatrix{0 {{\mathfrak{m}}athfrak{a}}r[r] & {\Lambda}^t {{\mathfrak{m}}athfrak{a}}r[r] & {\Lambda} {{\mathfrak{m}}athfrak{a}}r[r] & {\mathbb{F}}F={\Lambda}/{\Lambda}^t {{\mathfrak{m}}athfrak{a}}r[r] & 0 \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & \varinjlim_e \tilde{{\Lambda}}_e^t {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & \varinjlim_e \tilde{{\Lambda}}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u]_{\simeq} & \varinjlim_e {\mathbb{F}}F_e {{\mathfrak{m}}athfrak{a}}r[u] {{\mathfrak{m}}athfrak{a}}r[r] & 0}$$ Note that ${\Lambda}^t \simeq \varinjlim_e \tilde{{\Lambda}}_e^t$. Indeed, an element ${\epsilon}ta \in {\Lambda}^t \hookrightarrow {\Lambda}$ lifts to a class $[\tilde{{\epsilon}ta}_e] \in \varinjlim \tilde{{\Lambda}}_e = {\Lambda}$ and this class maps to 0 under the composition $\varinjlim \tilde{{\Lambda}}_e {\rightarrow} {\mathbb{F}}F$, so it lies in $\varinjlim \tilde{{\Lambda}}_e^t$ (by exactness of the direct limit). By the 5-lemma, we have that the right vertical map is also an isomorphism. {\mathfrak{m}}edskip
We are now ready to state our main technical result: {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on a $g$-dimensional abelian variety satisfying the Mittag-Leffler condition. Let ${\Lambda}ambda_e=RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$. Denote as above $\tilde{{\Lambda}}_e=\textrm{Im}({\Lambda}_e {\rightarrow} {\Lambda})$ and define $\tilde{{\mathcal{O}}mega}_e = RS_{\hat{A},A} (D_{\hat{A}} (\tilde{{\Lambda}}_e))$. Suppose that $\{{{\mathcal{O}}mega}_e\}$ is a GV-inverse system, in the sense that ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$. If ${\mathcal{H}}^0({\Lambda})$ has a torsion point $P$ of maximal dimension $g-k$, then $$\varprojlim \left(R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0.$$ Equivalently, the maps $$\varprojlim \left(R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) {\rightarrow} R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)$$ are non-zero for every $e\gg0$. \label{torsion-non-zero-map}
{\epsilon}nd{thm}
{{\mathfrak{m}}athfrak{b}}egin{pf}
We start by performing a sequence of reductions. {\mathfrak{m}}edskip
\textbf{Reduction 1}: With the notation introduced in diagram (\ref{main-thm-notation}), in order to show that $$\varprojlim \left(R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0$$ it is sufficient to show that $$\varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right] \neq 0.$$
Indeed, consider the long exact sequence for ${\mathcal{E}} xt$ induced by the short exact sequence $$0 {\longrightarrow} \tilde{{\Lambda}}_e^t {\longrightarrow} \tilde{{\Lambda}}_e {\longrightarrow} {\mathbb{F}}F_e {\longrightarrow} 0$$ namely $${{\mathfrak{m}}athfrak{c}}dots {\longrightarrow} {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e, {\mathcal{O}}_{\hat{A}}) {\longrightarrow} {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) {\longrightarrow} {\mathcal{E}} xt^{k+1}({\mathbb{F}}F_e, {\mathcal{O}}_{\hat{A}}) {\longrightarrow} {{\mathfrak{m}}athfrak{c}}dots$$ By Lemma 6.3 in ~{{\mathfrak{m}}athfrak{c}}ite{pp08b} it follows that ${\mathcal{E}} xt^{k+1}({\mathbb{F}}F_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) = 0$ for all $e$, so since $\otimes k(P)$ is right-exact, we obtain a surjection $${\mathcal{E}} xt^k(\tilde{{\Lambda}}_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) {\longrightarrow} {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P)$$ and hence\footnote{Note that the system $\left\{ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P)\right\}_e$ satisfies the ML-condition.} a surjection $$\varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right] {\longrightarrow} \varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right].$$
Therefore, if $\varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right] \neq0$, then $$\varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right] \stackrel{[{{\mathfrak{m}}athfrak{a}}st]}{\simeq} \varprojlim \left(R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0$$ as claimed, where the isomorphism $[{{\mathfrak{m}}athfrak{a}}st]$ follows from the following computation
{{\mathfrak{m}}athfrak{b}}egin{eqnarray} {\mathcal{E}} xt^p(\tilde{{\Lambda}}_e,{\mathfrak{m}}athcal{O}_{\hat{A}}) &\simeq&
{\mathfrak{m}}athcal{H}^{p-g} \left( D_{\hat{A}}(\tilde{{\Lambda}}_e) \right) \simeq
{\mathfrak{m}}athcal{H}^{p-g} ( (-1_{\hat{A}})^{{{\mathfrak{m}}athfrak{a}}st} RS_{A,\hat{A}} \overbrace{RS_{\hat{A},A} (D_{\hat{A}}(\tilde{{\Lambda}}_e))}^{:=\tilde{{{\mathcal{O}}mega}}_e} [g]) \nonumber \\
&\simeq& {\mathfrak{m}}athcal{H}^p \left( (-1_{\hat{A}})^* RS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e) \right). {\epsilon}nd{eqnarray}
\textbf{Reduction 2}: Denoting $i:Z:=\overline{\{P\}}\hookrightarrow \hat{A}$, we next reduce to showing that $$\varprojlim_e \left[ {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}^{t}_e,{\mathcal{O}}_Z) \otimes k(P) \right] \neq0$$
In order to see this, it suffices to show that for every $e$ there is an isomorphism {{\mathfrak{m}}athfrak{b}}egin{equation} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P) \simeq {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t, {\mathcal{O}}_Z) \otimes k(P). \label{key-iso} {\epsilon}nd{equation}
But observe that {{\mathfrak{m}}athfrak{b}}egin{multline*} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}})_{|Z} \otimes k(P) \simeq L^0i^{{{\mathfrak{m}}athfrak{a}}st} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P) \stackrel{[1]}{\simeq} {\mathcal{H}}^k\left( Li^{{{\mathfrak{m}}athfrak{a}}st} R{\mathcal{H}} om(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \right) \otimes k(P) \\ \stackrel{[2]}{\simeq} {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}^t_e, {\mathcal{O}}_Z) \otimes k(P) {\epsilon}nd{multline*}
where:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item The isomorphism in [1] follows from Grothendieck's spectral sequence\footnote{C.f. equation (3.10) in ~{{\mathfrak{m}}athfrak{c}}ite{huy06}. Also note that since tensoring by $k(P)$ is exact on $D(Z)$, there is a spectral sequence as written.} $$E_2^{p,q} = L^pi^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^q(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) {\Lambda}ongrightarrow L^{p+q}i^{{{\mathfrak{m}}athfrak{a}}st}R{\mathcal{H}} om(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P)$$ Note in the first place that ${\mathcal{E}} xt^q(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}})=0$ near $P$ for all $q<k$. Indeed, since $\tilde{{\Lambda}}_e^t$ is supported on $Z$\footnote{Note that if $Z$ is an irreducible component of ${{\mathfrak{m}}athbb S}upp {\Lambda}$, then it is also an irreducible component of ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}_e)$ for every $e>>0$. Let $Z$ be one such component, denote by ${\mathcal{I}}_Z$ its ideal sheaf and take a section ${\epsilon}ta=\{{\epsilon}ta_e\}\in {\Lambda}$ supported on $Z$. It is then clear that $Z\subset {{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}_e)$ for every $e>>0$. Now, if $f\in {\mathcal{I}}_Z$ is in $\operatorname{ann}({\epsilon}ta)$, it is clear that $f\in \operatorname{ann}({\epsilon}ta_e)$ for $e>>0$, so that $Z={{\mathfrak{m}}athbb S}upp {\epsilon}ta_e$ for all $e>>0$, as claimed.} near $P$, which has codimension $k$, our claim follows from the fact that ${\mathcal{E}} xt^q({{\mathfrak{m}}athfrak{b}}ullet,{\mathcal{O}}_{\hat{A}})=0$ for all $q<{{\mathfrak{m}}athfrak{c}}odim{{\mathfrak{m}}athbb S}upp({{\mathfrak{m}}athfrak{b}}ullet)$ (c.f. Lemma 6.1 in ~{{\mathfrak{m}}athfrak{c}}ite{pp08b}). {\mathfrak{m}}edskip
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{{\mathfrak{m}}athfrak{b}}egin{eqnarray} {\mathcal{E}} xt^p(\tilde{{\Lambda}}_e,{\mathfrak{m}}athcal{O}_{\hat{A}}) &\simeq&
{\mathfrak{m}}athcal{H}^{p-g} \left( D_{\hat{A}}(\tilde{{\Lambda}}_e) \right) \simeq
{\mathfrak{m}}athcal{H}^{p-g} ( (-1_{\hat{A}})^{{{\mathfrak{m}}athfrak{a}}st} RS_{A,\hat{A}} \overbrace{RS_{\hat{A},A} (D_{\hat{A}}(\tilde{{\Lambda}}_e))}^{:=\tilde{{{\mathcal{O}}mega}}_e} [g]) \nonumber \\
&\simeq& {\mathfrak{m}}athcal{H}^p \left( (-1_{\hat{A}})^* RS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e) \right). {\epsilon}nd{eqnarray}
\textbf{Reduction 2}: Denoting $i:Z:=\overline{\{P\}}\hookrightarrow \hat{A}$, we next reduce to showing that $$\varprojlim_e \left[ {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}^{t}_e,{\mathcal{O}}_Z) \otimes k(P) \right] \neq0$$
In order to see this, it suffices to show that for every $e$ there is an isomorphism {{\mathfrak{m}}athfrak{b}}egin{equation} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P) \simeq {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t, {\mathcal{O}}_Z) \otimes k(P). \label{key-iso} {\epsilon}nd{equation}
But observe that {{\mathfrak{m}}athfrak{b}}egin{multline*} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}})_{|Z} \otimes k(P) \simeq L^0i^{{{\mathfrak{m}}athfrak{a}}st} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P) \stackrel{[1]}{\simeq} {\mathcal{H}}^k\left( Li^{{{\mathfrak{m}}athfrak{a}}st} R{\mathcal{H}} om(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \right) \otimes k(P) \\ \stackrel{[2]}{\simeq} {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}^t_e, {\mathcal{O}}_Z) \otimes k(P) {\epsilon}nd{multline*}
where:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item The isomorphism in [1] follows from Grothendieck's spectral sequence\footnote{C.f. equation (3.10) in ~{{\mathfrak{m}}athfrak{c}}ite{huy06}. Also note that since tensoring by $k(P)$ is exact on $D(Z)$, there is a spectral sequence as written.} $$E_2^{p,q} = L^pi^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^q(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) {\Lambda}ongrightarrow L^{p+q}i^{{{\mathfrak{m}}athfrak{a}}st}R{\mathcal{H}} om(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P)$$ Note in the first place that ${\mathcal{E}} xt^q(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}})=0$ near $P$ for all $q<k$. Indeed, since $\tilde{{\Lambda}}_e^t$ is supported on $Z$\footnote{Note that if $Z$ is an irreducible component of ${{\mathfrak{m}}athbb S}upp {\Lambda}$, then it is also an irreducible component of ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}_e)$ for every $e>>0$. Let $Z$ be one such component, denote by ${\mathcal{I}}_Z$ its ideal sheaf and take a section ${\epsilon}ta=\{{\epsilon}ta_e\}\in {\Lambda}$ supported on $Z$. It is then clear that $Z\subset {{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}_e)$ for every $e>>0$. Now, if $f\in {\mathcal{I}}_Z$ is in $\operatorname{ann}({\epsilon}ta)$, it is clear that $f\in \operatorname{ann}({\epsilon}ta_e)$ for $e>>0$, so that $Z={{\mathfrak{m}}athbb S}upp {\epsilon}ta_e$ for all $e>>0$, as claimed.} near $P$, which has codimension $k$, our claim follows from the fact that ${\mathcal{E}} xt^q({{\mathfrak{m}}athfrak{b}}ullet,{\mathcal{O}}_{\hat{A}})=0$ for all $q<{{\mathfrak{m}}athfrak{c}}odim{{\mathfrak{m}}athbb S}upp({{\mathfrak{m}}athfrak{b}}ullet)$ (c.f. Lemma 6.1 in ~{{\mathfrak{m}}athfrak{c}}ite{pp08b}). {\mathfrak{m}}edskip
The differentials coming out of $E_2^{0,k} = L^0i^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}})$ are hence trivial and the differential targeting $E_2^{0,k}$ is $$d_2^{-2,k+1}: L^{-2}i^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^{k+1}(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) {\longrightarrow} L^0i^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P)$$ which is also trivial since there is an open neighborhood $U$ of $P$ over which $$\left[L^{-2}i^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^{k+1}(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}})\right]_{|U} \simeq L^{-2} i_{U}^{{{\mathfrak{m}}athfrak{a}}st} {\mathcal{E}} xt^{k+1}(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}})_{U} = 0$$ where $i_U: Z {{\mathfrak{m}}athfrak{c}}ap U \hookrightarrow U$ and where the last vanishing follows from the coherence of $\tilde{{\Lambda}}^t_e$ and the fact that $P$ has codimension $k$.
\item The isomorphism in [2] follows from the $Lf^{{{\mathfrak{m}}athfrak{a}}st}R{\mathcal{H}} om({\mathbb{F}}F^{{{\mathfrak{m}}athfrak{b}}ullet},{\mathfrak{m}}athcal{G}^{{{\mathfrak{m}}athfrak{b}}ullet}) \simeq R{\mathcal{H}} om(Lf^{{{\mathfrak{m}}athfrak{a}}st}{\mathbb{F}}F^{{{\mathfrak{m}}athfrak{b}}ullet}, Lf^{{{\mathfrak{m}}athfrak{a}}st}{\mathfrak{m}}athcal{G}^{{{\mathfrak{m}}athfrak{b}}ullet})$ (c.f. equation (3.17) in ~{{\mathfrak{m}}athfrak{c}}ite{huy06}).
{\epsilon}nd{enumerate}
Finally, in order to show that $$\varprojlim_e \left( {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st} \tilde{{\Lambda}}_e^{t},{\mathcal{O}}_Z)\otimes k(P) \right)$$ is non-zero, we will use the isomorphism $$\varprojlim_e \left( {\mathcal{E}} xt^k_{{\mathcal{O}}_Z} (Li^{{{\mathfrak{m}}athfrak{a}}st} \tilde{{\Lambda}}_e^{t},{\mathcal{O}}_Z)\otimes k(P) \right) = \varprojlim_e {\mathcal{E}} xt^k_{k(P)} \left( Li^{{{\mathfrak{m}}athfrak{a}}st} \tilde{{\Lambda}}_e^{t} \otimes k(P), k(P) \right) \neq 0$$
where we used that $k(P)\simeq {\mathcal{O}}_P$ and Proposition III.6.8 in ~{{\mathfrak{m}}athfrak{c}}ite{har77}. Denote by $i:Z=\overline{\{P\}} \hookrightarrow \hat{A}$ the inclusion. By Grothendieck duality (c.f. discussion in section 2.3 from ~{{\mathfrak{m}}athfrak{c}}ite{bst12}), since all the higher direct images of a closed immersion are zero, we have a functorial isomorphism {{\mathfrak{m}}athfrak{b}}egin{equation} R{\mathcal{H}} om_{\hat{A}} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e \otimes k(P) \right], k(P)[-k] \right) \simeq Ri_{{{\mathfrak{m}}athfrak{a}}st} R{\mathcal{H}} om_{Z} \left( Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e \otimes k(P), Li^! k(P) \right) \label{GD} {\epsilon}nd{equation}
Taking k-th cohomology, we obtain {{\mathfrak{m}}athfrak{b}}egin{eqnarray}{\mathcal{H}} om_{k(P)} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[ Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P) \right], k(P) \right) &\simeq& {\mathcal{H}}^k\left(Ri_{{{\mathfrak{m}}athfrak{a}}st}R{\mathcal{H}} om_{k(P)}(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P), Li^! k(P))\right) \nonumber \\ &\simeq& i_{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^k_{k(P)}(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P), Li^! k(P)). \label{GD2} {\epsilon}nd{eqnarray}
Note that the inverse limit of the left hand side is $$\varprojlim {\mathcal{H}} om_{\hat{A}} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[ Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P) \right], k(P) \right) = {\mathcal{H}} om_{\hat{A}} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[ Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} \otimes k(P) \right], k(P) \right).$$
We claim that the latter sheaf is non-zero. Indeed, note that {{\mathfrak{m}}athfrak{b}}egin{equation} H om_{\hat{A}} \left( i^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} \otimes k(P) , k(P) \right) \neq0 \label{non-zero-hom} {\epsilon}nd{equation}
since it is simply the $k(P)$-dual of the non-zero $k(P)$-vector space ${\Lambda}_{|Z} \otimes k(P)$. The natural map $Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} {\rightarrow} L^0 i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda}$ induces $$R^0i_{{{\mathfrak{m}}athfrak{a}}st} \left[ Li^{{{\mathfrak{m}}athfrak{a}}st}({\Lambda}) \otimes k(P) \right] \stackrel{\simeq}{{\longrightarrow}} R^0i_{{{\mathfrak{m}}athfrak{a}}st} \left[ i^{{{\mathfrak{m}}athfrak{a}}st}({\Lambda}) \otimes k(P) \right] \stackrel{\neq0}{{\longrightarrow}} k(P)$$ where the last map is just a non-zero morphism from (\ref{non-zero-hom}) with the source sheaf extended by zero. Since ${\Lambda}$ is locally free in a neighborhood of $P$ and closed immersions have no higher direct images, the fact that the first map is an isomorphism follows from the degeneration of the spectral sequence (c.f. equation (3.10) in ~{{\mathfrak{m}}athfrak{c}}ite{huy06}) $$R^s i_{{{\mathfrak{m}}athfrak{a}}st} \left( L^t i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right) \simeq R^s i_{{{\mathfrak{m}}athfrak{a}}st} \left( {\mathcal{H}}^t \left( L i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right)\right) \stackrel{s+t=p}{{\Lambda}ongrightarrow} R^pi_{{{\mathfrak{m}}athfrak{a}}st} \left( Li^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right)$$ since $L^t i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P)=0$ for all $t<0$. Hence, the inverse limit on the right hand side of (\ref{GD2}) is also non-zero, and in particular that $$\varprojlim {\mathcal{E}} xt^k_Z(Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda}_e \otimes k(P), Li^! k(P)) \neq0$$
But recall that $Z$ is a torsion translate of an abelian subvariety of $\hat{A}$, so ${\omega}_Z\simeq {\mathcal{O}}_Z \simeq {\mathcal{O}}_P \simeq k(P)$, so $Li^! k(P)\simeq k(P)$ and we may hence conclude that $$\varprojlim {\mathcal{E}} xt^k_Z(Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda}_e \otimes k(P), k(P)) \neq0$$ as claimed.
{\epsilon}nd{pf} {{\mathfrak{m}}athfrak{b}}igskip
{{\mathfrak{m}}athfrak{b}}egin{rmk}
Theorem \ref{torsion-non-zero-map} has shown that $$\varprojlim \left(R^k S_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0.$$ where recall, we defined $\tilde{{\mathcal{O}}mega}_e = RS_{\hat{A},A} (D_{\hat{A}} (\tilde{{\Lambda}}_e))$ where $\tilde{{\Lambda}}_e=\textrm{Im}({\Lambda}_e {\rightarrow} {\Lambda})$. In what follows we will drop the tildes in order to ease de notation: all we need is a projective system of coherent sheaves satisfying the generic vanishing property and inducing a non-zero limit as stated in the theorem.
{\epsilon}nd{rmk} {\mathfrak{m}}edskip
If $A$ has no super-singular factors, we know by Proposition 3.3.5 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that for every $e\geq0$, the top dimensional components of the set of points $P\in \hat{A}$ such that the map ${\mathcal{H}}^0({\Lambda}_e)_P{\rightarrow} {\mathcal{H}}^0({\Lambda})_P$ is non-zero is a torsion translate of an abelian subvariety of $\hat{A}$. {\mathfrak{m}}edskip
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Note that the inverse limit of the left hand side is $$\varprojlim {\mathcal{H}} om_{\hat{A}} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[ Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P) \right], k(P) \right) = {\mathcal{H}} om_{\hat{A}} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[ Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} \otimes k(P) \right], k(P) \right).$$
We claim that the latter sheaf is non-zero. Indeed, note that {{\mathfrak{m}}athfrak{b}}egin{equation} H om_{\hat{A}} \left( i^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} \otimes k(P) , k(P) \right) \neq0 \label{non-zero-hom} {\epsilon}nd{equation}
since it is simply the $k(P)$-dual of the non-zero $k(P)$-vector space ${\Lambda}_{|Z} \otimes k(P)$. The natural map $Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} {\rightarrow} L^0 i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda}$ induces $$R^0i_{{{\mathfrak{m}}athfrak{a}}st} \left[ Li^{{{\mathfrak{m}}athfrak{a}}st}({\Lambda}) \otimes k(P) \right] \stackrel{\simeq}{{\longrightarrow}} R^0i_{{{\mathfrak{m}}athfrak{a}}st} \left[ i^{{{\mathfrak{m}}athfrak{a}}st}({\Lambda}) \otimes k(P) \right] \stackrel{\neq0}{{\longrightarrow}} k(P)$$ where the last map is just a non-zero morphism from (\ref{non-zero-hom}) with the source sheaf extended by zero. Since ${\Lambda}$ is locally free in a neighborhood of $P$ and closed immersions have no higher direct images, the fact that the first map is an isomorphism follows from the degeneration of the spectral sequence (c.f. equation (3.10) in ~{{\mathfrak{m}}athfrak{c}}ite{huy06}) $$R^s i_{{{\mathfrak{m}}athfrak{a}}st} \left( L^t i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right) \simeq R^s i_{{{\mathfrak{m}}athfrak{a}}st} \left( {\mathcal{H}}^t \left( L i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right)\right) \stackrel{s+t=p}{{\Lambda}ongrightarrow} R^pi_{{{\mathfrak{m}}athfrak{a}}st} \left( Li^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right)$$ since $L^t i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P)=0$ for all $t<0$. Hence, the inverse limit on the right hand side of (\ref{GD2}) is also non-zero, and in particular that $$\varprojlim {\mathcal{E}} xt^k_Z(Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda}_e \otimes k(P), Li^! k(P)) \neq0$$
But recall that $Z$ is a torsion translate of an abelian subvariety of $\hat{A}$, so ${\omega}_Z\simeq {\mathcal{O}}_Z \simeq {\mathcal{O}}_P \simeq k(P)$, so $Li^! k(P)\simeq k(P)$ and we may hence conclude that $$\varprojlim {\mathcal{E}} xt^k_Z(Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda}_e \otimes k(P), k(P)) \neq0$$ as claimed.
{\epsilon}nd{pf} {{\mathfrak{m}}athfrak{b}}igskip
{{\mathfrak{m}}athfrak{b}}egin{rmk}
Theorem \ref{torsion-non-zero-map} has shown that $$\varprojlim \left(R^k S_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0.$$ where recall, we defined $\tilde{{\mathcal{O}}mega}_e = RS_{\hat{A},A} (D_{\hat{A}} (\tilde{{\Lambda}}_e))$ where $\tilde{{\Lambda}}_e=\textrm{Im}({\Lambda}_e {\rightarrow} {\Lambda})$. In what follows we will drop the tildes in order to ease de notation: all we need is a projective system of coherent sheaves satisfying the generic vanishing property and inducing a non-zero limit as stated in the theorem.
{\epsilon}nd{rmk} {\mathfrak{m}}edskip
If $A$ has no super-singular factors, we know by Proposition 3.3.5 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that for every $e\geq0$, the top dimensional components of the set of points $P\in \hat{A}$ such that the map ${\mathcal{H}}^0({\Lambda}_e)_P{\rightarrow} {\mathcal{H}}^0({\Lambda})_P$ is non-zero is a torsion translate of an abelian subvariety of $\hat{A}$. {\mathfrak{m}}edskip
Let $P\in \hat{A}$ be a torsion point of maximal dimension (namely, ${\textrm{dim }}(P)$ is maximal such that ${\mathcal{H}}^0({\Lambda})_P$ has torsion) and consider $W=\overline{\{P\}}$. In particular, $W$ is a component of ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda})$, and we already argued earlier that $W$ must also be an irreducible component of ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}_e)$ for $e>>0$, so $W$ is also a top dimensional component of the support of the image of the map ${\mathcal{H}}^0({\Lambda}_e){\rightarrow} {\mathcal{H}}^0({\Lambda})$. We thus conclude that if $P\in \hat{A}$ is a torsion point of maximal dimension, then $W=\overline{\{P\}}$ is a torsion translate of an abelian subvariety of $\hat{A}$. {\mathfrak{m}}edskip
In this context, Theorem \ref{torsion-non-zero-map} yields the following.
{{\mathfrak{m}}athfrak{b}}egin{corol}
Let $\{{\mathcal{O}}mega_e\}$ be a Mittag-Leffler inverse system of coherent sheaves on a
$g$-dimensional abelian variety $A$ with no supersingular factors, and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let
${\Lambda}ambda_e=RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$.
Suppose that $\{{{\mathcal{O}}mega}_e\}$ is a GV-inverse system, in the sense that ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$. If $P$ is a torsion point of ${\mathcal{H}}^0({\Lambda})$ of maximal dimension (so that $W=\overline{\{P\}}$ is a torsion translate of an abelian subvariety of $\hat{A}$), then there are non-zero maps $$\varprojlim \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {\rightarrow} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P)$$ for $e\gg0$, where the Fourier-Mukai kernel of $S_{A,\hat{W}}$ is given by ${{\mathfrak{m}}athbb P}P^{A\times \hat{W}}=(id\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{A\times \hat{A}}$, ${{\mathfrak{m}}athbb P}P^{A\times \hat{A}}$ being the normalized Poincar{\'e} bundle of $A\times \hat{A}$. \label{torsion-non-zero-map2}
{\epsilon}nd{corol}
{{\mathfrak{m}}athfrak{b}}egin{pf}
By Theorem \ref{torsion-non-zero-map} we have a non-zero map $$\varprojlim \left(R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {\rightarrow} R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)$$ for some $e>0$. Consider the base-change maps $$R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P) {\rightarrow} H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}), {{\mathfrak{m}}athfrak{q}}uad R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P) {\rightarrow} H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}})$$ Note that the second map is an isomorphism by flat base change: indeed, denoting by $\iota:\hat{W} \hookrightarrow \hat{A}$ the inclusion, we have {{\mathfrak{m}}athfrak{b}}egin{eqnarray*} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P) &\stackrel{def}{=}& R^{g-k}p_{\hat{W}{{\mathfrak{m}}athfrak{a}}st}\left(p_A^{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}} \right)\otimes k(P) \\ &\stackrel{FBC}{\simeq}& R^{g-k}p_{\hat{W}{{\mathfrak{m}}athfrak{a}}st}'\left(p_A^{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}\otimes k(P) \right) \\ &\stackrel{def}{\simeq}& R^{g-k}p_{\hat{W}{{\mathfrak{m}}athfrak{a}}st}'\left(p_A^{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_e \otimes (id\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{A\times \hat{A}}\otimes k(P) \right) \\ &\stackrel{[{{\mathfrak{m}}athfrak{a}}st]}{\simeq}& H^{g-k}(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}) {\epsilon}nd{eqnarray*} where in $[{{\mathfrak{m}}athfrak{a}}st]$ we used Proposition III.8.5 in ~{{\mathfrak{m}}athfrak{c}}ite{har77} and where $p_{\hat{W}}'$ is the base change of the projection, as illustrated in the diagram $$\xymatrix{A \times_{\hat{W}} {{\mathfrak{m}}athbb S}pec k(P) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{p_{\hat{W}}'} & A \times \hat{W} {{\mathfrak{m}}athfrak{a}}r[d]^{p_{\hat{W}}} \\ {{\mathfrak{m}}athbb S}pec k(P) {{\mathfrak{m}}athfrak{a}}r[r]^{\textrm{flat}} & \hat{W}}$$
However note that we have {{\mathfrak{m}}athfrak{b}}egin{equation} H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}) \simeq H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}) \label{obvious-iso} {\epsilon}nd{equation} so we can write both base change maps in the following diagram $$\xymatrix{\varprojlim\left(R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {{\mathfrak{m}}athfrak{a}}r[r]^-{\neq 0} {{\mathfrak{m}}athfrak{a}}r[d] & R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P) {{\mathfrak{m}}athfrak{a}}r[d]^-{[{{\mathfrak{m}}athfrak{a}}st]} \\ \varprojlim H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{\simeq} & H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[d]^{\simeq} \\ \varprojlim H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[r] & H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}\right) \\ \varprojlim\left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {{\mathfrak{m}}athfrak{a}}r[u]^{\simeq} {{\mathfrak{m}}athfrak{a}}r[r] & R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P) {{\mathfrak{m}}athfrak{a}}r[u]_{\simeq}}$$ where the top horizontal map is non-zero for $e\gg0$ by Theorem \ref{torsion-non-zero-map}, the middle isomorphisms are the ones in (\ref{obvious-iso}), and where the isomorphisms at the bottom follow from flat base change as described above. {\mathfrak{m}}edskip
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However note that we have {{\mathfrak{m}}athfrak{b}}egin{equation} H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}) \simeq H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}) \label{obvious-iso} {\epsilon}nd{equation} so we can write both base change maps in the following diagram $$\xymatrix{\varprojlim\left(R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {{\mathfrak{m}}athfrak{a}}r[r]^-{\neq 0} {{\mathfrak{m}}athfrak{a}}r[d] & R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P) {{\mathfrak{m}}athfrak{a}}r[d]^-{[{{\mathfrak{m}}athfrak{a}}st]} \\ \varprojlim H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{\simeq} & H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[d]^{\simeq} \\ \varprojlim H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[r] & H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}\right) \\ \varprojlim\left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {{\mathfrak{m}}athfrak{a}}r[u]^{\simeq} {{\mathfrak{m}}athfrak{a}}r[r] & R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P) {{\mathfrak{m}}athfrak{a}}r[u]_{\simeq}}$$ where the top horizontal map is non-zero for $e\gg0$ by Theorem \ref{torsion-non-zero-map}, the middle isomorphisms are the ones in (\ref{obvious-iso}), and where the isomorphisms at the bottom follow from flat base change as described above. {\mathfrak{m}}edskip
We seek to show that the bottom horizontal map is non-zero for some $e$. Nevertheless, note that if it this were not the case, then the top horizontal map could not possibly be non-zero, since in any case the base change maps $[{{\mathfrak{m}}athfrak{a}}st]$ are injective by the proof of Proposition III.12.5 in ~{{\mathfrak{m}}athfrak{c}}ite{har77}\footnote{In a nutshell, let $f:X{\rightarrow} Y={{\mathfrak{m}}athbb S}pec A$ be a projective morphism and let ${\mathbb{F}}F$ be a coherent sheaf on $X$. For any $A$-module $M$, define $T^i(M):=H^i(X,{\mathbb{F}}F\otimes_A M)$, which is a covariant additive functor from $A$-modules to $A$-modules which is exact in the middle (by Proposition III.12.1 in ~{{\mathfrak{m}}athfrak{c}}ite{har77}). Writing $$A^r{\rightarrow} A^s {\rightarrow} M {\rightarrow} 0$$ we have a diagram $$\xymatrix{T^i(A)\otimes A^r {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{\simeq} & T^i(A)\otimes A^s {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{\simeq} & T^i(A) \otimes M {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]^{\varphi} & 0 \\ T^i(A^r) {{\mathfrak{m}}athfrak{a}}r[r] & T^i(A^s) {{\mathfrak{m}}athfrak{a}}r[r] & T^i(M)}$$ where $\varphi:T^i(A)\otimes M {\rightarrow} T^i(M)$ is the base change map and where the two first vertical arrows are isomorphisms. A straight-forward diagram chase then shows that $\varphi$ is injective.}.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{rmk}
We will be using two different Fourier-Mukai kernels on $A\times \hat{W}$. If $\iota:\hat{W}\hookrightarrow \hat{A}$ denotes the inclusion and ${{\mathfrak{m}}athfrak{p}}i:A{{\mathfrak{m}}athfrak{p}}roj W$ is the dual projection, we have a diagram $$\xymatrix{A\times \hat{W} {{\mathfrak{m}}athfrak{a}}r[r]^{id_A\times \iota} {{\mathfrak{m}}athfrak{a}}r[d]_{{{\mathfrak{m}}athfrak{p}}i \times id_{\hat{W}}} & A \times \hat{A} \\ W \times \hat{W} & }$$ If ${{\mathfrak{m}}athbb P}P^{A\times \hat{A}}$ and ${{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$ denote the normalized Poincar{\'e} bundles on $A\times \hat{A}$ and $W\times \hat{W}$ respectively, on $A\times \hat{W}$ we may consider the locally-free sheaves $(id_A\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{A\times \hat{A}}$ and $({{\mathfrak{m}}athfrak{p}}i\times id_{\hat{W}})^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$. {\mathfrak{m}}edskip
In Corollary \ref{torsion-non-zero-map2} we proved a non-vanishing statement for the derived Fourier-Mukai transform with respect to the former kernel and in what follows we need an analogous statement for the transform with respect to the latter. Nevertheless, note that we are simply looking at fibers over points $w\in \hat{W}\subset \hat{A}$ (concretely over the generic point of $\hat{W}$), and over these points both sheaves are isomorphic. Indeed, $w\in \hat{W}\subset \hat{A}$ determines ${{\mathfrak{m}}athbb P}P^{W\times \hat{W}}_{|W\times \{w\}}\in {{\mathfrak{m}}athbb P}ic(W)$ and ${{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{w\}}\in {{\mathfrak{m}}athbb P}ic(A)$, with ${{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{w\}}\simeq {{\mathfrak{m}}athfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athbb P}P^{W\times \hat{W}}_{|W\times \{w\}}$, and therefore $$\overbrace{\left[(id_A\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{A\times \hat{A}}\right]_{|A\times \{w\}}}^{\simeq {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{w\}}} \simeq \overbrace{\left[({{\mathfrak{m}}athfrak{p}}i\times id_{\hat{W}})^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}\right]_{|A\times \{w\}}}^{\simeq {{\mathfrak{m}}athfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athbb P}P^{W\times \hat{W}}_{|W\times \{w\}}}$$ \label{different-FM-kernels}
{\epsilon}nd{rmk} {\mathfrak{m}}edskip
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\section{Fibering of the Albanese image}
In ~{{\mathfrak{m}}athfrak{c}}ite{el97}, Ein and Lazarsfeld showed the following statement:
{{\mathfrak{m}}athfrak{b}}egin{thm}[see ~{{\mathfrak{m}}athfrak{c}}ite{el97}, Theorem 3]
$X$ is a smooth projective variety of maximal Albanese dimension over a field of characteristic zero and ${{\mathfrak{m}}athfrak{c}}hi({\omega}_X)=0$, then the image of the Albanese map is fibered by subtori of $A$. \label{EL-fibered-by-tori}
{\epsilon}nd{thm}
{{\mathfrak{m}}athfrak{b}}egin{skpf} We sketch the proof given in ~{{\mathfrak{m}}athfrak{c}}ite{pp08} (Theorem E). {\mathfrak{m}}edskip
If ${{\mathfrak{m}}athfrak{c}}hi({\omega}_X)=0$, it follows that $V^0({\omega}_X)\subset \hat{A}$ is a proper subvariety (c.f. Lemma 1.12(b) in ~{{\mathfrak{m}}athfrak{c}}ite{par11}). Choose an irreducible component $W\subset V^0({\omega}_X)$ of codimension $p>0$, which is a torsion translate of an abelian subvariety of $\hat{A}$ that we also denote by $W$. Let ${{\mathfrak{m}}athfrak{p}}i:A{{\mathfrak{m}}athfrak{p}}roj \hat{W}$ denote the dual projection and consider the diagram $$\xymatrix{X {{\mathfrak{m}}athfrak{a}}r[r]^-a & Y:=a(X) {{\mathfrak{m}}athfrak{a}}r[d]_-{h={{\mathfrak{m}}athfrak{p}}i_{|Y}} {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[r] & A {{\mathfrak{m}}athfrak{a}}r@{->>}[dl]^{{{\mathfrak{m}}athfrak{p}}i} \\ & \hat{W}}$$
Since the fibers of the projection $A{{\mathfrak{m}}athfrak{p}}roj \hat{W}$ are abelian subvarieties of dimension $p$, the conclusion of the theorem will follow provided that $f\geq p$, where $f$ denotes the dimension of the generic fiber of $h$. In order to see this, recall the following standard facts:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(a)]
\item $a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X$ is a GV-sheaf on $Y=a(X)$ and $V^0({\omega}_X)=V^0(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X)$.
\item If $W$ is an irreducible component of $V^0(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X)$ of codimension $p$, then it is also a component of $V^p(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X)$.
\item $a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X$ is a $GV_{-f}$-sheaf with respect the the Fourier-Mukai functor with kernel $({{\mathfrak{m}}athfrak{p}}i \times 1_W)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$, so in particular ${{\mathfrak{m}}athfrak{c}}odim_W V^p(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X) \geq p-f$ for every $p\geq0$.
{\epsilon}nd{enumerate} {\mathfrak{m}}edskip
By (b) we have $W \subseteq V^p(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X) \subseteq W$ so that ${{\mathfrak{m}}athfrak{c}}odim_W V^p(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X)=0$, and finally (c) yields $f\geq p$, which is what we sought to show.
{\epsilon}nd{skpf} {\mathfrak{m}}edskip
Our goal in this section is to prove a positive characteristic analogue of Theorem \ref{EL-fibered-by-tori}. {\mathfrak{m}}edskip
Let $X$ be a smooth projective variety of maximal Albanese dimension and denote by $a:X{\rightarrow} A$ the Albanese map. Then $a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X$ is a Cartier module and we may consider the inverse system $\{{{\mathcal{O}}mega}_e=F_{{{\mathfrak{m}}athfrak{a}}st}^e S^0a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X\}_e$. Define ${\Lambda}_e=RS_{A,\hat{A}}D_A({{\mathcal{O}}mega}_e)$ and set ${\Lambda}={\textrm{hocolim}_{\rightarrow}} {\Lambda}_e$. By Corollary 3.3.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} we have that $H^i({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$ for every $i>0$ and very general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. Thus, defining as above ${{\mathfrak{m}}athfrak{c}}hi({{\mathcal{O}}mega}):={{\mathfrak{m}}athfrak{c}}hi({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ for very general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$, we see that ${{\mathfrak{m}}athfrak{c}}hi({{\mathcal{O}}mega})=h^0({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ and it seems that in trying to extend Theorem \ref{EL-fibered-by-tori} to positive characteristic, one should assume that $h^0({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$. {\mathfrak{m}}edskip
This leaves us in a setting which is similar to the one we encountered in the proof of Theorem \ref{Main-Theorem}. If $rk({\Lambda})=h^0({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$, then in particular ${\Lambda}ambda$ must be a torsion sheaf. In light of this observation, we show the following:
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $X$ be a smooth projective variety of maximal Albanese dimension and let $a:X{\rightarrow} A$ be a generically finite map to an abelian variety $A$ with no supersingular factors. Let $g={\textrm{dim }} A$. Consider the inverse system $\{{{\mathcal{O}}mega}_e=F_{{{\mathfrak{m}}athfrak{a}}st}^eS^0a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X\}_e$ and denote ${{\mathcal{O}}mega}=\varprojlim {{\mathcal{O}}mega}_e$. Define ${\Lambda}_e=RS_{A,\hat{A}}D_A({{\mathcal{O}}mega}_e)$ and assume that the sheaf ${\mathcal{H}}^0({\Lambda})=\varinjlim {\mathcal{H}}^0({\Lambda}_e)$ has torsion. Then the image of the Albanese map is fibered by linear subvarieties of $\hat{A}$. \label{alb-fibered-by-tori}
{\epsilon}nd{thm}
{{\mathfrak{m}}athfrak{b}}egin{pf}
Let $w\in \hat{A}$ be a torsion point of ${\mathcal{H}}^0({\Lambda})$ of maximal dimension $k$; by the remark preceding Corollary \ref{torsion-non-zero-map2}, we have that $\hat{W}:=\overline{\{w\}}\subset \hat{A}$ is a torsion translate of an abelian subvariety of $\hat{A}$, which we still denote by $\hat{W}$. Denote by ${{\mathfrak{m}}athfrak{p}}i:A{\rightarrow} W$ the projection dual to the inclusion $\hat{W}\hookrightarrow \hat{A}$. $${{\mathfrak{m}}athfrak{b}}egin{diagram} \node{Y:=a(X)} {{\mathfrak{m}}athfrak{a}}rrow{e,t,J}{} {{\mathfrak{m}}athfrak{a}}rrow{s,l}{h={{\mathfrak{m}}athfrak{p}}i_{|Y}} \node{A} {{\mathfrak{m}}athfrak{a}}rrow{sw,r}{{{\mathfrak{m}}athfrak{p}}i} \\ \node{W} {\epsilon}nd{diagram}$$
By Corollary \ref{torsion-non-zero-map2} we know that the map $$\varprojlim \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(w)\right) {\rightarrow} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(w)$$ is non-zero for every $e\gg 0$, where the Fourier-Mukai kernel of $S_{A,\hat{W}}$ is given by $(id\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P$, ${{\mathfrak{m}}athbb P}P$ being the normalized Poincar{\'e} bundle of $A\times \hat{A}$. {\mathfrak{m}}edskip
Recall that, in general, even though the system $\{{{\mathcal{O}}mega}_e\}$ satisfies the Mittag-Leffler condition, the inverse system $\{R^tS({{\mathcal{O}}mega}_e)\}_e$ may fail to do so (c.f. Example 3.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}). We handle the Mittag-Leffler case first, however, since the proof is neater and the subsequent generalization does not rely on new ideas. {\mathfrak{m}}edskip
\textbf{Case in which $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the ML-condition}. The proof in this case goes along the lines of that of Theorem E in ~{{\mathfrak{m}}athfrak{c}}ite{pp08}. Note that if $w$ is the generic point of $\hat{W}\hookrightarrow \hat{A}$, we have the following:
{{\mathfrak{m}}athfrak{b}}egin{eqnarray*} w &\stackrel{[1]}{\in}& \left\{w\in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w) \right\} \\ &\stackrel{[2]}{=}& \left\{ w \in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w) \right\} \\ &\subseteq& \left\{ w \in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \stackrel{\neq0}{{\longrightarrow}} \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \right\} \\ &\subseteq& \hat{W} {\epsilon}nd{eqnarray*}
where $w$ lies in the first set by Corollary \ref{torsion-non-zero-map2} and where the equality [2] follows from Lemma \ref{inverse-limit-tensor-product-commute}, since we are under the assumption that the system $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the Mittag-Leffler condition. {\mathfrak{m}}edskip
This implies that the codimension (in $\hat{W}$) of the support of image of the map $$\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) {\longrightarrow} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)$$ is zero (this support is closed - under the Mittag-Leffler assumption - by Proposition 4.3 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}). At the same time, by Proposition \ref{GV-k} we know that this codimension must be $\geq g-k-f$, where $f$ is the dimension of a general fiber of $h$, so in particular $f\geq g-k$ and this concludes the proof under the Mittag-Leffler assumption (indeed, the fibers of the projection $A{{\mathfrak{m}}athfrak{p}}roj W$ are abelian subvarieties of dimension $g-k$). {\mathfrak{m}}edskip
\textbf{General case}. We finally observe that, in our setting, we can actually do without the Mittag-Leffler assumption on $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$. We only used this assumption in order to guarantee the closedness of the support of the image of the map $\left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \stackrel{\neq0}{{\longrightarrow}} \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w$ and in order to ensure that the inverse limit commutes with $\otimes k(w)$. {\mathfrak{m}}edskip
Note in the first place that we don't need the support of the image of the above map to be closed for the previous argument to work. Proposition \ref{GV-k} shows that in order for $w$ to belong to the support, we need ${{\mathfrak{m}}athfrak{c}}odim \overline{\{w\}}\geq g-k-f$, and this suffices in order to conclude that $f\geq g-k$. {\mathfrak{m}}edskip
With regards to the commutation of the inverse limit and $\otimes k(w)$, note in the first place that there is always an inclusion $\supseteq$ induced by the natural map $$\varprojlim_e \left( R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) \otimes k(w) {\longrightarrow} \varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right)$$ Moreover, in our setting, the opposite inclusion $\subseteq$ follows from the flatness of $k(w)$ as an ${\mathcal{O}}_{\hat{W}}$-module and the fact that the projection formula and its consequences still hold in the category of quasi-coherent sheaves under some perfection assumptions (c.f. Lemma 71 in ~{{\mathfrak{m}}athfrak{c}}ite{mur06}). We state this below as a lemma, and the proof is hence complete.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
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\textbf{Case in which $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the ML-condition}. The proof in this case goes along the lines of that of Theorem E in ~{{\mathfrak{m}}athfrak{c}}ite{pp08}. Note that if $w$ is the generic point of $\hat{W}\hookrightarrow \hat{A}$, we have the following:
{{\mathfrak{m}}athfrak{b}}egin{eqnarray*} w &\stackrel{[1]}{\in}& \left\{w\in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w) \right\} \\ &\stackrel{[2]}{=}& \left\{ w \in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w) \right\} \\ &\subseteq& \left\{ w \in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \stackrel{\neq0}{{\longrightarrow}} \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \right\} \\ &\subseteq& \hat{W} {\epsilon}nd{eqnarray*}
where $w$ lies in the first set by Corollary \ref{torsion-non-zero-map2} and where the equality [2] follows from Lemma \ref{inverse-limit-tensor-product-commute}, since we are under the assumption that the system $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the Mittag-Leffler condition. {\mathfrak{m}}edskip
This implies that the codimension (in $\hat{W}$) of the support of image of the map $$\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) {\longrightarrow} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)$$ is zero (this support is closed - under the Mittag-Leffler assumption - by Proposition 4.3 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}). At the same time, by Proposition \ref{GV-k} we know that this codimension must be $\geq g-k-f$, where $f$ is the dimension of a general fiber of $h$, so in particular $f\geq g-k$ and this concludes the proof under the Mittag-Leffler assumption (indeed, the fibers of the projection $A{{\mathfrak{m}}athfrak{p}}roj W$ are abelian subvarieties of dimension $g-k$). {\mathfrak{m}}edskip
\textbf{General case}. We finally observe that, in our setting, we can actually do without the Mittag-Leffler assumption on $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$. We only used this assumption in order to guarantee the closedness of the support of the image of the map $\left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \stackrel{\neq0}{{\longrightarrow}} \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w$ and in order to ensure that the inverse limit commutes with $\otimes k(w)$. {\mathfrak{m}}edskip
Note in the first place that we don't need the support of the image of the above map to be closed for the previous argument to work. Proposition \ref{GV-k} shows that in order for $w$ to belong to the support, we need ${{\mathfrak{m}}athfrak{c}}odim \overline{\{w\}}\geq g-k-f$, and this suffices in order to conclude that $f\geq g-k$. {\mathfrak{m}}edskip
With regards to the commutation of the inverse limit and $\otimes k(w)$, note in the first place that there is always an inclusion $\supseteq$ induced by the natural map $$\varprojlim_e \left( R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) \otimes k(w) {\longrightarrow} \varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right)$$ Moreover, in our setting, the opposite inclusion $\subseteq$ follows from the flatness of $k(w)$ as an ${\mathcal{O}}_{\hat{W}}$-module and the fact that the projection formula and its consequences still hold in the category of quasi-coherent sheaves under some perfection assumptions (c.f. Lemma 71 in ~{{\mathfrak{m}}athfrak{c}}ite{mur06}). We state this below as a lemma, and the proof is hence complete.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{lemma}
With the same notations as above, if $$\varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)$$ then $$\varprojlim_e \left( R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) \otimes k(w) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)$$ In particular, equality [2] in the above chain still holds.
{\epsilon}nd{lemma}
{{\mathfrak{m}}athfrak{b}}egin{pf}
As in the proof of Proposition \ref{GV-k}, let $\tilde{{\Lambda}}_e = R\tilde{S}_{A,\hat{W}}(D_A({{\mathcal{O}}mega}_e))$, where $\tilde{S}_{A,\hat{W}}$ denotes the Fourier-Mukai transform with kernel ${\mathfrak{m}}athcal{P}^{\vee}$, with ${\mathfrak{m}}athcal{P}=\left({{\mathfrak{m}}athfrak{p}}i \times 1_{\hat{W}}\right)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$. Note in the first place that we have the following isomorphisms of ${\mathcal{O}}_{\hat{W}}$-modules: {{\mathfrak{m}}athfrak{b}}egin{eqnarray*} \varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right) &\stackrel{[1]}{\simeq}& \varprojlim_e \left({\mathcal{E}} xt^{g-k}_{{\mathcal{O}}_{\hat{W}}}(\tilde{{\Lambda}}_e,{\mathcal{O}}_{\hat{W}}) \otimes k(w)\right) \\ &\simeq& \varprojlim_e \left({\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}}_e,{\mathcal{O}}_{\hat{W}}) \right) \otimes k(w)\right) \\ &\stackrel{[2]}{\simeq}& \varprojlim_e {\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}}_e,{\mathcal{O}}_{\hat{W}}) \otimes k(w) \right) \\ &\stackrel{[3]}{\simeq}& {\mathcal{H}}^{g-k}\left(\varprojlim_e \left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}}_e,{\mathcal{O}}_{\hat{W}}) \otimes k(w) \right)\right) \\ &\stackrel{[4]}{\simeq}& {\mathcal{H}}^{g-k}\left(\varprojlim_e R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}}_e,k(w)) \right) \\ &\stackrel{[5]}{\simeq}& {\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}},k(w)) \right) \\ &\stackrel{[6]}{\simeq}& {\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}},{\mathcal{O}}_{\hat{W}}) \otimes k(w) \right) \\ &\stackrel{[7]}{\simeq}& {\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}},{\mathcal{O}}_{\hat{W}})\right) \otimes k(w) \\ &\stackrel{[8]}{\simeq}& {\mathcal{H}}^{g-k}\left({\textrm{holim}_{\leftarrow}} RS_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) {\epsilon}nd{eqnarray*}
where [1] and [8] follow from the computations (\ref{comp1}) and (\ref{comp2}) in the proof of Proposition \ref{GV-k}, [2] and [7] follow from the flatness of $\otimes k(w)$ as an ${\mathcal{O}}_{\hat{W}}$-module, [3] follows from Proposition \ref{inverse-limits-commute-functor}, since the system $\left\{{\mathcal{E}} xt^{p-1}({\Lambda}_e,{\mathcal{O}}_{\hat{W}})\otimes k(w)\right\}_e$ satisfies the ML-condition, and [4] and [6] follow from the isomorphism\footnote{This isomorphism holds for complexes of sheaves of modules ${\mathfrak{m}}athcal{F},{\mathfrak{m}}athcal{G},{\mathfrak{m}}athcal{H}$ provided that either ${\mathfrak{m}}athcal{F}$ or ${\mathfrak{m}}athcal{H}$ are perfect (c.f. Lemma 71 in ~{{\mathfrak{m}}athfrak{c}}ite{mur06}). Note that $k(w)$ is a perfect complex, being a coherent sheaf.} $$R{\mathcal{H}} om({\mathbb{F}}F,{\mathfrak{m}}athcal{G}) \otimes {\mathfrak{m}}athcal{H} \simeq R{\mathcal{H}} om({\mathbb{F}}F,{\mathfrak{m}}athcal{G} \otimes {\mathfrak{m}}athcal{H}).$$
Hence, by assumption we have a non-zero map $${\mathcal{H}}^{g-k}\left({\textrm{holim}_{\leftarrow}} RS_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)$$ and the conclusion of the lemma then follows from the following commutative diagram $$\xymatrix{\varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) \otimes k(w) {{\mathfrak{m}}athfrak{a}}r[r] & R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w) \\ {\mathcal{H}}^{g-k}\left({\textrm{holim}_{\leftarrow}} RS_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) {{\mathfrak{m}}athfrak{a}}r[u] {{\mathfrak{m}}athfrak{a}}r[ur]_{\neq0}}$$
{\epsilon}nd{pf} {\mathfrak{m}}edskip
In particular, within the context of principally polarized abelian varieties, the same argument yields the following statement:
{{\mathfrak{m}}athfrak{b}}egin{corol}
Let $(A,\Theta)$ is a principally polarized abelian variety with no supersingular factors defined over an algebraically closed field of characteristic $p>0$. Assume further that $\Theta$ is irreducible. Consider the inverse system $\{{{\mathcal{O}}mega}_e:=F_{{{\mathfrak{m}}athfrak{a}}st}^e({\omega}_{\Theta}\otimes \tau_{\Theta})\}_e$ on $A$ and set ${\Lambda}={\textrm{hocolim}_{\rightarrow}}_e RS_{A,\hat{A}}D_A({{\mathcal{O}}mega}_e)$. Then ${\Lambda}$ is a torsion-free quasi-coherent sheaf concentrated in degree 0. \label{theta-divisors-not-ruled}
{\epsilon}nd{corol}
{{\mathfrak{m}}athfrak{b}}egin{pf}
The fact that ${\Lambda}={\mathcal{H}}^0({\Lambda})$ is a quasi-coherent sheaf concentrated in degree zero follows from Theorem \ref{generic-vanishing-char-p}(i), since ${\omega}_{\Theta}$ is a Cartier module. {\mathfrak{m}}edskip
Assume for a contradiction that ${\mathcal{H}}^0({\Lambda})$ is not torsion-free and fix an irreducible component $\hat{W}:=\overline{\{w\}}\hookrightarrow \hat{A}$ of maximal dimension of the closure of the set of torsion points of ${\mathcal{H}}^0({\Lambda})$. Denote by ${{\mathfrak{m}}athfrak{p}}i:A{{\mathfrak{m}}athfrak{p}}roj W$ the dual projection. $$\xymatrix{\Theta {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[r] {{\mathfrak{m}}athfrak{a}}r[d]_{h} & A {{\mathfrak{m}}athfrak{a}}r@{->>}[dl]_{{{\mathfrak{m}}athfrak{p}}i} \\ W & }$$
We may then argue as in the proof of Theorem \ref{alb-fibered-by-tori} to conclude that $\Theta$ is fibered by abelian subvarieties of $A$, but this is not possible given that $\Theta$ is irreducible (and hence of general type) in light of Abramovich's work (c.f. Section 2.3 or ~{{\mathfrak{m}}athfrak{c}}ite{abr95}).
{\epsilon}nd{pf} {\mathfrak{m}}edskip
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\section{Singularities of Theta divisors}
We now focus on the singularities of Theta divisors and embark on the proof of Theorem \ref{Main-theorem}. As a warm-up, we focus on simple abelian varieties to start with, namely those which do not contain smaller dimensional abelian varieties.
\subsection{Case of simple abelian varieties}
The crux of the argument resides in the construction of sections of ${\mathcal{O}}_A(\Theta)$ which vanish along the test ideal $\tau(\Theta)$ and, in the case of simple abelian varieties, it is a direct consequence of the results in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}. The proof of the general case will follow the same pattern, albeit further work will be required to prove that the required sections exist. {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $(A,\Theta)$ be a PPAV over an algebraically closed field $K$ of characteristic $p>0$ such that $A$ is simple and ordinary. Then $\Theta$ is strongly F-regular. (In particular, $\Theta$ is F-rational, and by Lemma 2.34 in ~{{\mathfrak{m}}athfrak{c}}ite{bst12}, it is normal and Cohen-Macaulay).
\label{Main-theorem-simple}
{\epsilon}nd{thm} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{pf}
On $\hat{A}$ consider the inverse system ${{\mathcal{O}}mega}_e=F_{{{\mathfrak{m}}athfrak{a}}st}^e{{\mathcal{O}}mega}_0$, where ${{\mathcal{O}}mega}_0=\omega_{\Theta}\otimes \tau(\Theta)$. This yields a direct system ${\Lambda}ambda_e= R\hat{S}D_A{{\mathcal{O}}mega}_e$ equipped with natural maps ${\mathcal{H}}^0({\Lambda}ambda_e) {\rightarrow} {\mathcal{H}}^0({\Lambda}ambda)={\Lambda}ambda={\textrm{hocolim}_{\rightarrow}} {\Lambda}ambda_e$. By \ref{generic-vanishing-char-p}, we know that ${\Lambda}ambda$ is quasi-coherent sheaf in degree 0. {\mathfrak{m}}edskip
Consider the set $$Z=\left\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}: {{\mathfrak{m}}athfrak{q}}uad Im \left({\mathcal{H}}^0({\Lambda}ambda_0)\otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} {\longrightarrow} {\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \right) \neq 0\right\}$$ By Proposition \ref{GV-corollary2} we know that $Z$ is a finite union of torsion translates of subtori. Since $\hat{A}$ is simple by assumption, this implies that either $Z=\hat{A}$ or $Z$ is a finite set. {\mathfrak{m}}edskip
Assume for a contradiction that $Z$ is finite. By Proposition \ref{GV-corollary3} we have $t_x^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{O}}mega={\mathcal{O}}mega$ for every $x\in {\omega}idehat{\hat{A}/Z}={\omega}idehat{\hat{A}}$, so that ${{\mathfrak{m}}athbb S}upp {\mathcal{O}}mega=A$. Since the maps in the inverse system $F_{{{\mathfrak{m}}athfrak{a}}st}^e{\mathcal{O}}mega_0=F_{{{\mathfrak{m}}athfrak{a}}st}^e(\omega_{\Theta}\otimes \tau(\Theta))$ are surjective, we know by Proposition \ref{GV-corollary3} that ${{\mathfrak{m}}athbb S}upp {\mathcal{O}}mega={{\mathfrak{m}}athbb S}upp {\mathcal{O}}mega_0={{\mathfrak{m}}athbb S}upp {\omega}_{\Theta}\otimes \tau(\Theta)=A$, which is absurd. {\mathfrak{m}}edskip
We must thus have $Z=\hat{A}$, so that ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}ambda_0)=\hat{A}$, and hence cohomology and base change yields $H^0(A,{\omega}_{\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$ for all ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. {\mathfrak{m}}edskip
Consider the following commutative diagram: $$\xymatrix{ H^0(A,P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] & H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] & H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta} \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) \\ H^0(A,K\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau}) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & }$$
where $K$ is defined so that the diagram commutes. In the top row we have $H^1(A,P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$ for ${{\mathfrak{m}}athfrak{a}}lpha\neq 0$ since $P_{{{\mathfrak{m}}athfrak{a}}lpha}$ is topologically trivial. The polarization induced by $\Theta$ is principal, so $h^0({\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=1$. Since by the above discussion $H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}\otimes \tau(\Theta))\neq 0$, it follows that $H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$ and hence both the right inclusion and the top right restriction are equalities. {\mathfrak{m}}edskip
The ideal sheaf $\tilde{\tau}$ on $A$ is defined as follows: fix an open subset $U={{\mathfrak{m}}athbb S}pec R\subseteq A$ and assume that $\Theta$ is given by an ideal sheaf $I=I(\Theta)$. Let $J=\tau_{\Theta}(U)$ be the test ideal of $\Theta$ and let $\tilde{J}\subset R$ be an ideal such that $J=\tilde{J}/I$. Omitting the twist by $P_{{{\mathfrak{m}}athfrak{a}}lpha}$, the diagram above locally boils down to
$$\xymatrix{ && R/\tilde{J} {{\mathfrak{m}}athfrak{a}}r[r]^-{\simeq} & (R/I)/(\tilde{J}/I) \simeq R/\tilde{J} & \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & I {{\mathfrak{m}}athfrak{a}}r[r] & R {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & R/I {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & 0 \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & I {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & \tilde{J} {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & J {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & 0 }$$
so $\tilde{\tau}(U)=\tilde{J}$. Now taking cohomology, a section $s\in H^0(J)$ embeds as ${{\mathfrak{m}}athfrak{b}}ar{s}\in H^0(R/I)$ and maps to zero in $H^0(R/\tilde{J})$ by exactness. By exactness of the second row, ${{\mathfrak{m}}athfrak{b}}ar{s}$ lifts to $\tilde{{{\mathfrak{m}}athfrak{b}}ar{s}}\in H^0(R)$, which still projects to zero in $H^0(R/\tilde{J})$ by commutativity of the top square, so $\tilde{{{\mathfrak{m}}athfrak{b}}ar{s}}$ must lift to a non-zero section of $H^0(\tilde{J})$. {\mathfrak{m}}edskip
Finally, since $H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}\otimes \tau(\Theta))\neq 0$ for every ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$ and these sections lift to sections of $H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ vanishing along $\tilde{\tau}$, we conclude that $h^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}\otimes \tilde{\tau})=1$. Hence if $\tilde{\tau}$ were not trivial, we would have $Zeros(\tilde{\tau})\subset \Theta+{{\mathfrak{m}}athfrak{a}}lpha_P$ for every ${{\mathfrak{m}}athfrak{a}}lpha_P\in A$ (where ${{\mathfrak{m}}athfrak{a}}lpha_P\in A$ is the point corresponding to $P_{{{\mathfrak{m}}athfrak{a}}lpha}\in {{\mathfrak{m}}athbb P}ic^0(A)$), which is absurd since these translates of $\Theta$ don't have any points in common. We thus conclude that $\tilde{\tau}={\mathcal{O}}_A$, and hence $\tau(\Theta)={\mathcal{O}}_{\Theta}$ so that $\Theta$ is strongly F-regular.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
In the proof of Theorem \ref{Main-theorem-simple} we used the simplicity of $A$ in order to show that $H^0(A,{\omega}_{\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$ for all ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. The same argument we employed above will work in the general case provided that we can show the existence of sections in $H^0(A,{\omega}_{\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$, and it turns out that this is somewhat more involved.
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\subsection{General case}
We finally study singularities of Theta divisors in the general setting. As we mentioned earlier, the argument will be analogous to the one employed to prove the theorem in the case of simple abelian varieties, although additional work will be required to prove that there exist sections in $H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$. {\mathfrak{m}}edskip
The main ingredient in Ein and Lazarsfeld's proof over fields of characteristic zero was that given a smooth projective variety $X$ of maximal Albanese dimension such that ${{\mathfrak{m}}athfrak{c}}hi(X,{\omega}_X)=0$, the image of its Albanese morphism is fibered by tori (c.f. Theorem 3 in ~{{\mathfrak{m}}athfrak{c}}ite{el97}). Our proof will rely on Corollary \ref{theta-divisors-not-ruled}, where we proved that if the sheaf ${\mathcal{H}}^0({\Lambda})$ associated to the inverse system $\{F_{{{\mathfrak{m}}athfrak{a}}st}^e({\omega}_{\Theta}\otimes \tau_{\Theta})\}$ was not torsion-free, then $\Theta$ would be fibered by tori, which is impossible since $\Theta$ is irreducible. {\mathfrak{m}}edskip
In a nutshell, and as in the case of simple abelian varieties, $\Theta$ will be strongly F-regular provided that there exist non-trivial sections in $H^0(\Theta,{\mathcal{O}}_A(\Theta)\otimes {\mathcal{O}}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ and we will show that if that was not the case, then the sheaf ${\mathcal{H}}^0({\Lambda})$ would have torsion, a contradiction. {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $(A,\Theta)$ be an ordinary principally polarized abelian variety over an algebraically closed field $k$ of characteristic $p>0$. If $\Theta$ is irreducible, then $\Theta$ is strongly F-regular. \label{Main-Theorem}
{\epsilon}nd{thm}
{{\mathfrak{m}}athfrak{b}}egin{pf}
The proof goes along the lines of Theorem \ref{Main-theorem-simple}: consider again the commutative diagram: $$\xymatrix{ H^0(A,P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] & H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] & H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta} \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) \\ H^0(A,K\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau}) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & }$$
In the proof of Theorem \ref{Main-theorem-simple} we used the simplicity of $A$ to conclude easily that $$H^0(\Theta,\overbrace{{\mathcal{O}}_A(\Theta)\otimes {\mathcal{O}}_{\Theta}}^{{\omega}_{\Theta}} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$$ It then followed from the commutative diagram above that $H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau})\neq 0$ and this in turn forced $\tilde{\tau}$ to be trivial, whence $\tau(\Theta)={\mathcal{O}}_{\Theta}$. {\mathfrak{m}}edskip
We shall now use the previous results in order to conclude that $H^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$. In a nutshell, assuming for a contradiction that $\tau(\Theta)$ is not trivial we will show that $0\neq S^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) \subseteq H^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ for general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. The diagram above then yields $H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau})\neq 0$ and since by assumption $\tau(\Theta)\ne {\mathcal{O}}_{\Theta}$, we conclude that $\textrm{Zeroes}(\tilde{\tau})\subset \Theta+{{\mathfrak{m}}athfrak{a}}lpha_P$ for general ${{\mathfrak{m}}athfrak{a}}lpha_P\in A$ (as before, ${{\mathfrak{m}}athfrak{a}}lpha_P$ denotes the point in $A$ corresponding to $P_{{{\mathfrak{m}}athfrak{a}}lpha}\in {{\mathfrak{m}}athbb P}ic^0A$), but this is not possible since general translates of $\Theta$ do not have points in common. Therefore we must have $\tau(\Theta)={\mathcal{O}}_{\Theta}$, and hence $\Theta$ is strongly F-regular. {\mathfrak{m}}edskip
We now argue by contradiction: let ${{\mathcal{O}}mega}=\varprojlim F_{{{\mathfrak{m}}athfrak{a}}st}^eS^0{\omega}_{\Theta}= \varprojlim F_{{{\mathfrak{m}}athfrak{a}}st}^e({\omega}_{\Theta}\otimes \tau(\Theta))$. By Corollary 3.2.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} and its proof, for all closed ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$ we have that $${\mathcal{H}}^0({\Lambda})\otimes k({{\mathfrak{m}}athfrak{a}}lpha) \simeq H^0(\Theta,\varprojlim {{\mathcal{O}}mega}_e \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}$$ so assuming for a contradiction that $S^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$ for general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$, it follows that $rk ({\Lambda}ambda)=0$, and hence that ${\mathcal{H}}^0({\Lambda}ambda)$ has torsion. Nevertheless, this is not possible by Corollary \ref{theta-divisors-not-ruled}, since we are assuming that $\Theta$ is irreducible, and this concludes the proof.
{\epsilon}nd{pf}
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In the proof of Theorem \ref{Main-theorem-simple} we used the simplicity of $A$ to conclude easily that $$H^0(\Theta,\overbrace{{\mathcal{O}}_A(\Theta)\otimes {\mathcal{O}}_{\Theta}}^{{\omega}_{\Theta}} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$$ It then followed from the commutative diagram above that $H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau})\neq 0$ and this in turn forced $\tilde{\tau}$ to be trivial, whence $\tau(\Theta)={\mathcal{O}}_{\Theta}$. {\mathfrak{m}}edskip
We shall now use the previous results in order to conclude that $H^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$. In a nutshell, assuming for a contradiction that $\tau(\Theta)$ is not trivial we will show that $0\neq S^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) \subseteq H^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ for general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. The diagram above then yields $H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau})\neq 0$ and since by assumption $\tau(\Theta)\ne {\mathcal{O}}_{\Theta}$, we conclude that $\textrm{Zeroes}(\tilde{\tau})\subset \Theta+{{\mathfrak{m}}athfrak{a}}lpha_P$ for general ${{\mathfrak{m}}athfrak{a}}lpha_P\in A$ (as before, ${{\mathfrak{m}}athfrak{a}}lpha_P$ denotes the point in $A$ corresponding to $P_{{{\mathfrak{m}}athfrak{a}}lpha}\in {{\mathfrak{m}}athbb P}ic^0A$), but this is not possible since general translates of $\Theta$ do not have points in common. Therefore we must have $\tau(\Theta)={\mathcal{O}}_{\Theta}$, and hence $\Theta$ is strongly F-regular. {\mathfrak{m}}edskip
We now argue by contradiction: let ${{\mathcal{O}}mega}=\varprojlim F_{{{\mathfrak{m}}athfrak{a}}st}^eS^0{\omega}_{\Theta}= \varprojlim F_{{{\mathfrak{m}}athfrak{a}}st}^e({\omega}_{\Theta}\otimes \tau(\Theta))$. By Corollary 3.2.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} and its proof, for all closed ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$ we have that $${\mathcal{H}}^0({\Lambda})\otimes k({{\mathfrak{m}}athfrak{a}}lpha) \simeq H^0(\Theta,\varprojlim {{\mathcal{O}}mega}_e \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}$$ so assuming for a contradiction that $S^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$ for general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$, it follows that $rk ({\Lambda}ambda)=0$, and hence that ${\mathcal{H}}^0({\Lambda}ambda)$ has torsion. Nevertheless, this is not possible by Corollary \ref{theta-divisors-not-ruled}, since we are assuming that $\Theta$ is irreducible, and this concludes the proof.
{\epsilon}nd{pf}
\small
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\begin{document}
\title{Cavity QED with Multiple Hyperfine Levels}
\author{K.~M.~Birnbaum}
\altaffiliation[Permanent address: ]{Jet Propulsion Laboratory, California Institute of Technology, Mail Stop 161-135, 4800 Oak Grove Drive, Pasadena, CA 91109, U.S.A.}
\affiliation{Norman Bridge Laboratory of Physics 12-33, California Institute of Technology,
Pasadena, CA 91125, U.S.A.}
\author{A.~S.~Parkins}
\altaffiliation[Permanent address: ]{Department of Physics, University of Auckland,
Private Bag 92019, Auckland, New Zealand}
\affiliation{Norman Bridge Laboratory of Physics 12-33, California Institute of Technology,
Pasadena, CA 91125, U.S.A.}
\author{H. J. Kimble}
\affiliation{Norman Bridge Laboratory of Physics 12-33, California Institute of Technology,
Pasadena, CA 91125, U.S.A.}
\date{June 8, 2006}
\begin{abstract}
We calculate the weak-driving transmission of a linearly polarized cavity mode strongly coupled to the D2 transition of a single Cesium atom. Results are relevant to future experiments with microtoroid cavities, where the single-photon Rabi frequency $g$ exceeds the excited-state hyperfine splittings, and photonic bandgap resonators, where $g$ is greater than both the excited- and ground-state splitting.
\end{abstract}
\pacs{42.50.Pq, 42.50.-p, 32.10.Fn}
\maketitle
\section{Introduction}
The Jaynes-Cummings model of cavity QED treats an atom as a two-level system. This is appropriate for a realistic atom when that atom has a cycling transition, typically reached by optical pumping with circularly polarized light \cite{hood98,hood00}. However, new types of optical resonators such as microtoroids \cite{toroid} and photonic band gap cavities \cite{pbg} do not support circularly polarized modes. Though these structures with extremely low critical atom and photon numbers show great promise for strong coupling, a more detailed model of the atom must be employed when calculating the properties of these atom-cavity systems \cite{kmb,iqec}. A linearly polarized mode may couple multiple Zeeman states of the atom. Additionally, for these very small resonators, the single photon Rabi frequency ($2g$) can be comparable to or larger than the hyperfine splitting of the atom, so that multiple hyperfine levels must be considered when calculating the excitations of the system. We will consider a linearly polarized single-mode resonator coupled to the D2 ($6S_{1/2} \to 6P_{3/2}$) transition of a single Cesium atom. However, this may also give some intuition for other multilevel scatterers, such as molecules and excitons \cite{exciton1, exciton2}.
\section{Coupling to Multiple Excited Levels}
\label{sec:toroid}
In order to describe the interaction of the atom with various light fields, it is useful to define the atomic dipole transition operators
\begin{multline}
\label{dipole}
D_{q}(F,F')= \\
\sum_{m_{F}=-F}^{F}|F,m_{F}\rangle \langle F,m_{F}| \mu_{q} |F',m_{F}+q \rangle \langle F',m_{F}+q|
\end{multline}
where $q=\{-1,0,1\}$ and $\mu_{q}$ is the dipole operator for $\{\sigma_{-},\pi, \sigma_{+}\}$-polarization, normalized such that for a cycling transition $\langle \mu \rangle =1$. We will approximate all atom-field interactions to be dipole interactions.
First, let us consider the case when $g$ is comparable to the hyperfine splitting of the excited states, but still small compared to the ground-state splitting. This limit is appropriate for the proposed parameters of microtoroid resonators \cite{toroid} and small Fabry-Perot cavities~\cite{hood01}. If the cavity is tuned near the $F=4 \to F'$ transitions, then the Hamiltonian for the atom cavity system can be written using the rotating wave approximation as
\begin{eqnarray}
\label{H_0}
H_{0} &=& \omega_c a^{\dag} a + \sum_{F'=2}^{5}\omega_{F'}|F'\rangle \langle F'| \nonumber \\
&+& g\Big(\sum_{F'=3}^{5} a^{\dag}D_{0}(4,F') + D_{0}^{\dag}(4,F')a\Big),
\end{eqnarray}
where $\omega_{F'}$ is the frequency of the $F=4 \to F'$ transition, $\omega_c$ is the frequency of the cavity, and $a$ is the annihilation operator for the cavity mode. The operator $|F'\rangle \langle F'|$ projects onto the manifold of excited states with hyperfine number $F'$, and may be written more explicitly as $|F'\rangle \langle F'| = \sum_{m_F'} |F',m_F'\rangle \langle F',m_F'|$. We use units such that $\hbar=1$ and energy has the same dimensions as frequency. Note that we are treating the cavity as a single-mode resonator with linear polarization. Fabry-Perot cavities have two modes with orthogonal polarizations, so this model is only appropriate if there is a birefringent splitting which makes one of the modes greatly detuned (compared to $g$) from the atomic resonance.
In the weak-driving limit of an atom-cavity system in the regime of strong coupling, we expect that high transmission will occur when the probe light is resonant with a transition from a ground state of the system to a state in the $N=1$ lowest excitation manifold. Furthermore, we expect a higher transmission when resonantly exciting an eigenstate which is ``cavity-like,'' i.e., an eigenstate which has larger weight in the field excitation rather than the atomic dipole.
\begin{figure}\label{toroid_eigen}
\end{figure}
In Fig.~\ref{toroid_eigen}, we plot the eigenfrequencies $\{\epsilon_k^{(1)}\}$ of $H_0$ determined by the equation $H_0|\psi_k^{(N)}\rangle = \epsilon_k^{(N)} |\psi_k^{(N)}\rangle$. Here $N$ is the excitation manifold, where $\epsilon_k^{(N+1)}-\epsilon_k^{(N)} \sim \omega_c$ and $\epsilon_k^{(0)}=0$. Also displayed is $\langle \psi_k^{(1)} | a^{\dag} a |\psi_k^{(1)}\rangle$ for each eigenstate $|\psi_k^{(1)}\rangle$ corresponding to each eigenfrequency $\epsilon_k^{(1)}$, which is a measure of how ``cavity-like'' that state is. This should give some indication of what cavity and probe detunings yield high transmission.
In order to study the system properties more precisely, we can find the Hamiltonian of the driven system, write the Liouvillian that describes the time-evolution including damping, and calculate the steady state of the system. We will assume that the cavity resonance is tuned near the $F=4\to F'$ atomic transitions. We expect that, absent any repumping fields, atomic decays to the $F=3$ ground state will leave the atom uncoupled to the resonator. To avoid this, we will assume that a classical (coherent-state) driving field tuned near the $F=3\to F'$ transitions is applied to the atom in addition to the probe field which drives the cavity. In the rotating wave approximation, the Hamiltonian of this driven atom-cavity system in the frame rotating with the cavity probe is
\begin{eqnarray}
\label{H_1}
H_1 &=&\sum_{F'=2}^{5}\Delta_{F'}|F'\rangle \langle F'| + \Delta_r |F=3\rangle\langle F=3| + \Delta_c a^{\dag} a \nonumber \\
&+& g\sum_{F'=2}^{5} \Big(a^{\dag}D_{0}(4,F') + D_{0}^{\dag}(4,F')a\Big) \nonumber \\
&+& \Omega_r \sum_{F'=2}^{5} \Big( D_{0}(3,F') + D_{0}^{\dag}(3,F') \Big)\nonumber \\
&+&\varepsilon a^{\dag} +\varepsilon^{*} a,
\end{eqnarray}
where $\Delta_{F'} = \omega_{4\to F'} - \omega_p$, $\Delta_r = \omega_r - \omega_{GSS} - \omega_p$, and $\Delta_c = \omega_c-\omega_p$. Here $\omega_p$ is the probe frequency, $\omega_r$ is the repump frequency, and $\omega_{GSS}\approx 9.2$~GHz is the ground-state splitting of Cs. The cavity is driven at a rate $\varepsilon$ so that in the absence of an atom the intracavity photon number would be $N_{no~atom} = |\varepsilon|^2 /(\kappa^2 + \Delta_c^2)$, and the atom is driven by the repump field with Rabi frequency $2\Omega_r$. Here, we have assumed that there is no off-resonant coupling of the cavity mode to the $F=3$ ground states, nor is there off-resonant coupling of the repump light to the $F=4$ states. We expect that corrections due to those terms will be small when $g,\Omega_r \ll \omega_{GSS}$.
\begin{figure}\label{toroidplot}
\end{figure}
The time evolution of the density matrix $\rho$ of the atom-cavity system is given by the master equation,
\begin{eqnarray}
\label{master_1}
\dot{\rho} = -i[H_1,\rho] +
\kappa \mathcal{D}[a]\rho +
\gamma \sum_{q,F}\mathcal{D} \Big[\sum_{F'}D_{q}(F,F')\Big]\rho ,
\end{eqnarray}
where $\kappa$ and $\gamma$ are the cavity field and atomic dipole decay rates, respectively,
and the zero-temperature decay superoperator $\mathcal{D}$ acts on the density matrix such that $\mathcal{D}[c]\rho \equiv 2c\rho c^{\dag} - c^{\dag}c\rho -\rho c^{\dag}c$ for any operator $c$. Note that in the atomic spontaneous emission term (proportional to $\gamma$), we have assumed that all $F'\to F=4$ transitions of the same polarization couple to a common reservoir of vacuum electromagnetic field modes and similarly for all $F'\to F=3$ transitions (but the reservoirs for $F'\to F=4$ and $F'\to F=3$ transitions are independent). This assumption arises from the fact that level shifts due to the atom-cavity coupling will be comparable to the atomic excited state hyperfine splittings (but small compared to the ground state splitting)
and, therefore, there exists the possibility for coherence, or quantum interference effects between
transitions of the same polarization from different $F'$ states to a single, common ground-state level
\cite{cardimona82,cardimona83,kmb}.
Such a possibility is described in the common-reservoir master equation (\ref{master_1}) by
generalized atomic damping terms which couple such transitions.
Note that the choice of independent reservoirs for transitions to the different hyperfine ground states is consistent with our assumption that there is no off-resonant coupling between transitions from different hyperfine ground-state manifolds.
From the steady-state solution to Eq.~\ref{master_1}, $\dot{\rho}_{ss}=0$, we can compute steady-state expectation values of an operator $c$ by evaluating $\mathrm{Tr}(\rho_{ss}c)$. We define the normalized cavity transmission $T = \mathrm{Tr}(\rho_{ss}a^{\dag}a) \kappa^2/|\varepsilon|^2$, where $T=1$ for an empty cavity on resonance. $T$ is plotted in Fig.~\ref{toroidplot} versus cavity and probe detunings. Notice the similarity to Fig.~\ref{toroid_eigen}, which demonstrates that the qualitative features of the transmission are indeed determined by the eigenvalues and eigenstates of the Hamiltonian.
\begin{figure}\label{toroid_slice}
\end{figure}
Fig.~\ref{toroid_slice} shows $T$ as a function of probe detuning for fixed cavity frequency along with atomic ground-state populations $\langle F=4,m_F|\rho_{ss}|F=4,m_F\rangle$. The large swings in the relative populations of various Zeeman states demonstrate the importance of optical pumping in understanding the steady-state behavior of the transmission. The rapid variation of the populations that occurs near the transmission peaks can be understood by noting in Fig.~\ref{toroid_eigen} that each transmission peak is associated with multiple eigenstates with similar eigenvalues. These eigenstates have different amplitudes of the Zeeman states and therefore lead to different optical pumping effects. It should be noted that the width of the transmission peaks are therefore not simply determined by $\kappa$ and $\gamma$ but also by the separation of the various eigenvalues contributing to each peak, making the peaks wider than would be naively expected.
\begin{figure}\label{toroid_4_5}
\end{figure}
Fig.~\ref{toroid_4_5} demonstrates the importance of incorporating multiple hyperfine levels into the model of the atom when calculating the cavity transmission for the large values of $g$ expected in upcoming experiments \cite{toroid}. The solid red curve denotes the transmission $T$ from Fig.~\ref{toroidplot} for a cavity fixed to be resonant with the $F=4\to F'=5'$ transition. The dashed black curve indicates the transmission calculated using a model of the atom which includes all Zeeman states of the $F=4$ and $F'=5'$ manifolds, but no other hyperfine levels. The substantial differences between the curves indicates that although the other hyperfine transitions are not resonant, the large coupling $g$ causes these transitions to have a significant effect on the atom-cavity system.
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\section{Coupling to the Entire D2 Transition}
\label{sec:pbg}
\begin{figure}\label{pbg_eigen}
\end{figure}
\begin{figure}\label{pbg_levels}
\end{figure}
Now we will turn to the regime where $g$ is larger than both the ground- and excited-state hyperfine splittings. This case is applicable for the expected parameters of cavity QED with photonic band gap cavities \cite{pbg}. In this regime, the cavity mode couples to both ground-state hyperfine manifolds, and the Hamiltonian of the atom-cavity system in the absence of a driving field can be written
\begin{eqnarray}
\label{H_2}
H_2 &=&\sum_{F'}\omega_{F'}|F'\rangle \langle F'| - \omega_{GSS} |F=3\rangle\langle F=3| + \omega_c a^{\dag} a \nonumber \\
&+& g\sum_{F,F'}\Big( a^{\dag}D_{0}(F,F') + D_{0}^{\dag}(F,F')a \Big)
\end{eqnarray}
As we did earlier for $H_0$, we find the eigenvalues and eigenvectors of this Hamiltonian determined by the condition $H_2|\phi_k^{(N)}\rangle = \eta_k^{(N)} |\phi_k^{(N)}\rangle$. In Fig.~\ref{pbg_eigen}, we plot the the frequencies $\eta_k^{(1)}$ of the lowest lying excitations, as well as how ``cavity-like'' the corresponding eigenmodes are, $\langle\phi_k^{(1)}| a^{\dag}a |\phi_k^{(1)}\rangle$. The eigenvalues in the first excitation manifold separate into five bands. The lowest and second-highest of these bands have eigenstates which are superpositions of $F=3$ atomic ground states with one photon in the cavity and $F'=\{2',3',4'\}$ atomic excited states with zero photons; the highest and second-lowest bands have eigenstates which are superpositions of $F=4$ states with one photon and $F'=\{3',4',5'\}$ states with zero photons.
The central band is occupied by eigenstates the composition of which is dominated by atomic excited states. In particular, these eigenstates have a greatly suppressed coupling to the cavity mode as a result of quantum interference between transition amplitudes from atomic excited states with the same $m_F$ number but different values of $F'$. Similarly, with the assumption of a common reservoir for atomic spontaneous
emission from the various hyperfine states (see below),
these eigenstates also exhibit strongly suppressed spontaneous emission via $\pi$-polarized dipole transitions. It should be noted that coupling to the D1 transition ($6S_{1/2} \to 6P_{1/2}$) does not result in a similar set of eigenstates with suppressed coupling; the absence of $F'=2',5'$ states precludes the possibility of the
required destructive quantum interference between $\pi$-polarized transitions.
\begin{figure}\label{pbgplot}
\end{figure}
We expect high transmission when a probe is tuned to be resonant with a transition from a ground state of the atom-cavity system to one of the states in the first excitation manifold. The eigenvalues of the ground states are $\eta^{(0)}=0$ for states with the atom in the $F=4$ manifold and $\eta^{(0)}=-\omega_{GSS}$ for states with the atom in $F=3$. In Fig.~\ref{pbg_levels}, we plot the difference frequencies for transitions between ground and first excited states, $\eta_k^{(1)}-\eta_j^{(0)}$, where $k,j$ are restricted to single-quantum transitions that can be excited by the cavity probe.
Notice that although the eigenvalues of the Hamiltonian do not cross, the differences of eigenvalues between the ground and first excitation manifolds do have crossings. These crossings correspond to a dual resonance condition, in which a transition from one hyperfine ground state to an excited state is resonant with a transition from the other hyperfine ground state to a different excited state. As we will show in a moment, this can lead to some distinctive features in the probe transmission spectrum.
\begin{figure}\label{pbg_slice_m13}
\end{figure}
We will now calculate the steady state of the driven, damped system. We will consider the cavity to be driven by a single coherent-state field at the frequency $\omega_p$. Since the cavity mode can couple to all of the atomic ground states, a repump field is not needed. The Hamiltonian of the driven atom-cavity system under the rotating wave approximation, in the frame rotating with the probe, is
\begin{eqnarray}
\label{H_3}
H_3 &=&\sum_{F'}\Delta_{F'}|F'\rangle \langle F'| - \omega_{GSS} |F=3\rangle\langle F=3| + \Delta_c a^{\dag} a \nonumber \\
&+& g\sum_{F,F'}\Big( a^{\dag}D_{0}(F,F') + D_{0}^{\dag}(F,F')a \Big)\nonumber \\
&+&\varepsilon a^{\dag} +\varepsilon^{*} a ,
\end{eqnarray}
and the master equation for the evolution of the density matrix is
\begin{eqnarray}
\label{master_2}
\dot{\rho} = -i[H_3,\rho] +
\kappa \mathcal{D}[a]\rho +
\gamma \sum_{q}\mathcal{D}\Big[\sum_{F,F'}D_{q}(F,F')\Big]\rho .
\end{eqnarray}
Note that in this limit, in which level shifts produced by the atom-field coupling may yield transitions
of similar frequencies to and from {\it different} hyperfine ground states (i.e., $F=3$ and $F=4$),
we assume that all atomic decays of a given polarization are into a common reservoir (without regard for the initial $F'$ and final $F$) \cite{cardimona83}.
From the master equation (\ref{master_2}),
we find the steady-state density matrix $\rho_{ss}$ and the steady-state normalized transmission $T$, plotted versus probe and cavity detunings in Fig.~\ref{pbgplot}.
\begin{figure}\label{pbg_slice_p20}
\end{figure}
These transmission spectra reflect the structure of the eigenvalues plotted in Fig.~\ref{pbg_levels}, although since $\kappa$ is not substantially smaller than $g$ for the parameter set considered, the correspondence is perhaps not as pronounced as for the previous section.
While the transmission spectra are dominated by a pair of broad peaks with widths of the order of $\kappa$, of particular interest are sharp features at $\omega_p \approx \omega_{4\to5'}-0.3$~GHz and $\omega_p \approx \omega_{4\to5'}+8.9$~GHz. These transmission features are particularly strong at the cavity detunings where transition frequencies of $H_2$ cross, i.e., where the dual resonance condition is satisfied, which for the parameters $(g,\kappa,\gamma)=(17,4.4,0.0026)$GHz occurs at $\omega_c \approx \omega_{4\to5'}+20$~GHz and $\omega_c \approx \omega_{4\to5'}-13$~GHz. The steady-state transmission at these cavity detunings is plotted in Figs.~\ref{pbg_slice_m13}(a) and \ref{pbg_slice_p20}(a) versus probe detuning.
Also plotted, in Figs.~\ref{pbg_slice_m13}(b) and \ref{pbg_slice_p20}(b), are the total populations in the $F=3$ and $F=4$ ground-state manifolds, which illustrate that
the sharp peaks in cavity transmission are associated with significant optical pumping effects.
For the case illustrated in Fig.~\ref{pbg_slice_p20},
the transitions which satisfy the dual resonance condition are between the $F=3$ ground-state manifold and a manifold of excited eigenstates which have a significant photon component, and between the $F=4$ ground-state manifold and the central band of atom-like eigenstates. Weak dissipative channels (primarily atomic spontaneous emission) can transfer population between the two transitions in a manner that depends sensitively on the probe field detuning and the atomic state compositions of the excited eigenstates.
Pronounced optical pumping effects between the different $m_F$ levels also occur as the probe field is tuned to the various atom-like eigenstates as
a result of the suppression of $\pi$-polarized spontaneous emission from each of
these states.
In Fig.~\ref{pbg_slice}(a), we plot the normalized steady-state transmission versus probe detuning with the cavity frequency fixed between the frequencies of the $F=3\to F'$ and $F=4\to F'$ transitions. Two small narrow peaks associated with the dual resonance effect are still apparent, and the atomic populations, plotted in Fig.~\ref{pbg_slice}(b), now show very strong and abrupt pumping into the $F=3$ or $F=4$ manifolds around these peaks.
\begin{figure}\label{pbg_slice}
\end{figure}
Future experiments with single atoms coupled to photonic band gap cavities should be able to study these sharp features. They should be relatively easy to measure because although they are narrow in probe frequency (which is easily controlled), they are robust against changes in cavity frequency (which is harder to control experimentally) of the order of $\kappa$.
\section{Conclusion}
We have presented results of the calculation of the weak-field steady-state transmission of a single-mode linearly polarized optical resonator coupled to the D2 transition of a single Cesium atom.
Our results are for a regime of single-photon dipole coupling strength not previously
considered, but of relevance to planned experiments with microtoroid and photonic bandgap cavities, as well as with other recently-implemented atom-chip microcavity systems \cite{treutlein06,barclay06}.
They necessarily take into account the entire atomic hyperfine
structure and comparison with simpler models highlights the importance of doing so.
In addition to features expected from a strongly coupled atom-cavity system, they also
reveal interesting and significant quantum interference phenomena associated with
the coupling of different atomic transitions to the same mode or modes of
the electromagnetic field.
\begin{acknowledgments}
This research is supported by the National Science Foundation, by the Caltech MURI Center for Quantum Networks, and by the Advanced Research and Development Activity (ARDA). ASP acknowledges support from the Marsden Fund of the Royal Society of New Zealand.
\end{acknowledgments}
\end{document}
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\begin{document}
\begin{abstract} \noindent
We study the stable pairs theory of local curves in 3-folds
with descendent insertions.
The rationality of the partition function
of descendent invariants is established
for the full local curve geometry (equivariant
with respect
to the scaling 2-torus) including relative
conditions and odd degree insertions for higher genus curves.
The capped 1-leg descendent vertex (equivariant
with respect to the 3-torus)
is also proven to be rational.
The results are obtained by combining geometric constraints
with a detailed analysis of the poles of the descendent
vertex.
\end{abstract}
\title{ extbf{Descendents on local curves: Rationality}
\setcounter{tocdepth}{1}
\tableofcontents
\setcounter{section}{-1}
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\section{Introduction}
\subsection{Descendents}\label{dess}
Let $X$ be a nonsingular 3-fold, and let
$$\beta \in H_2(X,\mathbb{Z})$$ be a nonzero class. We will study here the
moduli space of stable pairs
$$[\OO_X \stackrel{s}{\rightarrow} F] \in P_n(X,\beta)$$
where $F$ is a pure sheaf supported on a Cohen-Macaulay subcurve of $X$,
$s$ is a morphism with 0-dimensional cokernel, and
$$\chi(F)=n, \ \ \ [F]=\beta.$$
The space $P_n(X,\beta)$
carries a virtual fundamental class obtained from the
deformation theory of complexes in
the derived category \cite{pt}. A review can be found in Section \ref{ooo}.
Since $P_n(X,\beta)$ is a fine moduli space, there exists a universal sheaf
$$\FF \rightarrow X\times P_{n}(X,\beta),$$
see Section 2.3 of \cite{pt}.
For a stable pair $[\OO_X\to F]\in P_{n}(X,\beta)$, the restriction of
$\FF$
to the fiber
$$X \times [\OO_X \to F] \subset
X\times P_{n}(X,\beta)
$$
is canonically isomorphic to $F$.
Let
$$\pi_X\colon X\times P_{n}(X,\beta)\to X,$$
$$\pi_P\colon X\times P_{n}(X,\beta)
\to P_{n}(X,\beta)$$
be the projections onto the first and second factors.
Since $X$ is nonsingular
and
$\FF$ is $\pi_P$-flat, $\FF$ has a finite resolution
by locally free sheaves.
Hence, the Chern character of the universal sheaf $\FF$ on
$X \times P_n(X,\beta)$ is well-defined.
By definition, the operation
$$
\pi_{P*}\big(\pi_X^*(\gamma)\cdot \text{ch}_{2+i}(\FF)
\cap(\pi_P^*(\ \cdot\ )\big)\colon
H_*(P_{n}(X,\beta))\to H_*(P_{n}(X,\beta))
$$
is the action of the descendent $\tau_i(\gamma)$, where
$\gamma \in H^*(X,\Z)$.
For nonzero $\beta\in H_2(X,\Z)$ and arbitrary $\gamma_i\in H^*(X,\Z)$,
define the stable pairs invariant with descendent insertions by
\begin{eqnarray*}
\left\langle \prod_{j=1}^k \tau_{i_j}(\gamma_j)
\right\rangle_{\!n,\beta}^{\!X}& = &
\int_{[P_{n}(X,\beta)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_j) \\
& = &
\int_{P_n(X,\beta)} \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\Big( [P_{n}(X,\beta)]^{vir}\Big).
\end{eqnarray*}
The partition function is
$$
\ZZ^X_{\beta}\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)
=\sum_{n}
\left\langle \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right\rangle_{\!n,\beta}^{\!X}q^n.
$$
Since $P_n(X,\beta)$ is empty for sufficiently negative
$n$,
$\ZZ^X_{\beta}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)$
is a Laurent series in $q$. The following conjecture was made in
\cite{pt2}.
\begin{conj}
\label{111}
The partition function
$\ZZ_{\beta}^X\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)$ is the
Laurent expansion of a rational function in $q$.
\end{conj}
If only primary field insertions $\tau_0(\gamma)$ appear,
Conjecture \ref{111} is known for
toric $X$ by \cite{moop, mpt} and for Calabi-Yau $X$ by
\cite{bridge,toda} together with \cite{joy}.
In the presence of descendents $\tau_{i>0}(\gamma)$,
very few results have been obtained.
The central result of the present paper is the proof of
Conjecture 1 in case $X$ is the total space of
an rank 2 bundle over a curve, a {\em local curve}.
In fact, the rationality
of the stable pairs descendent theory of relative local curves
is proven.
\subsection{Local curves} \label{lc1}
Let $N$ be a split
rank 2 bundle on a nonsingular projective curve $C$ of genus $g$,
\begin{equation}\label{ffg}
N=L_1\oplus L_2.
\end{equation}
The splitting determines a scaling action of a 2-dimensional torus
$$T=\C^* \times \C^*$$ on $N$.
The {\em level} of the splitting is the pair of integers
$(k_1,k_2)$ where,
$$k_i= {\text {deg}}(L_i).$$
Of course, the scaling action and the level
depend upon the choice of splitting \eqref{ffg}.
Let $s_1,s_2 \in H^*_\mathbf{T}(\bullet)$ be the first Chern classes
of the standard representations of the first and second
$\C^*$-factors of $T$ respectively.
We define
\begin{equation}\label{lwww}
\left\langle \prod_{j=1}^k \tau_{i_j}(\gamma_j)
\right\rangle_{\!n,d}^{\!N} =
\int_{[P_{n}(N,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_j) \ \ \ \in \mathbb{Q}(s_1,s_2)\ .
\end{equation}
Here,
the curve class is $d$ times the zero section $C \subset N$ and
$$\gamma_j \in H^*(C,\mathbb{Z})\ .$$
The right side of \eqref{lwww} is defined by $T$-equivariant
residues as in \cite{BryanP,lcdt}. Let
$$
\ZZ_{d}^{N}\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^T
=\sum_{n}
\left\langle \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right\rangle_{\!n,d}^{\! N}q^n.
$$
\begin{thm}
\label{onnn}
$\ZZ_{d}^{N}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)^T$ is the
Laurent expansion in $q$ of a rational function in $\mathbb{Q}(q,s_1,s_2)$.
\end{thm}
The rationality of Theorem \ref{onnn} holds even when
$\gamma_j \in H^1(C,\mathbb{Z})$.
Theorem \ref{onnn} is proven via
the stable pairs theory of relative
local curves and the 1-leg descendent vertex.
The proof provides a method to
compute $\ZZ_{d}^{N}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)^T$.
\subsection{Relative local curves} \label{lc2}
\label{relgeom}
The fiber of $N$ over a point $p\in C$ determines a $T$-invariant
divisor
$$N_p \subset N$$
isomorphic to $\com^2$ with the standard $T$-action.
For $r>0$, we will consider the local theory of $N$
relative to the divisor
$$S= \bigcup_{i=1}^r N_{p_i} \subset N$$
determined by the fibers over $p_1,\ldots,p_r\in C$.
Let $P_n(N/S,d)$ denote the relative moduli space of stable pairs, see
\cite{pt} for a discussion.
For each $p_i$, let $\eta^i$
be a partition of $d$ weighted
by the equivariant Chow ring,
$$A_T^*(N_{p_i},{\mathbb Q})\stackrel{\sim}{=} {\mathbb Q}[s_1,s_2],$$
of the fiber $N_{p_i}$.
By Nakajima's construction,
a weighted partition $\eta^i$ determines a $T$-equivariant class
$$\CC_{\eta^i} \in A_T^*(\text{Hilb}(N_{p_i},d), \mathbb{Q})$$
in the
Chow ring of the Hilbert scheme of points.
In the theory of stable pairs, the weighted partition $\eta^i$
specifies relative
conditions via the boundary map
$$\epsilon_i: P_n(N/S,d)\rightarrow \text{Hilb}(N_{p_i},d).$$
An element $\eta\in {\mathcal P}(d)$ of the set of
partitions of $d$ may be
viewed as a
weighted partition with all weights set to the identity class
$$1\in H^*_T(N_{p_i},{\mathbb Q})\ .$$
The Nakajima basis of $A_T^*(\text{Hilb}(N_{p_i},d), \mathbb{Q})$ consists of
identity weighted partitions indexed by ${\mathcal P}(d)$.
The $T$-equivariant intersection pairing in the Nakajima basis is
$$g_{\mu\nu}=\int_{\text{Hilb}(N_{p_i},d)} \CC_\mu \cup \CC_\nu =
\frac{1}{(s_1s_2)^{\ell(\mu)}}
\frac{(-1)^{d-\ell(\mu)}}
{{\mathfrak{z}}(\mu)}\ {\delta_{\mu,\nu}},$$
where
$${\mathfrak z}(\mu) = \prod_{i=1}^{\ell(\mu)} \mu_i \cdot
|\text{Aut}(\mu)|.$$
Let $g^{\mu\nu}$ be the inverse matrix.
The notation $\eta([0])$ will be used to set all
weights to $[0]\in A^*_T(N_{p_i},{\mathbb Q} )$.
Since
$$[0]= s_1s_2 \in A^*_T(N_{p_i}, {\mathbb Q} ),$$
the weight choice has only a mild effect.
Following the
notation of \cite{BryanP,lcdt},
the relative stable pairs partition function with
descendents,
\begin{equation*}
{\mathsf Z}^{N/S}_{d,\eta^1,\dots,\eta^r}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^T
=\sum _{n\in \Z }q^{n}
\int _{[P_{n} (N/S,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_{j})\
\prod_{i=1}^r \epsilon_i^*(\CC_{\eta^i}),
\end{equation*}
is well-defined for local curves.
\begin{thm}
\label{tnnn}
$\ZZ_{d,\eta^1,\dots,\eta^r}
^{N/S}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)^T$ is the
Laurent expansion in $q$ of a rational function in $\mathbb{Q}(q,s_1,s_2)$.
\end{thm}
Theorem \ref{tnnn} implies Theorem \ref{onnn} by the degeneration formula.
The
proof of Theorem \ref{tnnn} uses the TQFT formalism exploited in
\cite{BryanP,lcdt} together with an analysis of
the capped 1-leg descendent vertex.
\subsection{Capped 1-leg descendent vertex} \label{legger}
The capped 1-leg geometry concerns
the trivial bundle,
$$N = \mathcal{O}_{\PP^1} \oplus \mathcal{O}_{\PP^1} \rightarrow \PP^1\ ,$$
relative to the fiber
$$N_\infty \subset N$$
over $\infty \in \PP^1$.
Capped geometries have been studied (without descendents)
in \cite{moop}.
The total space $N$ naturally carries an action of a
3-dimensional torus $$\mathbf{T} = T \times \com^*\ .$$
Here, $T$ acts as before by scaling the
factors of $N$ and preserving the relative divisor $N_\infty$.
The $\com^*$-action
on the base $\PP^1$ which fixes the points $0, \infty\in \PP^1$
lifts to an additional $\com^*$-action on $N$ fixing
$N_\infty$.
The equivariant cohomology
ring $H_{\mathbf{T}}^*(\bullet)$ is generated by
the Chern classes $s_1$, $s_2$, and $s_3$
of the standard representation of the three $\com^*$-factors.
We define
\begin{equation}\label{pppw}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}
=\sum _{n\in \Z }q^{n}
\int _{[P_{n} (N/N_\infty,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_{j})\
\cup \epsilon_\infty^*(\mathsf{C}_{\eta}),
\end{equation}
by $\mathbf{T}$-equivariant residues.\footnote{The $T$-equivariant
series associated to the cap will be denoted
$${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^T \ ,
$$
for $\gamma_j\in H^*(\Pp,\mathbb{Z})$.}
Here, $\gamma_j \in H^*_{\mathbf{T}}(\PP^1,\mathbb{Z})$.
By definition, the partition function \eqref{pppw} is
a Laurent series in $q$ with coefficients in the field
$\mathbb{Q}(s_1,s_2,s_3)$.
\begin{thm}
\label{cnnn}
$
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}$
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{thm}
Theorem \ref{cnnn} is the main contribution of the paper.
The result relies upon
a delicate cancellation of
poles in the vertex formula of \cite{pt2}
for stable pairs invariants.
Theorem \ref{tnnn} is derived as a consequence.
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\subsection{Capped 1-leg descendent vertex} \label{legger}
The capped 1-leg geometry concerns
the trivial bundle,
$$N = \mathcal{O}_{\PP^1} \oplus \mathcal{O}_{\PP^1} \rightarrow \PP^1\ ,$$
relative to the fiber
$$N_\infty \subset N$$
over $\infty \in \PP^1$.
Capped geometries have been studied (without descendents)
in \cite{moop}.
The total space $N$ naturally carries an action of a
3-dimensional torus $$\mathbf{T} = T \times \com^*\ .$$
Here, $T$ acts as before by scaling the
factors of $N$ and preserving the relative divisor $N_\infty$.
The $\com^*$-action
on the base $\PP^1$ which fixes the points $0, \infty\in \PP^1$
lifts to an additional $\com^*$-action on $N$ fixing
$N_\infty$.
The equivariant cohomology
ring $H_{\mathbf{T}}^*(\bullet)$ is generated by
the Chern classes $s_1$, $s_2$, and $s_3$
of the standard representation of the three $\com^*$-factors.
We define
\begin{equation}\label{pppw}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}
=\sum _{n\in \Z }q^{n}
\int _{[P_{n} (N/N_\infty,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_{j})\
\cup \epsilon_\infty^*(\mathsf{C}_{\eta}),
\end{equation}
by $\mathbf{T}$-equivariant residues.\footnote{The $T$-equivariant
series associated to the cap will be denoted
$${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^T \ ,
$$
for $\gamma_j\in H^*(\Pp,\mathbb{Z})$.}
Here, $\gamma_j \in H^*_{\mathbf{T}}(\PP^1,\mathbb{Z})$.
By definition, the partition function \eqref{pppw} is
a Laurent series in $q$ with coefficients in the field
$\mathbb{Q}(s_1,s_2,s_3)$.
\begin{thm}
\label{cnnn}
$
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}$
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{thm}
Theorem \ref{cnnn} is the main contribution of the paper.
The result relies upon
a delicate cancellation of
poles in the vertex formula of \cite{pt2}
for stable pairs invariants.
Theorem \ref{tnnn} is derived as a consequence.
\subsection{Stationary theory}
In \cite{parttwo}, we prove reduction rules for
stationary descendents in the
$T$-equivariant local theory of curves. Let
$\mathsf{p}\in H^2(C,\mathbb{Z})$ be the class of a point on a
nonsingular curve $C$. The stationary descendents are
$\tau_i(\mathsf{p})$.
For the degree $d$ local theory of $C$, we
find universal formulas expressing the descendents
$\tau_{i>d}(\mathsf{p})$ in terms of the descendents
$\tau_{i\leq d}(\mathsf{p})$.
The reduction rules provide an alternative (and more effective)
approach
to
the rationality of Theorem \ref{tnnn} in the
stationary case.
The exact calculation in \cite{parttwo} of the basic stationary
descendent series
$$\mathsf{Z}^{\mathsf{cap}}_{d,(d)}( \tau_d(\mathsf{p}))^T =
\frac{q^d}{d!}\left(\frac{s_1+s_2}{s_1s_2}\right)
\frac{1}{2}\sum_{i=1}^d \frac{ 1+(-q)^{i}}{1-(-q)^i} \ $$
plays a special role.
The coefficient of $q^d$,
$$ \left\langle \tau_d, (d) \right\rangle_{\text{Hilb}(\com^2,d)}=
\frac{1}{2\cdot (d-1)!} \left(\frac{s_1+s_2}{s_1s_2}\right),$$
is the classical $T$-equivariant pairing on the
Hilbert scheme of $d$ points in $\C^2$.
The $T$-equivariant stationary descendent theory is simpler
than the full descendent theories studied here. We do not know an
alternative approach to the rationality of the
full $T$-equivariant descendent theory
of local curves. Even the rationality of the $\mathbf{T}$-equivariant
stationary theory of the cap does not appear to
be accessible via \cite{parttwo}.
The methods of \cite{parttwo} also prove a functional equation for
the partition function for stationary descendents which is
a special case of the following conjecture we make here.
\begin{conj}
\label{33345}
Let $\ZZ_{d,\eta^1,\dots,\eta^r}
^{N/S}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)^T$ be the Laurent expansion in $q$ of
$F(q,s_1,s_2) \in \mathbb{Q}(q,s_1,s_2)$. Then, $F$
satisfies the functional equation
\[
F(q^{-1},s_2,s_2) = (-1)^{\Delta+|\eta|-\ell(\eta)
+
\sum_{j=1}^k
i_j}q^{-\Delta}F(q,s_1,s_2),
\]
where the constants are defined by
$$\Delta = \int_{\beta}c_1(T_N),\ \ \
|\eta|=\sum_{i=1}^r |\eta^i|,\ \ \ \text{and} \ \ \
\ell(\eta)=\sum_{i=1}^r \ell(\eta^i)
\ .$$
\end{conj}
Here, $T_N$ is the tangent bundle of the 3-fold $N$, and
$\beta$ is the curve class given by $d$ times the $0$-section.
We believe the straightforward generalization of Conjecture \ref{33345}
to all descendent partition functions for the stable
pairs theories of relative 3-folds (equivariant and non-equivariant) holds.
If there are no descendents, the functional equation is known
to hold in the toric case \cite{moop}.
The strongest evidence with descendents
is the stationary result of Theorem 2
of \cite{parttwo}.
\subsection{Denominators}
The descendent partition
functions for the stable pairs theory of local
curves have very restricted
denominators when considered as rational functions in $q$
with coefficients in $\Q(s_1,s_2)$ for Theorems \ref{onnn}-\ref{tnnn}
and rational functions in $q$
with coefficients in $\Q(s_1,s_2,s_3)$ for Theorem \ref{cnnn}.
\begin{conj}
\label{222}
The denominators of the degree $d$ descendent partition functions
$\ZZ$ of Theorems \ref{onnn}, \ref{tnnn}, and \ref{cnnn}
are products of factors of the form $q^k$ and
$$1-(-q)^r$$
for $1\leq r \leq d$.
\end{conj}
In other words, the poles in $-q$ are conjectured to occur
only at 0 and
$r^{th}$ roots for $r$ at most $d$ (and have no
dependence on the variables $s_i$).
Conjecture \ref{222} is proven in Theorem \ref{2222} of
Section \ref{ennd}
for descendents
of even cohomology.
The denominator restriction yields new
results about the 3-point functions of the Hilbert scheme of
points of $\mathbb{C}^2$ stated as a Corollary to Theorem \ref{2222}.
\subsection{Descendent theory of toric 3-folds}
Calculation of the descendent theory of stable pairs on nonsingular
toric 3-folds requires
knowledge of the capped 3-leg descendent vertex.{\footnote{The
capped 2-leg descendent vertex is, of course, a specialization
of the 3-leg vertex.}}
The rationality of the capped 3-leg descendent vertex is proven
in \cite{part3} via a geometric reduction to the 1-leg case
of Theorem \ref{cnnn}.
As a result, Conjecture \ref{111} is
established for all nonsingular toric 3-folds. The rationality
of the descendent theory of
several log Calabi-Yau geometries
is also proven in \cite{part3}.
\subsection{Plan of the paper}
After a brief review of the theory of
stable pairs in Section \ref{ooo}, the
vertex formalism of \cite{pt2} is summarized in
Section \ref{ttt}.
The proof of Theorem \ref{cnnn} is presented in Section \ref{333}
for descendents of the nonrelative $\mathbf{T}$-fixed point
$0\in \PP^1$ modulo the
pole cancellation property established
in Section \ref{polecan}.
Depth and the rubber calculus for stable pairs of local curves
are discussed in Sections \ref{depp} and \ref{rubc}.
The full statement of Theorem \ref{cnnn}
is obtained in Section
\ref{444}. In fact,
the rationality of the $\mathbf{T}$-equivariant descendent
theories of all twisted caps and tubes is established
in Section \ref{444}.
Theorems \ref{onnn} and \ref{tnnn} are proven as a
consequence of Theorem \ref{cnnn} in Section \ref{555} using the
methods of \cite{BryanP,vir,lcdt}. Denominators
are studied in Section \ref{ennd}.
\subsection{Other directions}
Whether parallel results can be obtained for the
local Gromov-Witten theory of curves \cite{BryanP} is
an interesting question. Although conjectured to be equivalent,
the descendent theory of stable pairs on $3$-folds appears more accessible
than descendents in Gromov-Witten theory. The direct vertex analysis undertaken here for
Theorem \ref{cnnn} must be replaced
in Gromov-Witten theory with a deeper
understanding of Hodge integrals \cite{FP}.
Another advantage of stable pairs, at least for Calabi-Yau
geometries, is the possibility of using
motivic integrals with respect to Beh\-rend's $\chi$-function \cite{Beh},
see \cite{pt3} for an early use. Recently, D. Maulik and R. P. Thomas
have been pursuing $\chi$-functions in the log Calabi-Yau setting.
Applications to the rationality of descendent series in
Fano geometries might be possible.
A principal motivation of studying descendents for stable
pairs is the perspective of \cite{mptop}.
Descendents constrain relative invariants. With the
degeneration formula, the possibility emerges of
studying stable pairs on arbitrary (non-toric) 3-folds.
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\section{Stable pairs on $3$-folds}
\label{ooo}
\subsection{Definitions}
Let $X$ be a nonsingular quasi-projective $3$-fold over $\mathbb{C}$
with
polarization $L$.
Let $\beta\in H_2(X,\mathbb{Z})$ be a nonzero class.
The moduli space $P_n(X,\beta)$ parameterizes \emph{stable pairs}
\begin{equation}\label{vqq2}
\OO_X \stackrel{s}{\rightarrow} F
\end{equation}
where $F$ is a sheaf with Hilbert polynomial
$$ \chi(F\otimes L^k) = k\int_\beta c_1(L) + n$$
and $s\in H^0(X,F)$ is a section.
The two stability conditions are:
\begin{enumerate}
\item[(i)]
the sheaf $F$ is {pure} with proper support,
\item[(ii)] the section $\OO_X \stackrel{s}{\rightarrow} F$ has 0-dimensional
cokernel.
\end{enumerate}
By definition, {\em purity} (i) means
every nonzero
subsheaf of $F$ has support of dimension 1 \cite{HLShaves}. In particular,
purity implies the (scheme-theoretic)
support $C_F$ of $F$ is a Cohen-Macaulay curve.
A quasi-projective moduli space of stable pairs
can be constructed by a standard GIT analysis of Quot scheme
quotients \cite{LPPairs1}.
For convenience, we will often refer to the stable pair
\eqref{vqq2} on $X$ simply by $(F,s)$.
\subsection{Virtual class}
A central result of \cite{pt} is the construction of a
virtual class on $P_n(X,\beta)$.
The standard approach to the deformation theory of pairs
fails to yield an appropriate 2-term deformation theory
for $P_n(X,\beta)$.
Instead, $P_n(X,\beta)$ is viewed in \cite{pt}
as a moduli space
of complexes in the derived category.
Let $D^b(X)$ be the bounded derived category of coherent
sheaves on $X$.
Let
$${I}\udot = \left\{ \OO_X \rightarrow F \right\}\in
D^b(X)$$
be the complex determined by a stable pair.
The tangent-obstruction theory obtained by deforming ${I}\udot$
in $D^b(X)$ while fixing its determinant is 2-term and governed by the
groups{\footnote{The subscript 0 denotes traceless $\Ext$.}}
$$\Ext^1({I}\udot, {I}\udot)_0, \ \
\Ext^2({I}\udot, {I}\udot)_0.$$
The virtual class
$$[P_n(X,\beta)]^{vir} \in A_{\text{dim}^{vir}}
\left(P_n(X,\beta),\mathbb{Z}\right)$$
is then obtained by standard methods \cite{BehFan,LiTian}.
The virtual dimension is
$$\text{dim}^{vir} = \int_\beta c_1(T_X).$$
Apart from the derived category deformation theory,
the construction of the virtual class of $P_n(X,\beta)$
is parallel to virtual class construction in DT theory \cite{Thomas}.
\subsection{Characterization}
Consider the kernel/cokernel exact sequence associated to a stable
pair $(F,s)$,
\beq \label{IOFQ}
0\to\I_{C_F}\to\OO_X\Rt{s}F\to Q\to0.
\eeq
The kernel is the ideal sheaf of the Cohen-Macaulay support
curve $C_F$ by Lemma 1.6 of \cite{pt}. The cokernel
$Q$ has dimension 0 support by stability.
The {\em reduced} support scheme, $\text{Support}^{red}(Q)$, is
called the {\em zero locus} of the pair.
The zero locus lies on $C_F$.
Let $C\subset X$ be a fixed Cohen-Macaulay curve.
Stable pairs with support $C$ and bounded zero locus are characterized
as follows.
Let $$\m\subset\OO_C$$
be the ideal in $\OO_C$ of a 0-dimensional subscheme.
Since $$\hom(\m^r/\m^{r+1},\OO_C)=0$$ by
the purity of $\OO_C$, we obtain an inclusion $$\hom(\m^r,\OO_C)\subset
\hom(\m^{r+1},\OO_C).$$
The inclusion $\m^r\into\OO_C$ induces a canonical section
$$\OO_C\into\hom(\m^r,\OO_C).$$
\begin{prop} \label{descl}
A stable pair $(F,s)$ with support $C$ satisfying
$$\text{\em Support}^{red}(Q) \subset \text{\em Support}(\OO_C/\m)$$
is equivalent to a subsheaf
of $\hom(\m^r,\OO_C)/\OO_C,\ r\gg0.$
\end{prop}
Alternatively, we may work with coherent subsheaves of the quasi-coherent sheaf
\begin{equation}\label{infhom}
\lim\limits_{\mathbf{T}o}\hom(\m^r,\OO_C)/\OO_C
\end{equation}
Under the equivalence of Proposition \ref{descl}, the
subsheaf of \eqref{infhom} corresponds to $Q$, giving a subsheaf $F$ of
$\lim\limits_{\mathbf{T}o}\hom(\m^r,\OO_C)$ containing the canonical subsheaf $\OO_C$
and the sequence
$$0\to\OO_C\stackrel{s}{\rightarrow} F \rightarrow Q \rightarrow 0.$$
Proposition \ref{descl} is proven in \cite{pt}.
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\section{$\mathbf{T}$-fixed points with one leg} \label{ttt}
\subsection{Affine chart}
Let $N$ be the 3-fold total space of
$$\OO_{\PP^1} \oplus \OO_{\PP^1} \rightarrow \PP^1 \ $$
carrying the action of the 3-dimensional torus $\mathbf{T}$ as in Section
\ref{legger}.
Let
\begin{equation}\label{vqaa}
[\OO_N \stackrel{s}{\rightarrow} F] \in P_n(N,d)^\mathbf{T}
\end{equation}
be a $\mathbf{T}$-fixed stable pair. The curve class is
$d[\PP^1]$.
Let $U\subset N$ be the $\mathbf{T}$-invariant affine chart
associated to the
$\mathbf{T}$-fixed point of $N$ lying over $0\in \PP^1$.
The restriction of the stable pair \eqref{vqaa} to
the chart $U$,
\begin{equation}\label{vvvt}
\OO_{U} \stackrel{s_U}{\rightarrow} F_U\ ,
\end{equation}
determines an invariant section $s_U$ of an
equivariant sheaf $F_U$.
Let $x_1,x_2,x_3$ be coordinates on the affine chart $U$
in which the $\mathbf{T}$-action takes the diagonal form,
$$(t_1,t_2,t_3) \cdot x_i = t_i x_i.$$
By convention, $x_1$ and $x_2$ are coordinates on the fibers of
$N$ and $x_3$ is a coordinate on the base $\PP^1$.
We will characterize the restricted data $(F_U,s_U)$
in the coordinates $x_i$ closely following the presentation
of \cite{pt2}.
\subsection{Monomial ideals and partitions}
Let $x_1,x_2$ be coordinates on the plane
$\C^2$.
A subscheme $S\subset \C^2$ invariant under the
action of the diagonal torus,
$$(t_1,t_2)\cdot x_i = t_ix_i$$
must be defined by a monomial ideal
$\I_S \subset \C[x_1,x_2]$.
If
$$\dim_\C \C[x_1,x_2]/\I_S < \infty$$
then $\I_S$ determines a finite partition $\mu_S$
by considering lattice points corresponding
to monomials of $\C[x_1,x_2]$
{\em not} contained in $\I_S$.
Conversely, each partition $\mu$ determines a monomial ideal
$$\mu[x_1,x_2]\subset \C[x_1,x_2].$$
Similarly, the subschemes $S\subset \C^3$ invariant under
the diagonal $\mathbf{T}$-action are in bijective correspondence with
$3$-dimensional partitions.
\subsection{Cohen-Macaulay support}
The first step in the characterization of the restricted
data
\eqref{vvvt}
is to determine the scheme-theoretic support
$C_U$ of $F_U$. If nonempty, $C_U$ is a
$\mathbf{T}$-invariant, Cohen-Macaulay subscheme of pure dimension 1.
The $\mathbf{T}$-fixed subscheme $C_U \subset \C^3$ is defined by a
monomial ideal $$\I_C \subset \C[x_1,x_2,x_3].$$
associated to the 3-dimensional partition $\pi$.
The localisation
$$(\I_C)_{x_3} \subset \C[x_1,x_2,x_3]_{x_3},$$
is $T$-fixed and corresponds to a 2-dimensional partition
$\mu$.
Alternatively, the 2-dimensional partitions $\mu$ can be defined
as the infinite limit of
the $x_3$-constant cross-sections of $\pi$.
In order for $C_U$ to have dimension 1,
$\mu$ can not be empty.
There exists
a unique {\em minimal} $\mathbf{T}$-fixed subscheme
$$C_{\mu}\subset \C^3$$
with outgoing partition $\mu$.
The $3$-dimensional partition corresponding to $C_\mu$ is
the infinite cylinder on
the $x_3$-axis determined by the $2$-dimensional
partitions $\mu$.
Let
\begin{eqnarray*}
\I_{\mu}= \mu[x_1,x_2] \cdot \C[x_1,x_2,x_3],& \ \ &
C_{\mu}= \OO_{\C^3}/\I_{\mu}\ .
\end{eqnarray*}
\label{cmmm}
\subsection{Module $M_3$}
The kernel/cokernel sequence associated to the
$\mathbf{T}$-fixed restricted data \eqref{vvvt} takes the form
\begin{equation}\label{cvrw}
0 \rightarrow \I_{C_\mu} \rightarrow \OO_{U} \stackrel{s}
{\rightarrow} F_U \rightarrow Q_U \rightarrow 0\
\end{equation}
for an outgoing partition $\mu$.
Since the
support of the quotient $Q_U$ in \eqref{cvrw} is 0-dimensional
by stability and $\mathbf{T}$-fixed,
$Q_U$
must be supported at the origin.
By Proposition \ref{descl}, the pair $(F_U,s_U)$
corresponds to a $\mathbf{T}$-invariant subsheaf of
$$\lim\limits_{\mathbf{T}o}\hom(\m^r,\OO_{C_\mu})/\OO_{C_\mu}
,$$
where
$\m$ is the
ideal sheaf of the origin in $C_\mu\subset\C^3$.
Let
$$M_3 = (\OO_{C_{\mu}})_{x_3}$$
be the $\C[x_1,x_2,x_3]$-module obtained by localisation. Explicitly
$$
M_3=\C[x_3,x_3^{-1}]\otimes\frac{\C[x_1,x_2]}{\mu[x_1,x_2]}\,.
$$
By elementary algebraic arguments,
\begin{eqnarray*}
\lim\limits_{\mathbf{T}o}\hom(\m^r,\OO_{C_\mu}) &
\cong & M_3\ .
\end{eqnarray*}
The $\mathbf{T}$-equivariant
$\C[x_1,x_2,x_3]$-module $M_3$ has a canonical $\mathbf{T}$-invariant
element 1.
By Proposition \ref{descl}, the $\mathbf{T}$-fixed pair $(F_U,s_U)$
corresponds to a finitely generated $\mathbf{T}$-invariant
$\C[x_1,x_2,x_3]$-submodule
\beq\label{datum}
Q_U\subset M_3/\langle 1 \rangle.
\eeq
Conversely, {\em every} finitely generated{\footnote{Here, finitely generated
is equivalent to finite dimensional or Artinian.}}
$\mathbf{T}$-invariant $\C[x_1,x_2,x_3]$-sub\-module
$$Q \subset M_3/\langle 1 \rangle$$
occurs as the restriction to $U$ of
a $\mathbf{T}$-fixed stable pair on $N$.
\subsection{The 1-leg stable pairs vertex}
\label{vc}
Let $R$ be the coordinate ring,
$$
R = \C[x_1,x_2,x_3] \cong \Gamma(U).
$$
Following
the conventions of Section \ref{legger}, the $\mathbf{T}$-action on $R$ is
\begin{equation*}
(t_1,t_2,t_3)\cdot x_i = t_i x_i \,.
\end{equation*}
Since the tangent spaces are dual to the coordinate functions,
the tangent weight of $\mathbf{T}$ along the third axis is $-s_3$.
Let $Q_U \subset M/\langle 1 \rangle$ be a $\mathbf{T}$-invariant
submodule viewed as a stable pair on $U$.
Let ${\mathbb{I}}_U\udot$ denote the universal complex on
$[Q_U] \times U$.
Consider a $\mathbf{T}$-equivariant free
resolution{\footnote{Here, ${\mathbb{I}}_U\udot$ is viewed to
live in degrees 0 and -1.}}
of ${\mathbb{I}}_U\udot$,
\begin{equation}
\label{resol}
\{ \F_{s} \rightarrow \dots \rightarrow \F_{-1}\}
\cong
{\mathbb{I}}_U\udot \ \in D^b([{Q}_U] \times U).
\end{equation}
Each term in \eqref{resol}
can be taken to have the form
$$
\F_i = \bigoplus_j
R(d_{ij})\,, \quad d_{ij} \in \Z^3.$$
The Poincar\'e polynomial
$$
P_U = \sum_{i,j} (-1)^{i+1} \ t^{d_{ij}}
\
\in \Z[t_1^\pm,t_2^\pm,t_3^\pm]$$
does not depend on the choice of the resolution
\eqref{resol}.
We denote the $\mathbf{T}$-character of $F_U$ by $\FFF_U$.
By the
sequence
$$0 \rightarrow \OO_{C_U} \rightarrow F_U \rightarrow Q_U
\rightarrow 0,$$
we have a complete understanding of
the representation $\FFF_U$.
The $\mathbf{T}$-eigenspaces of $F_U$ correspond to the
$\mathbf{T}$-eigenspaces of $\OO_{C_U}$ and
$Q_U$.
The result determines
$$\FFF_U \in \Z(t_1,t_2,t_3).$$
The rational dependence on the $t_i$ is elementary.
From
the resolution \eqref{resol}, we see
that the Poincar\'e polynomial
$P_U$ is related to the $\mathbf{T}$-character
of $F_U$ as follows:
\begin{equation}
\FFF_U =
\frac{1+P_U}{(1-t_1)(1-t_2)(1-t_3)} \label{PQ}
\,.
\end{equation}
The virtual represention
$\chi({\mathbb{I}}_U\udot,{\mathbb{I}}_U\udot)$ is given by the
following alternating sum
\begin{align*}
\chi({\mathbb{I}}_U\udot,{\mathbb{I}}_U\udot) &= \sum_{i,j,k,l} (-1)^{i+k}
\Hom_R(R(d_{ij}), R(d_{kl}))
\\
&= \sum_{i,j,k,l} (-1)^{i+k}
R(d_{kl}-d_{ij})\,.
\end{align*}
Therefore, the $\mathbf{T}$-character is
$$
\tr_{\chi({\mathbb{I}}_U,{\mathbb{I}}_U)} =
\frac{P_U \,\overline{P}_U}
{(1-t_1)(1-t_2)(1-t_3)} \,.
$$
The bar operation
$$\gamma \in \Z(\!(t_1,t_2,t_3)\!) \mapsto
\Z(\!(t_1^{-1},t_2^{-1},
t_3^{-1})\!)$$
is
$t_i \mapsto t_i^{-1}$
on the variables.
We find the $\mathbf{T}$-character of
the $U$ summand of virtual tangent
space $\mathcal{T}_{\left[{I}\udot\right]}$
of the moduli space of stable pairs of the 1-leg cap is
$$ \tr_{R-\chi({\mathbb{I}}\udot_U,{\mathbb{I}}_U\udot)} =
\frac{1-P_U \, \overline{P}_U}
{(1-t_1)(1-t_2)(1-t_3)} \, ,
$$
see \cite{pt2}.
Using \eqref{PQ}, we may express the answer in terms of
$\FFF_U$,
\begin{equation}\label{vertexchar}
\tr_{R-\chi({\mathbb{I}}\udot_U,{\mathbb{I}}_U\udot)}
= \FFF_{U} -
\frac{\overline{\FFF}_U}{t_1t_2t_3} + \FFF_{U}
\overline{\FFF}_U \frac{(1-t_1)(1-t_2)(1-t_3)}{t_1 t_2 t_3} \,.
\end{equation}
On the right side of
\eqref{vertexchar}, the rational functions
should be expanded
in ascending powers in the $t_i$.
The stable pairs vertex is obtained from \eqref{vertexchar}
after a redistribution of edge terms following \cite{pt2}.
Let
$$
\FFF_{\mu} = \sum_{(k_1,k_2) \in \mu} t_1^{k_1}
t_2^{k_2}\
$$
correspond to the outgoing partition $\mu$.
Define
$$
\GGG_{\mu} = - \FFF_{\mu} -
\frac{\overline{\FFF}_{\mu}}{t_1 t_2} + \FFF_{\mu}
\overline{\FFF}_{\mu} \frac{(1-t_1)(1-t_2)}{t_1 t_2} \,.
$$
Define the vertex character $\mathsf{V}_U$ by the following
modification,
\begin{equation}\label{gx34}
\mathsf{V}_U = \tr_{R-\chi({\mathbb{I}}\udot_U,{\mathbb{I}}_U\udot)}
+
\frac{\GGG_{\mu}(t_{1},t_{2})}{1-t_3}\, .
\end{equation}
The character $\mathsf{V}_U$ depends
{\em only on the local data ${Q}_U$}.
By the results of \cite{pt2}, $\mathsf{V}_U$ is a
Laurent polynomial in $t_1$, $t_2$, and $t_3$.
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\subsection{Descendents}
Let $[0]\in H^*_\mathbf{T}(\PP^1,\Z)$ be the class of the
$\mathbf{T}$-fixed point
$0\in \PP^1$. Consider the $\mathbf{T}$-equivariant
descendent (with value in the $\mathbf{T}$-equivariant cohomology
of a point),
\begin{multline}\label{vpzz}
\left\langle \tau_{i_1}([0]) \cdots \tau_{i_k}([0]) \right
\rangle_{n,d}^N =\\
\int_{P_n(N,d)} \prod_{j=1}^k \tau_{i_j}([0])
\Big( [P_{n}(N,d)]^{vir}\Big)\in \Q(s_1,s_2,s_3)\ ,
\end{multline}
following the notation of Section \ref{dess}.
In order to calculate \eqref{vpzz} by $\mathbf{T}$-localization, we
must determine the action of
the operators $\tau_{i}([0])$ on the $\mathbf{T}$-equivariant
cohomology of the $\mathbf{T}$-fixed loci.
The calculation of \cite{pt2} yields a formula for the
descendent weight,
\begin{multline}
\mathsf{w}_{{i_1},\cdots, {i_m}}
(Q_U) =\\
e(-\mathsf{V}_{U})
\cdot \prod_{j=1}^m
\text{ch}_{2+i_j}\big(\FFF_{U}\cdot
(1-t_1)(1-t_2)(1-t_3)\big) \ .
\end{multline}
The {\em descendent vertex} $\bW_\mu^{\mathsf{Vert}}(\tau_{i_1}([0]) \cdots
\tau_{i_m}([0]))$
is obtained from the descendent weight,
\begin{multline}\label{vvped}
\bW_\mu^{\mathsf{Vert}} (\tau_{i_1}([0]) \cdots \tau_{i_k}([0]))
= \\
\left(\frac{1}{s_1s_2}\right)^k
\sum_{Q_U}
\mathsf{w}_{{i_1}, \cdots, {i_k}}
(Q_U)\ q^{\ell({Q}_U)+|\mu|}\ \in
\Q(s_1,s_2,s_3)(\!(q)\!)\ .
\end{multline}
\label{heyle}
Here, $\ell(Q_U)$ is the length of $Q_U$.
\subsection{Edge weights}
The edge weight in the cap geometry is
$$\bW^{(0,0)}_\mu = e(\GGG_\mu)\ \in \Q(s_1,s_2).$$
In fact, $\bW^{(0,0)}_\mu$ is simply the inverse product
of the tangent weights of the Hilbert scheme of points of $\C^2$
at the $T$-fixed point corresponding to the partition $\mu$.
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\section{Capped 1-leg descendents: stationary} \label{333}
\subsection{Overview}
Consider the capped geometry of Section \ref{legger}. As before, let
$0\in \PP^1$ be the $\mathbf{T}$-fixed point away
from the relative divisor over $\infty \in \PP^1$, and
let
$$[0] \in H^*_{\mathbf{T}}(\PP^1, \mathbb{Z})$$
be the associated class.
The $\mathbf{T}$-weight on the tangent space to $\PP^1$ at 0
is $-s_3$.
We study here the stationary{\footnote{Stationary
refers to descendents of point classes.}} series
\begin{equation}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}\ .
\label{gbn}
\end{equation}
Our main result is a special case of Theorem \ref{cnnn}.
\begin{prop}
\label{cttt}
$
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{prop}
\subsection{Dependence on $s_3$}
The function \eqref{gbn} is the generating series of the
integrals
\begin{equation} \label{krt}
\left\langle \prod_{j=1}^k \tau_{i_j}([0])
\right\rangle_{\!n,\eta}^{\mathsf{cap},{\mathbf{T}}}
=
\int _{[P_{n} (N/N_\infty,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}([0])\
\cup \epsilon_\infty^*(C_{\eta})\ ,
\end{equation}
following the notation of Section \ref{legger}.
Let $\ell(\eta)$ denote the length of the partition
$\eta$ of $d$, and let \begin{equation}\label{ktgg}
\delta=\sum_{j=1}^k i_j + d-\ell(\eta)\ .
\end{equation}
The dimension of
$[P_{n} (N/N_\infty,d)]^{vir}$ after applying
the integrand of \eqref{krt} is $2d-\delta$.
\begin{lem} The
integral $\left\langle \prod_{j=1}^k \tau_{i_j}([0])
\right\rangle_{\!n,\eta}^{\mathsf{cap},\mathbf{T}}$ \label{rq2}
is a {\em polynomial} in $s_3$ of degree
$\delta$
with coefficients in the subring
$${\mathbb Q}[s_1,s_2]_{(s_1s_2)}\subset {\mathbb Q}(s_1,s_2).$$
\end{lem}
\begin{proof}
Let $N=\mathcal{O}_{\Pp} \oplus \mathcal{O}_{\Pp}$.
Let
${\mathbb{F}} \rightarrow {\mathcal N}$
denote the universal sheaf over the universal total space
$${\mathcal N} \rightarrow P_n(N/N_\infty,d).$$
Since $N=\PP^1 \times \com^2$, there is a proper morphism
$${\mathcal N} \rightarrow P_n(N/N_\infty,d) \times \com^2.$$
The locations and multiplicities of the supports of the universal
sheaf determine a
morphism of Hilbert-Chow type,
$$\iota:P_n(N/N_\infty,d) \rightarrow \text{Sym}^{d}(\com^2).$$
A $\mathbf{T}$-equivariant, proper
morphism,
$$\widehat{\iota}: \text{Sym}^{d}(\com^2) \rightarrow \oplus_{1}^{d}
(\com^2),$$
is obtained via the higher moments,
\begin{multline*}
\widehat{\iota}\Big( \ \{(x_i,y_i)\} \ \Big) = \\
\Big(\sum_i x_i, \sum_i y_i\Big) \oplus
\Big(\sum_i x^2_i, \sum_i y^2_i\Big) \oplus \cdots \oplus
\Big(\sum_i x^d_i, \sum_i y^d_i\Big).
\end{multline*}
Let $\rho=\widehat{\iota}\circ \iota$.
Since $\rho$ is a $\mathbf{T}$-equivariant, proper morphism,
there is a $\mathbf{T}$-equivariant push-forward
$$\rho_*: A^{\mathbf{T}}_*(P_n(N/N_\infty,d), {\mathbb Q}) \rightarrow
A^{\mathbf{T}}_*( \oplus_1^{d}(\com^2) , {\mathbb Q}).$$
Descendent invariants
are defined via the $\mathbf{T}$-equivariant residue of
$$\left(\prod_{j=1}^k \tau_{i_j}([0]) \cup \epsilon^*_{\infty}(C_\eta)
\right) \ \cap
[P_n(N/S,d)]^{vir} \ \in A^{\mathbf{T}}_*(P_n(N/N_\infty,d), {\mathbb Q}).$$
We may instead calculate the $\mathbf{T}$-equivariant residue of
\begin{equation}\label{ress}
\rho_*\left( \left(
\prod_{j=1}^k \tau_{i_j}([0]) \cup \epsilon^*_{\infty}(C_\eta)\right) \ \cap
[P_n(N/N_\infty,d)]^{vir}\right)
\end{equation} in
$A^{\mathbf{T}}_*(
\oplus_1^{d}(\com^2) , {\mathbb Q})$.
The codimension of the class \eqref{ress} in
$\oplus_1^{d}(\com^2)$
is $\delta$.
Since the third factor of $\mathbf{T}$ acts trivially on
$\oplus_1^{d}(\com^2)$, the class \eqref{ress} may be written as
\begin{equation}\label{htty3}
\gamma_0 s_3^0 + \gamma_1 s_3^1 + \ldots + \gamma_{\delta} s_3^{\delta}
\end{equation}
where $\gamma_i \in A^{T}_{2d-\delta+i}(
\oplus_1^{d}(\com^2) , {\mathbb Q})$.
Since the space $\oplus_1^{d}
(\com^2)$ has a unique $T$-fixed point
with tangent weights,
$$-s_1,-s_2,-2s_1,-2s_2, \ldots, -ds_1, -ds_2,$$
we conclude the localization of $\gamma_i$
has only monomial poles in the variables $t_1$ and $t_2$.
\end{proof}
As a consequence of Lemma \ref{rq2}, we may write
\begin{equation} \label{krtt}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
=\sum_{r=0}^\delta s_3^r \cdot \Gamma_r(q,s_1,s_2)
\end{equation}
where $\Gamma_r \in \mathbb{Q}(s_1,s_2)((q))$.
\subsection{Localization: rubber contribution} \label{rubcon}
The $\mathbf{T}$-equivariant
localization formula for the series
${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
has three parts:
\begin{enumerate}
\item[(i)] vertex terms over $0\in \PP^1$,
\item[(ii)] edge terms,
\item[(iii)] rubber integrals over $\infty \in \PP^1$.
\end{enumerate}
The vertex and edge terms have been explained already in Section \ref{ttt}.
We discuss the rubber integrals here.
The stable pairs theory of {\em rubber}{\footnote{We
follow the terminology and conventions of the
parallel rubber discussion for the local Donaldson-Thomas
theory of curves treated in \cite{lcdt}.}} naturally arises at the
boundary of $P_n(N/N_\infty,d)$.
Let $R$ be a rank 2 bundle of level $(0,0)$ over $\Pp$. Let
$$R_0, R_\infty\subset R$$
denote the fibers over $0, \infty\in \Pp$.
The 1-dimensional torus $\C^*$ acts on $R$ via the symmetries of
$\Pp$.
Let $P_n(R/R_0\cup R_\infty,d)$ be the relative moduli space
of stable pairs, and let
$$P_n(R/R_0 \cup R_\infty,d)^\circ \subset P_n(R/R_0\cup R_\infty,d)$$
denote the open set with finite stabilizers for the $\C^*$-action
and {\em no} destabilization over $\infty\in \Pp$.
The rubber moduli space,
$${P_n(R/R_0\cup R_\infty,d)}^\sim
= P_n(R/R_0 \cup R_\infty,d)^\circ/\C^*,$$
denoted by a superscripted tilde,
is determined by the (stack) quotient. The moduli space is
empty unless $n>d$.
The rubber theory of $R$ is defined by integration against the
rubber virtual class,
$$[{P_n(R/R_0\cup R_\infty,d)}^\sim ]^{vir}.$$
All of the above rubber constructions are $T$-equivariant for the
scaling action on the fibers of $R$ with weights $s_1$ and $s_2$.
The rubber moduli space $P_n(R/R_0\cup R_\infty, d)^\sim$ carries
a cotangent line at the dynamical point $0 \in \Pp$. Let
$$\psi_0 \in A^1_T({P_n(R/R_0\cup R_\infty,d)}^\sim, {\mathbb Q})$$
denote the associated cotangent line class.
Let $$\mathsf{P}_\mu \in A^{2d}_T(\text{Hilb}(\C^2,d),\mathbb{Z})$$
be the class corresponding to the $T$-fixed point determined
by the monomial ideal $\mu[x_1,x_2]\subset \C[x_1,x_2]$.
In the localization formula for the cap, special
rubber integrals with relative conditions $\mathsf{P}_\mu$ over $0$ and $\CC_\eta$
(in the Nakajima basis) over $\infty$ arise. Let
\begin{equation*}
\mathsf{S}^\mu_\eta =
\sum_{n\geq d} q^{n}
\left\langle \mathsf{P}_\mu \ \left| \ \frac{1}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}\ \in \Q(s_1,s_2,s_3)((q)) \ .
\end{equation*}
The bracket on the right is the rubber
integral defined by $T$-equivariant
residues. If $n=d$, the rubber moduli space in undefined ---
the bracket is then taken to be the $T$-equivariant intersection pairing
between the classes $\mathsf{P}_\mu$ and $\CC_\eta$ in
$\text{Hilb}(\C^2,d)$.
The $s_3$ dependence of the rubber integral
$$ \left\langle \mathsf{P}_\mu \ \left| \ \frac{1}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}\ \in \Q(s_1,s_2,s_3)$$
enter {\em only} through the term $s_3-\psi_0$.
On the $T$-fixed loci of the moduli space $P_n(R/R_0\cup R_\infty, d)^\sim$, the
cotangent line class $\psi_0$ is either equal to
a weight of $\text{Tan}_\mu$ (if $0$ lies on a twistor component)
or is nilpotent (if $0$ lies on a non-twistor component).
We conclude the following result.
\begin{lem} The evaluation of $\mathsf{S}_\eta^\mu$ at \label{plle}
$$s_3= n_1 s_1 + n_2 s_2,\ \ \ \ n_1,n_2\in \Q$$
is well-defined if $(n_1,n_2) \neq (0,0)$
and $n_1 s_1 + n_2 s_2$ is not a weight of $\text{\em Tan}_\mu$.
\end{lem}
The weights of $\text{Tan}_\mu$ are either proportional to
$s_1$ or $s_2$ or of the form
$$n_1s_1+n_2s_2 ,\ \ \ \ n_1,n_2\neq 0$$
where $n_1$ is the {\em opposite} sign of $n_2$.
\subsection{Localization: full formula}
The localization formula \cite{GraberP} for the capped 1-leg descendent
vertex is the following:
\begin{equation}\label{fred}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}} = \sum_{|\mu|=d}
\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right) \cdot
{\bW_\mu^{(0,0)}} \cdot \mathsf{S}^{\mu}_{\eta}\ .
\end{equation}
The form is the same as the Donaldson-Thomas localization formulas
used in \cite{moop,lcdt}.
\subsection{Proof of Proposition \ref{cttt}} \label{ggtt2}
We will consider the evaluations of ${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
( \prod_{j=1}^k \tau_{i_j}([0]))^{\mathbf{T}}$ at the values
\begin{equation}\label{gthh4}
s_3 = \frac{1}{a}(s_1+s_2)
\end{equation}
for all integers $a>0$.
By Theorem \ref{canpole}, the main cancellation of poles result
of Section \ref{polecan}, the evaluation \eqref{gthh4} of
$\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right)$
is well-defined and yields a Laurent {\em polynomial} in $q$
with coefficients in $\Q(s_1,s_2)$.
The edge term $\bW_\mu^{(0,0)}$ has no $s_3$ dependence (and
$q$ dependence given by $q^{-d}$).
The evaluation \eqref{gthh4} of
$\mathsf{S}^\mu_\eta$ is well-defined by Lemma \ref{plle} and
is the Laurent series
associated to a rational function in $\Q(q,s_1,s_2)$
by Lemma \ref{hyy3} below.
We have proven the evalution of ${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
at \eqref{gthh4} for all integers $a>0$ is well-defined and yields
a rational function in $\Q(q,s_1,s_2)$.
By \eqref{krtt} and the invertibility of the
Vandermonde matrix, we see
$$\Gamma_r(q,s_1,s_2) \in \Q(q,s_1,s_2)$$
for all $0 \leq r \leq \delta$.
\qed
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\subsection{Localization: full formula}
The localization formula \cite{GraberP} for the capped 1-leg descendent
vertex is the following:
\begin{equation}\label{fred}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}} = \sum_{|\mu|=d}
\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right) \cdot
{\bW_\mu^{(0,0)}} \cdot \mathsf{S}^{\mu}_{\eta}\ .
\end{equation}
The form is the same as the Donaldson-Thomas localization formulas
used in \cite{moop,lcdt}.
\subsection{Proof of Proposition \ref{cttt}} \label{ggtt2}
We will consider the evaluations of ${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
( \prod_{j=1}^k \tau_{i_j}([0]))^{\mathbf{T}}$ at the values
\begin{equation}\label{gthh4}
s_3 = \frac{1}{a}(s_1+s_2)
\end{equation}
for all integers $a>0$.
By Theorem \ref{canpole}, the main cancellation of poles result
of Section \ref{polecan}, the evaluation \eqref{gthh4} of
$\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right)$
is well-defined and yields a Laurent {\em polynomial} in $q$
with coefficients in $\Q(s_1,s_2)$.
The edge term $\bW_\mu^{(0,0)}$ has no $s_3$ dependence (and
$q$ dependence given by $q^{-d}$).
The evaluation \eqref{gthh4} of
$\mathsf{S}^\mu_\eta$ is well-defined by Lemma \ref{plle} and
is the Laurent series
associated to a rational function in $\Q(q,s_1,s_2)$
by Lemma \ref{hyy3} below.
We have proven the evalution of ${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
at \eqref{gthh4} for all integers $a>0$ is well-defined and yields
a rational function in $\Q(q,s_1,s_2)$.
By \eqref{krtt} and the invertibility of the
Vandermonde matrix, we see
$$\Gamma_r(q,s_1,s_2) \in \Q(q,s_1,s_2)$$
for all $0 \leq r \leq \delta$.
\qed
\subsection{Evaluation of $\mathsf{S}_\eta^\mu$}
\label{dfv}
The following result is well-known from the study of the
quantum differential equation of the Hilbert scheme of
points \cite{hilb1,hilb2}. We include the proof
for the reader's convenience.
\begin{lem} For all integers $a\neq 0$, the evaluation
$$\mathsf{S}_\eta^\mu
|_{s_3=\frac{1}{a}(s_1+s_2)}$$
yields the Laurent series associated to a rational
function in $\Q(q,s_1,s_2)$. \label{hyy3}
\end{lem}
\begin{proof}
Let $\com^*$ act on $\PP^1$ with tangent weights $-s_3$ and $s_3$
at $0,\infty \in \PP^1$ respectively.
Lift the $\com^*$-action to $\mathcal{O}_{\Pp}(-a)$
with fiber weights{\footnote{Remember, weights on the
coordinate functions are the opposite of the weights on the fibers.}} $as_3$ and $0$ over $0,\infty\in \PP^1$.
Lift $\com^*$ to $\mathcal{O}_{\Pp}$ with fiber
weights $0$ and $0$ over $0,\infty\in \PP^1$.
The $(-a,0)$-tube is the geometry of total
space of
\begin{equation} \label{gttr}
\mathcal{O}_{\Pp}(-a) \oplus \mathcal{O}_{\Pp} \rightarrow \Pp
\end{equation}
relative to the fibers over both $0,\infty \in \Pp$.
The 2-dimensional torus $T$ acts on the $(-a,0)$-tube
as before by scaling the line summands. For
$$\mathbf{T}=T \times \com^* ,$$
we obtain a $\mathbf{T}$-action on the $(-a,0)$-tube.
Define the generating series of $\mathbf{T}$-equivariant integrals
\begin{equation} \label{hllw}
{\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty}
= \sum_{n} q^{n}
\Big\langle \CC_{\eta^0} \ \Big| \ 1 \ \Big|\ \CC_{\eta^\infty}
\Big\rangle_{n,d}^{(-a,0)}\ \in \Q(s_1,s_2,s_3)((q))\
\end{equation}
where the superscript $(-a,0)$ refers to the geometry \eqref{gttr}.
The series ${\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty} $
has no insertions. Hence, the results of
\cite{moop,mpt} show
${\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty}$
is actually the Laurent series associated to a rational function
in $\Q(q,s_1,s_2,s_3)$.
The $\mathbf{T}$-equivariant localization formula yields
$${\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty} =
\sum_{|\mu|=d} \mathsf{S}_{\eta^0}^\mu\Big|_{s_1=s_1-as_3,s_2,s_3=-s_3} \cdot \bW^{(-a,0)}_\mu \cdot \mathsf{S}_{\eta^\infty}^\mu
\ .$$
The formula for the edge term $\bW^{(-a,0)}_\mu$ can be found in Section 4.6
of \cite{pt2}.
Next, we consider the evaluation of the three terms of the
localization formula at
\begin{equation}\label{jjttf}
s_3= \frac{1}{a}({s_1+s_2})\ .
\end{equation}
After evaluation, the first term becomes
\begin{equation} \label{oldd}
\mathsf{S}_{\eta^0}^\mu\Big|_{s_1=-s_2,s_2,s_3=-s_3}
\end{equation}
which only has $q^d$ terms by holomorphic symplectic vanishing \cite{mpt,lcdt}.
The evaluation of $\bW^{(-a,0)}_\mu$ at \eqref{jjttf} is easily
seen to be well-defined and nonzero by inspection of the formulas in
Section 4.6 of \cite{pt2}.
The $q$ dependence of $\bW^{(-a,0)}_\mu$ is monomial.
The evaluation of the third term
$ \mathsf{S}_{\eta^\infty}^\mu\ $
at \eqref{jjttf} is well-defined by Lemma \ref{plle}.
We conclude the evaluation of ${\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty} $
at \eqref{jjttf} is a well-defined rational function in
$\Q(q,s_1,s_2)$.
By the invertibility of \eqref{oldd} and the edge terms,
$\mathsf{S}_{\eta^\infty}^\mu$ must also be a rational function in
$\Q(q,s_1,s_2)$ after the evaluation \eqref{jjttf}.
\end{proof}
\subsection{Twisted cap}
The twisted $(a_1,a_2)$-cap is the geometry of the total space of
\begin{equation} \label{gttrr}
\mathcal{O}_{\Pp}(a_1) \oplus \mathcal{O}_{\Pp}(a_2) \rightarrow \Pp
\end{equation}
relative to the fiber over $\infty \in \Pp$.
We lift the $\com^*$-action on $\Pp$ to $\mathcal{O}_{\Pp}(a_i)$
with fiber weights $0$ and $-a_is_3$ over $0,\infty\in \PP^1$.
The 2-dimensional torus $T$ acts on the $(a_1,a_2)$-cap
by scaling the line summands, so
we obtain a $\mathbf{T}$-action on the $(a_1,a_2)$-cap.
Define the generating series of $\mathbf{T}$-equivariant integrals
\begin{multline*}
{\mathsf Z}^{(a_1,a_2)}_{d,\eta} \left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
= \\
\sum_{n} q^{n}
\left\langle \prod_{j=1}^k \tau_{i_j}([0]) \ \Bigg|\ \CC_{\eta}
\right\rangle_{n,d}^{(a_1,a_2)}\ \in \Q(s_1,s_2,s_3)((q))\
\end{multline*}
where the superscript $(a_1,a_2)$ refers to the geometry \eqref{gttrr}.
\begin{prop}
\label{ctttt}
$
{\mathsf Z}^{(a_1,a_2)}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{prop}
\begin{proof}
The twisted $(a_1,a_2)$-cap admits a $\mathbf{T}$-equivariant
degeneration to a standard $(0,0)$-cap and an $(a_1,a_2)$-tube by
bubbling off $0\in \Pp$.
The insertions $\tau_{i_j}([0])$ are sent $\mathbf{T}$-equivariantly
to the non-relative point of the $(0,0)$-cap.
The rationality of $
{\mathsf Z}^{(a_1,a_2)}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
then follows from Proposition \ref{cttt}, the
$\mathbf{T}$-equivariant
rationality
results for the $(a_1,a_2)$-tube without insertions \cite{mpt,lcdt}, and
the degeneration formula.
\end{proof}
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\section{Cancellation of poles} \label{polecan}
\subsection{Overview}
Our goal here is to prove the following result.
\begin{thm}\label{canpole}
For all integers $a>0$, the evaluation
\begin{equation*}
\bW^{\mathbf{V}er}_\mu\left(\prod_{j=1}^k\tau_{i_j}([0])\right)\bigg|_{s_3=\frac{1}{a}(s_1+s_2)}
\end{equation*}
is well-defined and yields a
Laurent polynomial in $q$ with coefficients in $\Q(s_1,s_2)$.
\end{thm}
We regard the partition $\mu$, the
descendent factor $\prod_{j=1}^k\tau_{i_j}([0])$, and
the integer $a$ as fixed throughout Section \ref{polecan}.
Recall $\bW^{\mathbf{V}er}_\mu\left(\prod_{j=1}^k\tau_{i_j}([0])\right)$ is
defined as an infinite sum over the fixed loci $Q_U$,
\begin{equation}\label{infinite sum}
\bW^{\mathbf{V}er}_\mu\left(\prod_{j=1}^k\tau_{i_j}([0])\right) = \left(\frac{1}{s_1s_2}\right)^k\sum_{Q_U}\mathsf{w}_{\tau_{i_1},\ldots,\tau_{i_k}}(Q_U) q^{l(Q_U)+|\mu|}.
\end{equation}
The $Q_U$ are determined by $\FFF_U$, the weight of the corresponding box configuration. Although $\FFF_U$ is just a Laurent series in $t_1,t_2,t_3$, the
product
$(1-t_3)\FFF_U$
is a Laurent polynomial.
Our approach to proving Theorem~\ref{canpole}
is to break (\ref{infinite sum}) into finite sums
based on the Laurent polynomial
$$(1-t_3)\FFF_U|_{t_3=(t_1t_2)^{\frac{1}{a}}}\ .$$
For any Laurent polynomial $f\in\Z[t_1,t_2,(t_1t_2)^{-\frac{1}{a}}]$, define
\begin{equation*}
\mathcal{S}_f = \left\{Q_U \ \bigg| \ (1-t_3)\FFF_U|_{t_3=(t_1t_2)^{\frac{1}{a}}} = f
\right\} \ .
\end{equation*}
Theorem~\ref{canpole} follows from the following result
regarding the subsums of (\ref{infinite sum})
corresponding to the sets $\mathcal{S}_f$.
\begin{prop}\label{vanishing}
Let $f\in\Z[t_1,t_2,(t_1t_2)^{-\frac{1}{a}}]$ be a Laurent polynomial. The evaluation
\begin{equation*}
\left(\sum_{Q_U\in\mathcal{S}_f}\mathsf{w}_{i_1,\ldots,i_k}(Q_U)\right)\bigg|_{s_3=\frac{1}{a}(s_1+s_2)}
\end{equation*}
is well-defined. Moreover, the evaluation vanishes for all but finitely many choices of $f$.
\end{prop}
\subsection{Notation and Preliminaries}
We introduce here the notation and conventions required
to analyze the sums appearing in Proposition~\ref{vanishing}.
First, we view the partition $\mu$ as a subset of $\Z_{\ge 0}^2$.
The lattice points,
for which we use the coordinates $(i,j)\in\mu$,
correspond to the lower left corners of
the boxes of $\mu$.
We also write
$$(\delta; j) = (i,j)$$ for $\delta = i-j$.
The points $(\delta; j)\in\mu$ for fixed $\delta$ lie on a single diagonal.
The diagonals will play an important role. Let
$\mu_\delta = \{j \mid (\delta; j)\in\mu\}$, and
define
\begin{equation*}
\Sym_\mu = \prod_{\delta\in\Z}\Sym(\mu_\delta),
\end{equation*}
where $\Sym(S)$ is the group of permutations of a set $S$. Thus,
$\Sym_\mu$ may be viewed
as the group of permutations of $\mu$ which move points only
inside their diagonals.
Let $$\sgn:\Sym_\mu\to\{\pm 1\}$$ be the sign of the permutation of $\mu$.
Recall the Laurent polynomials $(1-t_3)\FFF_U$ are of the form
\begin{equation*}
(1-t_3)\FFF_U = \sum_{(i,j)\in\mu}t_1^it_2^jt_3^{-h_U(i,j)},
\end{equation*}
where $h_U(i,j)$ is the depth of the box arrangement below $(i,j)$.
Because of our reparametrization of the partition $\mu$ and the evaluation $t_3=(t_1t_2)^{\frac{1}{a}}$, the
following change of variables will be convenient:
\begin{equation*}
v_1=t_1, \quad v_2=t_1t_2, \quad v_3=t_1t_2t_3^{-a}
\end{equation*}
and $u_i=e(v_i)$, so
\begin{equation*}
u_1 = s_1, \quad u_2=s_1+s_2, \quad u_3=s_1+s_2-as_3.
\end{equation*}
The evaluations under consideration are then simply $v_3=1$ and $u_3=0$.
From now on we will assume $\mathcal{S}_f$ to be nonempty, so
$$f = (1-t_3)\FFF_U|_{t_3=(t_1t_2)^{\frac{1}{a}}}$$
for some $Q_U$ and thus $f$ can be written in the form
\begin{equation*}
f = \sum_{(\delta; j)\in\mu}v_1^{\delta}v_2^{e_\delta(j)}
\end{equation*}
for some exponents $e_\delta(j)$. These exponents are made unique by requiring that $e_\delta(j)$ is a weakly decreasing function of $j$, for each $\delta$. We generally regard $f$ as fixed and thus do not indicate the $f$-dependence in $e_\delta(j)$.
We now classify all $Q_U\in\mathcal{S}_f$. Given any $\sigma=(\sigma_\delta)\in\Sym_\mu$, we define a function $h_\sigma: \mu \to \Z$ by
\begin{equation*}
h_\sigma(\delta; j) = a\cdot(j - e_\delta(\sigma_\delta^{-1}(j))).
\end{equation*}
When $h_\sigma$ defines a valid box arrangement, we say
$\sigma$ is {\it admissible}. Admissibility
is equivalent to the following conditions on $\sigma$:
\begin{align*}
\sigma_0(j)&\ne 0 \text{ if } e_0(j)>0 \\
\sigma_{\delta+1}(j)&\ne\sigma_{\delta}(k) \text{ if } e_{\delta+1}(j)>e_{\delta}(k) \\
\sigma_{\delta}(j)&\ne\sigma_{\delta+1}(k)+1 \text{ if } e_{\delta}(j)>e_{\delta+1}(k)+1.
\end{align*}
For admissible $\sigma$, let $Q_\sigma$ denote the corresponding $\mathbf{T}$-fixed locus.
Unraveling the definitions, we compute
\begin{align*}
(1-t_3)\FFF_\sigma &= \sum_{(i,j) \in \mu}t_1^it_2^jt_3^{-h_\sigma(i-j,j)} \\
&= \sum_{(\delta; j)\in \mu}v_1^\delta v_2^j v_2^{-\frac{1}{a}h_\sigma(\delta; j)}v_3^{\frac{1}{a}h_\sigma(\delta; j)} \\
&= \sum_{(\delta; j)\in \mu}v_1^\delta v_2^{e_\delta(\sigma_\delta^{-1}(j))} v_3^{j - e_\delta(\sigma_\delta^{-1}(j))} \\
&= \sum_{(\delta; j)\in \mu}v_1^\delta v_2^{e_\delta(j)} v_3^{\sigma_\delta(j) - e_\delta(j)}.
\end{align*}
We conclude $(1-t_3)\FFF_\sigma|_{v_3=1} = f$ and $Q_\sigma\in\mathcal{S}_f$. In fact,
a direct examination shows
every $Q_{U'}\in\mathcal{S}_f$ can be obtained as $Q_\sigma$
for some admissible
$\sigma\in \Sym_\mu$.
If we let $\Sym_\mu^0$ be the subgroup of $\Sym_\mu$ consisting of elements $\tau$ such that $e_\delta(\tau_\delta(j))=e_\delta(j)$, then $Q_\sigma = Q_{\sigma'}$ if and only if $\sigma^{-1}\sigma'\in \Sym_\mu^0$.
We thus can replace the sum over $Q_U\in\mathcal{S}_f$ with a sum over admissible $\sigma\in\Sym_\mu$:
\begin{multline}\label{kk449}
\left(\sum_{Q_U\in\mathcal{S}_f}\mathsf{w}_{i_1,\ldots,i_k}(Q_U)\right)\bigg|_{s_3=\frac{1}{a}(s_1+s_2)} = \\
\frac{1}{|\Sym_\mu^0|}\left(\sum_{\sigma\in\Sym_\mu\text{ admissible}}\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)\right)\bigg|_{s_3=\frac{1}{a}(s_1+s_2)}.
\end{multline}
We will show the evaluation is well-defined
by choosing $\kappa_0$ such that each
term $\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)$ in the above sum has order of vanishing along $u_3=0$ at least $-\kappa_0$, and then showing
\begin{equation}\label{differentiated}
\sum_{\sigma\in\Sym_\mu\text{ admissible}}\left(\frac{\partial}{\partial u_3}\right)^\kappa(u_3^{\kappa_0}\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma))\bigg|_{u_3=0} = 0
\end{equation}
for $0\le\kappa<\kappa_0$.
The second part of Proposition \ref{vanishing},
the vanishing of the evaluation \eqref{kk449}
for all but finitely many $f$, is
then equivalent to proving that (\ref{differentiated}) holds for $\kappa=\kappa_0$ (for all but finitely many $f$).
In order to prove these vanishing results, we will need to analyze the dependence of the terms $\left(\frac{\partial}{\partial u_3}\right)^\kappa(u_3^{\kappa_0}\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma))\bigg|_{u_3=0}$ on the permutation $\sigma\in\Sym_\mu$.
For each $\kappa$, we will find
the corresponding term is equal to a polynomial in the values $\sigma_\delta(j)$ of relatively low degree which
vanishes at all inadmissible permutations $\sigma$.
Let $\Q[\sigma]$ and $\Q(\sigma)$ denote the
ring of polynomials and the field of rational functions respectively
in the variables $\sigma_\delta(j)$. For a polynomial $P\in\Q[\sigma]$, let $\deg(P)$ be the (total) degree of $P$. For rational functions $\frac{P}{Q}\in\Q(\sigma)$, we set
$$\deg\left(\frac{P}{Q}\right) = \deg(P)-\deg(Q).$$
We observe that if $P\in\Q[\sigma]$ has degree $\deg(P)<\sum_{\delta}\frac{1}{2}|\mu_\delta|(|\mu_\delta|-1)$, then
\begin{equation*}
\sum_{\sigma\in\Sym_\mu}\sgn(\sigma)P(\sigma) = 0,
\end{equation*}
since a nonzero alternating polynomial with respect to $\Sym_\mu$ would have to have greater degree.
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\subsection{Proof of Proposition~\ref{vanishing}}
We need to study the $\sigma$-dependence of
\begin{equation*}
\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma) = e(-\mathsf{V}_\sigma)\prod_{j=1}^k\ch_{2+i_j}(\FFF_\sigma\cdot(1-t_1)(1-t_2)(1-t_3)).
\end{equation*}
We begin by explicitly writing $\mathsf{V}_\sigma$ in terms of $\sigma$ and the numbers $e_\delta(j)$. Recall
\begin{equation*}
\mathsf{V}_\sigma = \frac{\FFF'_\sigma -\FFF'_0}{1-t_3}
+ \frac{\overline{\FFF'_\sigma} -\overline{\FFF'_0}}{t_1t_2(1-t_3)}
- \frac{\FFF'_\sigma\overline{\FFF'_\sigma} - \FFF'_0\overline{\FFF'_0}}{1-t_3}(1-t_1^{-1})(1-t_2^{-1}),
\end{equation*}
where $\FFF'_\sigma = (1-t_3)\FFF_\sigma$ and $$\FFF'_0 =
\sum_{(i,j)\in\mu}t_1^it_2^j.$$
In particular, $\mathsf{V}_\sigma|_{v_3=1}$ does not
depend on $\sigma$. Hence,
the order of vanishing of $e(-\mathsf{V}_\sigma)$ along $u_3=0$ is an
integer $-\kappa_0$ independent of $\sigma$.
Since the descendent factor is a polynomial in $u_1,u_2,u_3$,
the order of vanishing of $\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)$ along $u_3=0$
is at least $-\kappa_0$. If $\kappa_0\le 0$,
then the evaluation is well-defined on each $\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)$
and thus on their sum. If $\kappa_0<0$, then
the evaluation in fact yields zero.
So we may assume $\kappa_0\ge 0$.
We now rewrite $\mathsf{V}_\sigma$ in terms of $v_1,v_2,v_3$. We find
$\mathsf{V}_\sigma$ equals
\footnotesize
\begin{align*}
&\ \ \ \sum_{(\delta; j)\in\mu}\frac{v_1^\delta v_2^{e_\delta(j)}v_3^{\sigma_\delta(j) - e_\delta(j)}-v_1^\delta v_2^j}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}\\
& + \sum_{(\delta; j)\in\mu}\frac{v_1^{-\delta} v_2^{-e_\delta(j)-1}v_3^{-\sigma_\delta(j) + e_\delta(j)}-v_1^{-\delta} v_2^{-j-1}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}} \\
&- \sum_{(\delta_1; j_1),(\delta_2; j_2)\in\mu}\frac{v_1^{\delta_1-\delta_2} v_2^{e_{\delta_1}(j_1)-e_{\delta_2}(j_2)}v_3^{\sigma_{\delta_1}(j_1)-\sigma_{\delta_2}(j_2)-e_{\delta_1}(j_1)+e_{\delta_2}(j_2)}-v_1^{\delta_1-\delta_2} v_2^{j_1-j_2}}{(1-(\frac{v_2}{v_3})^{\frac{1}{a}}) \cdot(1-v_1^{-1})^{-1}(1-v_1v_2^{-1})^{-1}}.
\end{align*}
\normalsize
Let $C > 2\max(e_\delta(j))$ be a large positive integer. We break up each of the three above sums above using $C$. Then, $\mathsf{V}_\sigma$ equals
\footnotesize
\begin{align*}
&\ \ \
\sum_{(\delta; j)\in\mu}\frac{v_1^\delta v_2^{e_\delta(j)}v_3^{\sigma_\delta(j) - e_\delta(j)}-v_1^\delta v_2^{-C}v_3^{\sigma_\delta(j)+C}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}
\\ &+ \sum_{(\delta; j)\in\mu}\frac{v_1^\delta v_2^{-C}v_3^{j+C}-v_1^\delta v_2^j}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}} \\
&+ \sum_{(\delta; j)\in\mu}\frac{v_1^{-\delta} v_2^{-e_\delta(j)-1}v_3^{-\sigma_\delta(j) + e_\delta(j)}-v_1^{-\delta} v_2^{-C-1}v_3^{-\sigma_\delta(j)+C}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}} \\ &
+ \sum_{(\delta; j)\in\mu}\frac{v_1^{-\delta} v_2^{-C-1}v_3^{-j+C}-v_1^{-\delta} v_2^{-j-1}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}} \\
&- \sum_{(\delta_1; j_1),(\delta_2; j_2)\in\mu} \Bigg(
\frac{v_1^{\delta_1-\delta_2} v_2^{e_{\delta_1}(j_1)-e_{\delta_2}(j_2)}v_3^{\sigma_{\delta_1}(j_1)-\sigma_{\delta_2}(j_2)-e_{\delta_1}(j_1)+e_{\delta_2}(j_2)}}
{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}
\\&
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\frac{v_1^{\delta_1-\delta_2} v_2^{-C}v_3^{\sigma_{\delta_1}(j_1)-\sigma_{\delta_2}(j_2)+C}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}\Bigg)
\cdot(1-v_1^{-1})(1-v_1v_2^{-1}) \\
&- \sum_{(\delta_1; j_1),(\delta_2; j_2)\in\mu}\frac{v_1^{\delta_1-\delta_2} v_2^{-C}v_3^{j_1-j_2+C}-v_1^{\delta_1-\delta_2} v_2^{j_1-j_2}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}\cdot(1-v_1^{-1})(1-v_1v_2^{-1}).
\end{align*}
\normalsize
We now expand out the above sums into monomials:
all of the resulting terms will be of the form
$$\pm v_1^x v_2^y v_3^{z(\sigma)},$$
where $x$ and $y$ have no dependence on the permutation $\sigma = (\sigma_\delta)$ and $z\in\Q[\sigma]$ is a linear function of the values $\sigma_\delta(j)$.
After separating out the monomials with $x=y=0$, we write
\begin{equation*}
\mathsf{V}_\sigma = \sum_{(c,0,0,z)\in S}c v_3^{z(\sigma)}+\sum_{\substack{(c,x,y,z)\in S \\ (x,y)\ne(0,0)}}c v_1^x v_2^y v_3^{z(\sigma)},
\end{equation*}
where $S$ is a finite set containing the data of the monomials which
appear
(with
coefficients $c \in \mathbb{Z}$).
Then
\begin{equation*}
e\left(-\sum_{(c,0,0,z)\in S}c v_3^{z(\sigma)}\right) = \phi(\sigma)u_3^{-\kappa_0},
\end{equation*}
for a rational function $\phi=\phi_f\in\Q(\sigma)$
which will be explicitly described below.
We analyze first
the descendent factors in $\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)$.
The descendent terms can be expressed in the form
\begin{multline*}
\prod_{j=1}^k\ch_{2+i_j}\left(\FFF_\sigma\cdot(1-t_1)(1-t_2)(1-t_3)\right)
=\\
\prod_{j=1}^k\sum_{(c',x,y,z)\in S'_j}c'(xu_1+yu_2+z(\sigma)u_3)^{2+i_j},
\end{multline*}
where the $S'_j$ are more fixed finite sets containing the data of the
terms which appear. As before, $z\in\Q[\sigma]$ is linear.
We then find
\begin{multline*}
u_3^{\kappa_0}\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma) = \phi(\sigma)\prod_{\substack{(c,x,y,z)\in S \\
(x,y)\ne(0,0)}}(xu_1+yu_2+z(\sigma)u_3)^{-c}
\\
\cdot \prod_{j=1}^k\sum_{(c',x,y,z)\in S'_j}c'(xu_1+yu_2+z(\sigma)u_3)^{2+i_j}.
\end{multline*}
Differentiating the above
product $\kappa$ times with respect to $u_3$ and then setting $u_3$ equal to $0$ is easily done. We obtain
\begin{equation*}
\left(\frac{\partial}{\partial u_3}\right)^\kappa(u_3^{\kappa_0}
\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma))
|_{u_3=0}
= \sum_{i\in\mathcal{I}}\phi(\sigma)Z_i(\sigma)R_i(u_1,u_2),
\end{equation*}
where $\mathcal{I}$ is an indexing set, $Z_i\in\Q[\sigma]$ has degree at most $\kappa$, and $R_i(u_1,u_2)\in\Q(u_1,u_2)$ does not depend on $\sigma$.
Proposition~\ref{vanishing} will follow from the claim that
\begin{equation}\label{cancel}
\sum_{\sigma\in\Sym_\mu\text{ admissible}}\phi(\sigma)Z(\sigma) = 0
\end{equation}
for any polynomial $Z$ of degree $\kappa < \kappa_0$
(or degree $\kappa=
\kappa_0$ for all but finitely many $f$). The
vanishing property \eqref{cancel}
is purely a property of the rational function $\phi\in\Q(\sigma)$.
We will now study $\phi$ in more detail.
The goal is to find a polynomial $\psi\in\Q[\sigma]$ of
sufficiently low degree satisfying
$$\phi(\sigma) = \sgn(\sigma)\psi(\sigma)$$
for every admissible $\sigma\in\Sym_\mu$ and
satisfying
$\psi(\sigma) = 0$ for every inadmissible $\sigma\in\Sym_\mu$.
From the formula for $\mathsf{V}_\sigma$,
we can describe $\phi\in\Q(\sigma)$ explicitly as a product of linear factors:
\footnotesize
\begin{align*}
\phi(\sigma) =
&\left(\prod_{\substack{(0; j)\in\mu \\ e_0(j)>0}}\sigma_0(j)\right)
\left(\prod_{\substack{(0; j)\in\mu \\ j>0}}j\right)^{-1}
\left(\prod_{\substack{(0; j)\in\mu \\ e_0(j)<-1}}(-\sigma_0(j)-1)\right)
\\ &
\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ e_\delta(j_1)>e_\delta(j_2)}}(\sigma_\delta(j_1)-\sigma_\delta(j_2))\right)^{-1}
\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ j_1>j_2}}(j_1-j_2)\right)
\\ &
\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ e_\delta(j_1)>e_\delta(j_2)+1}}(\sigma_\delta(j_1)-\sigma_\delta(j_2)-1)\right)^{-1}
\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ j_1>j_2+1}}(j_1-j_2-1)\right) \\
&\left(\prod_{\substack{(\delta+1; j_1),(\delta; j_2)\in\mu \\ e_{\delta+1}(j_1)>e_\delta(j_2)}}(\sigma_{\delta+1}(j_1)-\sigma_{\delta}(j_2))\right)
\left(\prod_{\substack{(\delta+1; j_1),(\delta; j_2)\in\mu \\ j_1>j_2}}(j_1-j_2)\right)^{-1} \\
&\left(\prod_{\substack{(\delta; j_1),(\delta+1; j_2)\in\mu \\ e_{\delta}(j_1)>e_{\delta+1}(j_2)+1}}(\sigma_{\delta}(j_1)-\sigma_{\delta+1}(j_2)-1)\right)
\left(\prod_{\substack{(\delta; j_1),(\delta+1; j_2)\in\mu \\ j_1>j_2+1}}(j_1-j_2-1)\right)^{-1}.
\end{align*}
\normalsize
The degree of $\phi$ is easily computed to be $-\kappa_0$, since there are the same number of constant factors appearing on the numerator and denominator in the above expression.
\begin{lem} \label{frrg} We have
\begin{equation*}
\frac{\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ j_1>j_2}}(j_1-j_2)}{\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ e_\delta(j_1)>e_\delta(j_2)}}(\sigma_\delta(j_1)-\sigma_\delta(j_2))} = \pm\sgn(\sigma)\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ e_\delta(j_1)=e_\delta(j_2) \\ j_1>j_2}}(\sigma_\delta(j_1)-\sigma_\delta(j_2))
\end{equation*}
for every $\sigma\in\Sym_\mu$.
\end{lem}
\begin{proof}
The formula is obtained by cancelling equal terms on the left side.
\end{proof}
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Suppose that $\{\delta \mid \mu_\delta \ne \emptyset\} = \{\delta \mid a\le\delta\le b\}$. By using the identity of Lemma \ref{frrg}
and grouping terms appropriately, we find
$$\phi(\sigma) = \sgn(\sigma)\phi_0(\sigma)$$
for $\phi_0\in\Q(\sigma)$ given by
\begin{equation}
\phi_0 = XPQ\frac{\prod_{a\le \delta\le b-1}R_\delta}{\prod_{a+1 \le \delta \le b-1}S_\delta},
\end{equation}
where
\begin{equation*}
P = \prod_{\substack{j\in\mu_0 \\ e_{0}(j)<-1}}(-\sigma_{0}(j)-1),\ \ \
Q = \prod_{\substack{j\in\mu_0 \\ e_{0}(j)>0}}\sigma_{0}(j),
\end{equation*}
\footnotesize
\begin{equation*}
R_\delta = \left(\prod_{\substack{j_1\in\mu_{\delta+1}, j_2\in\mu_{\delta} \\ e_{\delta+1}(j_1)>e_\delta(j_2)}}(\sigma_{\delta+1}(j_1)-\sigma_{\delta}(j_2))\right)\left(\prod_{\substack{j_1\in\mu_{\delta}, j_2\in\mu_{\delta+1} \\ e_{\delta}(j_1)>e_{\delta+1}(j_2)+1}}(\sigma_{\delta}(j_1)-\sigma_{\delta+1}(j_2)-1)\right),
\end{equation*}
\normalsize
\begin{equation*}
S_\delta = \prod_{\substack{j_1,j_2\in\mu_{\delta} \\ e_{\delta}(j_1)>e_\delta(j_2)+1}}(\sigma_{\delta}(j_1)-\sigma_{\delta}(j_2)-1),
\end{equation*}
and $X\in\Q[\sigma]$ is a polynomial. The total degree of the
rational function $\phi_0$ is
\begin{equation*}
\deg(\phi)+\deg\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ j_1>j_2}}(\sigma_\delta(j_1)-\sigma_\delta(j_2))\right) = -\kappa_0 + \sum_{\delta}\frac{1}{2}|\mu_\delta|(|\mu_\delta|-1).
\end{equation*}
We now require an algebraic result in order to
convert $\phi_0$ into a polynomial.
Let $m,n\geq 0$ be integers, and
let $$A=\Q[x_1,\ldots,x_n,y_1,\ldots,y_m].$$
Let $P$ be the collection of $n!m!$
points $$(x_1,\ldots,x_n,y_1,\ldots,y_m)\in \Q^{n+m}$$ satisfying
$\{x_1,\ldots,x_n\} = \{1,\ldots,n\}$ and
$\{y_1,\ldots,y_m\} = \{1,\ldots,m\}$.
Let $a_1\le a_2\le\cdots\le a_n$ be integers with $0\le a_i < i$, and set
\begin{equation*}
F = \prod_{1\le j\le a_i}(x_j-x_i+1) \in A.
\end{equation*}
The following Proposition will be proven in Section \ref{division}.
\begin{prop}\label{division}
If $G\in A$ vanishes when evaluated at every point of $P$ at which $F$ vanishes,
then there exists $H\in A$ with $$\deg(H) \le \deg(G)-\deg(F)$$ satisying
$G=FH$ for every point of $P$.
\end{prop}
If $S_{\delta+1}(\sigma)=0$ for a given $\sigma\in\Sym_\mu$ (which is then
necessarily inadmissible), then
\begin{equation}
\label{y34}
R_\delta(\sigma)=R_{\delta+1}(\sigma)=0.
\end{equation}
By reindexing the permutation sets $\mu_\delta$ and $\mu_{\delta+1}$ as necessary, we can apply Proposition~\ref{division} with $G = R_\delta$ and $F = S_\delta$, since $S_{\delta}$ is of the appropriate form.\footnote{By definition,
$e_\delta(j)$ is a weakly decreasing function of $j$. We use the {\emph{opposite}} ordering on the variables $\sigma_\delta(j)$ to write $S_\delta$ in the desired form. Explicitly, if $$\mu_\delta = \{A, A+1, \ldots, B\},$$ then we take $x_i = \sigma_\delta(B-i+1)-A+1$.} Thus for
$a+1\le\delta\le b-1$, there exist polynomials $T_\delta\in\Q[\sigma]$
with $\deg(T_\delta)\le \deg(R_\delta) - \deg(S_\delta)$
satisfying
$$T_\delta(\sigma) = \frac{R_\delta(\sigma)}{S_\delta(\sigma)}$$
for all $\sigma$ for which
which $S_\delta(\sigma)\neq 0$. Then
\begin{equation*}
\psi = XPQR_a\prod_{a+1 \le \delta \le b-1}T_\delta \in \Q[\sigma]
\end{equation*}
has degree at most equal to that of $\phi_0$ and satisfies
$$\sgn(\sigma)\psi(\sigma) = \sgn(\sigma)\phi_0(\sigma)=\phi(\sigma)$$
for any admissible $\sigma$.
For a polynomial $\theta\in\Q[\sigma]$, let $V(\theta)$
denote the set of $\sigma\in\Sym_\mu$ such that $\theta(\sigma)=0$.
We see
\begin{align*}
V(\psi) &\supseteq V(Q)\cup V(R_a) \cup \left(\bigcup_{a+1 \le \delta \le b-1}V(T_\delta)\right) \\
&\supseteq V(Q)\cup V(R_a) \cup \left(\bigcup_{a+1 \le \delta \le b-1}(V(R_\delta)-V(S_\delta))\right) \\
&\supseteq V(Q)\cup \left(\bigcup_{a \le \delta \le b-1}V(R_\delta)\right) \\
&= \{\sigma\in\Sym_\mu \mid \sigma\text{ is not admissible}\}.
\end{align*}
The third inclusion is by repeated application of \eqref{y34}.
We conclude $\psi$ vanishes when evaluated at any inadmissible $\sigma$.
We are finally able to evaluate the sum \eqref{cancel}. We have
\begin{equation*}
\sum_{\sigma\in\Sym_\mu\text{ admissible}}\phi(\sigma)Z(\sigma) = \sum_{\sigma\in\Sym_\mu}\sgn(\sigma)\psi(\sigma)Z(\sigma).
\end{equation*}
If $\deg(Z)<\kappa_0$, then $\deg(\psi Z)<\sum_{\delta}\frac{1}{2}|\mu_\delta|(|\mu_\delta|-1)$, and thus
\begin{equation*}
\sum_{\sigma\in\Sym_\mu}\sgn(\sigma)\psi(\sigma)Z(\sigma) = 0.
\end{equation*}
We have proven the evaluation of Proposition \ref{vanishing}
is well-defined.
The second part of Proposition~\ref{vanishing} asserts the
vanishing of the evaluation for all but finitely many $f$. We will use a combination of two
ideas to prove the assertion.
First, if $\mathcal{S}(f) = \emptyset$, then the evaluation is trivially zero.
Second, we replace
the polynomial $\psi$ above with another polynomial
$\psi'$ which assumes the same values but has lower degree. Then
$$\deg(\psi')<\deg(\phi_0)
= -\kappa_0 + \sum_{\delta}\frac{1}{2}|\mu_\delta|(|\mu_\delta|-1).$$
So for $\deg(Z)\leq\kappa_0$,
\begin{equation*}
\sum_{\sigma\in\Sym_\mu}\sgn(\sigma)\psi'(\sigma)Z(\sigma) = 0.
\end{equation*}
As we have seen, a choice of $f$ such that $\mathcal{S}(f)\ne\emptyset$ uniquely determines
constants $e_\delta(j)$ weakly decreasing in $j$. We use linear inequalities in
the constants $e_\delta(j)$
to describe four cases in which either $\mathcal{S}(f) = \emptyset$ or $\psi$ can be
replaced by $\psi'$ as above. In the end, we will check that only finitely many
possibilities avoid all four cases. The finiteness will come from giving upper
and lower bounds for the $e_\delta(j)$.
For the lower bound, since
$e_\delta(j)$ is weakly decreasing in $j$, we introduce the notation
\[
m_\delta = \max(\mu_\delta)
\]
and focus on the values $e_\delta(m_\delta)$.
\noindent {\bf Case I.}
Let $J = \max\{j\mid (\delta; j)\in\mu\text{ for some }\delta\}$ and suppose
$e_\delta(j) > J$ for some $(\delta; j)\in\mu$.
Then for any $\sigma\in\Sym_\mu$,
\[
h_\sigma(\delta; \sigma_\delta(j)) = a\cdot(\sigma_\delta(j)-e_\delta(j)) < 0,
\]
so $\sigma$ is not admissible. Thus $\mathcal{S}(f)=\emptyset$.
\noindent {\bf Case II.}
Consider the sequence
$$e_0(0)\ge e_0(1)\ge \cdots \ge e_0(m_0).$$
Suppose there exists $i\in\{0,\ldots,m_0\}$ for which
the conditions
\begin{enumerate}
\item[$\bullet$]
$e_0(i)<-1$
\item[$\bullet$]
$i=0$ or $e_0(i)<e_0(i-1)-1$
\end{enumerate}
hold.
Then, for
admissible $\sigma\in\Sym_\mu$, the factor $\sigma_0$ must map
$\{i,\ldots,m_0\}$ to itself, as the box configuration function
$$h_\sigma(\delta; j) = a(j-e_\delta(\sigma_\delta^{-1}(j)))$$
must be weakly increasing in $j$.
The factor $P$ of $\psi$ is a multiple of
\begin{equation*}
\prod_{j=i}^{m_0}(-\sigma_0(j)-1).
\end{equation*}
Since $\frac{\psi}{P}$ vanishes at all inadmissible $\sigma$, we can take
\begin{equation*}
\psi' = \frac{\prod_{j=i}^{m_0}(-j-1)}{\prod_{j=i}^{m_0}(-\sigma_0(j)-1)}\psi,
\end{equation*}
and then $\psi'(\sigma)=\psi(\sigma)$ at all $\sigma\in\Sym_\mu$.
We have $\deg(\psi')<\deg(\psi)$, as desired.
\noindent {\bf Case III.}
Suppose $\delta \ge 0$ and $e_{\delta+1}(m_{\delta+1}) +1 < e_{\delta}(m_{\delta})$.
Then, either $m_{\delta+1} = m_{\delta} - 1$ or $m_{\delta+1} = m_{\delta}$. We consider the two options separately.
\noindent({\bf{i}})
If $m_{\delta+1} = m_{\delta} - 1$, then for any $\sigma\in\Sym_\mu$, we can take
$$i = \sigma_\delta^{-1}(\sigma_{\delta+1}(m_{\delta+1})+1)\ .$$ Then,
$
\sigma_\delta(i) = \sigma_{\delta+1}(m_{\delta+1})+1$
and
$e_\delta(i) \ge e_\delta(m_\delta) > e_{\delta+1}(m_{\delta+1})+1$,
so $\sigma$ is not admissible. Thus $\mathcal{S}(f)=\emptyset$.
\noindent({\bf{ii}})
If $m_{\delta+1} = m_{\delta}$, then we have
$e_{\delta+1}(m_{\delta}) +1 < e_{\delta}(m_{\delta}) \le e_\delta(j)$
for $0\le j \le m_\delta$, so $R_\delta$ is a multiple of
\begin{equation}\label{jj45}
\prod_{j=0}^{m_\delta}(\sigma_\delta(j)-\sigma_{\delta+1}(m_\delta)-1).
\end{equation}
The product \eqref{jj45} vanishes unless $\sigma_{\delta+1}(m_\delta)=m_\delta$.
Hence
\begin{equation*}
(-m_\delta-1)\prod_{j=1}^{m_\delta}(j-\sigma_{\delta+1}(m_\delta)-1)
\end{equation*}
equals \eqref{jj45}
for all $\sigma\in\Sym_\mu$ and is of lower degree, so we may replace $\psi$ with $\psi'$ of lower degree.
\pagebreak
\noindent {\bf Case IV.}
Suppose $\delta < 0$ and $e_{\delta}(m_\delta) < e_{\delta+1}(m_{\delta+1})$.
The situation is parallel to Case III. As before, either $\mathcal{S}(f)=\emptyset$ or we can replace a divisor of $R_\delta$ with a polynomial of lower degree.
To complete the proof of Proposition \ref{vanishing}, we
must check there are only
finitely many $f$
which avoid Cases I-IV.
If $f$ does not fall into Case I, then $e_\delta(j)\le J$ for all $(\delta; j)\in\mu$. If $f$ does not fall into Case II, then $e_0(j)\ge -j-1$ for each $j$, and in particular $e_0(m_0) \ge -m_0-1$. If $f$ also does not fall
into either of the other two cases,
we can extend the inequality to obtain
$$e_\delta(m_\delta) \ge -m_0 - 1 - \max\{\delta \mid \mu_\delta\ne\emptyset\}$$
for all $\delta$.
Since $e_\delta(j)$ is a weakly decreasing function of $j$, the
bounds imply bounds for all of the $e_\delta(j)$. Since the $e_\delta(j)$ belong to $\frac{1}{a}\Z$, we conclude
there are only a finite number of possibilities for each if $f$ does not fall into any of the Cases I-IV. \qed
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\subsection{Proof of Proposition~\ref{division}}
Let $R=\Q[x_1,\ldots,x_n]$, and let
$$e_1, e_2,\ldots,e_n\in R$$ be the elementary
symmetric polynomials with $c_1, c_2, \ldots, c_n \in \Z$ their evaluations at $x_i=i$.
Let $$I = (e_1-c_1,\ldots,e_n-c_n) \subset R$$ denote the ideal of polynomials
vanishing on every permutation of $(1,\ldots,n)$.
For a polynomial $f\in R$, let $f_0$ denote the homogeneous part
of $f$ of highest degree.
For an ideal $J\subset R$, let $J_0$ denote the homogeneous
ideal generated by the top-degree parts,
$$J_0 = \langle \ f_0 \ | \ f \in J \ \rangle \ .$$
Using the regularity of $e_1,\ldots, e_n$,
we easily see $I_0 = (e_1,\ldots,e_n)$.
We define $R'=\Q[y_1,\ldots,y_m]$ and ideals $I',I'_0\subset R'$
as above with respect to the permutations of $(1,\ldots,m)$.
We have
$$A = R\otimes_\Q R'= \Q[x_1,\ldots, x_n,y_1,\ldots,y_m]\ .$$
For notational convenience, we let
$$I,I_0,I',I'_0 \subset A$$
denote the extensions of the respective ideals of $R$ and $R'$ in $A$.
The ideal of $A$ vanishing on the set $P\subset \Q^{n+m}$
of Proposition \ref{division} is precisely $I+I'$.
The basic equality
$$(I+I')_0 = I_0 + I_0'$$
holds.
Let $\widehat{P} = \{p\in P \mid F(p)\ne 0\}$. Let $H\in A$ be a polynomial with
the prescribed values
$$H(p) = \frac{G(p)}{F(p)}$$ for
$p\in \widehat{P}$, of minimum possible degree $d = \deg(H)$.
We must show $d\le \deg(G)-\deg(F)$.
For contradiction, assume $d > \deg(G)-\deg(F)$.
Then, the polynomial $G - FH$ vanishes at every $p\in P$ and has top degree part $F_0H_0$.
Since $F_0\in R$, we verify the following equality
\begin{equation*}
H_0\in\{f\in A \mid F_0f \in (I+I')_0\} = \{r\in R \mid F_0r \in I_0\} + I'_0 \ \ \subset A.
\end{equation*}
We claim the above ideal is equal to
\begin{equation*}
\{f\in A \mid Ff \in I+I'\}_0 = \{r\in R \mid Fr \in I\}_0 + I'_0 \ \ \subset A,
\end{equation*}
Assuming the equality,
there exists $H'\in A$ with top degree part $H_0$ and $FH' \in I+I'$ vanishing at
every $p\in P$.
But then $H_0-H'$ has degree less than that of $H_0$ and still interpolates the desired values, so we have a contradiction.
To complete the proof of Proposition \ref{division},
we must show
\begin{equation*}
\{r\in R \mid F_0r \in I_0\} + I'_0 = \{r\in R \mid Fr \in I\}_0 + I'_0,
\end{equation*}
or equivalently
\begin{equation} \label{befff}
\{r\in R \mid F_0r \in I_0\} = \{r\in R \mid Fr \in I\}_0.
\end{equation}
The left hand side contains the right hand side.
The equality \eqref{befff} is thus a
consequence of the following Lemma which implies
the two sides have equal (and finite) codimension in $R$.
\begin{lem}\label{ranks}
Let $n\geq 0$ be an integer, and
let $a_1\le a_2\le\cdots\le a_n$ be integers satisfying $0\le a_i < i$. Let
\begin{equation*}
F = \prod_{1\le j\le a_i}(x_j-x_i+1) \ \ \ \ {\text and} \ \ \ \
F_0 = \prod_{1\le j\le a_i}(x_j-x_i).
\end{equation*}
Then, we have
\begin{eqnarray*}
\rk_\Q(m_F: R/I \to R/I) & =& \rk_\Q(m_{F_0}:R/I_0 \to R/I_0) \\
& = & \prod_{i=1}^n(i-a_i),
\end{eqnarray*}
where $m_F$ and $m_{F_0}$ denote multiplication operators by $F$ and $F_0$ respectively.
\end{lem}
\begin{proof}
We first show $\rk_\Q(m_F) = \prod_{i=1}^n(i-a_i)$.
Since $R/I$ is the coordinate ring of the set of $n!$ permutations of $(1,\ldots,n)$,
the rank is simply the number of permutations at which $F$ does not vanish.
We must count the number of permutations $\sigma\in \Sym_n$
satisfying $$\sigma(i)-1\ne\sigma(j)$$
for $1\le j\le a_i$.
We view the permutation $\sigma$ (extended by $\sigma(0)=0$)
as a directed path on vertices labeled $0,1,\ldots,n$ with an edge from $i$ to $j$
if $\sigma(i)-1=\sigma(j)$.
We are then counting permutations which
do not have an edge from $i$ to $j$ if $1\le j\le a_i$.
We count the number of ways of building such a path by first choosing an edge leading out of $n$, then an edge leading out of $n-1$, and so on. The edge leading out of $n$ can go to $0$ or to any $j$ with $a_n < j < n$; there are $n-a_n$ choices. After placing the edges leading out of $n,n-1,\ldots,k+1$, the digraph will be a disjoint union of $k+1$ paths. One of these paths will end at $k$ and $a_k$ of the other paths will end at $1,\ldots,a_k$, so the choices for the edge leading out of $k$ are to go to the start of one of the $k-a_k$ other paths. Thus,
the number of such permutations is indeed the product $(n-a_n)\cdots(1-a_1)$.
Proving $\rk_\Q(m_{F_0}) = \prod_{i=1}^n(i-a_i)$ will require more work.
Let $$J = \{f\in R \mid F_0f\in I_0\},$$
so multiplication by $F_0$ induces an isomorphism between $R/J$ and
$\text{Image}(m_{F_0})\subset R/I_0$.
We will show
\begin{equation}\label{uu23}
\rk_\Q(R/J) = \prod_{i=1}^n(i-a_i)\ .
\end{equation}
In fact, we claim $R/J$ is a 0-dimensional complete intersection of
multidegree $(1-a_1,\ldots,n-a_n)$. The
dimension \eqref{uu23}
will then follow from Bezout's Theorem.
For $1\le k \le n$, let
\begin{equation*}
f_k = \sum_{i=k}^n x_i\prod_{j=a_k+1}^{k-1}(x_j-x_i).
\end{equation*}
We claim $J = (f_1,\ldots,f_n)$. Note $f_k$ has degree $k-a_k$ as desired.
We will prove this claim by induction on the sequence $(a_i)_{i=1}^n$. The base case is $a_i=0$ for all $i$
where
$$F=1,\ \ \ J = I_0,\ \ \ \text{and}\ \ \
f_k = \sum_{i=k}^nx_i\prod_{j=1}^{k-1}(x_j-x_i)\ .$$
We must show $(f_1,\ldots,f_n) = (e_1,\ldots,e_n)$.
First, suppose $f_1=f_2=\cdots=f_n=0$
at some point $$(t_1,\ldots,t_n)\in\overline{\Q}^n.$$
From $f_n=0$, we find either $t_n=0$ or $t_n=t_i$ for some $i<n$.
Since $f_{n-1}=0$, either $t_{n-1}=0$ or $t_{n-1}=t_i$ for some $i<n-1$.
Continuing, we conclude for every $k$, either $t_k=0$ or $t_k=t_i$ for some $i<k$.
Thus, $t_k=0$ for all $k$.
Therefore $R/(f_1,\ldots,f_n)$ is a complete intersection and has $\Q$-rank
$$(\deg f_1)\cdots(\deg f_n) = n! = \rk_\Q(R/(e_1,\ldots,e_n))\ .$$
By the rank computation, we
need only show
\begin{equation}\label{httyy}
(f_1,\ldots,f_n) \subseteq (e_1,\ldots,e_n)
\end{equation}
to complete the base case of the induction.
But the inclusion \eqref{httyy} is easily seen. For every $k$, we have
\begin{align*}
f_k &= \sum_{i=1}^n x_i\prod_{j=1}^{k-1}(x_j-x_i) \\
&= \sum_{i=1}^n \sum_{e=1}^n c_ex_i^e \\
&=\sum_{e=1}^n c_e \left(\sum_{i=1}^n x_i^e\right),
\end{align*}
where $c_e\in R$. The power sum
$\sum_{i=1}^n x_i^e$ is symmetric and can be written as a polynomial in the elementary symmetric functions $e_1,\ldots,e_n$.
The base case is now established.
We now consider two sets of indices $a_1,\ldots, a_n$ and $a'_1,\ldots,a'_n$
for which such that $a'_i=a_i$ except when $i=l$ and
\begin{equation}
a'_l=a_l+1.
\label{grtt}
\end{equation}
We moreover require either $l = n$ or $a_{l+1}=a_l+1$. We assume
inductively our claim holds for $a_1,\ldots, a_n$ and show
the claim for $a'_1,\ldots,a'_n$.
Every $(a'_i)_{1\le i \le n}$ which is not identically zero can be reached by
taking $l = \min\{l \mid a'_l=a'_n\}$, so the inductive step will imply the Lemma.
Let $J$,$J'$ be the corresponding ideals and let $f_1,\ldots,f_n$ and $f'_1\ldots,f'_n$ be the claimed generators.
We are assuming $J = (f_1,\ldots,f_n)$ and want to prove $J' = (f'_1\ldots,f'_n)$.
From the definition of $J$ and $J'$, we easily see
$$J' = \{g\in R \mid (x_{a_l+1}-x_l)g\in J\}.$$
Also note $f'_k = f_k$ for $k\ne l$.
If $l=n$, then
$$f'_l = \frac{f_l}{x_{a_l+1}-x_l}$$
and otherwise
$$f'_l = \frac{f_l-f_{l+1}}{x_{a_l+1}-x_l}$$
by condition \eqref{grtt}.
Let $\overline{R} = R/(x_{a_l+1}-x_l)$. For an element $r\in R$,
let $\overline{r}$ denote the projection in $\overline{R}$. Consider
the $\overline{R}$-module homomorphism
$$\psi: \overline{R}^n\to \overline{R}$$
defined by $\psi(\overline{r}_1,\ldots,\overline{r}_n) = \overline{f}_1\overline{r}_1+\cdots+\overline{f}_n\overline{r}_n$. Let $s_i^{(j)}$ for $1\le i\le n$ and $1\le j\le m$ be such that the $m$ elements $(\overline{s}_1^{(j)},\ldots, \overline{s}_n^{(j)})\in\overline{R}^n$ generate the kernel of $\psi$. Clearly,
$J'$ is the ideal generated by $J$ and the $m$ elements
$$\frac{1}{x_{a_l+1}-x_l}\sum_{i=1}^nf_is^{(j)}_i \ .$$
In other words, we must find all the relations between the elements
$$\overline{f}_1,\ldots,\overline{f}_n.$$
Now $\overline{f}_l = \overline{f}_{l+1}$ if $l\ne n$, or $\overline{f}_l=0$ if $l = n$, so we need only consider relations between the $n-1$ elements with $\overline{f}_l$ removed.
These $n-1$ elements in $\overline{R}$ form a complete intersection,
so the relations are generated by the trivial ones
$\overline{f}_i\overline{f}_j-\overline{f}_j\overline{f}_i = 0$.
We have proven that
$J'$ is the ideal generated by $J = (f_1,\ldots,f_n)$, either $\frac{f_l}{x_{a_l+1}-x_l}$ if $l=n$ or $\frac{f_l-f_{l+1}}{x_{a_l+1}-x_l}$ otherwise, and the elements $$\frac{f_if_j-f_jf_i}{x_{a_l+1}-x_l} = 0.$$
Thus $J' = (f'_1\ldots,f'_n)$, as desired.
\end{proof}
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\section{Descendent depth}\label{depp}
\subsection{$T$-Depth}
Let $N$ be a split rank 2 bundle on
a nonsingular projective curve $C$ of genus $g$.
Let $S\subset N$ be the relative divisor associated
to the points $p_1,\ldots, p_r\in C$.
We consider the $T$-equivariant stable pairs theory of $N/S$
with respect to the scaling action.
The $T$-{\em depth} $m$ theory of $N/S$ consists of all
$T$-equivariant series
\begin{equation}
\label{hkkq}
{\mathsf Z}^{N/S}_{d,\eta^1,\dots,\eta^r}
\left( \prod_{{j'}=1}^{k'}
\tau_{i'_{j'}}(\mathsf{1}) \ \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T
\end{equation}
where $k' \leq m$.
As before, $\mathsf{p}\in H^2(C,\mathbb{Z})$ is the class of a point.
The $T$-depth $m$ theory has at most $m$ descendents
of $1$ and arbitrarily many descendents of $\mathsf{p}$ in the integrand.
The $T$-depth $m$ theory of $N/S$ is {\em rational} if all
$T$-depth $m$ series \eqref{hkkq} are Laurent expansions in $q$
of rational functions in $\Q(q,s_1,s_2)$.
The $T$-depth 0 theory concerns only descendents of $\mathsf{p}$.
By taking the specialization $s_3=0$ of Proposition \ref{cttt},
$$
\ZZ_{d,\eta}
^{\mathsf{cap}}\left( \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T=
\ZZ_{d,\eta}
^{\mathsf{cap}}\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}\Big|_{s_3=0}\ , $$
we see
the depth 0 theory of the cap is rational.
\begin{lem} The $T$-depth 0 theory of $N/S$ over a curve $C$ is rational.
\label{ht99}
\end{lem}
\begin{proof} By the degeneration formula, all the descendents
$\tau_{i_j}(\mathsf{p})$ can be degenerated on to a $(0,0)$-cap.
The $T$-depth 0 theory of the cap is rational.
The pairs
theory of local curves without any insertions is rational by \cite{mpt,lcdt}.
Hence, the result follows by the degeneration formula.
\end{proof}
\subsection{Degeneration}
We have already used the degeneration formula in simple cases
in Proposition \ref{ctttt}
and Lemma \ref{ht99} above. We review here the
full $T$-equivariant formula for descendents of
$\mathsf{1},\mathsf{p}\in H^*(C,\mathbb{Z})$.
Let $C$ degenerate to a union $C_1\cup C_2$ of nonsingular
projective curves $C_i$
meeting at a node $p'$.
Let $N$ degenerate to split bundles
$$N_1 \rightarrow C_1, \ \ \ \ N_2 \rightarrow C_2 \ .$$
The levels of $N_i$ must sum to the level of $N$.
The relative points $p_i$, distributed to nonsingular points
of $C_1\cup C_2$, specify relative points $S_i\subset C_i$ away from
$p'$. Let $S_i^+= S_i \cup \{ p' \}$.
In order to apply the degeneration formula to the series
\eqref{hkkq}, we must also specify the distribution of
the point classes occuring in the descendents $\tau_{i_j}(\mathsf{p})$.
The disjoint union $$J_1\cup J_2 = \{1,\ldots, k\}$$
specifies the descendents $\tau_{i_j}(\mathsf{p})$
distribute to $C_i$ for $j\in J_i$.
The degeneration formula for \eqref{hkkq} is
\begin{multline*}
\sum_{J'_1\cup J'_2=\{1,\ldots, k'\}}
{\mathsf Z}^{N_1/S^+_1}_{d,\eta^1,\dots,\eta^{|S_1|}, \mu}
\left( \prod_{{j'}\in J'_1}
\tau_{i'_{j'}}(\mathsf{1}) \ \prod_{j\in J_1} \tau_{i_j}(\mathsf{p})
\right)^T\ \frac{g^{\mu\widehat{\mu}}}{q^d} \\ \cdot
{\mathsf Z}^{N_2/S^+_2}_{d,\eta^{|S_1|+1},\dots,\eta^{|S_2|}, \widehat{\mu}}
\left( \prod_{{j'}\in J'_2}
\tau_{i'_{j'}}(\mathsf{1}) \ \prod_{j\in J_2} \tau_{i_j}(\mathsf{p})
\right)^T
\end{multline*}
A crucial point in the derivation of the degeneration formula
is the pre-deformability condition (ii) of Section 3.7 of \cite{pt}.
The condition insures the existence of finite resolutions of
the universal sheaf $\mathbb{F}$ in the relative geometry (needed
for the definition of the descendents) and guarantees the
splitting of the descendents under pull-back via the gluing
maps of the relative geometry. The foundational treatment for
stable pairs is essentially the same as for ideal sheaves \cite{liwu}.
\subsection{Induction I}
To obtain the rationality of the $T$-depth $m$ theory of
$N/S$ over a curve $C$, further knowledge of the descendent
theory of twisted caps is required.
\begin{lem} The rationality of the
$T$-depth $m$ theories of all twisted caps implies
the rationality of the
$T$-depth $m$ theory
of $N/S$ over a curve $C$. \label{rtt5}
\end{lem}
\begin{proof}
We start by proving rationality for the $T$-depth $m$ theories of
all $(0,0)$ geometries,
\begin{equation} \label{hxxz}
\mathcal{O}_{\C} \oplus \mathcal{O}_{\C} \rightarrow \Pp\ ,
\end{equation}
relative to $p_1,\ldots, p_r \in \Pp$.
If $r=1$, the geometry is the cap and rationality of the
$T$-depth $m$ theory is given.
Assume rationality holds for $r$. We will show rationality holds for
$r+1$.
Let $p(d)$ be the number of partitions of size $d>0$.
Consider the $\infty \times p(d)$ matrix $M_d$, indexed by
monomials
$$L= \prod_{i\geq 0} \tau_i (\mathsf{p})^{n_i} $$
in the descendents of $\mathsf{p}$ and partitions $\mu$ of $d$,
with
coefficient
$
{\mathsf Z}^{\mathsf{cap}}_{d,\mu}
\left( L
\right)^T$
in position $(L,\mu)$.
The lowest Euler characteristic for a degree $d$
stable pair on the cap is $d$.
The leading $q^d$
coefficients of $M_d$ are well-known to be of maximal
rank.{\footnote{ The leading $q^d$ coefficients
are obtained from the Chern characters
of the tautological rank $d$ bundle
on $\text{Hilb}(N_\infty,d)$.
The Chern characters generate the ring
$H^*_T(\text{Hilb}(N_\infty,d),\mathbb{Q})$ after
localization as can easily
be seen in the $T$-fixed point basis.
A more refined result is
discussed in Section \ref{ennd}.}}
Hence, the full matrix $M_d$ is also of maximal rank.
Consider the level $(0,0)$ geometry
over $\Pp$ relative to $r+1$ points in $T$-depth $m$,
\begin{equation}
\label{yone}
{\mathsf Z}^{(0,0)}_{d,\eta^1, \ldots, \eta^r,\mu}
\left( \prod_{j'=1}^{k'}
\tau_{i'_{j'}}(1) \ \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T\ .
\end{equation}
We will determine the series \eqref{yone} from the
$T$-depth $m$ series relative to $r$ points,
\begin{equation}
\label{yall}
{\mathsf Z}^{(0,0)}_{d,\eta^1, \ldots, \eta^r}
\left( L \ \prod_{j'=1}^{k'}
\tau_{i'_{j'}}(1) \ \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T
\end{equation}
defined by all monomials $L$ in the descendents of $\mathsf{p}$.
Consider the $T$-equivariant degeneration of
the $(0,0)$ geometry relative to $r$ points obtained
by bubbling off a single $(0,0)$-cap.
All the descendents of $\mathsf{p}$
remain on the original $(0,0)$ geometry in the degeneration except for those
in $L$ which distribute to the cap.
By induction on $m$, we need only analyze the terms of the degeneration formula
in which the descendents of the identity distribute away from the cap.
Then,
since $M_d$ has full rank,
the invariants \eqref{yone} are determined
by the invariants \eqref{yall}.
We have proven the rationality of the $T$-depth $m$ theory of the $(0,0)$-cap
implies the rationality of the $T$-depth $m$ theories of all
$(0,0)$ relative geometries over $\Pp$. By degenerations of
higher genus curves $C$ to rational curves with relative points,
the rationality of the $(0,0)$ relative geometries over
curves $C$ of arbitrary genus is established.
Finally, consider a relative geometry $N/S$ over $C$ of
level $(a_1,a_2)$. We can degenerate $N/S$ to the
union of a $(0,0)$ relative geometry over $C$ and
a twisted $(a_1,a_2)$-cap. Since the rationality of the
$T$-depth $m$ theory of
the twisted cap is given, we conclude the rationality of
$N/S$ over $C$.
\end{proof}
The proof of Lemma \ref{rtt5} yields a slightly refined
result which will be half of our induction argument
relating the descendent theory of the $(0,0)$-cap and
the $(0,0)$-tube.
\begin{lem}\label{nndd}
The rationality of the $T$-depth $m$ theory of the
$(0,0)$-cap implies the rationality of the $T$-depth $m$ theory of
the $(0,0)$-tube.
\end{lem}
\subsection{$\mathbf{T}$-depth}
The $\mathbf{T}$-{\em depth} $m$ theory of the $(a_1,a_2)$-cap
consists of all the
$\mathbf{T}$-equivariant series
\begin{equation}
\label{hkkqq}
{\mathsf Z}^{(a_1,a_2)}_{d,\eta}
\left(
\prod_{j=1}^k \tau_{i_j}([0]) \ \prod_{{j'}=1}^{k'}
\tau_{i'_{j'}}([\infty])
\right)^{\mathbf{T}}
\end{equation}
where $k' \leq m$.
Here, $0\in \Pp$ is the non-relative $\mathbf{T}$-fixed point and
$\infty\in \Pp$ is the relative point.
The $\mathbf{T}$-depth $m$ theory of the $(a_1,a_2)$-cap
is {\em rational} if all
$\mathbf{T}$-equivariant
depth $m$ series \eqref{hkkq} are Laurent expansions in $q$
of rational functions in $\Q(q,s_1,s_2,s_3)$.
\begin{lem} The rationality of the $\mathbf{T}$-depth $m$
theory of the $(a_1,a_2)$-cap implies the
rationality of the $T$-depth $m$ theory of the $(a_1,a_2)$-cap.
\end{lem}
\begin{proof}
The identity class
$1\in H^*_T(\Pp,\mathbb{Z})$ has a well-known
expression in terms of the $\mathbf{T}$-fixed point classes
$$1 = -\frac{[0]}{s_3} + \frac{[\infty]}{s_3}\ .$$
We can calculate at most $m$ descendents of $1$ in the $T$-equivariant
theory via at most
$m$ descendents of $[\infty]$ in the $\mathbf{T}$-equivariant
theory (followed the specialization $s_3=0$).
\end{proof}
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\section{Rubber calculus} \label{rubc}
\subsection{Overview}
We collect here results concerning
the rubber calculus which will be needed to complete the
proof of Theorem \ref{cnnn}. Our discussion of the
rubber calculus follows the treatment given in Section 4.8-4.9 of
\cite{lcdt}.
\subsection{Universal 3-fold $\mathcal{R}$}
Consider the moduli space of stable pairs on rubber
$P_n(R/R_0\cup R_\infty)^\sim$ discussed in Section \ref{rubcon}.
Let
$$\pi:\mathcal{R} \rightarrow {P_n(R/R_0\cup R_\infty,d)}^\sim$$
denote the universal 3-fold.
The space $\mathcal{R}$ can be viewed as a moduli space of stable pairs on
rubber {\em together} with a point $r$ of the 3-fold rubber.
The point $r$ is {\em not} permitted to lie on the relative divisors $R_0$ and
$R_\infty$. The stability condition is given by finiteness of the
associated automorphism group.
The virtual class of ${\mathcal R}$ is obtained via $\pi$-flat pull-back,
$$[{\mathcal R}]^{vir} = \pi^* \Big( [{P_n(R/R_0\cup R_\infty,d)}^\sim]^{vir}\Big).$$
As before, let
$$\mathbb{F} \rightarrow {\mathcal R}$$
denote the universal sheaf on ${\mathcal R}$.
The target point $r$ together with $R_0$ and $R_\infty$
specifies 3 distinct points of the destabilized $\Pp$ over which the rubber is fibered.
By viewing the target point as $1\in \Pp$,
we obtain a rigidification map to the tube,
$$\phi: \mathcal{R} \rightarrow P_n(N/N_0\cup N_\infty,d),$$
where $N=\mathcal{O}_\Pp \oplus \mathcal{O}_\Pp$ is the trivial bundle over $\Pp$.
By a comparison of deformation theories,
\begin{equation}\label{zek}
[{\mathcal R}]^{vir} = \phi^* \Big( [{P_n(N/N_0\cup N_\infty,d)}]^{vir}\Big).
\end{equation}
\subsection{Rubber descendents} \label{papap}
Rubber calculus transfers $T$-equivariant rubber descendent
integrals to $T$-equivariant descendent integrals
for the $(0,0)$-tube geometry via the maps $\pi$ and $\phi$.
Consider the rubber descendent
\begin{equation} \label{dref}
\Big\langle \mu \ \Big| \ \psi_0^\ell \ \tau_{c}\cdot \prod_{j=1}^k \tau_{i_j}
\ \Big|\ \nu \Big
\rangle_{n,d}^{\sim}\ .
\end{equation}
As before, $\psi_0$ is the cotangent line at the dynamical point
$0\in \Pp$. The action of the rubber descendent $\tau_{i}$ is defined
via the universal sheaf $\mathbb{F}$ by the operation
$$
\pi_{*}\big( \text{ch}_{2+i}(\FF)
\cap(\pi^*(\ \cdot\ )\big)\colon
H_*(P_{n}(N/N_0\cup N_\infty,d))\to H_*(P_{n}(N/N_0\cup N_\infty,d))\ .
$$
By the push-pull formula, the integral \eqref{dref} equals
\begin{equation}\label{zex}
\Big\langle \mu \ \Big|\ \text{ch}_{2+c}(\mathbb{F}) \ \pi^*\left(\psi_0^\ell
\cdot \prod_{j=1}^k \tau_{i_j}
\right) \ \Big|\ \nu \Big
\rangle_{n,d}^{{\mathcal R}\sim}.
\end{equation}
Next,
we compare the cotangent lines $\pi^*(\psi_0)$ and $\phi^*(\psi_0)$ on
${\mathcal R}$. A standard argument yields
$$\pi^*(\psi_0)=\phi^*(\psi_0) - \phi^*(D_0),$$
where
$$D_0 \subset I_n(N/N_0\cup N_\infty,d)$$
is the virtual boundary divisor
for which the rubber over $\infty$ carries Euler characteristic $n$.
We will apply the cotangent line comparisons to \eqref{zex}.
The basic vanishing
\begin{equation}\label{gbb6}
\psi_0|_{D_0}=0
\end{equation}
holds.
Consider the Hilbert scheme of points ${\text{Hilb}}(R_0,d)$ of the
relative divisor.
The boundary condition $\mu$ corresponds to a Nakajima basis element
of $A^*_T({\text{Hilb}}(R_0,d))$.
Let
${\mathbb{F}}_0$ be the universal quotient sheaf on
$$\text{Hilb}(R_0,d) \times R_0,$$ and define the descendent
\begin{equation}\label{mrr}
\tau_c=\pi_*\Big( {\text{ch}}_{2+c}({\mathbb{F}}_0)\Big)
\in A^c_T({\text{Hilb}}(R_0,d))
\end{equation}
where $\pi$ is the projection
$$\pi: \text{Hilb}(R_0,d) \times R_0 \rightarrow
\text{Hilb}(R_0,d)\ .
$$
The cotangent line comparisons, equation \eqref{zex},
and the vanishing \eqref{gbb6} together yield
the following result,
\begin{multline}\label{dx}
\Big\langle \mu \ \Big| \ \psi_0^\ell \ \tau_{c}\cdot \prod_{j=1}^k \tau_{i_j}
\ \Big|\ \nu \Big\rangle_{n,d}^{\sim} =
\\ \Big\langle \mu \ \Big|\
\psi_0^\ell \ \tau_{c}(\mathsf{p}) \cdot \prod_{j=1}^k \tau_{i_j}
\ \Big|\ \nu \Big
\rangle_{n,d}^{\mathsf{tube},T} \\
- \Big\langle \tau_c\cdot \mu \ \Big|
\ \psi_0^{\ell-1} \prod_{j=1}^k \tau_{i_j} \ \Big|\ \nu
\Big\rangle_{n,d}^{\sim} \ .
\end{multline}
Equation \eqref{dx} will be the main required property of the rubber
calculus.
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\section{Capped 1-leg descendents: full} \label{444}
\subsection{Overview} We complete the proof of Theorem \ref{cnnn}
using the interplay between the $\mathbf{T}$-equivariant localization of the cap
and the theory of rubber integrals. A similar strategy was used in \cite{vir}
to prove the Virasoro constraints for target curves.
As a consequence, we will also obtain a special case of Theorem
\ref{tnnn}.
Let $N$ be a split rank 2 bundle on
a nonsingular projective curve $C$ of genus $g$.
Let $S\subset N$ be the relative divisor associated
to the points $p_1,\ldots, p_r\in C$.
We consider the $T$-equivariant stable pairs theory of $N/S$
with respect to the scaling action.
\begin{prop}
If
$\gamma_j \in H^{2*}(C,\mathbb{Z})$ are {\em even} cohomology classes, then
\label{pnnn}
$$\ZZ_{d,\eta^1,\dots,\eta^r}
^{N/S}\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{{T}}$$ is the
Laurent expansion in $q$ of a rational function in $\mathbb{Q}(q,s_1,s_2)$.
\end{prop}
Proposition \ref{pnnn} is the restriction of Theorem \ref{tnnn}
to even cohomology. The proof is given in Section \ref{jj367}.
The proof of Theorem \ref{tnnn} will be completed with the
inclusion of descendents of odd cohomology in Section \ref{555}.
\subsection{Induction II}
The first half of our induction argument was established
in Lemma \ref{nndd}.
The second half relates the $(0,0)$-tube back to the
$(0,0)$-cap with an increase in depth.
\begin{lem} The rationality of \label{p45}
the ${T}$-depth $m$
theory of the $(0,0)$-tube implies the
rationality of $\mathbf{T}$-depth $m+1$ theory of the $(0,0)$-cap.
\end{lem}
\begin{proof}
The result follows from the $\mathbf{T}$-equivariant
localization
formula for the $(0,0)$-cap and the rubber calculus of Section
\ref{papap}.
To illustrate the method, consider first the $m=0$
case of Lemma \ref{p45}.
The localization formula for
$\mathbf{T}$-depth 1 series for the $(0,0)$-cap is the following:
\begin{multline*}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0]) \cdot \tau_{i'_1}([\infty])
\right)^{\mathbf{T}} =
\\
\sum_{|\mu|=d}
\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right) \cdot
\bW_\mu^{(0,0)} \cdot
\left( \mathsf{S}^{\tau_{i'_1}\cdot\mu}_\eta+
\mathsf{S}^{\mu}_{\eta}(\tau_{i'_1}) \right) \ ,
\end{multline*}
where the rubber terms on the right are
\begin{eqnarray*}
\mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta & = &
\sum_{n\geq d} q^{n}
\left\langle \tau_{i'_1}\cdot \mathsf{P}_\mu \ \left| \ \frac{1}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim} ,
\\
\mathsf{S}^\mu_\eta(\tau_{i'_1}) & = &
\sum_{n\geq d} q^{n}
\left\langle \mathsf{P}_\mu \ \left| \ \frac{ s_3 \tau_{i'_1}}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}. \\
\end{eqnarray*}
In the first rubber term, $\tau_{i'_1}$ acts
on the boundary condition $P_\mu$ via
\eqref{mrr}. The term arises from the
distribution of the Chern character
of the descendent
$\tau_{i'_1}([\infty])$
away from the rubber.
The second rubber term simplifies
via the
topological recursion relation for $\psi_0$ after
writing
\begin{equation}\label{nhhk}
\frac{s_3}{s_3-\psi_0} = 1 + \frac{\psi_0}{s_3-\psi_0}\
\end{equation}
and the rubber calculus relation \eqref{dx}. We find
\begin{eqnarray*}
\mathsf{S}^\mu_\eta(\tau_{i'_1})
& = &
\sum_{|\widehat{\eta}|=d} \mathsf{S}^\mu_{\widehat{\eta}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty]) \right)^T
\ -\ \mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta
.
\end{eqnarray*}
The leading $1$ on the right side of \eqref{nhhk} corresponds to
the degenerate leading term of $\mathsf{S}^\mu_{\widehat{\eta}}$.
The topological recursion applied to the $\psi_0$ prefactor
of the second term produces the rest of $\mathsf{S}^\mu_{\widehat{\eta}}$.
The superscript $\mathsf{tube}$ refers here to
the $(0,0)$-tube.
The rubber calculus produces the correction
$-\mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta$.
After reassembling the localization formula, we find
\begin{multline*}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0]) \cdot \tau_{i'_1}([\infty])
\right)^{\mathbf{T}} =
\\
\sum_{|\widehat{\eta}|=d}
{\mathsf Z}^{\mathsf{cap}}_{d,\widehat{\eta}}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty]) \right)^T
\end{multline*}
which implies the $m=0$ case of Lemma \ref{p45}.
The above method of expressing the $\mathbf{T}$-depth
$m+1$ theory of the $(0,0)$-cap in terms of
the $\mathbf{T}$-depth
$0$ theory of the $(0,0)$-cap and the
$T$-depth $m$ theory of the $(0,0)$-tube is valid for
all $m$.
Consider the $m=1$ case.
The localization formula for
$\mathbf{T}$-depth 2 series for the $(0,0)$-cap is the following:
\begin{multline*}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0]) \cdot \tau_{i'_1}([\infty])
\tau_{i_2}([\infty])
\right)^{\mathbf{T}} =
\\
\sum_{|\mu|=d}
\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right) \cdot
\bW_\mu^{(0,0)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \\
\cdot \left( \mathsf{S}^{\tau_{i'_1}\tau_{i'_2}\cdot\mu}_\eta+
\mathsf{S}^{\tau_{i'_1}\cdot \mu}_{\eta}(\tau_{i'_2})
+\mathsf{S}^{\tau_{i'_2}\cdot \mu}_{\eta}(\tau_{i'_1})
+\mathsf{S}^{\mu}_{\eta}(\tau_{i'_1}\tau_{i'_2})
\right) \ ,
\end{multline*}
where the rubber terms on the right are
\begin{eqnarray*}
\mathsf{S}^{\tau_{i'_1}\tau_{i'_2}\cdot \mu}_\eta & = &
\sum_{n\geq d} q^{n}
\left\langle \tau_{i'_1}\tau_{i'_2}\cdot \mathsf{P}_\mu \ \left| \ \frac{1}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim} ,
\\
\mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta(\tau_{i'_2}) & = &
\sum_{n\geq d} q^{n}
\left\langle \tau_{i'_1}\cdot \mathsf{P}_\mu \ \left| \ \frac{ s_3 \tau_{i'_2}}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}, \\
\\
\mathsf{S}^{\tau_{i'_2}\cdot \mu}_\eta(\tau_{i'_1}) & = &
\sum_{n\geq d} q^{n}
\left\langle \tau_{i'_2}\cdot \mathsf{P}_\mu \ \left| \ \frac{ s_3 \tau_{i'_1}}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}, \\
\\
\mathsf{S}^{\mu}_\eta(\tau_{i'_1}\tau_{i'_2}) & = &
\sum_{n\geq d} q^{n}
\left\langle \mathsf{P}_\mu \ \left| \ \frac{ s_3^2 \tau_{i'_1}\tau_{i'_2}
}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}. \\
\end{eqnarray*}
Using \eqref{nhhk} and the rubber calculus relation \eqref{dx}, we find
\begin{eqnarray*}
\mathsf{S}^\mu_\eta(\tau_{i'_1}\tau_{i'_2})
& = &\ \
\sum_{|\widehat{\eta}|=d} \mathsf{S}^\mu_{\widehat{\eta}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty])\cdot \tau_{i'_2}(1) \right)^T
\ -\ \mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta(\tau_{i'_2})\\
& & + \sum_{|\widehat{\eta}|=d} \mathsf{S}^\mu_{\widehat{\eta}}(\tau_{i'_2})
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d}
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty]) \right)^T
.
\end{eqnarray*}
As we have seen before,
\begin{eqnarray*}
\mathsf{S}^\mu_{\widehat{\eta}}(\tau_{i'_2})
& = &
\sum_{|\widehat{\mu}|=d} \mathsf{S}^\mu_{\widehat{\mu}}
\cdot \frac{g^{\widehat{\mu}\widehat{\mu}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\mu},\widehat{\eta}}
\left( \tau_{i_2'}([\infty]) \right)^T
\ -\ \mathsf{S}^{\tau_{i'_2}\cdot \mu}_{\widehat{\eta}}, \\
\mathsf{S}^{\tau_{i'_2}\cdot \mu}_\eta(\tau_{i'_1})
& = &
\sum_{|\widehat{\eta}|=d} \mathsf{S}^{\tau_{i'_2}\cdot\mu}_{\widehat{\eta}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty]) \right)^T
\ -\ \mathsf{S}^{\tau_{i'_1}\tau_{i'_2}\cdot \mu}_\eta
.
\end{eqnarray*}
After adding everything together, we have for $m=1$
the relation:
\begin{multline*}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0]) \cdot \prod_{j'=1}^2
\tau_{i'_{j'}}([\infty])
\right)^{\mathbf{T}} =
\\
+s_3 \sum_{|\widehat{\eta}|=d}
{\mathsf Z}^{\mathsf{cap}}_{d,\widehat{\eta}}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty])\cdot \tau_{i_2'}(1) \right)^T \\
+\sum_{|\widehat{\mu}|,|\widehat{\eta}|=d}
{\mathsf Z}^{\mathsf{cap}}_{d,\widehat{\mu}}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
\cdot \frac{g^{\widehat{\mu}\widehat{\mu}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\mu},\widehat{\eta}}
\left(
\tau_{i'_{2}}([\infty]) \right)^T
\\
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},{\eta}}
\left(
\tau_{i'_{1}}([\infty]) \right)^T
\ . \ \ \ \ \ \
\end{multline*}
We leave the derivation of the parallel formula for general $m$
(via elementary bookkeeping) to
the reader.
\end{proof}
An identical argument yields the twisted version of
Lemma \ref{p45} for the $(a_1,a_2)$-cap.
\begin{lem} The rationality of \label{p456}
the ${T}$-depth $m$
theory of the $(0,0)$-tube implies the
rationality of the $\mathbf{T}$-depth $m+1$ theory of the $(a_1,a_2)$-cap.
\end{lem}
\subsection{Proof of Theorem \ref{cnnn}}
Lemmas \ref{nndd} and \ref{p45}
together provide an induction which results in the
rationality of the $\mathbf{T}$-depth $m$ theory of
the $(0,0)$-cap for all $m$.
Since the classes of the $\mathbf{T}$-fixed points $0,\infty \in \Pp$
generate $H_{\mathbf{T}}^*(\Pp, \mathbb{Z})$
after localization, all partition functions
$$
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}},\ \ \ \ \gamma_{j}\in H^*_{\mathbf{T}}(\Pp,\mathbb{Z})$$
are Laurent series
in $q$ of rational functions in $\mathbb{Q}(q,s_1,s_2,s_3)$.
\qed
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\subsection{Proof of Theorem \ref{cnnn}}
Lemmas \ref{nndd} and \ref{p45}
together provide an induction which results in the
rationality of the $\mathbf{T}$-depth $m$ theory of
the $(0,0)$-cap for all $m$.
Since the classes of the $\mathbf{T}$-fixed points $0,\infty \in \Pp$
generate $H_{\mathbf{T}}^*(\Pp, \mathbb{Z})$
after localization, all partition functions
$$
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}},\ \ \ \ \gamma_{j}\in H^*_{\mathbf{T}}(\Pp,\mathbb{Z})$$
are Laurent series
in $q$ of rational functions in $\mathbb{Q}(q,s_1,s_2,s_3)$.
\qed
\subsection{Proof of Proposition \ref{pnnn}} \label{jj367}
Using Lemma \ref{p456}, we obtain the extension of
Theorem \ref{cnnn} to twisted $(a_1,a_2)$-caps.
\begin{prop}
\label{qnnn}
For $\gamma_{j}\in H^*_{\mathbf{T}}(\Pp,\mathbb{Z})$, the descendent series
$$
{\mathsf Z}^{(a_1,a_2)}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}$$
of the $(a_1,a_2)$-cap
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{prop}
By taking the $s_3=0$ specialization of Proposition
\ref{qnnn}, we obtain the rationality of the $T$-depth $m$
theory of the $(a_1,a_2)$-cap for all $m$.
Proposition \ref{pnnn} then follows from Lemma \ref{rtt5}. \qed
\subsection{$\mathbf{T}$-equivariant tubes}
The $(a_1,a_2)$-tube is the total space of
$$\mathcal{O}_{\Pp}(a_1) \oplus \mathcal{O}_{\Pp}(a_2) \rightarrow \Pp$$
relative to the fibers over both $0,\infty \in \Pp$.
We lift the $\com^*$-action on $\Pp$ to $\mathcal{O}_{\Pp}(a_i)$
with fiber weights $0$ and $a_is_3$ over $0,\infty\in \PP^1$.
The 2-dimensional torus $T$ acts on the $(a_1,a_2)$-tube
by scaling the line summands, so
we obtain a $\mathbf{T}$-action on the $(a_1,a_2)$-tube.
\begin{prop}
\label{qqnnn}
For $\gamma_{j}\in H^*_{\mathbf{T}}(\Pp,\mathbb{Z})$, the
descendent series
$$
{\mathsf Z}^{(a_1,a_2)}_{d,\eta_1\eta_2}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}$$
of the $(a_1,a_2)$-tube
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{prop}
\begin{proof}
Consider the descendent series
\begin{equation}\label{pw4}
{\mathsf Z}^{(a_1,a_2)}_{d,\eta_2}
\left( L \prod_{j'=1}^{k'} \tau_{i'_{j'}} (\mathsf{1})\ \prod_{j=1}^k \tau_{i_j}([\infty])
\right)^{\mathbf{T}}
\end{equation}
of the $(a_1,a_2)$-cap where $L$ is a monomial in the descendents of
$[0]$.
The $(a_1,a_2)$-cap admits a $\mathbf{T}$-equivariant
degeneration to a standard $(0,0)$-cap and an $(a_1,a_2)$-tube by
bubbling off $0\in \Pp$.
The insertions $\tau_{i_j}([0])$ of $L$ are sent $\mathbf{T}$-equivariantly
to the non-relative point of the $(0,0)$-cap.
Since \eqref{pw4} is rational by Proposition \ref{qnnn} and the
matix $M_d$ of Lemma \ref{rtt5} is full rank,
the rationality of
$$
{\mathsf Z}^{(a_1,a_2)}_{d,\eta_1\eta_2}
\left( \prod_{j'=1}^{k'} \tau_{i'_{j'}} (\mathsf{1})\ \prod_{j=1}^k \tau_{i_j}([\infty])
\right)^{\mathbf{T}}$$
follows by induction on $k'$ from the degeneration formula.
The classes $\mathsf{1}$ and $[\infty]$ generate $H_{\mathbf{T}}^*(\Pp,\mathbb{Z})$
after localization.
\end{proof}
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\section{Descendents of odd cohomology} \label{555}
\subsection{Reduction to $(0,0)$}
Let $N/S$ be the relative
geometry of
a split rank 2 bundle on a nonsingular projective curve $C$ of genus $g$.
Let
$$\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g \in H^1(C,\mathbb{Z})$$
be a standard symplectic basis of the odd cohomology of $C$.
Proposition \ref{pnnn} establishes Theorem \ref{tnnn}
in case only the descendents of the even classes
$\mathsf{1},\mathsf{p}\in H^*(C,\mathbb{Z})$ are present.
The descendents of $\alpha_i$ and $\beta_j$
will now be considered.
The relative geometry $N/S$ may be $T$-equivariantly degenerated to
$$\mathcal{O}_C \oplus \mathcal{O}_C \rightarrow C$$
and an $(a_1,a_2)$-cap. The relative points and
the
descendents $\tau_k(\alpha_i)$ and $\tau_k(\beta_j)$ in the
integrand remain on $C$. Since the rationality of the
$T$-equivariant descendent theory of the $(a_1,a_2)$-cap
has been proven, we may restrict our study of the descendents
of odd cohomology to the $(0,0)$ relative geometry over $C$.
\subsection{Proof of Theorem \ref{tnnn}}
The full descendent theories of $(0,0)$ relative geometries of
curves $C$
are uniquely determined by the even descendent theories of $(0,0)$ relative geometries
by the following four properties:
\begin{enumerate}
\item[(i)] Algebraicity of the virtual class,
\item[(ii)] Degeneration formulas for the relative theory in the
presence of odd cohomology,
\item[(iii)] Monodromy invariance of the relative theory,
\item[(iv)] Elliptic vanishing relations.
\end{enumerate}
The properties (i)-(iv) were used in \cite{vir} to determine the
full relative Gromov-Witten descendents of target curves in terms
of the descendents of even classes.
The results of Section 5 of \cite{vir} are entirely formal and
apply verbatim to the descendent theory of $(0,0)$ relative geometries of
curves. Moreover, the rationality of the even theory implies
the rationality of the full descendent theory. \qed
\section{Denominators} \label{ennd}
\subsection{Summary} We prove the
denominator claims of Conjecture \ref{222}
when only descendents of $\mathsf{1}$ and $\mathsf{p}$
are present.
\begin{thm}
\label{2222}
If only descendents of even cohomology are considered,
the denominators of the degree $d$ descendent partition functions
$\ZZ$ of Theorems \ref{onnn}, \ref{tnnn}, and \ref{cnnn}
are products of factors of the form $q^k$ and
$$1-(-q)^r$$
for $1\leq r \leq d$.
\end{thm}
Theorem \ref{2222} is proven by carefully tracing
the denominators through the proofs of Theorems \ref{onnn}-\ref{cnnn}.
When the descendents of odd cohomology are included,
the strategy of Section 5 of \cite{vir} requires
matrix inversions{\footnote{Specifically,
the matrix associated to Lemma 5.6 of \cite{vir}
has an inverse with denominators we cannot
at present constrain.}} for which we can not control the
denominators.
Theorem \ref{2222} is new even when {\em no} descendents
are present. For the trivial bundle
$$N = \OO_{\mathbb{P}^1} \oplus \OO_{\mathbb{P}^1} \rightarrow
\mathbb{P}^1\ ,$$
the $T$-equivariant partition $\mathsf{Z}^{N/S}_{d,\eta^1,\eta^2,\eta^3}$
of Theorem \ref{tnnn}
is (up to $q$ shifts) equal to the 3-point
function $$\langle \eta^1, \eta^2,\eta^3 \rangle$$
in the quantum cohomology of the Hilbert scheme of points
of $\mathbb{C}^2$, see \cite{hilb1,lcdt}.
\noindent{\bf Corollary.} {\em The 3-point functions in the $T$-equivariant
quantum cohomology of $\text{Hilb}(\mathbb{C}^2,d)$
have possible poles in -q only at the $r^{th}$ roots of
unity for $r$ at most $d$.}
\begin{proof}
By Theorem \ref{2222}, we see the possible poles in $-q$
of the 3-point
functions are at $0$ and the $r^{th}$ roots of
unity for $r$ at most $d$.
By definition, the 3-point functions have
no poles at 0.
\end{proof}
\subsection{Denominators for Proposition \ref{cttt}}
We follow here the notation used in the proof of Proposition
\ref{cttt} in Section \ref{333}.
The matrix $\mathsf{S}_\eta^\mu$ is a fundamental solution of a
linear differential equation with singularities only at
0 and $r^{th}$ roots of unity for $r$ at most $d$, see
\cite{hilb2}. Hence, the poles in $-q$ of the evaluation
$$\mathsf{S}_\eta^\mu
|_{s_3=\frac{1}{a}(s_1+s_2)}$$
can occur only at $0$ and $r^{th}$ roots of unity for $r$ at most $d$.
The denominator claim of Theorem \ref{2222} for
Proposition \ref{cttt} then follows directly from the proof
in Section \ref{ggtt2}.
While only the rationality of Theorem \ref{canpole} is needed
in the proof of Proposition \ref{cttt}, the much stronger Laurent
{polynomiality}
of Theorem \ref{canpole} is used here.
\subsection{Denominators for $T$-equivariant stationary theory}
Consider the denominators of
\begin{equation*}
{\mathsf Z}^{N/S}_{d,\eta^1,\dots,\eta^r}
\left( \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T \ .
\end{equation*}
The denominator result for the $T$-equivariant stationary theory of
the
$(0,0)$-cap is obtained from the denominator result for
Proposition \ref{cttt} by the specialization $s_3=0$.
By degenerating all the descendents $\tau_{i_j}(\mathsf{p})$
on to a $(0,0)$-cap, we need only study the denominators
of $T$-equivariant
partition functions ${\mathsf Z}^{N/S}_{d,\eta^1,\dots,\eta^r}$
with no descendent insertions.
The denominator result for
the $T$-equivariant $(a,b)$-tube with no descendents
is again a consequence
of the study of the fundamental solution in \cite{hilb2}.
By repeated degenerations (using the $(a,b)$-tube for the
twists in $N$), we need only study the
denominators
of $T$-equivariant
partition functions ${\mathsf Z}^{(0,0)}_{d,\eta^1,\eta^2,\eta^3}$
with 3 relative insertions.
\subsection{Relative/descendent correspondence}
Relative conditions in the theory of local curves
were exchanged for descendents
in the proof of Lemma \ref{rtt5}.
For the denominator result for
${\mathsf Z}^{(0,0)}_{d,\eta^1,\eta^2,\eta^3}$, we require a
more efficient correspondence.
\begin{prop}\label{gbb5}
Let $d>0$ be an integer.
The square matrix with coefficients
\begin{equation}\label{gredd}
\mathsf{Z}^{\mathsf{cap}}_{d,\lambda}\left( \tau_{\mu_1-1}([0])
\cdots \tau_{\mu_{\ell(\mu)}-1}([0]) \right)^T
\end{equation}
as $\lambda$ and $\mu$ vary among
partitions of $d$
\begin{enumerate}
\item[(i)] is triangular with respect to the partial ordering
by length,
\item[(ii)] has
diagonal entries given
by monomials in $q$,
\item[(iii)] and is of
maximal rank.
\end{enumerate}
\end{prop}
\begin{proof}
The Proposition follows from the
results of Section 4.6 of \cite{lcdt} applied to
the theory of stable pairs.
Our relative conditions $\lambda$ are defined
with identity weights in the $T$-equivariant cohomology
of $\mathbb{C}^2$. For the proof, we weight all the parts of
$\lambda$ with he $T$-equivariant class of the origin in
$\mathbb{C}^2$. Then, by compactness and dimension
constraints, the triangularity of the matrix is
immediate for partitions of different lengths.
On the diagonal, the expected dimension of
the integrals are 0. Using the compactification
\begin{equation}\label{jttm}
\mathbb{C}^2 \times \mathbb{P}^1 \subset \mathbb{P}^2 \times
\mathbb{P}^1
\end{equation}
as in Section 4.6 of \cite{lcdt}, we
obtain the triangularity of equal length partitions.
Consider the Hilbert scheme of points ${\text{Hilb}}(\com^2,d)$ of the
plane.
Let
${\mathbb{F}}$ be the universal quotient sheaf on
$$\text{Hilb}(\com^2,d) \times \com^2,$$ and define the
descendent{\footnote{The Chern character of $\mathbb{F}$
is properly supported over ${\text{Hilb}}(\com^2,d)$.} }
\begin{equation*}
\tau_k=\pi_*\Big( {\text{ch}}_{2+k}({\mathbb{F}})\Big)
\in A^{k}({\text{Hilb}}(\com^2,d),\mathbb{Q})
\end{equation*}
as before \eqref{mrr}.
Using the compactification \eqref{jttm}, we
reduce the calculation of the diagonal entries to
the pairing
\begin{equation}\label{appp}
s_1s_2\ \Big\langle \tau_{c-1} \ \Big| \ (c) \Big\rangle _
{{\text{Hilb}}(\com^2,d)}
= \frac{1}{c!}\
\end{equation}
which appears in \cite{parttwo}.
We conclude the diagonal entries do not vanish.
The diagonal entries are monomial in $q$ by the usual
vanishing obtained by the holomorphic symplectic form on $\mathbb{C}^2$.
\end{proof}
The denominator result holds for the
nonvanishing entries of the correspondence
matrix \eqref{gredd}. Since the matrix is triangular
with monomials in $q$ on the diagonal, the denominator
result holds for the {\em inverse} matrix.
We can now establish the denominator result for
the $T$-equivariant 3-point function
${\mathsf Z}^{(0,0)}_{d,\eta^1,\eta^2,\eta^3}$.
We start with the descendent series
\begin{equation}\label{jj87}
{\mathsf{Z}}^{0,0}_{d,\eta^3}\left(
\tau_{\mu_1-1}(\mathsf{p})
\cdots \tau_{\mu_{\ell(\mu)}-1}(\mathsf{p})\cdot \tau_{\widehat{\mu}_1-1}(\mathsf{p})
\cdots \tau_{\widehat{\mu}_{\ell(\widehat{\mu})}-1}(\mathsf{p})\right)
\end{equation}
for partitions $\mu$ and $\widehat{\mu}$ of $d$.
The denominator result holds for all
series \eqref{jj87}. By bubbling all the
descendents $\tau_{\mu_i-1}(\mathsf{p})$ off of the point $0\in \mathbb{P}^1$
and
bubbling all the
descendents $\tau_{\widehat{\mu}_i-1}(\mathsf{p})$ off of the point $1\in \mathbb{P}^1$, we conclude the
denominator result for ${\mathsf Z}^{(0,0)}_{d,\eta^1,\eta^2,\eta^3}$
from the denominator result for the
inverse of the correspondence matrix \eqref{gredd}.
\subsection{Denominators for Theorems \ref{tnnn}-\ref{cnnn}}
The denominator result for Theorem \ref{cnnn} is obtained by following
the proof given in Sections \ref{depp}-\ref{444}. An important point is
to replace the matrix $M_d$ appearing in the proof of
Lemma \ref{rtt5} with the correspondence matrix \eqref{gredd}. The required
matrix inversion then keeps the denominator form.
The rest of the proof of Theorem \ref{cnnn} respects the
denominators.
Proposition \ref{pnnn} is the statement of Theorem \ref{tnnn} for descendents
of even cohomology. Again, the proof respects the denominators.
The proof of Theorem \ref{2222} is complete.\qed
\noindent
Department of Mathematics\\
Princeton University\\
[email protected]
\noindent
Department of Mathematics\\
Princeton University\\
[email protected]
\end{document}
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\begin{document}
\title[Large-time asymptotics]{Large-time asymptotics for degenerate \\
cross-diffusion population models \\
with volume filling}
\author[X. Chen]{Xiuqing Chen}
\address{School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082,
Guang\-dong Province, China}
\email{[email protected]}
\author[A. J\"ungel]{Ansgar J\"ungel}
\address{Institute of Analysis and Scientific Computing, Technische Universit\"at Wien,
Wiedner Hauptstra\ss e 8--10, 1040 Wien, Austria}
\email{[email protected]}
\author[X. Lin]{Xi Lin}
\address{Department of Mathematics and Physics, Guangzhou Maritime University, Guangzhou 510765,
Guang\-dong Province, China}
\email{[email protected]}
\author[L. Liu]{Ling Liu}
\address{School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082,
Guang\-dong Province, China}
\email{[email protected]}
\date{\today}
\thanks{The first, third, and fourth authors acknowledge support from the National Natural
Science Foundation of China (NSFC), grant 11971072.
The second author acknowledges partial support from the Austrian Science Fund (FWF), grants P33010 and F65. This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, ERC Advanced Grant no.~101018153.}
\begin{abstract}
The large-time asymptotics of the solutions to a class of degenerate parabolic cross-diffusion systems is analyzed. The equations model the interaction of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. Compared to previous works, we allow for different diffusivities and degenerate nonlinearities. The proof is based on the relative entropy method, but in contrast to usual arguments, the relative entropy and entropy production are not directly related by a logarithmic Sobolev inequality. The key idea is to apply convex Sobolev inequalities to modified entropy densities including ``iterated'' degenerated functions.
\end{abstract}
\keywords{Degenerate parabolic equations, cross-diffusion systems, entropy method, large-time asymptotics.}
\subjclass[2000]{35K51, 35K59, 35K65, 35Q92, 92D25.}
\maketitle
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\section{Introduction}
The aim of this note is to extend the large-time asymptotics result of \cite{ZaJu17} on multi-species cross-diffusion systems with volume-filling effects to the degenerate case. Such systems describe, for instance, the spatial segregation of population species \cite{SKT79}, chemotactic cell migration in tissues \cite{Pai09}, motility of biological cells \cite{SLH09}, or ion transport in fluid mixtures \cite{BSW12}. The main difficulties of the cross-diffusion systems are the lack of positive semidefiniteness of the diffusion matrix and the nonstandard degeneracies. The first issue was overcome by applying the boundedness-by-entropy method \cite{Jue15}, which exploits the underlying entropy (or formal gradient-flow) structure. This allows for both a global existence analysis and the proof of lower and upper bounds, without the use of a maximum principle. The second issue was handled by extending the Aubin--Lions compactness lemma \cite{ZaJu17}. However, the large-time asymptotics in \cite{ZaJu17} only holds if the problem is not degenerate. In the present note, we remove this restriction.
The evolution of the volume fraction $u_i(x,t)$ of the $i$th species is given by
\begin{align}\label{1.eq}
& \pa_t u_i = \operatorname{div}\sum_{j=1}^n A_{ij}(u)\na u_j\quad\mbox{in }\Omega,\ t>0,\ i=1,\ldots,n \\
& \sum_{j=1}^n A_{ij}(u)\na u_j\cdot\nu = 0\quad\mbox{on }\pa\Omega, \quad
u_i(\cdot,0) = u_i^0\quad\mbox{in }\Omega, \label{1.bic}
\end{align}
where $u_0=1-\sum_{i=1}^n u_i$ is the solvent volume fraction or the proportion of unoccupied space (depending on the application), $\Omega\subset{\mathbb R}^d$ ($d\ge 1$) is a bounded domain with Lipschitz boundary, $\nu$ is the exterior unit normal vector to $\pa\Omega$, and the diffusion coefficients are given by
\begin{equation}\label{1.A}
A_{ij}(u) = D_ip_i(u)q(u_0)\delta_{ij} + D_iu_ip_i(u)q'(u_0)
+ D_iu_i q(u_0)\frac{\pa p_i}{\pa u_j}(u),
\end{equation}
where $i,j=1,\ldots,n$, $u=(u_1,\ldots,u_n)$ is the solution vector, $D_i>0$ are the diffusivities, $\delta_{ij}$ denotes the Kronecker symbol, and $p_i$ and $q$ are smooth functions. In particular, the bounds $0\le u_i\le 1$ should hold for all $i=0,\ldots,n$. The boundary condition in \eqref{1.bic} means that the physical or biological system is isolated.
We note that equations \eqref{1.eq} and \eqref{1.A} can be written as
\begin{align}\label{1.eq2}
\pa_t u_i = D_i\operatorname{div}\bigg(u_ip_i(u)q(u_0)\na\log\frac{u_ip_i(u)}{q(u_0)}\bigg)
= D_i\operatorname{div}\bigg(q(u_0)^2\na{\frac{u_ip_i(u)}{q(u_0)}}
\bigg).
\end{align}
In some applications, drift or reaction terms need to be added; see, e.g., \cite{BDPS10,GeJu18} for systems with drift terms and \cite{DJT20} for reaction rates.
Equations \eqref{1.eq} and \eqref{1.A} can be formally derived from a random-walk lattice model in the diffusion limit \cite[Appendix A]{ZaJu17}. The functions $p_i$ and $q$ are related to the transition rates of the lattice model with $p_i$ measuring the occupancy and $q$ measuring the non-occupancy. This class of systems contains the population model of Shigesada, Kawasaki, and Teramoto \cite{SKT79} (if $p_i$ is a linear function and $q=1$) and Nernst--Planck-type equations accounting for finite ion sizes (if $p_i=1$ and $q(u_0)=u_0$; see \cite{GeJu18}).
In this note, we consider the degenerate case $q'(0)=0$ and assume that there exists a smooth function $\chi$ such that $p_i=\exp(\pa\chi/\pa u_i)$ to guarantee an entropy structure via the entropy density
\begin{equation}\label{1.h}
h(u) = \sum_{i=1}^n (u_i(\log u_i-1)+1) + \int_1^{u_0}\log q(s)ds + \chi(u),
\end{equation}
where $u\in\mathcal{D}:=\{u\in(0,1)^n:\sum_{i=1}^n u_i<1\}$.
There exist other approaches to model volume filling. The finite particle size may be taken into account by adding cross-diffusion terms of the type $u_i\na\sum_{j=1}^n b_{ij}u_j$ to the standard Nernst--Planck flux \cite{Hsi19} or by using the Bikerman-type flux $J_i=-D_i(\na u_i-u_i\na\log u_0)$ in the mass conservation equation $\pa_t u_i+\operatorname{div} J_i=0$ \cite{Bik42}.
The global existence of bounded weak solutions to \eqref{1.eq}--\eqref{1.A} was shown in \cite[Theorem 1]{ZaJu17} assuming $D_i=1$ for $i=1,\ldots,n$ and the following conditions:
\begin{itemize}
\item[\bf (H1)] Domain: $\Omega\subset{\mathbb R}^d$ ($d\ge 1$) is a bounded convex domain with Lipschitz boundary, $T>0$. Set $\mathcal{D}=\{u\in(0,1)^n:\sum_{i=1}^n u_i<1\}$ and $\Omega_T=\Omega\times(0,T)$.
\item[\bf (H2)] Initial datum: $u^0(x)\in\mathcal{D}$ for a.e.\ $x\in\Omega$ and $h(u^0)\in L^1(\Omega)$.
\item[\bf (H3)] Functions $p_i$: $p_i=\exp(\pa\chi/\pa u_i)$, where $\chi\in C^3(\overline{\mathcal{D}})$ is convex.
\item[\bf (H4)] Function $q$: $q\in C^3([0,1])$ satisfies $q(0)=0$, $q(1)=1$, $q'(0)\ge 0$ and $q(s)>0$, $q'(s)>0$ for all $0<s\le 1$.
\end{itemize}
The convexity of $\Omega$ in Hypothesis (H1) is used for the convex Sobolev inequality; see Lemma \ref{lem.csi} below. For generalized Nernst--Planck systems with $p_i=\mbox{const.}$, we may choose $\chi(u)=\sum_{i=1}^n u_i$, which satisfies Hypothesis (H3). Moreover, if $p_i(u)=P_i(u_i)$ for some functions $P_i:[0,1]\to[0,\infty)$, condition $p_i=\exp(\pa\chi/\pa u_i)$ is satisfied with $\chi(u)=\sum_{i=1}^n\chi_i(u_i)$ and $\chi_i(s)=\int_0^s\log P_i(\tau)d\tau$. The functions $q(s)=s^\alpha$ with $\alpha\ge 1$ satisfy Hypothesis (H4).
We claim that the existence result also holds for arbitrary $D_i>0$. Indeed, it is sufficient to define $\widetilde\chi(u)=\chi(u)+\sum_{j=1}^n u_j\log D_j$, since $\exp(\pa\widetilde\chi/\pa u_i)=D_i\exp(\pa\chi/\pa u_i)=D_ip_i$, and we can apply Theorem 1 in \cite{ZaJu17} with $\widetilde\chi$. We observe that the condition $q'(s)/q(s)\ge c_1>0$ in \cite{ZaJu17} is not needed for the existence analysis.
The weak solution $u=(u_1,\ldots,u_n)$ to \eqref{1.eq}--\eqref{1.A} satisfies $u(x,t)\in\mathcal{D}$ for a.e.\ $(x,t)\in\Omega_T$, mass conservation, the regularity
\begin{align*}
& \sqrt{q(u_0)},\ \sqrt{q(u_0)u_i}\in L^2(0,T;H^1(\Omega)), \quad
\sqrt{q(u_0)}\na u_i\in L^2(\Omega_T), \\
& \pa_t u_i\in L^2(0,T;H^1(\Omega)') \quad\mbox{for }i=,1\ldots,n,
\end{align*}
and the weak formulation
\begin{align*}
\int_0^T\langle\pa_t u_i,\phi_i\rangle dt
= -\int_0^T\int_\Omega D_i \sqrt{q(u_0)}\big[\na\big(u_ip_i(u)\sqrt{q(u_0)}\big)
- 3u_ip_i(u)\na\sqrt{q(u_0)}\big]\cdot\na\phi_i dxdt
\end{align*}
for all $\phi_i\in L^2(0,T;H^1(\Omega))$, $i=1,\ldots,n$, where $\langle\cdot,\cdot\rangle$ denotes the duality product of $H^1(\Omega)'$ and $H^1(\Omega)$. Moreover, the initial datum in \eqref{1.bic} is satisfied in the sense of $H^1(\Omega)'$ and the entropy inequality
\begin{align}\label{1.ei}
\int_\Omega h(u(t))dx
+ c_0\int_s^t\int_\Omega\bigg(q(u_0)\sum_{i=1}^n|\na\sqrt{u_i}|^2
+ |\na\sqrt{q(u_0)}|^2\bigg)dxdr \le \int_\Omega h(u(s))dx,
\end{align}
holds for $0\le s<t$, $t>0$ for some $c_0>0$ depending on $D_i$, $p_i$, and $q$, recalling definition \eqref{1.h} of $h(u)$. The $L^\infty(\Omega_T)$ bound for $u_i$ and the $L^2(\Omega_T)$ for $\sqrt{q(u_0)}\na u_i$ imply that $\na(u_ip_i(u)\sqrt{q(u_0)})\in L^2(\Omega_T)$, so that the weak formulation is well defined.
Our main result is the convergence of the solutions to \eqref{1.eq}--\eqref{1.A} towards the constant steady state
$$
u_i^\infty = \frac{1}{|\Omega|}\int_\Omega u_i^0dx
\quad\mbox{for }i=1,\ldots,n,
\quad u_0^\infty= 1 - \sum_{i=1}^n u_i^\infty
$$
for large times under the following additional hypothesis:
\begin{itemize}
\item[\bf (H5)] $q$ is convex, $q/q'$ is concave, and there exist $\beta\in[0,1]$, $c_1>0$ such that
$$
\lim_{s\to 0} \frac{s^\beta q'(s)}{q(s)}=c_1>0.
$$
\end{itemize}
Examples of functions satisfying Hypothesis (H5) are $q(s)=s^\alpha$ with $\alpha\ge 1$.
The convergence (with exponential decay rate) was proved in \cite{ZaJu17} for the nondegenerate case $q'(0)>0$ only. In the degenerate situation $q'(0)=0$, the numerical results of \cite{GeJu18} indicate that exponential rates cannot be expected. Therefore, we show the convergence without rate.
\begin{theorem}[Large-time asymptotics]\label{thm.time}
Let Hypotheses (H1)--(H5) hold and let $u=(u_1,$ $\ldots,u_n)$ be a weak solution to \eqref{1.eq}--\eqref{1.A} satisfying the entropy inequality \eqref{1.ei}. Then $u_i(t)\to u_i^\infty$ strongly in $L^p(\Omega)$ as $t\to\infty$ for all $i=1,\ldots,n$ and $1\le p<\infty$.
\end{theorem}
The idea of the proof is to exploit, as in \cite{ZaJu17}, the relative entropy density (or Bregman distance)
\begin{equation}\label{1.hstar}
h^*(u|u^\infty) = h(u) - h(u^\infty) - h'(u^\infty)\cdot(u-u^\infty),
\end{equation}
where $u=(u_1,\ldots,u_n)$ is the weak solution to \eqref{1.eq}--\eqref{1.A}. The entropy inequality implies that
$$
\frac{dh^*}{dt}(u|u^\infty)
+ \frac{c_0}{2}\int_\Omega\sum_{i=1}^n|\na\sqrt{q(u_0)u_i}|^2 dx \le 0.
$$
Unfortunately, the entropy production integral cannot be estimated in terms of the relative entropy directly by applying a logarithmic Sobolev inequality to $u_i$. We overcome this issue by using two ideas.
First, we apply the logarithmic Sobolev inequality to $\sqrt{q(u_0)u_i}$,
$$
\int_\Omega q(u_0)u_i\log\frac{q(u_0)u_i}{|\Omega|^{-1}\int_\Omega q(u_0)u_idx}dx
\le C\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dx.
$$
The idea is to relate the integrand of the left-hand side to the relative entropy part
$h_1^*(u|u^\infty)=\sum_{i=1}^n(u_i\log(u_i/u_i^\infty)-u_i+u_i^\infty)dx$. For this, we define
$$
f_1(u) = \sum_{i=1}^n\bigg(q(u_0)u_i\log\frac{q(u_0)u_i}{|\Omega|^{-1}\int_\Omega q(u_0)u_idx}
- q(u_0)u_i + \frac{1}{|\Omega|}\int_\Omega q(u_0)u_idx\bigg).
$$
Since $\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dxdt<\infty$, we also have
$\int_0^\infty\int_\Omega f_1(u)dxdt<\infty$, and there exists a subsequence $t_k\to\infty$
such that $f_1(u(t_k))\to 0$. The key result is the limit (see Lemma \ref{lem.key})
$$
\lim_{t_k\to\infty}\bigg(\frac{f_1(u(t_k))}{|\Omega|^{-1}\int_\Omega q(u_0(t_k))dx}
- h_1^*(u(t_k)|u^\infty)\bigg) = 0.
$$
This result shows that $h_1^*(u(t_k)|u^\infty)\to 0$ as $t_k\to\infty$.
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Our main result is the convergence of the solutions to \eqref{1.eq}--\eqref{1.A} towards the constant steady state
$$
u_i^\infty = \frac{1}{|\Omega|}\int_\Omega u_i^0dx
\quad\mbox{for }i=1,\ldots,n,
\quad u_0^\infty= 1 - \sum_{i=1}^n u_i^\infty
$$
for large times under the following additional hypothesis:
\begin{itemize}
\item[\bf (H5)] $q$ is convex, $q/q'$ is concave, and there exist $\beta\in[0,1]$, $c_1>0$ such that
$$
\lim_{s\to 0} \frac{s^\beta q'(s)}{q(s)}=c_1>0.
$$
\end{itemize}
Examples of functions satisfying Hypothesis (H5) are $q(s)=s^\alpha$ with $\alpha\ge 1$.
The convergence (with exponential decay rate) was proved in \cite{ZaJu17} for the nondegenerate case $q'(0)>0$ only. In the degenerate situation $q'(0)=0$, the numerical results of \cite{GeJu18} indicate that exponential rates cannot be expected. Therefore, we show the convergence without rate.
\begin{theorem}[Large-time asymptotics]\label{thm.time}
Let Hypotheses (H1)--(H5) hold and let $u=(u_1,$ $\ldots,u_n)$ be a weak solution to \eqref{1.eq}--\eqref{1.A} satisfying the entropy inequality \eqref{1.ei}. Then $u_i(t)\to u_i^\infty$ strongly in $L^p(\Omega)$ as $t\to\infty$ for all $i=1,\ldots,n$ and $1\le p<\infty$.
\end{theorem}
The idea of the proof is to exploit, as in \cite{ZaJu17}, the relative entropy density (or Bregman distance)
\begin{equation}\label{1.hstar}
h^*(u|u^\infty) = h(u) - h(u^\infty) - h'(u^\infty)\cdot(u-u^\infty),
\end{equation}
where $u=(u_1,\ldots,u_n)$ is the weak solution to \eqref{1.eq}--\eqref{1.A}. The entropy inequality implies that
$$
\frac{dh^*}{dt}(u|u^\infty)
+ \frac{c_0}{2}\int_\Omega\sum_{i=1}^n|\na\sqrt{q(u_0)u_i}|^2 dx \le 0.
$$
Unfortunately, the entropy production integral cannot be estimated in terms of the relative entropy directly by applying a logarithmic Sobolev inequality to $u_i$. We overcome this issue by using two ideas.
First, we apply the logarithmic Sobolev inequality to $\sqrt{q(u_0)u_i}$,
$$
\int_\Omega q(u_0)u_i\log\frac{q(u_0)u_i}{|\Omega|^{-1}\int_\Omega q(u_0)u_idx}dx
\le C\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dx.
$$
The idea is to relate the integrand of the left-hand side to the relative entropy part
$h_1^*(u|u^\infty)=\sum_{i=1}^n(u_i\log(u_i/u_i^\infty)-u_i+u_i^\infty)dx$. For this, we define
$$
f_1(u) = \sum_{i=1}^n\bigg(q(u_0)u_i\log\frac{q(u_0)u_i}{|\Omega|^{-1}\int_\Omega q(u_0)u_idx}
- q(u_0)u_i + \frac{1}{|\Omega|}\int_\Omega q(u_0)u_idx\bigg).
$$
Since $\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dxdt<\infty$, we also have
$\int_0^\infty\int_\Omega f_1(u)dxdt<\infty$, and there exists a subsequence $t_k\to\infty$
such that $f_1(u(t_k))\to 0$. The key result is the limit (see Lemma \ref{lem.key})
$$
\lim_{t_k\to\infty}\bigg(\frac{f_1(u(t_k))}{|\Omega|^{-1}\int_\Omega q(u_0(t_k))dx}
- h_1^*(u(t_k)|u^\infty)\bigg) = 0.
$$
This result shows that $h_1^*(u(t_k)|u^\infty)\to 0$ as $t_k\to\infty$.
Second, instead of the part $h_2^*(u|u^\infty)=\int_{u_0^\infty}^{u_0}\log(q(s)/q(u_0^\infty))ds$ of the relative entropy density, we analyze the function
$$
f_2(u_0) = \int_{\bar{q}}^{q(u_0)}\log\frac{q(s)}{q(\bar{q})}ds,
$$
where $\bar{q}:=|\Omega|^{-1}\int_\Omega q(u_0)dx$, which can be seen as an ``iterated'' version of $h_2^*(u|u^\infty)$, since it involves $q\circ q$ instead of $q$. Then an application of the convex Sobolev inequality yields a bound for the integral over $|\na\sqrt{q(u_0)}|^2$ without the need of condition $q'(0)>0$; see Remark \ref{rem.f2} for details. It follows from $\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)}|^2 dxdt<\infty$ that $\int_0^\infty\int_\Omega f_2(u)dxdt<\infty$, and there exists a subsequence $t_k\to\infty$ such that $f_2(u(t_k))\to 0$.
The convergences $f_1(u(t_k))\to 0$ and $f_2(u(t_k))\to 0$ as well as the monotonicity of the entropy imply that $h^*(u(t_k)|u^\infty) \to 0$ pointwise. The monotonicity of $t\mapsto \int_\Omega h^*(u(t)|u^\infty)dx$ then implies the convergence for all sequences $t\to\infty$ and finally $u_i(t)\to u_i^\infty$ strongly in $L^2(\Omega)$.
To conclude the introduction, we mention some results on the large-time asymptotics for diffusion systems. Exponential equilibration rates in $L^p(\Omega)$ norms were shown for reaction-diffusion systems in \cite{DeFe06,DFM08}, for electro-reaction-diffusion systems in \cite{GlHu97}, and for Maxwell--Stefan systems for chemically reacting fluids in \cite{DJT20,JuSt13}. The convergence to equilibrium was proved for Shigesada--Kawasaki--Teramoto cross-diffusion systems without rate in \cite{Shi06}, for instance. All these results concern nondegenerate diffusion equations. The work \cite{BDPS10} is concerned with the large-time asymptotics for systems like \eqref{1.eq} with $D_i=p_i=1$ and $q(u_0)=u_0$ without rate.
The asymptotics for solutions to Poisson--Nernst--Planck-type equations with quadratic nonlinearity was investigated in \cite{Zin16} using Wasserstein techniques. Decay rates for degenerate diffusion systems without cross-diffusion terms were derived in \cite{CJMTU01}.
An extension of our results to cross-diffusion systems with drift or reactions seems delicate; see Remark \ref{rem.drift} for drift terms and \cite{DJT20} for cross-diffusion systems with reversible reactions.
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\section{Proof of Theorem \ref{thm.time}}
We first recall the convex Sobolev inequality; see \cite[Lemma 11]{ZaJu17}.
\begin{lemma}[Convex Sobolev inequality]\label{lem.csi}
Let $\Omega\subset{\mathbb R}^d$ ($d\ge 1$) be a convex domain and let $g\in C^4({\mathbb R})$ be convex such that $1/g''$ is concave. Then there exists $C_S>0$ such that for all $v\in L^1(\Omega)$ such that $g(v)$, $g''(v)|\na u|^2\in L^1(\Omega)$,
$$
\frac{1}{|\Omega|}\int_\Omega g(v)dx - g\bigg(\frac{1}{|\Omega|}\int_\Omega vdx\bigg)
\le \frac{C_S}{|\Omega|}\int_\Omega g''(v)|\na v|^2 dx.
$$
\end{lemma}
The logarithmic Sobolev inequality is obtained for the choice $g(v)=v(\log v-1)+1$:
\begin{equation}\label{3.lsi}
\int_\Omega v\log\frac{v}{|\Omega|^{-1}\int_\Omega vdx}dx
\le 4C_S\int_\Omega|\na\sqrt{v}|^2 dx
\end{equation}
and for functions $\sqrt{v}\in H^1(\Omega)$.
Since $h(u^\infty)$ is independent of time (because of mass conservation), the entropy inequality \eqref{1.ei} implies the relative entropy inequality
\begin{align}\label{3.ei}
\int_\Omega h^*(u(t)|u^\infty)dx &+ c_0\int_s^t\int_\Omega
\bigg(q(u_0)\sum_{i=1}^n|\na\sqrt{u_i}|^2 + |\na\sqrt{q(u_0)}|^2\bigg)dxdr \\
&\le \int_\Omega h^*(u(s)|u^\infty)dx \nonumber
\end{align}
for $0\le s<t$ and $t>0$, where $h^*(u|u^\infty)$ is defined in \eqref{1.hstar}. As mentioned in the introduction, we cannot apply the logarithmic Sobolev inequality \eqref{3.lsi} with $v=u_i$ since $q(u_0)=0$ for $u_0=0$. Instead we apply this inequality to $v=q(u_0)u_i$.
We split the relative entropy density $h^*$ into three parts, $h^*=h_1^* + h_2^* + h_3^*$, where
\begin{align*}
h_1^*(u|u^\infty) &= \sum_{i=1}^n\bigg(u_i\log\frac{u_i}{u_i^\infty}
- u_i + u_i^\infty\bigg), \\
h_2^*(u|u^\infty) &= \int_{u_0^\infty}^{u_0}\log\frac{q(s)}{q(u_0^\infty)}ds, \\
h_3^*(u|u^\infty) &= \chi(u) - \chi(u^\infty) - \sum_{i=1}^n(u_i-u_i^\infty)\log p_i(u^\infty),
\end{align*}
where $\chi$ is introduced in Hypothesis (H3).
\begin{lemma}
The functions $h_i^*(\cdot|u^\infty)$, $i=1,2,3$, are nonnegative and bounded on $\overline\mathcal{D}$.
\end{lemma}
\begin{proof}
The function $h_1^*$ is bounded since $u_i\mapsto u_i\log u_i$ is bounded for $0\le u_i\le 1$, and $h_3^*$ is bounded thanks to Hypothesis (H3) on $p_i$. Integrating by parts in $h_2^*(u|u^\infty)$ and observing that $u_0\log q(u_0)\le 0$, we find that
\begin{align}\label{3.aux}
h_2^*(u|u^\infty) = u_0\log\frac{q(u_0)}{q(u_0^\infty)} - \int_{u_0^\infty}^{u_0}
s\frac{q'(s)}{q(s)}ds \le -\log q(u_0^\infty) + \int_0^1 s\frac{q'(s)}{q(s)}ds.
\end{align}
By Hypothesis (H5), $\lim_{s\to 0} sq'(s)/q(s) = \lim_{s\to 0}s^{1-\beta}\cdot s^\beta q'(s)/q(s)$ is finite (here, we need $\beta\le 1$). Therefore, $s\mapsto sq'(s)/q(s)$ is bounded on $[0,\delta]$ for some $\delta>0$. On the other hand, $s\mapsto sq'(s)/q(s)$ is also bounded on $[\delta,1]$ since this function is continuous and $q(s)>0$ for $s>0$ is nondecreasing. This shows that $\int_0^1 (sq'(s)/q(s))ds$ is bounded, proving the claim.
\end{proof}
\subsection{Study of some auxiliary functions}
The study of the large-time behavior is based on the analysis of the two functions
\begin{align}\label{3.f}
f_1(u) = \sum_{i=1}^n\bigg(q(u_0)u_i\log\frac{q(u_0)u_i}{\bar{q}_i} - q(u_0)u_i
+ \bar{q}_i\bigg), \quad
f_2(u_0) = \int_{\bar{q}}^{q(u_0)}\log\frac{q(s)}{q(\bar{q})}ds,
\end{align}
for $u\in\overline{\mathcal{D}}$, where
\begin{equation}\label{3.qbar}
\bar{q} = \frac{1}{|\Omega|}\int_\Omega q(u_0)dx, \quad
\bar{q}_i = \frac{1}{|\Omega|}\int_\Omega q(u_0)u_idx.
\end{equation}
\begin{lemma}\label{lem.f}
The function $f_1$ is nonnegative, and the function $f_2$ is nonnegative and bounded on $\overline\mathcal{D}$.
\end{lemma}
\begin{proof}
Set $z=q(u_0)u_i/\bar{q}_i$ and let $u\in\overline{\mathcal{D}}$. Then
$$
f_1(u) = \sum_{i=1}^n\bar{q}_i(z\log z - z+1)\ge 0,
$$
proving the first claim. To show the nonnegativity of $f_2$, we distinguish two cases. If $q(u_0(x,t))\ge\bar{q}$ at some $(x,t)\in\Omega_T$, then $\log(q(s)/q(\bar{q}))\ge 0$ for any $\bar{q}\le s\le q(u_0(x,t))$ and consequently $f_2(u(x,t))\ge 0$. If $q(u_0(x,t))<\bar{q}$, we have
$\log(q(s)/q(\bar{q})) < 0$ for $q(u_0(x,t))\le s\le\bar{q}$ and
$f_2(u_0(x,t)) = \int_{q(u_0(x,t))}^{\bar{q}}\log(q(\bar{q})/q(s))ds \ge 0$.
It remains to show that $f_2$ is bounded. Since $q$ is convex, Jensen's inequality shows that $\bar{q}\ge q(|\Omega|^{-1}\int_\Omega u_0dx)=q(u_0^\infty)$.
Then, using integration by parts and arguing as in \eqref{3.aux},
\begin{align*}
f_2(u_0) &= q(u_0)\log\frac{q(q(u_0))}{q(\bar{q})}
- \int_{\bar{q}}^{q(u_0)} s\frac{q'(s)}{q(s)}ds
\le -q(u_0)\log q(\bar{q}) + \int_0^1 s\frac{q'(s)}{q(s)}ds \\
&\le -\log q(q(u_0^\infty)) + \int_0^1 s\frac{q'(s)}{q(s)}ds.
\end{align*}
We already showed above that the last integral is bounded. This finishes the proof.
\end{proof}
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\subsection{Convergence of $f_1$ and $f_2$}
\begin{lemma}\label{lem.convf}
It holds for a.e.\ $x\in\Omega$, $s\in(0,1]$ that
$$
\lim_{N\to\infty}f_1(u(x,s+N)) = 0, \quad \lim_{N\to\infty}f_2(u_0(x,s+N)) = 0.
$$
\end{lemma}
\begin{proof}
The idea is to exploit the boundedness of the entropy production integrated over $t\in(0,\infty)$. First, we consider $f_1$. We know from \eqref{3.ei} for $s=0$ and $t\to\infty$ that
\begin{equation}\label{3.infty}
c_0\int_0^\infty\int_\Omega\bigg(q(u_0)\sum_{i=1}^n|\na\sqrt{u_i}|^2
+ |\na\sqrt{q(u_0)}|^2\bigg)dxdt \le \int_\Omega h^*(u^0|u^\infty)dx.
\end{equation}
Thus, in view of $q(u_0)u_i\ge 0$ and
\begin{align*}
|\na\sqrt{q(u_0)u_i}|^2 &= q(u_0)|\na\sqrt{u_i}|^2
+ 2\sqrt{q(u_0)u_i}\na\sqrt{q(u_0)}\cdot\na\sqrt{u_i} + u_i|\na\sqrt{q(u_0)}|^2 \\
&\le 2q(u_0)|\na\sqrt{u_i}|^2 + 2|\na\sqrt{q(u_0)}|^2,
\end{align*}
it follows for a constant $C>0$ being independent of time that
\begin{equation*}
\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dxdt \le C.
\end{equation*}
Furthermore, by the logarithmic Sobolev inequality \eqref{3.lsi}, applied to $v=q(u_0)u_i$,
$$
\int_0^\infty\int_\Omega q(u_0)u_i\log\frac{q(u_0)u_i}{\bar{q}_i}dx
\le C\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dx \le C,
$$
recalling definition \eqref{3.qbar} of $\bar{q}_i$. Taking into account definition \eqref{3.f} of $f_1$, we see that
$$
\int_0^\infty\int_\Omega f_1(u(x,t))dxds
= \sum_{N=0}^{\infty}\int_0^1\int_\Omega f_1(u(x,s+N))dx ds < \infty.
$$
Therefore, the sequence $N\mapsto \int_0^1\int_\Omega f_1(u(\cdot,s+N))dx ds$ converges to zero,
$$
\lim_{N\to\infty}f_1(u(x,s+N)) = 0\quad\mbox{for a.e. }x\in\Omega,\ s\in(0,1].
$$
Next, we prove the limit for $f_2$. For any fixed $t>0$, we introduce the nonnegative function
$$
f(s;t) = \int_{\bar{q}(t)}^s\log\frac{q(\sigma)}{q(\bar{q}(t))}d\sigma, \quad 0<s\le 1.
$$
By Lemma \ref{lem.f}, $x\mapsto f(q(u_0(x,t));t) = f_2(u(x,t))$ is integrable in $\Omega$
for any fixed $t>0$. Moreover, $f(\cdot,t)$ is twice differentiable in $(0,1)$:
$$
\frac{df}{ds}(s;t) = \log\frac{q(s)}{q(\bar{q}(t))}, \quad
\frac{d^2 f}{ds^2}(s;t) = \frac{q'(s)}{q(s)} > 0.
$$
We infer from the positivity of $d^2f/ds^2$ that $f(\cdot,t)$ is convex. By Hypothesis (H5), $(d^2f/ds^2)^{-1} = q/q'$ is concave. Thus, the assumptions of the convex Sobolev inequality (Lemma \ref{lem.csi}) are satisfied for $f(q(u_0(x,t));t)$:
\begin{align*}
\frac{1}{|\Omega|}\int_\Omega & f(q(u_0(x,t));t)dx
- f\bigg(\frac{1}{|\Omega|}\int_\Omega q(u_0(x,t))dx;t\bigg) \\
&\le C(\Omega)\int_\Omega \frac{q'(q(u_0(x,t)))}{q(q(u_0(x,t)))}|\na q(u_0)|^2 dx.
\end{align*}
Hence, since $f(\bar{q}(t);t)=0$ by definition and recalling that $f(q(u_0(x,t));t) = f_2(u_0(x,t))$, the previous inequality becomes
\begin{equation}\label{2.f2est}
\int_\Omega f_2(u_0)dx
\le C(\Omega)\int_\Omega\frac{q(u_0)q'(q(u_0))}{q(q(u_0))}\frac{|\na q(u_0)|^2}{q(u_0)}dx
\le C\int_\Omega|\na\sqrt{q(u_0)}|^2 dx,
\end{equation}
where we used Hypothesis (H5) to infer that
$$
\frac{s q'(s)}{q(s)} = s^{1-\beta}\frac{s^\beta q'(s)}{q(s)}\quad\mbox{with }
s = q(u_0)
$$
is bounded in $[0,1]$. By \eqref{3.infty}, the integrated entropy dissipation is finite:
$$
\int_0^\infty\int_\Omega f_2(u_0)dxdt \le C\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)}|^2
dxdt \le C.
$$
Therefore, arguing as for the function $f_1$, we obtain
$\lim_{N\to\infty}f_2(u_0(x,s+N)) = 0$ for a.e.\ $x\in\Omega$, $s\in(0,1]$,
which finishes the proof.
\end{proof}
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\begin{remark}\label{rem.f2}\rm
In the nondegenerate case $q'(0)>0$, it was shown in \cite[Section 5]{ZaJu17} that $t\mapsto
h_2^*(u(t)|u^\infty)$ converges to zero exponentially fast. Indeed, applying the convex Sobolev inequality similarly as in the previous proof,
\begin{equation}\label{2.h2est}
\int_\Omega h_2^*(u|u^\infty)dx \le C\int_\Omega\frac{q'(u_0)}{q(u_0)}|\na u_0|^2 dx
= 4C\int_\Omega\frac{|\na\sqrt{q(u_0)}|^2}{q'(u_0)} dx,
\end{equation}
and we conclude from the entropy inequality \eqref{1.ei} and Gronwall's lemma. Since we allow for $q'(0)=0$, this argument cannot be used here. We solve this issue by considering the ``iterated'' function $f_2$ involving $q\circ q$ and assuming that $s\mapsto sq'(s)/q(s)$ is bounded; see \eqref{2.f2est}. The iterated use of $q$ gives the term $|\na\sqrt{q(u_0)}|^2$ in \eqref{2.f2est} without requiring the nondegeneracy condition $q'(0)>0$.
\qed\end{remark}
A consequence of the limit for $f_2$ is the following result.
\begin{lemma}\label{lem.conv1}
If $\lim_{N\to\infty}f_2(u_0(x,s+N))=0$ for some $x\in\Omega$, $s\in(0,1]$ then
$$
\lim_{N\to\infty}\frac{q(u_0(x,s+N))}{\bar{q}(s+N)} = 1.
$$
\end{lemma}
\begin{proof}
We write $u_i^N:=u_i(x,s+N)$ and $\bar{q}^N=\bar{q}(s+N)$
to simplify the notation. We recall from Lemma \ref{lem.f} that $f_2$ is nonnegative and change the variable $\sigma=s/\bar{q}^N$ in the integral:
\begin{align*}
f_2(u_0^N) &= \int_{\bar{q}^N}^{q(u_0^N)}\log\frac{q(s)}{q(\bar{q}^N)}ds
= \bar{q}^N\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma \\
&\ge q(u_0^\infty)\int_1^{q(u_0^N)/\bar{q}^N}
\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma,
\end{align*}
where we used Jensen's inequality to find that $\bar{q}^N\ge q(|\Omega|^{-1}\int_\Omega u_0^N dx) = q(u_0^\infty)$. Moreover, since $\bar{q}^N\le 1$,
$$
q(u_0^\infty)\int_1^{q(u_0^N)/\bar{q}^N}
\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma
\le f_2(u_0^N) \le \int_1^{q(u_0^N)/\bar{q}^N}
\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma.
$$
This shows that $\lim_{N\to\infty}f_2(u_0^N)=0$ if and only if
\begin{equation}\label{3.con}
\lim_{N\to\infty}\int_1^{q(u_0^N)/\bar{q}^N}
\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma = 0.
\end{equation}
Set $A:=\{(x,s)\in\Omega\times(0,1]:\lim_{N\to\infty} f_2(u_0(x,s+N))=0\}$. We want to show that $\lim_{N\to\infty} q(u_0^N)/\bar{q}^N=1$ for $(x,s)\in A$. If not, there exist $(x_0,s_0)\in A$ and $\eps_0>0$ such that either
$$
\frac{q(u_0^N)}{\bar{q}^N} > 1+\eps_0 \quad\mbox{or}\quad
\frac{q(u_0^N)}{\bar{q}^N} < 1-\eps_0\quad\mbox{for all }N\in{\mathbb N}.
$$
In the former case, we have $q(\bar{q}^N\sigma)\ge q(\bar{q}^N(1+\eps_0/2))$
for $\sigma\ge 1+\eps_0/2$, since $q$ is increasing, and therefore,
\begin{equation}\label{3.aux2}
\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma
\ge \int_{1+\eps_0/2}^{1+\eps_0}\log\frac{q(\bar{q}^N(1+\eps_0/2))}{q(\bar{q}^N)}d\sigma.
\end{equation}
Using the convexity of $q$, a Taylor expansion shows that
$q(\bar{q}^N + \bar{q}^N\eps_0/2) \ge q(\bar{q}^N) + q'(\bar{q}^N)\bar{q}^N\eps_0/2$.
Then the integrand of the previous integral can be estimated according to
$$
\log\bigg(\frac{q(\bar{q}^N(1+\eps_0/2))}{q(\bar{q}^N)}\bigg)
\ge \log\bigg(1 + \frac{q'(\bar{q}^N)}{q(\bar{q}^N)}\bar{q}^N\frac{\eps_0}{2}\bigg)
\ge \log\bigg(1 + c_0q(u_0^\infty)^{1-\beta}\frac{\eps_0}{2}\bigg),
$$
where we used Hypothesis (H5) and $\bar{q}^N\ge q(u_0^\infty)$ in the last step, and $c_0>0$ is some constant. As the right-hand side is independent of $\sigma$, we infer from \eqref{3.aux2} that
$$
\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma
\ge \frac{\eps_0}{2}\log\bigg(1 + c_0q(u_0^\infty)^{1-\beta}\frac{\eps_0}{2}\bigg).
$$
In the latter case $q(u_0^N)/\bar{q}^N<1-\eps_0$, we estimate as
\begin{align*}
\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma
&= \int_{q(u_0^N)/\bar{q}^N}^1\log\frac{q(\bar{q}^N)}{q(\bar{q}^N\sigma)}d\sigma \\
&\ge \int_{1-\eps_0}^{1-\eps_0/2}\log\frac{q(\bar{q}^N)}{q(\bar{q}^N(1-\eps_0/2)}d\sigma.
\end{align*}
We apply again a Taylor expansion, similarly as in the first case,
$$
q(\bar{q}^N) = q\bigg(\bar{q}^N\bigg(1-\frac{\eps_0}{2}\bigg)
+ \frac{\eps_0}{2}\bar{q}^N\bigg)
\ge q\bigg(\bar{q}^N\bigg(1-\frac{\eps_0}{2}\bigg)\bigg)
+ q'\bigg(\bar{q}^N\bigg(1-\frac{\eps_0}{2}\bigg)\bigg)\frac{\eps_0}{2}\bar{q}^N,
$$
which leads to
$$
\log\frac{q(\bar{q}^N)}{q(\bar{q}^N(1-\eps_0/2)}
\ge \log\bigg(1 + \frac{q'(\bar{q}^N(1-\eps_0/2))}{q(\bar{q}^N(1-\eps_0/2))}
\frac{\eps_0}{2}\bar{q}^N\bigg)
\ge \log\bigg(1 + c_0q(u_0^\infty)^{1-\beta}\frac{\eps_0}{2}\bigg).
$$
Thus, in both cases,
$$
\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma > 0
\quad\mbox{uniformly in }N\in{\mathbb N},
$$
which contradicts \eqref{3.con} and consequently $\lim_{N\to\infty}f_2(u_0^N)=0$.
\end{proof}
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\subsection{Key lemma}
We show that $f_1(u(\cdot,s+N))/\bar{q}(s+N)$ and $h_1^*(u(\cdot,s+N)|u^\infty)$ are close for sufficiently large $N\in{\mathbb N}$. The following lemma is the key of the proof.
\begin{lemma}\label{lem.key}
For a.e.\ $x\in\Omega$, $s\in(0,1]$, it holds that
$$
\lim_{N\to\infty}\bigg(\frac{f_1(u(x,s+N))}{\bar{q}(s+N)}
- h_1^*(u(x,s+N)|u^\infty)\bigg) = 0.
$$
\end{lemma}
\begin{proof}
We set $u^N:=u(\cdot,s+N)$, $\bar{q}^N=\bar{q}(s+N)$, and $\bar{q}_i^N=|\Omega|^{-1}\int_\Omega q(u_0^N)u_i^Ndx$. Inserting definition \eqref{3.f} of $f_1$, the lemma is proved if we can show that for any $i=1,\ldots,n$,
\begin{align}\label{3.aux3}
0 &= \lim_{N\to\infty}\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{q(u_0^N)u_i^N}{\bar{q}_i^N}
- \frac{q(u_0^N)}{\bar{q}^N}u_i^N + \frac{\bar{q}_i^N}{\bar{q}^N}
- u_i^N\log\frac{u_i^N}{u_i^\infty}
+ u_i^N - u_i^\infty\bigg) \\
&= \lim_{N\to\infty}\bigg\{\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{q(u_0^N)u_i^N}{\bar{q}_i^N} - u_i^N\log\frac{u_i^N}{u_i^\infty}\bigg)
- \bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N - u_i^N\bigg) \nonumber \\
&\phantom{xx}+ \bigg(\frac{\bar{q}_i^N}{\bar{q}^N} - u_i^\infty\bigg)\bigg\}. \nonumber
\end{align}
Fix $i\in\{1,\ldots,n\}$. We know from Lemmas \ref{lem.convf} and \ref{lem.conv1} that
$\lim_{N\to\infty} q(u_0^N)/\bar{q}^N=1$ a.e. Together with the boundedness of $u_i^N$,
this shows that
$$
\lim_{N\to\infty}\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N - u_i^N\bigg) = 0
$$
as well as
\begin{align*}
\lim_{N\to\infty}&\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N\log\frac{q(u_0^N)u_i^N}{\bar{q}_i^N}
- u_i^N\log\frac{u_i^N}{u_i^\infty}\bigg) \\
&= \lim_{N\to\infty}\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{(q(u_0^N)/\bar{q}^N)u_i^N}{\bar{q}_i^N/\bar{q}^N}
- u_i^N\log\frac{u_i^N}{u_i^\infty}\bigg) \\
&= \lim_{N\to\infty}\bigg\{\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{q(u_0^N)}{\bar{q}^N}
+ \bigg(\frac{q(u_0^N)}{\bar{q}^N}-1\bigg)u_i^N\log\frac{u_i^N}{u_i^\infty} \\
&\phantom{xx}{}
- \frac{q(u_0^N)}{\bar{q}^N}u_i^N\log\frac{\bar{q}_i^N/\bar{q}^N}{u_i^\infty}\bigg\}
= - \lim_{N\to\infty}\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{\bar{q}_i^N/\bar{q}^N}{u_i^\infty}.
\end{align*}
To show that the limit on the right-hand side equals zero, we observe that, because of mass conservation and dominated convergence,
\begin{align*}
&\lim_{N\to\infty}\bigg(\frac{\bar{q}_i^N}{\bar{q}^N} - u_i^\infty\bigg)
= \lim_{N\to\infty}\bigg(\frac{1}{|\Omega|}\int_\Omega\frac{q(u_0^N)}{\bar{q}^N}u_i^N dx
- u_i^\infty\bigg) \\
&\phantom{x}
= \lim_{N\to\infty}\bigg(\frac{1}{|\Omega|}\int_\Omega\frac{q(u_0^N)}{\bar{q}^N}u_i^N dx
- \frac{1}{|\Omega|}\int_\Omega u_i^0 dx\bigg)
= \lim_{N\to\infty}\frac{1}{|\Omega|}\int_\Omega\bigg(\frac{q(u_0^N)}{\bar{q}^N}-1\bigg)
u_i^N dx = 0,
\end{align*}
and this is equivalent to $\lim_{N\to\infty}\log((\bar{q}_i^N/\bar{q}^N)/u_i^\infty)=0$. We conclude that
$$
\lim_{N\to\infty}\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N\log\frac{q(u_0^N)u_i^N}{\bar{q}_i^N}
- u_i^N\log\frac{u_i^N}{u_i^\infty}\bigg) = 0.
$$
Putting together the previous limits, we have proved \eqref{3.aux3}.
\end{proof}
\subsection{Convergence of $h^*$}
We conclude from Lemmas \ref{lem.convf} and \ref{lem.key} that
$\lim_{N\to\infty}h_1^*(u^N|u^\infty) = 0$. We claim that also $h_2^*$ and $h_3^*$ converge to zero as $N\to\infty$. Since $u_i^N$ and $u_i^\infty$ are bounded in $[0,1]$, we have the estimate
\cite[Lemma 16]{HJT22}
$$
\frac12\sum_{i=1}^n(u_i^N-u_i^\infty)^2
\le \sum_{i=1}^n\bigg(u_i^N\log\frac{u_i^N}{u_i^\infty} - (u_i^N-u_i^\infty)\bigg)
= h_1^*(u^N|u^\infty)\to 0,
$$
showing that $u_i^N\to u_i^\infty$ a.e.\ in $\Omega\times(0,1]$ as $N\to\infty$ for $i=1,\ldots,n$. We deduce from the continuity of $\chi$ that also $\lim_{N\to\infty}h_3^*(u^N|u^\infty)=0$.
For the limit of $h_2^*$, we observe that $u_0^N=1-\sum_{i=1}^n u_i^N\to u_0^\infty$ a.e. Hence, for any fixed $(x,s)\in\Omega\times(0,1]$, there exists $N_0\in{\mathbb N}$ such that
$1/2\le u_0(x,s+N)/u_0^\infty\le 3/2$ for $N>N_0$. Next, we write $h_2^*$ as
$$
h_2^*(u^N|u^\infty) = \int_{u_0^\infty}^{u_0^N}\log\frac{q(s)}{q(u_0^\infty)}ds
= u_0^\infty\int_1^{u_0^N/u_0^\infty}\log\frac{q(u_0^\infty\sigma)}{q(u_0^\infty)}d\sigma.
$$
Since the integrand is a function in $L^1(1/2,3/2)$, it follows from the absolute continuity of the integral that $\lim_{N\to\infty}h_2^*(u^N|u^\infty)=0$ a.e.\ in $\Omega\times(0,1]$. By definition of $h^*$, we have proved that $\lim_{N\to\infty}h^*(u^N|u^\infty)=0$.
\subsection{Convergence in $L^p(\Omega)$}
We deduce from the relative entropy inequality \eqref{3.ei} that $t\mapsto \int_\Omega h^*(u(t)|u^\infty)dx$ is bounded and nonincreasing. Then it follows from the limit $\lim_{N\to\infty}h^*(u^N|u^\infty)=0$ that in fact we have the convergence for all sequences $t\to\infty$,
$\lim_{t\to \infty}\int_\Omega h^*(u(t)|u^\infty)dx=0$ and in particular, since $h_2^*\ge 0$ and $h_3^*\ge 0$,
$$
\lim_{t\to \infty}\int_\Omega h_1^*(u(t)|u^\infty)dx=0.
$$
Using \cite[Lemma 16]{HJT22} again, we have
$$
\lim_{N\to\infty}\frac12\sum_{i=1}^n\int_\Omega(u_i(t)-u_i^\infty)^2 dx
\le \lim_{N\to\infty}\int_\Omega h_1^*(u(t)|u^\infty)dx=0.
$$
The convergence in $L^p(\Omega)$ for any $p<\infty$ then follows from the uniform bound
for $(u_i(t))_{t>0}$, finishing the proof.
\begin{remark}[Drift terms]\label{rem.drift}\rm
Equations \eqref{1.eq2} with drift terms read as
$$
\pa_t u_i = D_i\operatorname{div}\bigg\{u_ip_i(u)q(u_0)\na\bigg(\log\frac{u_ip_i(u)}{q(u_0)}
+ \Phi_i\bigg)\bigg\}, \quad i=1,\ldots,n,
$$
where $\Phi_i=\Phi_i(x)$ are given (electric or environmental) potentials. Adding the associated energy to the entropy density \eqref{1.h},
$$
h_2(u) = \sum_{i=1}^n(u_i(\log u_i-1)+1) + \int_1^{u_0}\log q(s)ds + \chi(u)
+ \sum_{i=1}^n u_i\Phi_i,
$$
we can compute (formally) the entropy inequality, giving
$$
\frac{d}{dt}\int_\Omega h_2(u)dx + \int_\Omega\sum_{i=1}^n D_iu_ip_i(u)q(u_0)
\bigg|\na\bigg(\log\frac{u_ip_i(u)}{q(u_0)} + \Phi_i\bigg)\bigg|^2 dx = 0.
$$
It was shown in \cite[Section 3.2]{ZaJu17} that the entropy production term with $\Phi_i=0$ can be bounded from below by $p_i(u)(q(u_0)\sum_{i=1}^n|\na\sqrt{u_i}|^2+|\na\sqrt{q(u_0)}|^2)$. Such an estimate seems to be impossible in the presence of $\na\Phi_i$. Indeed, the entropy inequality shows that
\begin{align*}
4\int_0^\infty&\int_\Omega q(u_0)^2e^{-\Phi_i}
\bigg|\na\bigg(\frac{u_ip_i(u)e^{\Phi_i}}{q(u_0)}\bigg)^{1/2}
\bigg|^2 dx \\
&= \int_0^\infty\int_\Omega u_ip_i(u)q(u_0)\bigg|\na\bigg(\log\frac{u_ip_i(u)}{q(u_0)}
+ \Phi_i\bigg)\bigg|^2< \infty.
\end{align*}
Thus, in the special case $q(0)>0$ and if $\Phi_i$ is bounded from above,
we conclude the existence of a subsequence $t_k\to \infty$ such that $\na(u_ip_i(u)e^{\Phi_i}/q(u_0))^{1/2}(t_k)\to 0$ strongly in $L^2(\Omega)$ as $k\to\infty$, and one may proceed similarly as in \cite[Section 5]{BFS14}. However, the condition $q(0)=0$ is needed to model correctly the transition rate of nonoccupied cells in the lattice model \cite{BDPS10,ZaJu17}.
\qed\end{remark}
\end{document}
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\begin{document}
\author{M. I. Katsnelson\cite{mik}, V. V. Dobrovitski, and
B. N. Harmon}
\address{Ames Laboratory, Iowa State University, Ames, Iowa, 50011}
\title{Propagation of local decohering action in distributed quantum
systems}
\date{today}
\maketitle
\draft
\begin{abstract}
We study propagation of the decohering influence caused by a local
measurement performed on a distributed quantum system. As an
example, the gas of bosons forming a Bose-Einstein condensate is
considered. We demonstrate that the local decohering perturbation
exerted on the measured region propagates over the system in the
form of a decoherence wave, whose dynamics is governed by
elementary excitations of the system. We argue that the
post-measurement evolution of the system (determined by
elementary excitations) is of importance for transfer of
decoherence, while the initial collapse of the wave function
has negligible impact on the regions which are not directly
affected by the measurement.
\end{abstract}
\pacs{03.65.Bz, 05.30.Jp, 03.75.Fi}
\section*{Introduction}
The theory of quantum measurement begun
in the 1920s still remains an active topic of interest (see, e.g.
Ref.\ \onlinecite{meas1} and references therein). According to
von Neumann's theory of measurement \cite{neumann}, unitary
evolution of a system prepared initially in a pure quantum state
is interrupted by an instant decohering action of the measuring
apparatus, so that the density matrix describing an ensemble of
such systems changes radically (it ceases to be a projection
operator) and entropy rises. This view has been shown to describe
rather accurately the consequences of an act of measurement, but
the dynamics of the measurement process itself is lacking.
The contemporary theory of quantum measurements, which provides much
deeper analysis of the measurement process, is based on the
concept of decoherence \cite{meas}. To be measured, the system
has to interact with its environment, which consists of a large
number of degrees of freedom. The Hilbert space of the system
becomes divided into subspaces corresponding to the same
eigenvalue of the system-environment interaction Hamiltonian. As
a result of this interaction, coherence between different
subspaces is quickly lost, and after the measurement the system
appears in a mixed state. The concept of decoherence turned out
to be successful in many areas of fundamental physics, such as
the study of macroscopic quantum effects \cite{leg} and
consistent histories interpretation of quantum mechanics
\cite{omnes}, so that investigation of this process and related
effects is of considerable importance.
At present, decoherence and its consequences for point-like
quantum systems have been studied in detail (for review, see
Ref.\ \onlinecite{zurnew}), but distributed quantum systems
have received
significantly less attention. Mostly, linear systems have been
investigated, where separation into noninteracting modes is
possible, and each mode is considered as an independent
oscillator \cite{zurfield}. However, this approach is difficult to
apply to sufficiently
nonlinear systems (e.g., spin systems, or the Bose-Einstein
condensate as described by the Gross-Pitaevskii equation) possessing
localized soliton-like excitations. For systems where localized
excitations prevail, dealing explicitely with
real-space coordinates could be a more suitable strategy.
A real-space description of decoherence in distributed systems is a
very general and complicated issue. In this paper, we consider
only one aspect of the problem, namely, how {\it local\/}
properties of different regions in a distributed quantum system
are affected by a {\it local\/} measurement, that acts only on
some part of the system. Indeed, different regions in the system
are not isolated from each other, and correlations between them
exist (or can build up). Therefore, in spite of the fact that a
local measurement initially affects only one region, other regions
can ``acquire knowledge'' that some part of the system has been
measured. In this paper we explicitely show that the decohering
influence of the local measurement propagates through the system
in the form of a decoherence wave. Dynamics of the decoherence
wave is governed by elementary excitations, while the effect of
entanglement is very small for macroscopically large systems.
The consideration presented here can be applied to other similar
situations, so that a decoherence wave propagating with a
characteristic velocity of excitations is likely to be quite
common. This phenomenon, being a notable part of any real
measurement, is of fundamental interest. Moreover, propagation of
decoherence can be also of importance for the design of quantum
computers. Such a computer is a system of interacting quantum
entities, representing quantum bits (qubits). Fault-tolerant
quantum computations involve measurements performed on some
qubits and it is important to know how such measurement may
affect other qubits \cite{comput}. Moreover, decoherence is
introduced by a dissipative environment of qubits, so that
analysis of decoherence propagation may lead to strategies to
minimize influences detrimental to performance of the computer.
In this paper we consider a Bose-Einstein condensate of an ideal or
weakly non-ideal gas of bosons, which constitutes a good example
of a distributed system in a pure quantum state. It can be
implemented in reality as a gas of trapped atoms cooled down to
very low temperatures \cite{bose}. Suppose we measure the number
of particles in some region of space. If two such measurements are
done {\it simultaneously\/} at two different parts of the trap we
obtain the trivial result corresponding to the ground-state
wavefunction of the condensate. But if the second measurement is
carried out after some delay then the result is different and
provides information about the propagation of the perturbation
induced by the first measurement.
The situation considered here is related to the problem of broken
gauge symmetry and existence of a relative phase of two
interfering condensates \cite{phase}, which has been extensively
discussed recently. If we have a condensate with a definite
number of particles, its phase is spread uniformly between 0 and
$2\pi$, while a definite phase requires a non-conservation of the
number of particles in the condensate. It has been shown that a
well-defined phase (evidenced experimentally by appearance of the
interference fringes) builds up in the course of the measurement
(atoms detection), due to increasing uncertainty in the number of
particles in each of the interfering condensates: each detected
atom may well belong to either of them. For the circumstances
considered in this paper, we have a similar situation: the local
phase of the condensate is the same in every region. Identity of
the phase throughout the condensate is due to uncertainty in the
local number of the particles inside each region. However, when
the number of particles in some region is determined by a local
measurement, the phase coherence in the condensate as a whole is
partially destroyed, what leads to observable consequences,
propagation of the decoherence wave in the system. Note that
decoherence wave is the same both for the condensate with
definite number of particles (with uncertain global phase, the
case of non-interacting bosons) and for the condensate with
definite global phase (but with uncertain number of condensed
particles, the case of weakly interacting bosons): the results
for the latter case transform exactly to the results for the
former as interaction goes to zero.
We describe the dynamics of the condensate in a linear approximation,
i.e. we use the approximation of noninteracting quasiparticles to
study a weakly non-ideal Bose-gas. In so doing, we loose the
ability to investigate some interesting nonlinear effects, but
we gain in clarity of presentation: it is reasonable to
start from a simplified (and not totally unrealistic) case to
emphasize the main idea.
We do not specify the way of measuring the local density of
condensate, and the dynamics of the measurement process is not
considered here. Analysis of a specific experimental scheme is
a distinct problem, requiring separate study, while here we
focus on the post-measurement evolution of the condensate. In
principle, the local density of the Bose-condensate can be measured by
placing some probe into the trap, which interacts with the
condensate so that an entangled state is formed
\begin{equation}
|X\rangle = \sum C_n |n\rangle \otimes |\alpha_n\rangle
\end{equation}
where $|n\rangle$ is the state of condensate with the number of
particles $n$ in the measured region, and $|\alpha_n\rangle$ is
the state of the probe. If the probe interacts with a large
number of environmental degrees of freedom, so that
$|\alpha_n\rangle$ are the eigenstates corresponding to different
eigenvalues of the probe-environment interaction Hamiltonian,
then the coherence between different probe states is being lost,
and the condensate's state also becomes an incoherent mixture of
different states $|n\rangle$. If the probe (and, correspondingly,
the condensate) decoheres quickly enough (as is usually the case) we
can consider the measurement as instantaneous and safely use von
Neumann's theory to describe the condensate's state immediately
after the measurement.
Although the situation considered above is in many
respects too idealized to apply rigorously to a real experiment, it
is detailed enough to capture the essential processes of
concern in this paper.
\section*{Propagation of decoherence in Bose-Einstein condensate}
To study quantitatively the effect of decoherence propagation,
let us consider first an ideal Bose-gas confined by external fields and
described by the Hamiltonian
\begin{equation}
H = \sum_{\mu} E_{\mu} \alpha^{\dag}_{\mu} \alpha_{\mu},
\end{equation}
where $\alpha^{\dag}_{\mu}$ and $\alpha_{\mu}$ are the
boson creation and annihilation operators. $E_{\mu}$
are the one-particle energies, and we denote the
corresponding one-particle wavefunctions as
$\varphi_{\mu}({\bf r})$, where $\mu =0$ stands for
the ground state having minimal energy $E_0=0$.
Then, the
ground-state eigenfunction of the system of $M$ bosons
can be written as
\begin{equation}
\label{ground}
|\Psi\rangle =\frac{1}{\sqrt{M!}}
\left(\alpha_{0}^{\dag}\right)^{M} |0\rangle,
\end{equation}
where $|0\rangle$ is the vacuum state. For simplicity, we can consider
the trap as being divided into a large number $N_c$ of small
cells each having the
volume $V_0$ (it can be considered as the volume directly affected
by the measuring apparatus), satisfying the relation $V_0\ll V$,
where $V$ is the total volume of the trap. Then, the
coordinate ${\bf r}$ is understood as a discrete quantity
(the number of a cell). This is similar to a general practice
in solid-state theory, where $V_0$ is analogous to the volume
of an elementary cell of the crystal \cite{ziman}.
Note that in so doing, the
number of one-particle states taken into account becomes equal to
$N_c$, which is finite, though very large.
This corresponds to the fact that the number of states inside
the first Brillouin zone equals to the number of lattice cells.
At the instant $t=0$ we perform measurement
of the number of bosons in the cell ${\bf r}=0$. This observable
is represented by the operator $N=a^{\dag}(0) a(0)$,
where
\begin{equation}
\label{sum}
a({\bf r}) = \sum\limits_{\mu} \varphi_{\mu}({\bf r}) \alpha_{\mu}.
\end{equation}
is the boson field operator. Eigenvalues of the operator $N$
are $n=0,1,2...$ and, suppose, the measurement
has given us one of them. According to von
Neumann's theory, it corresponds to the action
of the operator $W_n$ on the system, where
\begin{equation}
\label{w}
W_n=\delta _{n,N}=\int\limits_{0}^{2\pi} \frac{d\phi}{2\pi}
\exp{\left[i\phi (n-N)\right]}
\end{equation}
is a projector onto the state with the number
of particles $n$ in the measured region. The operator $W_n$
has the value equal to unity on this state and it has
zero value on all others states.
Further development of the system is to be described by the density
matrix of the system $U(t)$, since the measurement interrupts unitary
evolution and casts the system into mixed quantum state.
According to the standard theory of measurement \cite{neumann,meas},
the density matrix at the time $t$ is
\begin{equation}
\label{evolution}
U(t)=\sum\limits_{n=0}^{\infty} \exp{(-iHt)} W_n U_{\text{in}}
W_n^{\dag} \exp{(iHt)} ,
\end{equation}
where $U_{\text{in}}=|\Psi\rangle \langle\Psi |$
is the density matrix before the measurement.
To trace propagation of decoherence in the system, we
study evolution of the one-particle density matrix
\begin{equation}
\label{ro}
\rho({\bf r}, {\bf r}', t)=\mathop{\rm Tr}\left[ U(t) a^{\dag}({\bf r}')
a({\bf r}) \right].
\end{equation}
This quantity describes local properties of the Bose-Einstein condensate;
in particular, the average number of particles resulting
from the second
measurement, which is performed at the point ${\bf r}$ at
the instant $t$, is given by the value $\rho({\bf r}, {\bf r}, t)$.
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We do not specify the way of measuring the local density of
condensate, and the dynamics of the measurement process is not
considered here. Analysis of a specific experimental scheme is
a distinct problem, requiring separate study, while here we
focus on the post-measurement evolution of the condensate. In
principle, the local density of the Bose-condensate can be measured by
placing some probe into the trap, which interacts with the
condensate so that an entangled state is formed
\begin{equation}
|X\rangle = \sum C_n |n\rangle \otimes |\alpha_n\rangle
\end{equation}
where $|n\rangle$ is the state of condensate with the number of
particles $n$ in the measured region, and $|\alpha_n\rangle$ is
the state of the probe. If the probe interacts with a large
number of environmental degrees of freedom, so that
$|\alpha_n\rangle$ are the eigenstates corresponding to different
eigenvalues of the probe-environment interaction Hamiltonian,
then the coherence between different probe states is being lost,
and the condensate's state also becomes an incoherent mixture of
different states $|n\rangle$. If the probe (and, correspondingly,
the condensate) decoheres quickly enough (as is usually the case) we
can consider the measurement as instantaneous and safely use von
Neumann's theory to describe the condensate's state immediately
after the measurement.
Although the situation considered above is in many
respects too idealized to apply rigorously to a real experiment, it
is detailed enough to capture the essential processes of
concern in this paper.
\section*{Propagation of decoherence in Bose-Einstein condensate}
To study quantitatively the effect of decoherence propagation,
let us consider first an ideal Bose-gas confined by external fields and
described by the Hamiltonian
\begin{equation}
H = \sum_{\mu} E_{\mu} \alpha^{\dag}_{\mu} \alpha_{\mu},
\end{equation}
where $\alpha^{\dag}_{\mu}$ and $\alpha_{\mu}$ are the
boson creation and annihilation operators. $E_{\mu}$
are the one-particle energies, and we denote the
corresponding one-particle wavefunctions as
$\varphi_{\mu}({\bf r})$, where $\mu =0$ stands for
the ground state having minimal energy $E_0=0$.
Then, the
ground-state eigenfunction of the system of $M$ bosons
can be written as
\begin{equation}
\label{ground}
|\Psi\rangle =\frac{1}{\sqrt{M!}}
\left(\alpha_{0}^{\dag}\right)^{M} |0\rangle,
\end{equation}
where $|0\rangle$ is the vacuum state. For simplicity, we can consider
the trap as being divided into a large number $N_c$ of small
cells each having the
volume $V_0$ (it can be considered as the volume directly affected
by the measuring apparatus), satisfying the relation $V_0\ll V$,
where $V$ is the total volume of the trap. Then, the
coordinate ${\bf r}$ is understood as a discrete quantity
(the number of a cell). This is similar to a general practice
in solid-state theory, where $V_0$ is analogous to the volume
of an elementary cell of the crystal \cite{ziman}.
Note that in so doing, the
number of one-particle states taken into account becomes equal to
$N_c$, which is finite, though very large.
This corresponds to the fact that the number of states inside
the first Brillouin zone equals to the number of lattice cells.
At the instant $t=0$ we perform measurement
of the number of bosons in the cell ${\bf r}=0$. This observable
is represented by the operator $N=a^{\dag}(0) a(0)$,
where
\begin{equation}
\label{sum}
a({\bf r}) = \sum\limits_{\mu} \varphi_{\mu}({\bf r}) \alpha_{\mu}.
\end{equation}
is the boson field operator. Eigenvalues of the operator $N$
are $n=0,1,2...$ and, suppose, the measurement
has given us one of them. According to von
Neumann's theory, it corresponds to the action
of the operator $W_n$ on the system, where
\begin{equation}
\label{w}
W_n=\delta _{n,N}=\int\limits_{0}^{2\pi} \frac{d\phi}{2\pi}
\exp{\left[i\phi (n-N)\right]}
\end{equation}
is a projector onto the state with the number
of particles $n$ in the measured region. The operator $W_n$
has the value equal to unity on this state and it has
zero value on all others states.
Further development of the system is to be described by the density
matrix of the system $U(t)$, since the measurement interrupts unitary
evolution and casts the system into mixed quantum state.
According to the standard theory of measurement \cite{neumann,meas},
the density matrix at the time $t$ is
\begin{equation}
\label{evolution}
U(t)=\sum\limits_{n=0}^{\infty} \exp{(-iHt)} W_n U_{\text{in}}
W_n^{\dag} \exp{(iHt)} ,
\end{equation}
where $U_{\text{in}}=|\Psi\rangle \langle\Psi |$
is the density matrix before the measurement.
To trace propagation of decoherence in the system, we
study evolution of the one-particle density matrix
\begin{equation}
\label{ro}
\rho({\bf r}, {\bf r}', t)=\mathop{\rm Tr}\left[ U(t) a^{\dag}({\bf r}')
a({\bf r}) \right].
\end{equation}
This quantity describes local properties of the Bose-Einstein condensate;
in particular, the average number of particles resulting
from the second
measurement, which is performed at the point ${\bf r}$ at
the instant $t$, is given by the value $\rho({\bf r}, {\bf r}, t)$.
To simplify calculations, we use
the fact that the total number of particles is large,
$M\gg 1$, so that operators $\alpha_0$ and $\alpha^{\dag}_0$ acting
on the state $|\Psi\rangle$ can be replaced by the number $\sqrt{M}$
with relative accuracy $1/\sqrt{M}$; this is a standard approximation
in the theory of Bose-Einstein condensation \cite{agd}. Therefore,
Eq. (\ref{sum}) can be rewritten as
\begin{equation}
\label{sum1}
a({\bf r})=\sqrt{n_B({\bf r})}+\bar a({\bf r}),
\qquad \bar a({\bf r)} = \sum\limits_{\mu \neq 0}
\varphi_{\mu}({\bf r}) \alpha_{\mu}
\end{equation}
where $n_B ({\bf r}) = M\varphi_0^2 ({\bf r})$ is the average number
of condensate particles contained in the volume $V_0$ at the cell
${\bf r}$. The expression for the one-particle density matrix
can be written as
\begin{eqnarray}
\label{rho1}
\rho({\bf r}, {\bf r}', t) &=& \sum\limits_{n=0}^{\infty }
\rho_n({\bf r},{\bf r}', t) ,\\
\nonumber
\rho_n({\bf r},{\bf r}', t) &=&
\langle\Psi |W_n^{\dag} a^{\dag}({\bf r}',t)
a({\bf r},t)W_n |\Psi\rangle,
\end{eqnarray}
where $a({\bf r}, t) = \exp{(iHt)} a({\bf r})\exp{(-iHt)}$. The
operator product in Eq.\ (\ref{rho1}) is to be ordered normally,
i.e. it is to be rewritten in such a way that
all $a^{\dag}$ stand to the left of all $a$ in each term
of the Taylor series expansion. In so doing, we take into account that
\begin{equation}
\label{commut}
\left[ a({\bf r}, t), \bar a^{\dag}(0)\right] = \sum\limits_{\mu\neq 0}
\varphi_{\mu}({\bf r}) \varphi^*_{\mu}(0)
{\rm e}^{-iE_{\mu}t} \equiv g({\bf r,}t).
\end{equation}
Note that for a system containing a large number of particles $M\gg 1$,
the function $g({\bf r}, t)$ can be replaced by the Green's function
\begin{equation}
G({\bf r},t) = \sum\limits_{\mu} \varphi_{\mu}({\bf r}) \varphi^*_{\mu}(0)
{\rm e}^{-iE_{\mu}t}
\end{equation}
with accuracy of order of $1/M$,
since $G({\bf r},t) = g({\bf r},t)+\varphi_0({\bf r})\varphi_0^*(0)$.
Performing the calculations, we obtain
\begin{eqnarray}
\label{answer}
\rho_n({\bf r},{\bf r}', t) &=& p_n \left[\sqrt{n_B({\bf r})}
- G({\bf r}, t) \sqrt{n_0} \right] \\
\nonumber
&&\times \left[ \sqrt{n_B({\bf r}')}
- G^*({\bf r}', t) \sqrt{n_0} \right] \\
\nonumber
&&+p_{n-1} n_0 G({\bf r}, t) G^*({\bf r}',t),
\end{eqnarray}
where $n_0=n_B(0)$, and $p_n = \mathop{\rm e}^{-n_0} n_0^n/(n!)$
is the Poisson distribution function.
Summation over $n$ can be performed
explicitly, yielding
\begin{eqnarray}
\label{answ1}
\rho({\bf r},{\bf r}',t) &=& \sqrt{n_B({\bf r}) n_B({\bf r}')}
- G^*({\bf r}',t) \sqrt{n_B({\bf r}) n_0}\\
\nonumber
&& - G({\bf r},t) \sqrt{n_B({\bf r}') n_0}
+ 2 n_0 G^*({\bf r}',t) G({\bf r},t).
\end{eqnarray}
This result shows that the measurement made at
the point ${\bf r=}0$
produces a decohering perturbation which propagates
over the trap in the form of a decoherence wave, and
this propagation is
governed by the Green's function $G({\bf r},t)$. It
can be explicitely demonstrated by considering
an example of the gas
consisting of free Bose-particles of mass $m$. The
corresponding Green's function at the distances $r\gg V_0^{1/3}$
and times $t\gg mV_0^{2/3}/\hbar$ is \cite{feynman}
\begin{equation}
\label{feyn1}
G({\bf r},t) = V_0 \left( \frac{m}{2\pi i\hbar t}\right)
^{3/2}\exp{\left( \frac{im{\bf r}^2}{2\pi \hbar t}\right)}.
\end{equation}
Local density of the
condensate after the measurement is given by the value
\begin{eqnarray}
\label{dens}
\rho({\bf r}, {\bf r}, t) &=& n_B + 2 n_B V_0^2
\left(\frac{m}{2\pi\hbar t}\right)^3 \\
\nonumber
&& - 2 n_B V_0 \left( \frac{m}{2\pi\hbar t}\right)^{3/2}
\cos{\left(\frac{m {\bf r}^2}{2\pi\hbar t}\right)},
\end{eqnarray}
where $n_B=M/V$ is density of the condensate before the
measurement, which is independent on position $\bf r$.
This is an observable effect, which, in principle, can
be detected experimentally.
The entropy of the system, being initially zero,
after the measurement is
\begin{equation}
S= -\mathop{\rm Tr} \left[ U(t)\ln{U(t)}\right] =
-\sum\limits_{n=0}^{\infty} p_n \ln{p_n} > 0,
\end{equation}
which is a clear indication of the decohering
effect of measurement. The increase of entropy of
condensate as a whole happens only at the instant of measurement
and further evolution, being unitary, keeps it constant (decoherence
only propagates in the system from one region to another). Note that
local entropy (in contrast to the one-particle density matrix, where
the decoherence propagation is clearly seen) can be hardly used
to track the decoherence wave. The value of the local entropy
is nonzero even in the initial pure state, while the total
entropy of the system is zero. It happens because of ``negative
entropy'' stored in the form of correlations between
different parts of the condensate (for more detailed discussion
see Ref.\ \cite{zurinfo}).
The results obtained can be qualitatively interpreted as follows.
The measurement performed at ${\bf r}=0$ leads to localization
of some number of particles
within the cell ${\bf r}=0$. The localized particles
acquire rather large momenta, of order
$\hbar/V_0^{1/3}$; the average number of such particles is
$n_0=n_B(0)$. Immediately after being localized, these
particles start to propagate over the trap, and their
propagation is governed by the Green's function (\ref{feyn1}).
Because of indistinguishability of particles in the trap,
we can not say that these are ``the same'' particles which
were measured at ${\bf r}=0$, so that the effect we consider
is not a physical motion of some separate particles in the trap,
but is the propagation of the decohering influence of
the measurement through the system.
An interesting feature of the decoherence propagation
can be illustrated by the gas of bosons trapped
in a parabolic external potential, so that each particle is
represented by an isotropic harmonic oscillator of
eigenfrequency $\Omega$. In this case, provided that
$r\gg V_0^{1/3}$ and $V_0\ll (\hbar /\Omega)^{3/2}\sim V$,
the Green's function has the form \cite{feynman}
\begin{equation}
\label{feynman}
G({\bf r},t) =V_0 \left(\frac{\Omega}{2\pi i\hbar
\sin{\Omega t}} \right)^{3/2} \exp{\left(\frac{i\Omega {\bf r}^2}
{2\pi\hbar}\cot{\Omega t}\right)}
\end{equation}
where the particles are assumed to have unitary mass. This function
is periodic in time with the period $2\pi/\Omega$.
Therefore, the decoherence propagation is also periodic
in time with the same period. In the general case of Bose-gas
trapped in a finite volume, the decoherence propagation becomes
a quasiperiodic process, according to Eq.\ (\ref{commut}).
And, last but not least, decoherence propagation is a wave process,
possessing both amplitude and phase. Existence of coherent waves
in the system without quantum coherence is not
unusual, the same property is shared, e.g., by the sound wave
propagating in the classical fluid. Therefore, in principle, an
interference of two decoherence waves is possible.
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The entropy of the system, being initially zero,
after the measurement is
\begin{equation}
S= -\mathop{\rm Tr} \left[ U(t)\ln{U(t)}\right] =
-\sum\limits_{n=0}^{\infty} p_n \ln{p_n} > 0,
\end{equation}
which is a clear indication of the decohering
effect of measurement. The increase of entropy of
condensate as a whole happens only at the instant of measurement
and further evolution, being unitary, keeps it constant (decoherence
only propagates in the system from one region to another). Note that
local entropy (in contrast to the one-particle density matrix, where
the decoherence propagation is clearly seen) can be hardly used
to track the decoherence wave. The value of the local entropy
is nonzero even in the initial pure state, while the total
entropy of the system is zero. It happens because of ``negative
entropy'' stored in the form of correlations between
different parts of the condensate (for more detailed discussion
see Ref.\ \cite{zurinfo}).
The results obtained can be qualitatively interpreted as follows.
The measurement performed at ${\bf r}=0$ leads to localization
of some number of particles
within the cell ${\bf r}=0$. The localized particles
acquire rather large momenta, of order
$\hbar/V_0^{1/3}$; the average number of such particles is
$n_0=n_B(0)$. Immediately after being localized, these
particles start to propagate over the trap, and their
propagation is governed by the Green's function (\ref{feyn1}).
Because of indistinguishability of particles in the trap,
we can not say that these are ``the same'' particles which
were measured at ${\bf r}=0$, so that the effect we consider
is not a physical motion of some separate particles in the trap,
but is the propagation of the decohering influence of
the measurement through the system.
An interesting feature of the decoherence propagation
can be illustrated by the gas of bosons trapped
in a parabolic external potential, so that each particle is
represented by an isotropic harmonic oscillator of
eigenfrequency $\Omega$. In this case, provided that
$r\gg V_0^{1/3}$ and $V_0\ll (\hbar /\Omega)^{3/2}\sim V$,
the Green's function has the form \cite{feynman}
\begin{equation}
\label{feynman}
G({\bf r},t) =V_0 \left(\frac{\Omega}{2\pi i\hbar
\sin{\Omega t}} \right)^{3/2} \exp{\left(\frac{i\Omega {\bf r}^2}
{2\pi\hbar}\cot{\Omega t}\right)}
\end{equation}
where the particles are assumed to have unitary mass. This function
is periodic in time with the period $2\pi/\Omega$.
Therefore, the decoherence propagation is also periodic
in time with the same period. In the general case of Bose-gas
trapped in a finite volume, the decoherence propagation becomes
a quasiperiodic process, according to Eq.\ (\ref{commut}).
And, last but not least, decoherence propagation is a wave process,
possessing both amplitude and phase. Existence of coherent waves
in the system without quantum coherence is not
unusual, the same property is shared, e.g., by the sound wave
propagating in the classical fluid. Therefore, in principle, an
interference of two decoherence waves is possible.
Above, we have considered the system of noninteracting bosons.
Now, let us investigate the case of weakly interacting particles,
i.e. a weakly non-ideal Bose-gas contained in a trap of
large volume $V$. We assume no external potential acting on
the particles, so that the one-particle states
are simple plane waves
\begin{equation}
\varphi_{\bf k}({\bf r}) = \sqrt{\frac{V_0}{V}} \exp{(i{\bf kr})},
\end{equation}
where the normalization reflects the fact that the
trap is divided into cells of volume $V_0\ll V$.
This system is described by the Hamiltonian
\begin{eqnarray}
\label{nonideal}
H&=&\sum\limits_{\bf k} E_{\bf k} \alpha_{\bf k}^{\dag} \alpha_{\bf k}\\
\nonumber
&&+ \frac 1{2V} \sum\limits_{{\bf k}_1+{\bf k}_2={\bf k}'_1+{\bf k}'_2}
v ({\bf k}_1-{\bf k}'_1) \alpha_{{\bf k}'_1}^{\dag}
\alpha_{{\bf k}'_2}^{\dag} \alpha_{{\bf k}_2} \alpha_{{\bf k}_1}
\end{eqnarray}
where $v({\bf k})$ is the Fourier transform of the interaction
potential (which is assumed to be repulsive). Since the interaction
is small, new Bose operators can be introduced according to
Bogoliubov transformation
\begin{eqnarray}
\label{bogol}
\alpha_{\bf k} &=& \xi_{\bf k} \cosh{\chi_{\bf k}} +
\xi_{-{\bf k}}^{\dag} \sinh{\chi_{\bf k}}, \\
\nonumber
\alpha_{-{\bf k}}^{\dag} &=& \xi_{\bf k} \sinh{\chi_{\bf k}} +
\xi_{-{\bf k}}^{\dag} \cosh{\chi_{\bf k}},
\end{eqnarray}
with the parameters $\chi_{\bf k}$ defined as
\begin{equation}
\label{bogol1}
\tanh{2\chi_{\bf k}} = -\frac{v({\bf k})n_B}{E_{\bf k} +
v({\bf k})n_B},
\end{equation}
where $n_B$ is the average number of particles belonging to Bose-Einstein
condensate contained in the volume $V_0$. Provided that the interaction
is small (or the gas density $M/V$ is small), almost all
particles belong to the condensate, so we can take
$n_B= M V_0/V$ with relative accuracy of order of $\sqrt{v(0) M/V}$
\cite{agd}.
By using the Bogoliubov transformation, we pass to the ideal gas
of new excitations with the dispersion law
\begin{equation}
\label{bogol2}
\omega_{\bf k}=\sqrt{E_{\bf k}^2 + 2 E_{\bf k} v({\bf k}) n_B}.
\end{equation}
Again, we consider dynamical behavior of the one-particle density matrix.
The calculation procedure remains essentially the same as for
the ideal Bose-gas. In so doing, we obtain the result:
\begin{eqnarray}
\label{answnew}
\rho_n ({\bf r},{\bf r}',t) &=& \frac{n_B}{(n!)^2}
\frac{\partial^{2n}}{\partial z^n \partial z^{\prime n}}
\Bigl\{ [1+(z-1) G({\bf r},t)] \\
\nonumber
&&\times [1+(z'-1) G^*({\bf r}',t)] \\
\nonumber
&&\times\exp{[n_B X(z,z')]}\Bigr\}_{z=z'=0}
\end{eqnarray}
where the following notations were used,
\begin{eqnarray}
\label{answlast}
X(z,z') &=& B(zz'-1) + (1-B)(z+z'-2) \\
&& + A\left[ (z-1)^2+(z'-1)^2\right], \\
\nonumber
A &=& \frac{V_0}{2V} \sum\limits_{\bf k} \frac{v({\bf k})n_B}
{\omega_{\bf k}}, \\
\nonumber
B &=& \frac{V_0}{2V} \sum\limits_{\bf k} \left[1+\frac{E_{\bf k}
+v({\bf k}) n_B} {\omega_{\bf k}}\right],
\end{eqnarray}
and $G({\bf r},t)$ is the Green's function of the weakly interacting
Bose-gas:
\begin{eqnarray}
\label{green}
G({\bf r},t) &=& \sum\limits_{\bf k} \exp{(i{\bf kr})}\\
\nonumber
&&\times \left\{ \cos{\omega_{\bf k}t}
-i\frac{E_{\bf k} + v({\bf k}) n_B}{\omega_{\bf k}}
\sin{\omega_{\bf k}t} \right\}.
\end{eqnarray}
Again, we see that the decoherence wave propagating in the
system follows the dynamics of the
Green's function (\ref{green}). Dynamic behavior of $G({\bf r},t)$
at large times $t$ and large distances $r$
can be analyzed by the
method of stationary phase \cite{witham}. According to
this method, the value
of the function $G(r,t)$ at the point $\bf r$ at the
instant $t$ is determined by those excitations which have
a group velocity ${\bf u}({\bf k})\equiv d\omega_{\bf k}/ d{\bf k}$
obeying the requirement ${\bf u}({\bf k}) = {\bf r}/t$.
The excitations with large wavevectors ${\bf k}$ are
subject to considerable damping
\cite{agd}, so that at large distances only the undamped long-wavelength
excitations determine the dynamics of the Green's function.
These excitations represent
sound propagating in the Bose-gas with the velocity $c=\sqrt{n_B v(0)/m}$,
so the decoherence wave
in a system of weakly interacting bosons propagates with the
sound velocity $c$.
This result can be interpreted in the same way as the
decoherence wave in an ideal Bose-gas. The measurement
affects the particles situated at ${\bf r}=0$. Due to
the interparticle interaction, the decohering perturbation
is transferred to other regions of the system.
The decoherence transfer is governed by the
undamped excitations present in the system, i.e. by the
long-wavelength excitations traveling with
the sound velocity $c$.
\section*{Discussion}
Summarizing, we have studied the decohering
influence of a local measurement performed on a distributed
quantum system. We show that the decohering perturbation exerted
on the measured region propagates over the system by forming a
{\it decoherence wave\/}, whose dynamics is determined by the
Green's function of the system. This result, although
not totally unexpected, is not as trivial as it might seem, since
decoherence is a rather peculiar effect, and the decohering impact
of a measurement can be quite different from other physical
influences (see, e.g. the discussion in Ref.\ \cite{scully}).
The usual scenario for few-particle systems is
based on the Einstein-Podolsky-Rosen (EPR) situation \cite{epr}
of strong entanglement, when, e.g. two
particles with spins $1/2$ form a singlet state
\begin{equation}
|\psi\rangle = \frac 1{\sqrt{2}}\left(\,|\!\uparrow\downarrow\rangle
- |\!\downarrow\uparrow\rangle\, \right).
\end{equation}
If the first spin has been measured, and as a result of this
measurement has been cast in the state $|\!\uparrow\rangle$ (here
again we use von Neumann's theory of instant measurement), then
the transfer of decoherence is instant: the second spin
immediately occurs in the state $|\!\downarrow\rangle$. In
distributed systems this effect is also present: the wave function of
the system collapses immediately after the measurement. But the
impact of the collapse upon the one-particle density matrix
(and even $s$-particle density matrix, for $s\ll M$) is
practically unobservable for the system of macroscopic size
(where $M\gg 1$): the change in the density matrix element
$\rho({\bf r},{\bf r}',t)$ immediately after the measurement is
of order of $n_0/M$ (provided, of course, that ${\bf r},{\bf r}
'\neq 0$), and the same is true for the $k$-particle density
matrix
$\rho({\bf r}_1,\dots{\bf r}_k; {\bf r}'_1,\dots{\bf r}'_k)$
if $k\ll M$. This result is rather obvious: localization of the
number $n_0$ of particles in some cell can not affect noticeably
other cells if the total number of particles is macroscopically
large. Therefore, the post-measurement evolution of the system,
which is governed by the Green's function, becomes important since it
provides much more noticeable changes in the density
matrix elements: in Eq.\ (\ref{dens}) the term corresponding to
the decoherence wave does not go to zero as $M\to\infty$.
Obviously, it happens because in the EPR-like situation the
entanglement is very ``stiff'', so that each state of one
particle determines completely the state of the other. But in the
many-particle system there is no one-to-one correspondence,
since the total number of degrees of freedom is much larger than
the number of degrees of freedom fixed during the measurement.
This difference is the reason for the different dynamics of
decoherence propagation.
Finally, we remark that another aspect of decoherence in
distributed systems has been studied within the context of
decoherent quantum histories \cite{gellmann,brun}. Although the
effects studied there, as well as systems considered and methods
used, are different from those investigated here, it is
interesting to note that local properties of distributed quantum
systems are often ``intrinsically'' decoherent \cite{brun} if a
coarse enough description is used. For the effects considered
here, sufficient coarse graining leads to averaging of the
oscillating Green's function over the spatial scale of several
oscillations, so that the details of the decoherence wave becomes
negligible. Therefore, the intrinsic structure of the decoherence
wave can be distinguished only at fine scales, where coherence of
the Green's function holds.
This work was partially carried out at the Ames Laboratory, which
is operated for the U.\ S.\ Department of Energy by Iowa State
University under Contract No.\ W-7405-82 and was supported by
the Director for Energy Research, Office of Basic Energy Sciences
of the U.\ S.\ Department of Energy.
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This result can be interpreted in the same way as the
decoherence wave in an ideal Bose-gas. The measurement
affects the particles situated at ${\bf r}=0$. Due to
the interparticle interaction, the decohering perturbation
is transferred to other regions of the system.
The decoherence transfer is governed by the
undamped excitations present in the system, i.e. by the
long-wavelength excitations traveling with
the sound velocity $c$.
\section*{Discussion}
Summarizing, we have studied the decohering
influence of a local measurement performed on a distributed
quantum system. We show that the decohering perturbation exerted
on the measured region propagates over the system by forming a
{\it decoherence wave\/}, whose dynamics is determined by the
Green's function of the system. This result, although
not totally unexpected, is not as trivial as it might seem, since
decoherence is a rather peculiar effect, and the decohering impact
of a measurement can be quite different from other physical
influences (see, e.g. the discussion in Ref.\ \cite{scully}).
The usual scenario for few-particle systems is
based on the Einstein-Podolsky-Rosen (EPR) situation \cite{epr}
of strong entanglement, when, e.g. two
particles with spins $1/2$ form a singlet state
\begin{equation}
|\psi\rangle = \frac 1{\sqrt{2}}\left(\,|\!\uparrow\downarrow\rangle
- |\!\downarrow\uparrow\rangle\, \right).
\end{equation}
If the first spin has been measured, and as a result of this
measurement has been cast in the state $|\!\uparrow\rangle$ (here
again we use von Neumann's theory of instant measurement), then
the transfer of decoherence is instant: the second spin
immediately occurs in the state $|\!\downarrow\rangle$. In
distributed systems this effect is also present: the wave function of
the system collapses immediately after the measurement. But the
impact of the collapse upon the one-particle density matrix
(and even $s$-particle density matrix, for $s\ll M$) is
practically unobservable for the system of macroscopic size
(where $M\gg 1$): the change in the density matrix element
$\rho({\bf r},{\bf r}',t)$ immediately after the measurement is
of order of $n_0/M$ (provided, of course, that ${\bf r},{\bf r}
'\neq 0$), and the same is true for the $k$-particle density
matrix
$\rho({\bf r}_1,\dots{\bf r}_k; {\bf r}'_1,\dots{\bf r}'_k)$
if $k\ll M$. This result is rather obvious: localization of the
number $n_0$ of particles in some cell can not affect noticeably
other cells if the total number of particles is macroscopically
large. Therefore, the post-measurement evolution of the system,
which is governed by the Green's function, becomes important since it
provides much more noticeable changes in the density
matrix elements: in Eq.\ (\ref{dens}) the term corresponding to
the decoherence wave does not go to zero as $M\to\infty$.
Obviously, it happens because in the EPR-like situation the
entanglement is very ``stiff'', so that each state of one
particle determines completely the state of the other. But in the
many-particle system there is no one-to-one correspondence,
since the total number of degrees of freedom is much larger than
the number of degrees of freedom fixed during the measurement.
This difference is the reason for the different dynamics of
decoherence propagation.
Finally, we remark that another aspect of decoherence in
distributed systems has been studied within the context of
decoherent quantum histories \cite{gellmann,brun}. Although the
effects studied there, as well as systems considered and methods
used, are different from those investigated here, it is
interesting to note that local properties of distributed quantum
systems are often ``intrinsically'' decoherent \cite{brun} if a
coarse enough description is used. For the effects considered
here, sufficient coarse graining leads to averaging of the
oscillating Green's function over the spatial scale of several
oscillations, so that the details of the decoherence wave becomes
negligible. Therefore, the intrinsic structure of the decoherence
wave can be distinguished only at fine scales, where coherence of
the Green's function holds.
This work was partially carried out at the Ames Laboratory, which
is operated for the U.\ S.\ Department of Energy by Iowa State
University under Contract No.\ W-7405-82 and was supported by
the Director for Energy Research, Office of Basic Energy Sciences
of the U.\ S.\ Department of Energy.
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\end{document}
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\begin{document}
\title{Arithmetic formulas for the Fourier coefficients of Hauptmoduln of level 2, 3, and 5}
\begin{abstract}
We give arithmetic formulas for the coefficients of Hauptmoduln of higher levels as analogues of Kaneko's formula for the elliptic modular $j$-invariant. We also obtain their asymptotic formulas by employing Murty-Sampath's method.
\end{abstract}
\section{Introduction}
For the elliptic modular function $j(\tau)$, let $\textbf{t}_m(d)$ be the modular trace function (the precise definition will be given later) and $c_n$ ($n \geq 1$) the $n$th Fourier coefficient of $j(\tau)$, that is, $j(\tau) = q^{-1} + 744 + \sum_{n=1}^{\infty} c_n q^n$. Zagier \cite{Zag02} studied the traces of singular moduli and showed that the generating function of $\textbf{t}_m(d)$ is a meromorphic modular form of weight $3/2$ on the right group for each $m$. Multiplying it by theta function and observing the modular forms of weight 2, Kaneko \cite{Kan95} gave the following arithmetic formula for $c_n$ experimentally, and showed it.
\begin{eqnarray*}
c_n &=& \frac{1}{n} \biggl\{ \sum_{r \in \mathbb{Z}} \textbf{t}_1(n - r^2) + \sum_{\substack{r \geq 1,\ odd}} ((-1)^n \textbf{t}_1(4n - r^2) - \textbf{t}_1(16n - r^2)) \biggr\} \\
&=& \frac{1}{2n} \sum_{r \in \mathbb{Z}} \textbf{t}_2(4n - r^2) .
\end{eqnarray*}
On the other hand, by using the circle method, Petersson \cite{Pet32} and later Rademacher \cite{Rad} independently derived the asymptotic formula for $c_n$:
\begin{eqnarray*}
c_n \sim \frac{e^{4 \pi \sqrt{n}}}{\sqrt{2} n^{3/4}}\ as\ n \to \infty.
\end{eqnarray*}
The circle method is introduced by Hardy and Ramanujan \cite{HR18} to prove the asymptotic formula for the partition function
\begin{eqnarray*}
p(n) \sim \frac{e^{\pi \sqrt{2n/3}}}{4\sqrt{3}n}\ as\ n \to \infty,
\end{eqnarray*}
where $p(n)$ is defined by $\sum_{n=0}^{\infty} p(n)q^n = \prod_{n=1}^{\infty}(1-q^n)^{-1}$. In 2013, Bruinier and Ono \cite{BO13} considered certain traces of singular moduli for weak Maass forms and derived the algebraic formula for $p(n)$. Combining this formula with Laplace's method, Dewar and Murty \cite{DM13, DM132} proved the asymptotic formulas for $p(n)$ and $c_n$ without the circle method. More recently, Murty and Sampath \cite{Sam15} derived the asymptotic formula for $c_n$ from Kaneko's arithmetic formula with Laplace's method.\\
In this article, we generalize these formulas to Hauptmoduln (defined in section 2) for the congruence subgroups $\Gamma_0(p)$ and $\Gamma_0^*(p)$ (the extension of $\Gamma_0(p)$ by the Atkin-Lehner involution) with $p = 2, 3,$ and $5$. \\
Let $j_p(\tau)$ and $j_p^*(\tau)$ be the corresponding Hauptmoduln for $\Gamma_0(p)$ and $\Gamma_0^*(p)$, respectively. Ohta \cite{Ohta09} gave the arithmetic formulas for the Fourier coefficients of $j_2(\tau)$ and $j_2^*(\tau)$, and a part of those of $j_3(\tau)$. She also treated the cases of $j_4(\tau)$ and $j_4^*(\tau)$. Let $c_n^{(p)}$ and $c_n^{(p*)}$ be the $n$th Fourier coefficients of $j_p(\tau)$ and $j_p^*(\tau)$, respectively. We express these coefficients in terms of the modular trace functions $\textbf{t}_m^{(p*)}(d)$.
\begin{thm}
\label{main1}
For any $n \geq 1$, we have
\begin{eqnarray*}
c_n^{(2)} &=& \frac{1}{2n} \times \left\{ \begin{array}{ll}
- \sum_{r \equiv 0 (2)} \textbf{t}_2^{(2*)}(4n - r^2) + 24\sigma_1^{(2)}(n) & (n \equiv 0 \bmod 2), \\
\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(2*)}(4n - r^2) + 24\sigma_1(n) & (n \not\equiv 0 \bmod 2), \\ \end{array} \right. \\
c_n^{(3)} &=& \frac{1}{2n} \times \left\{ \begin{array}{ll}
- \sum_{r \equiv 0 (3)} \textbf{t}_2^{(3*)}(4n - r^2) + 36\sigma_1^{(3)}(n) & (n \equiv 0 \bmod 3), \\
\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(3*)}(4n - r^2) + 36\sigma_1(n) & (n \not\equiv 0 \bmod 3), \\ \end{array} \right. \\
c_n^{(5)} &=& \frac{1}{2n} \times \left\{ \begin{array}{ll}
-\sum_{r \equiv 0 (5)} \textbf{t}_2^{(5*)}(4n - r^2) + 18\sigma_1^{(5)}(n) & (n \equiv 0 \bmod 5), \\
\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(5*)}(4n - r^2) +18\sigma_1(n) & (n \not\equiv 0 \bmod 5), \\ \end{array} \right. \\
c_n^{(p*)} &=& c_n^{(p)} - p c_{pn}^{(p)} \ \ (p = 2, 3, 5)\\
\end{eqnarray*}
where $\sigma_1(n) = \sum_{d | n}d$, and $\sigma_1^{(p)}(n) = \sum_{\substack{d | n \\ p \nmid d}}d$.
\end{thm}
\begin{rmk}
These formulas are different from those in Ohta \cite{Ohta09}. In \cite{Ohta09}, the definition of $\textbf{t}_m^{(p)}(d)$ was mixed with that of $\textbf{t}_m^{(p*)}(d)$, and used the values of $\textbf{t}_m^{(p)}(d)$ instead of $\textbf{t}_m^{(p*)}(d)$.
\end{rmk}
Combining these formulas with Laplace's method as in \cite{Sam15}, we obtain the asymptotic formulas of $c_n^{(p)}$.
\begin{thm}
\label{main2}
We have
\begin{eqnarray*}
c_n^{(2)} &\sim& \frac{e^{2\pi \sqrt{n}}}{2n^{3/4}} \times \left\{ \begin{array}{ll}
-1 & (n \equiv 0 \bmod 2), \\ 1 & (n \equiv 1 \bmod 2), \\
\end{array} \right.\\
c_n^{(3)} &\sim& \frac{e^{4\pi \sqrt{n}/3}}{\sqrt{6}n^{3/4}} \times \left\{ \begin{array}{ll}
-1 & (n \equiv 0, 2 \bmod 3), \\ 2 & (n \equiv 1\ \ \ \bmod 3), \\
\end{array} \right.\\
c_n^{(5)} &\sim& \frac{e^{4\pi \sqrt{n}/5}}{\sqrt{10}n^{3/4}} \times \left\{ \begin{array}{ll}
-1 & (n \equiv 0 \bmod 5), \\ (3 + \sqrt{5})/2 & (n \equiv 1 \bmod 5), \\ -1 + \sqrt{5} & (n \equiv 2 \bmod 5), \\ -1 - \sqrt{5} & (n \equiv 3 \bmod 5), \\ (3 - \sqrt{5})/2 & (n \equiv 4 \bmod 5)\\
\end{array} \right.
\end{eqnarray*}\\
as n $\to \infty$.
\end{thm}
\section{Preliminaries}
In this section, we shall define the Hauptmoduln and the modular trace functions.
\begin{dfn}
Let $\Gamma$ be a congruence subgroup of $SL_2(\mathbb{R})/{\pm I}$ containing $(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix})$. If the genus of $\Gamma$ is equal to 0, there is a unique modular function f of weight 0 satisfying the following conditions. We call this f the Hauptmodul with respect to $\Gamma$.\\
$(1)$ f is holomorphic in the upper half plane $\mathfrak{H}$,\\
$(2)$ f has a Fourier expansion of the form $f(\tau) = q^{-1} + \sum_{n = 1}^{\infty} H_n q^n \ (q := e^{2\pi i \tau})$,\\
$(3)$ f is holomorphic at cusps of $\Gamma$ except i$\infty$.
\end{dfn}
For $\Gamma_0(p) := \{(\begin{smallmatrix}a & b \\c & d \end{smallmatrix}) \in \mathrm{PSL}_2(\mathbb{Z})\ |\ c \equiv 0 \pmod{p} \}$ and $\Gamma_0^*(p) := \Gamma_0(p) \cup \Gamma_0(p)(\begin{smallmatrix}0 & -1/\sqrt{p} \\ \sqrt{p} & 0 \end{smallmatrix})$ ($p$ = 2, 3, 5), the corresponding Hauptmoduln $j_p(\tau)$ and $j_p^*(\tau)$ can be described by means of the Dedekind $\eta$-function $\eta(\tau):= q^{1/24}\prod_{n=1}^{\infty}(1-q^n)$;
\begin{eqnarray*}
j_2(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(2\tau)} \biggr)^{24} + 24 = \frac{1}{q} + 276q - 2048q^2 + 11202q^3 + \cdots,\\
j_2^*(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(2\tau)} \biggr)^{24} + 24 + 2^{12} \biggl( \frac{\eta(2\tau)}{\eta(\tau)} \biggr)^{24} = \frac{1}{q} + 4372q + 96256q^2 + 1240002q^3 + \cdots,\\
j_3(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(3\tau)} \biggr)^{12} + 12 = \frac{1}{q} + 54q - 76q^2 - 243q^3 + \cdots,\\
j_3^*(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(3\tau)} \biggr)^{12} + 12 + 3^6 \biggl( \frac{\eta(3\tau)}{\eta(\tau)} \biggr)^{12} = \frac{1}{q} + 783q + 8672q^2 + 65367q^3 + \cdots,\\
j_5(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(5\tau)} \biggr)^6 + 6 = \frac{1}{q} + 9q + 10q^2 - 30q^3 + \cdots,\\
j_5^*(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(5\tau)} \biggr)^6 + 6 + 5^3 \biggl( \frac{\eta(5\tau)}{\eta(\tau)} \biggr)^6 = \frac{1}{q} + 134q + 760q^2 + 3345q^3 + \cdots.
\end{eqnarray*}
For $p$ = 2, 3, and 5, let $d$ be a positive integer such that $-d$ is congruent to a square modulo 4$p$, and $\mathcal{Q}_{d,p}$ the set of positive definite binary quadratic forms $Q(X,Y) = [a,b,c] = a X^2 + b X Y + c Y^2\ (a,b,c \in \mathbb{Z})$ of discriminant $-d$ with $a \equiv 0$ (mod $p$). Moreover, we fix an integer $\beta$ (mod $2p$) with $\beta^2 \equiv -d$ (mod $4p$) and denote by $\mathcal{Q}_{d,p,\beta}$ the set of quadratic forms $[a,b,c] \in \mathcal{Q}_{d,p}$ such that $b \equiv \beta$ (mod $2p$). For every positive integer $m$, let $\varphi_m(j_p^*)$ be a unique polynomial of $j_p^*$ satisfying $\varphi_m(j_p^*(\tau)) = q^{-m} + O(q)$. We define two modular trace functions:
\begin{eqnarray*}
\textbf{t}_m^{(p)}(d) &:=& \sum_{Q \in \mathcal{Q}_{d,p,\beta} / \Gamma_0(p)} \frac{1}{|\Gamma_0(p)_Q|} \varphi_m(j_p^*(\alpha_Q)),\\
\textbf{t}_m^{(p*)}(d) &:=& \sum_{Q \in \mathcal{Q}_{d,p} / \Gamma_0^*(p)} \frac{1}{|\Gamma_0^*(p)_Q|} \varphi_m(j_p^*(\alpha_Q)),
\end{eqnarray*}
where $\alpha_Q$ is the root of $Q(X,1) = 0$ in $\mathfrak{H}$. The definition of $\textbf{t}_m^{(p)}(d)$ is independent of $\beta$. In addition, we set $\textbf{t}_2^{(2*)}(0) := 5, \, \textbf{t}_2^{(3*)}(0) = \textbf{t}_2^{(5*)}(0) := 3, \, \textbf{t}_2^{(p*)}(-1) := -1, \, \textbf{t}_2^{(p*)}(-4) := -2, \, \textbf{t}_2^{(p*)}(d) := 0$ for $d < -4$ or $-d \not\equiv$ square (mod $4p$) ($p$ = 2, 3, 5). For the relation between two modular trace functions, see $\cite{Kim08}$.
\begin{rmk}
For $p = 1$, we put $j_1^*(\tau) := j(\tau) - 744 = \{(\eta(\tau)/\eta(2\tau))^8 + 2^8 (\eta(2\tau)/\eta(\tau))^{16}\}^3-744$ and $\textbf{t}_m(d) := \textbf{t}_m^{(1*)}(d)$.
\end{rmk}
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\section{Proof of Theorem \ref{main1}}
We give a proof only for the case $p = 3$; the other cases are proved in the same way.
\begin{dfn}
For every positive integer $t$, we define the operator $U_t$ by
\begin{eqnarray*}
\biggl(\sum a_n q^n\biggr) \biggr|U_t := \sum a_{tn} q^n.
\end{eqnarray*}
\end{dfn}
Then $U_t$ sends a modular form to a modular form of the same weight but raises the level in general. To prove Theorem \ref{main1}, we need the following theorem, which is a special case $f = \varphi_m(j_p^*(\tau))$ of Theorem 1.1 in \cite{BF06}.
\begin{thm}
\label{BF}
The function
\begin{eqnarray*}
g_m^{(p*)}(\tau) := \sum_{d > 0} \textbf{t}_m^{(p*)}(d) q^d + (\sigma_1(m) + p \sigma_1(m/p)) - \sum_{k |m}k q^{-k^2}
\end{eqnarray*}
$($where $\sigma_1(x) = 0$ if $x \not\in \mathbb{Z})$ is a meromorphic modular form of weight $3/2$, holomorphic outside the cusps, with respect to $\Gamma_0(4p)$, that is,
\begin{eqnarray*}
g_m^{(p*)}(\tau) \in M_{3/2}^{mer}(\Gamma_0(4p)).
\end{eqnarray*}
Here $M_k^{mer}(\Gamma)$ denotes the space of meromorphic modular forms of weight $k$ with respect to $\Gamma$.
\end{thm}
We prove Theorem\ref{main1}. For the modular form $f(\tau) = \sum a_n q^n$, we define the functions $\tilde{f}_0, \ \tilde{f}_1 \ and\ \tilde{f}_2$ by
\begin{eqnarray*}
\tilde{f}_0(\tau) &:=& \frac{1}{3} \left\{f(\tau) + f(\tau + \frac{1}{3}) + f(\tau + \frac{2}{3})\right\},\\
\tilde{f}_1(\tau) &:=& \frac{1}{3} \left\{f(\tau) + \zeta^{-1}f(\tau + \frac{1}{3}) + \zeta f(\tau + \frac{2}{3})\right\},\\
\tilde{f}_2(\tau) &:=& \frac{1}{3} \left\{f(\tau) + \zeta f(\tau + \frac{1}{3}) + \zeta^{-1}f(\tau + \frac{2}{3})\right\}
\end{eqnarray*}
where $\zeta = e^{2\pi i/3}$. For each $k \pmod{3}$, then $\tilde{f}_k$ has a Fourier expansion of the form $\tilde{f}_k(\tau) = \sum_{n \equiv k (3)}a_n q^n$, and it is also a modular form of the same weight. By Theorem \ref{BF}, we have
\begin{eqnarray*}
g_2^{(3*)}(\tau) = \sum_{d = -4}^{\infty} \textbf{t}_2^{(3*)}(d) q^d \in M_{3/2}^{mer}(\Gamma_0(12)).
\end{eqnarray*}
Now consider the modular form $g_2^{(3*)}(\tau) \cdot \theta_0(\tau)$ where $\theta_0(\tau) := \sum_{n \in \mathbb{Z}} q^{n^2} \in M_{1/2}(\Gamma_0(4))$. This form is of weight 2 and we have
\begin{eqnarray*}
g_2^{(3*)}(\tau) \cdot \theta_0(\tau) = \biggl(\sum_{d = -4}^{\infty}\textbf{t}_2^{(3*)}(d) q^d \biggr) \cdot \biggl(\sum_{r \in \mathbb{Z}}q^{r^2} \biggr) = \sum_{n = -4}^{\infty} \biggl(\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(3*)}(n - r^2) \biggr) q^n \in M_2^{mer}(\Gamma_0(12)).
\end{eqnarray*}
Similarly, the product $g_2^{(3*)}(\tau) \cdot \theta_0(9\tau)$ is also a modular form of weight 2 and its Fourier expansion is
\begin{eqnarray*}
g_2^{(3*)}(\tau) \cdot \theta_0(9\tau) = \sum_{n = -4}^{\infty} \biggl(\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(3*)}(n - (3r)^2) \biggr) q^n = \sum_{n = -4}^{\infty} \biggl(\sum_{r \equiv 0 (3)} \textbf{t}_2^{(3*)}(n - r^2) \biggr) q^n \in M_2^{mer}(\Gamma_0(36)).
\end{eqnarray*}
We put
\begin{eqnarray*}
F(\tau) &:=& \biggl(g_2^{(3*)}(\tau) \cdot \theta_0(\tau)\biggr) \bigg|U_4 = \sum_{n = -1}^{\infty} \biggl(\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(3*)}(4n - r^2) \biggr) q^n \in M_2^{mer}(\Gamma(12)), \\
G(\tau) &:=& \biggl(g_2^{(3*)}(\tau) \cdot \theta_0(9\tau)\biggr) \bigg|U_4 = \sum_{n = -1}^{\infty} \biggl(\sum_{r \equiv 0 (3)} \textbf{t}_2^{(3*)}(4n - r^2) \biggr) q^n \in M_2^{mer}(\Gamma(36)).
\end{eqnarray*}
Then $F(\tau)$ and $G(\tau)$ are meromorphic modular forms of weight 2. Moreover, for
\begin{eqnarray*}
j'_3(\tau) &=& \sum_{n = -1}^{\infty} n c_n^{(3)} q^n \in M_2^{mer}(\Gamma_0(3)),\\
E_2^{(3)}(\tau) &:=& \frac{1}{2}(3E_2(3\tau) - E_2(\tau)) = 1 + 12\sum_{n = 1}^{\infty} \sigma_1^{(3)}(n) q^n \in M_2(\Gamma_0(3)),
\end{eqnarray*}
(where the prime denotes $(2\pi i)^{-1} d/d\tau$ and $E_2(\tau) := 1 - 24\sum_{n = 1}^{\infty} \sigma_1(n) q^n$ is the Eisenstein series of weight 2), we put
\begin{eqnarray*}
H(\tau) := j'_3(\tau) - \frac{3}{2} E_2^{(3)}(\tau) = -\frac{1}{q} - \frac{3}{2} + \sum_{n = 1}^{\infty} (n c_n^{(3)} - 18\sigma_1^{(3)}(n)) q^n \in M_2^{mer}(\Gamma_0(3)).
\end{eqnarray*}
Then, the theorem in the case of $p = 3$ is equivalent to the following identities of modular forms:
\begin{eqnarray*}
2\tilde{H}_0(\tau) = -\tilde{G}_0(\tau), \ \ 2\tilde{H}_1(\tau) = \tilde{F}_1(\tau), \ \ 2\tilde{H}_2(\tau) = \tilde{F}_2(\tau).
\end{eqnarray*}
Since these modular forms are of weight 2 on $\Gamma(36)$, we see that, by the Riemann-Roch theorem, it is enough to check the coincidence of Fourier coefficients on both sides of the equalities up to $q^{3960}$. We checked this by using Mathematica and Pari-GP.\\
Similarly, we can show the equation $j_3^*(\tau) = j_3(\tau) - 3(j_3|U_3)(\tau)$, and we obtain $c_n^{(3*)} = c_n^{(3)} - 3c_{3n}^{(3)}$.
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\section{Proof of Theorem \ref{main2}}
In this section, we give an overview of a proof. Since we can prove any case in the same way as \cite{Sam15}, we give a proof only for the case $p =$ 3. First, we prepare for a proof.
\begin{dfn}
The binary quadratic forms
\begin{eqnarray*}
\left\{ \begin{array}{ll}
[3, 0, d/12] & (-d \equiv 0 \pmod{12}), \\
\lbrack 3, 1, (d+1)/12 \rbrack \ ,\ \lbrack 3, -1, (d+1)/12 \rbrack & (-d \equiv 1 \pmod{12}), \\
\lbrack 3, 2, (d+4)/12 \rbrack \ ,\ \lbrack 3, -2, (d+4)/12 \rbrack & (-d \equiv 4 \pmod{12}), \\
\lbrack 3, 3, (d+9)/12 \rbrack & (-d \equiv 9 \pmod{12}) \\
\end{array} \right.
\end{eqnarray*}
are forms with discriminant $-d$ and are called the principal form of discriminant $-d$.
\end{dfn}
\begin{lem}
The following conditions are equivalent for a form $Q \in \mathcal{Q}_{d, 3}$:\\
$(1)$ There are $x, y \in \mathbb{Z}$ such that $Q(x, y) = 3$.\\
$(2)$ $Q$ is $\Gamma_0^*(3)$-equivalent to $[3, B, C]$ for some $B, C \in \mathbb{Z}$.\\
$(3)$ $Q$ is $\Gamma_0^*(3)$-equivalent to a principal form of discriminant $-d$.
\end{lem}
This lemma can be proved in the same way as Lemma 2.2 in \cite{Sam15}. The key theorem for the proof of Theorem\ref{main2} is the following.
\begin{thm}
$($Laplace's method$)$. Suppose that $h(t)$ is a real-valued $C^2$-function defined on the interval $(a, b)$ $($with $a, b \in \mathbb{R}$$)$. If we further suppose that $h$ has a unique maximum at $t = c$ with $a < c < b$ so that $h'(c) = 0$ and $h''(c) < 0$, then, we have
\begin{eqnarray*}
\int_a^b e^{\lambda h(t)} dt \sim e^{\lambda h(c)} \biggl(\frac{-2\pi}{\lambda h''(c)} \biggr)^{1/2}
\end{eqnarray*}
as $\lambda \to \infty$.
\end{thm}
We prove Theorem \ref{main2}. By definition,
\begin{eqnarray*}
\textbf{t}_2^{(3*)}(d) &:=& \sum_{Q \in \mathcal{Q}_{d,3} / \Gamma_0^*(3)} \frac{1}{|\Gamma_0^*(3)_Q|} \varphi_2(j_3^*(\alpha_Q)).
\end{eqnarray*}
If $Q = [a, b, c]$ is the element of $\mathcal{Q}_{d, 3}$, we have
\begin{eqnarray*}
e^{2\pi i \alpha_Q} = \exp \biggl(2\pi i \biggl(\frac{-b + i \sqrt{d}}{2a} \biggr) \biggr) = \exp \biggl(-\frac{\pi i b}{a} \biggr) \exp \biggl(-\frac{\pi \sqrt{d}}{a} \biggr)
\end{eqnarray*}
and consequently;
\begin{eqnarray*}
\varphi_2(j_3^*(\alpha_Q)) &=& q^{-2} + O(q)\\
&=& \exp \biggl(\frac{2\pi i b}{a} \biggr) \exp \biggl(\frac{2\pi \sqrt{d}}{a} \biggr) + O\biggl(\exp \biggl(-\frac{\pi \sqrt{d}}{a} \biggr)\biggr).
\end{eqnarray*}
By this calculation, the contribution to $\textbf{t}_2^{(3*)}(d)$ comes only from classes of forms with $a = 3$. By Lemma 4.2, any such form is equivalent to a principal form, so that we have
\begin{eqnarray*}
\textbf{t}_2^{(3*)}(d) = O\biggl(\exp \biggl(-\frac{\pi \sqrt{d}}{3}\biggr)\biggr) + \exp \biggl(\frac{2\pi \sqrt{d}}{3} \biggr) \times \left\{ \begin{array}{ll} 1 & (d \equiv 0, 3\ \bmod 12), \\
-1 & (d \equiv 8, 11 \bmod 12). \\
\end{array} \right.
\end{eqnarray*}
Combining this formula with Theorem\ref{main1}, we obtain
\begin{eqnarray*}
c_n^{(3)} \sim \frac{1}{2n} \times \left\{ \begin{array}{ll}
-\sum_{\substack{r \equiv 0 (3) \\ 4n \geq r^2}} \exp \bigl(2\pi \sqrt{4n - r^2}/3 \bigr) & (n \equiv 0 \bmod 3), \\
\sum_{\substack{r \equiv 1, 2 (3) \\ 4n \geq r^2}} \exp \bigl(2\pi \sqrt{4n - r^2}/3 \bigr) & (n \equiv 1 \bmod 3), \\
-\sum_{\substack{r \equiv 0 (3) \\ 4n \geq r^2}} \exp \bigl(2\pi \sqrt{4n - r^2}/3 \bigr) & (n \equiv 2 \bmod 3). \\
\end{array} \right.
\end{eqnarray*}
For each $k = 0, 1, 2$, we consider the sum
\begin{eqnarray*}
S_n^{(k)} := \frac{3}{2\sqrt{n}} \sum_{\substack{r \equiv k (3) \\ 4n \geq r^2}} e^{\frac{4}{3}\pi \sqrt{n} \sqrt{1 - \frac{r^2}{4n}}}
= \frac{3}{2\sqrt{n}} \sum_{\substack{l \in \mathbb{Z} \\ 4n \geq (3l + k)^2}} e^{\frac{4}{3}\pi \sqrt{n} \sqrt{1 - \frac{(3l + k)^2}{4n}}},
\end{eqnarray*}
and view this sum as a Riemann sum for the function $t \mapsto e^{4\pi \sqrt{n} \sqrt{1 - t^2}/3}$ : $(-1, 1) \to \mathbb{R}$. We can show that $S_n^{(k)}$ is asymptotic to the corresponding Riemann integral $J_n$ where
\begin{eqnarray*}
J_n := \int_{-1}^1 e^{4\pi \sqrt{n} \sqrt{1 - t^2}/3} dt.
\end{eqnarray*}
(For further detail, see \cite{Sam15}). Moreover, applying Laplace's method to the case $\lambda = \sqrt{n}$ and $h(t) = 4\pi \sqrt{1 - t^2}/3$ on $(-1, 1)$, we have
\begin{eqnarray*}
J_n \sim e^{\sqrt{n} \cdot 4 \pi/3} \cdot \biggl(\frac{-2\pi}{-4\pi \sqrt{n}/3} \biggr)^{1/2} = \frac{\sqrt{3}}{\sqrt{2} n^{1/4}} e^{4\pi \sqrt{n}/3}.
\end{eqnarray*}
Putting these asymptotic formulas together, we obtain
\begin{eqnarray*}
c_n^{(3)} &\sim& \frac{1}{3 \sqrt{n}} \times \left\{ \begin{array}{ll}
-S_n^{(0)} & (n \equiv 0 \bmod 3), \\
S_n^{(1)} + S_n^{(2)} & (n \equiv 1 \bmod 3), \\
-S_n^{(0)} & (n \equiv 2 \bmod 3), \\
\end{array} \right. \\
&\sim& \frac{e^{4\pi \sqrt{n}/3}}{\sqrt{6} n^{3/4}} \times \left\{ \begin{array}{ll}
-1 & (n \equiv 0 \bmod 3), \\
2 & (n \equiv 1 \bmod3), \\
-1 & (n \equiv 2 \bmod3) \\
\end{array} \right.
\end{eqnarray*}
as $n \to \infty$.
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\section{Tables of $\textbf{t}_m^{(p*)}(d)$ and $\textbf{t}_m^{(p)}(d)$ \ $(-4 \leq d \leq 50)$}
\begin{table}[h]
\begin{minipage}{9cm}
\begin{tabular}[t]{|c|r|r|r|r|} \hline
$d$ & $\textbf{t}_1^{(2*)}(d)$ & $\textbf{t}_2^{(2*)}(d)$ & $\textbf{t}_1^{(2)}(d)$ & $\textbf{t}_2^{(2)}(d)$\\ \hline \hline
$-$4 & 0 & $-$2 & 0 & $-$4 \\ \hline
$-$1 & $-$1 & $-$1 & $-$1 & $-$1 \\ \hline
0 & 1 & 5 & 2 & 10 \\ \hline
4 & $-$26 & 518 & $-$52 & 1036 \\ \hline
7 & $-$23 & $-$8215 & $-$23 & $-$8215 \\ \hline
8 & 76 & 7180 & 152 &14360 \\ \hline
12 & $-$248 & 52760 & $-$496 & 105520 \\ \hline
15 & $-$1 & $-$385025 & $-$1 & $-$385025 \\ \hline
16 & 518 & 287710 & 1036 & 575420 \\ \hline
20 & $-$1128 &1263640 & $-$2256 & 2527280 \\ \hline
23 & $-$94 & $-$6987870 & $-$94 & $-$6987870 \\ \hline
24 & 2200 & 4831256 & 4400 & 9662512 \\ \hline
28 & $-$4096 & 16572370 & $-$8192 & 33144740 \\ \hline
31 & 93 & $-$78987171 & 93 & $-$78987171 \\ \hline
32 & 7180 & 52263100 & 14360 & 104526200 \\ \hline
36 & $-$12418 & 153553438 & $-$24836 & 307106876 \\ \hline
39 & $-$236 & $-$663068908 & $-$236 & $-$663068908 \\ \hline
40 & 20632 & 425670680 & 41264 & 851341360 \\ \hline
44 & $-$33512 & 1122593352 & $-$67024 & 2245186704 \\ \hline
47 & 235 & $-$4515675925 & 235 & $-$4515675925 \\ \hline
48 & 53256 & 2835914280 & 106512 & 5671828560 \\ \hline
\end{tabular}
\end{minipage}
\begin{minipage}{9cm}
\begin{tabular}[t]{|c|r|r|r|r|} \hline
$d$ & $\textbf{t}_1^{(3*)}(d)$ & $\textbf{t}_2^{(3*)}(d)$ & $\textbf{t}_1^{(3)}(d)$ & $\textbf{t}_2^{(3)}(d)$\\ \hline \hline
$-$4 & 0 & $-$2 & 0 & $-$2 \\ \hline
$-$1 & $-$1 & $-$1 & $-$1 & $-$1 \\ \hline
0 & 1 & 3 & 2 & 6 \\ \hline
3 & $-$7 & 33 & $-$14 & 66 \\ \hline
8 & $-$34 & $-$410 & $-$34 & $-$410 \\ \hline
11 & 22 & $-$1082 & 22 &$-$1082 \\ \hline
12 & 26 & 1428 & 52 & 2856 \\ \hline
15 & $-$69 & 3195 & $-$138 & 6390 \\ \hline
20 & $-$116 & $-$11892 & $-$116 & $-$11892 \\ \hline
23 & 115 & $-$22797 & 115 & $-$22797 \\ \hline
24 & 174 & 28710 & 348 & 57420 \\ \hline
27 & $-$241 & 53223 & $-$482 & 106446 \\ \hline
32 & $-$410 & $-$140222 & $-$410 & $-$140222 \\ \hline
35 & 492 & $-$240500 & 492 & $-$240500 \\ \hline
36 & 492 & 287244 & 984 & 574488 \\ \hline
39 & $-$705 & 477567 & $-$1410 & 955134 \\ \hline
44 & $-$1060 & $-$1081096 & $-$1060 & $-$1081096 \\ \hline
47 & 1272 & $-$1718792 & 1272 & $-$1718792 \\ \hline
48 & 1442 & 2004918 & 2884 & 4009836 \\ \hline
\end{tabular}
\end{minipage}
\end{table}
\begin{table}[h]
\begin{tabular}{|c|r|r|r|r|} \hline
$d$ & $\textbf{t}_1^{(5*)}(d)$ & $\textbf{t}_2^{(5*)}(d)$ & $\textbf{t}_1^{(5)}(d)$ & $\textbf{t}_2^{(5)}(d)$\\ \hline \hline
$-$4 & 0 & $-$2 & 0 & $-$2 \\ \hline
$-$1 & $-$1 & $-$1 & $-$1 & $-$1 \\ \hline
0 & 1 & 3 & 2 & 6 \\ \hline
4 & $-$8 & $-$6 & $-$8 & $-$6 \\ \hline
11 & $-$12 & $-$124 & $-$12 & $-$124 \\ \hline
15 & $-$19 & 93 & $-$38 & 186 \\ \hline
16 & $-$6 & $-$270 & $-$6 & $-$270 \\ \hline
19 & 20 & 132 & 20 & 132 \\ \hline
20 & 6 & 268 & 12 & 536 \\ \hline
24 & $-$44 & 216 & $-$44 & 216 \\ \hline
31 & $-$39 & $-$1863 & $-$39 & $-$1863 \\ \hline
35 & $-$44 & 1668 & $-$88 & 3336 \\ \hline
36 & 20 & $-$3054 & 20 & $-$3054 \\ \hline
39 & 53 & 1653 & 53 & 1653 \\ \hline
40 & 56 & 2868 & 112 & 5736 \\ \hline
44 & $-$136 & 2416 & $-$136 & 2416 \\ \hline
\end{tabular}
\end{table}
\end{ack}
\noindent
T. Matsusaka: Graduate School of Mathematics, Kyushu University, Motooka 744, Nishi-ku Fukuoka 819-0395, Japan\\
e-mail: [email protected]\\
\noindent
R. Osanai:\\
e-mail: [email protected]
\end{document}
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\begin{document}
\title{A Tube-based MPC Scheme for Interaction Control of Underwater Vehicle Manipulator Systems\\
\thanks{This work was supported by the H2020 ERC Grant BUCOPHSYS, the EU H2020 Co4Robots project, the Swedish Foundation for Strategic Research (SSF), the Swedish Research Council (VR) and the Knut och Alice Wallenberg Foundation (KAW).}
}
\author{\IEEEauthorblockN{Alexandros Nikou, Christos K. Verginis and Dimos V. Dimarogonas}
\IEEEauthorblockA{Department of Automatic Control \\
School of Electrical Engineering and Computer Science\\
KTH Royal Institute of Technology, Stockholm, Sweden \\
{\tt \{anikou,cverginis,dimos\}@kth.se}}
}
\maketitle
\begin{abstract}
Over the last years, the development of Autonomous Underwater Vehicles (AUV) with attached robotic manipulators, the so-called Underwater Vehicle Manipulator System (UVMS), has gained significant research attention, due to the ability of interaction with underwater environments. In such applications, force/torque controllers which guarantee that the end-effector of the UVMS applies desired forces/torques towards the environment, should be designed in a way that state and input constraints are taken into consideration. Furthermore, due to their complicated structure, unmodeled dynamics as well as external disturbances may arise. Motivated by this, we proposed a robust Model Predicted Control Methodology (NMPC) methodology which can handle the aforementioned constraints in an efficient way and it guarantees that the end-effector is exerting the desired forces/torques towards the environment. Simulation results verify the validity of the proposed framework.
\end{abstract}
\section{Introduction}
Most of the underwater manipulation tasks, such as maintenance of ships, underwater weld inspection, surveying oil/gas searching, require the manipulator mounted on the vehicle to be in contact with the underwater object or environment (see \cite{antonelli, cieslak2015autonomous}). The aforementioned tasks are usually complex due to highly nonlinear dynamics, the presence of uncertainties, external disturbances as well as state and control input (actuation) constraints. Thus, these constraints should be taken into account in the force control design process in an efficient way.
Motivated by the aforementioned, this paper considers the modeling of a general UVMS in compliant contact with a planar surface, and the development of a constrained Nonlinear Model Predictive Control (NMPC) scheme for force/torque control. NMPC for manipulation of nominal system dynamics has been proposed in \cite{alex_med} for stabilization of ground vehicles with attached manipulators to pre-defined positions. In this work, we propose a novel robust tube-based NMPC force control approach that efficiently deals with state and input constraints and achieves a desired exerted force from the UVMS to the environment. In particular, the controller consists of two terms: a nominal control input, which is computed on-line and is the outcome of a Finite Horizon Optimal Control Problem (FHOCP) that is repeatedly solved at every sampling time, for its nominal system dynamics; and an additive state feedback law which is computed off-line and guarantees that the real trajectory of the closed-loop system will belong to a hyper-tube centered along the nominal trajectory. The volume of the hyper-tube depends on the upper bound of the disturbances, the bounds of the Jacobian matrix as well as Lipschitz constants of the UVMS dynamics. Under the assumption that the FHOCP is feasible at time $t = 0$, we guarantee the boundedness of the closed-loop system states.
The rest of this manuscript is structured as follows: Section \ref{sec:notation_preliminaries} provides the notation that will be used as well as necessary background knowledge; in Section \ref{sec:problem_formulation}, the problem treated in this paper is formally defined; Section \ref{sec:main_results} contains the main results of the paper; Section \ref{sec:simulation_results} is devoted to numerical simulations; and in Section \ref{sec:conclusions}, conclusions and future research directions are discussed.
\section{Notation and Preliminaries} \label{sec:notation_preliminaries}
Define by $\mathbb{N}$ and $\mathbb{R}$ the sets of positive integers and real numbers, respectively. Given the set $\mathcal{S}$, define by $S^n \coloneqq S \times \dots \times S$, its $n$-fold Cartesian product. Given vector $z \in \mathbb{R}^{n}$ define by $$\|z\|_{2} \coloneqq \sqrt{z^\top z}, \ \ \|z\|_{P} \coloneqq \sqrt{z^\top P z},$$ its Euclidean and weighted norm, with $P \ge 0$. Given vectors $z_1$, $z_2 \in \mathbb{R}^3$, $\mathcal{S}: \mathbb{R}^3 \to \mathfrak{so}(3)$ stands for the skew-symmetric matrix defined according to $\mathcal{S}(z_1) z_2 = z_1 \times z_2$ where $$\mathfrak{so}(3) \coloneqq \left\{\mathcal{S} \in \mathbb{R}^{3\times 3} : z^\top \mathcal{S}(\cdot) z = 0, \forall z \in \mathbb{R}^{3} \right\}.$$ $\lambda_{\scriptscriptstyle \min}(P)$ stands for the minimum absolute value of the real part of the eigenvalues of $P \in \mathbb{R}^{n \times n}$; $0_{m \times n} \in \mathbb{R}^{m \times n}$ and $I_n \in \mathbb{R}^{n \times n}$ stand for the $m \times n$ matrix with all entries zeros and the identity matrix, respectively. Given coordination frames $\Sigma_i$, $\Sigma_j$, denote by $R^j_i$ the transformation from $\Sigma_i$ to $\Sigma_j$. Given~sets~$\mathcal{S}_1$, $\mathcal{S}_2$~$\subseteq \mathbb{R}^n$, $\mathcal{S} \subseteq \mathbb{R}^{m}$~and~matrix $B \in \mathbb{R}^{n \times m}$,~the \emph{Minkowski addition}, the~\emph{Pontryagin~difference} and the \emph{matrix-set multiplication} are respectively defined by:
\begin{align*}
\mathcal{S}_1 \oplus \mathcal{S}_2 & \coloneqq \{s_1 + s_2 : s_1 \in \mathcal{S}_1, s_2 \in \mathcal{S}_2\}, \\
\mathcal{S}_1 \ominus \mathcal{S}_2 & \coloneqq \{s_1 : s_1+s_2 \in \mathcal{S}_1, \forall s_2 \in \mathcal{S}_2\}, \\
B \circ \mathcal{S} & \coloneqq \{b: b = Bs, s \in \mathcal{S} \}.
\end{align*}
\begin{lemma} \cite{alex_IJRNC_2018} \label{lemma:basic_ineq}
For any constant $\rho > 0$, vectors $z_1$, $z_2 \in \mathbb{R}^n$ and matrix $P \in \mathbb{R}^{n \times n}$, $P > 0$ it holds that $$z_1 P z_2 \le \tfrac{1}{4 \rho} z_1^\top P z_1 + \rho z_2^\top P z_2.$$
\end{lemma}
\begin{definition} \label{def:RPI_set} \cite{alex_IJRNC_2018}
Consider a dynamical system $\dot{\chi} = f(\chi,u,d)$ where: $\chi \in \mathcal{X}$, $u \in \mathcal{U}$, $d \in \mathcal{D}$ with initial condition $\chi(0) \in \mathcal{X}$. A set $\mathcal{X}' \subseteq \mathcal{X}$ is a \emph{Robust Control Invariant (RCI) set} for the system, if there exists a feedback control law $u \coloneqq \kappa(\chi) \in \mathcal{U}$, such that for all $\chi(0) \in \mathcal{X}'$ and for all $d \in \mathcal{D}$ it holds that $\chi(t) \in \mathcal{X}'$ for all $t \ge 0$, along every solution $\chi(t)$.
\end{definition}
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\section{Problem Formulation} \label{sec:problem_formulation}
\subsection{Kinematic Model}
Consider a UVMS which is composed of an AUV and a $n$ Degree Of Freedom (DoF) manipulator mounted on the base of the vehicle. The AUV can be considered as a six DoF rigid body with position and orientation vector $\eta \coloneqq [x, y, z~\vline~\phi, \theta, \psi]^\top \in \mathbb{R}^6$, where the components of the vectors have been named according to SNAME \cite{SNAME} as surge, sway, heave, roll, pitch and yaw respectively. The joint angular position state vector of the manipulator is defined by $q \coloneqq [ q_1,\dots,q_n]^\top \in \mathbb{R}^n$. Define by $\dot{q} \coloneqq [\dot{q}_1,\dots,\dot{q}_n]^\top \in \mathbb{R}^n$ the corresponding joint velocities.
In order to describe the motion of the combined system, the earth-fixed inertial frame $\Sigma_I$, the body-fixed frame $\Sigma_B$ and the end-effector fixed frame $\Sigma_E$ are introduced (see Fig. \ref{fig:uvms_frames}). Moreover, without loss of generality, the reference frame $\Sigma_0$ is chosen to be located at the manipulator's base, and the frames $\Sigma_1, \ldots, \Sigma_n$ are located to the $1$-st$,\ldots,n$-th link of the manipulator, respectively, under the Denavit-Hartenberg convention \cite{sciavicco2012modelling}. The translational and rotational kinematic equations for the AUV system (see \cite{antonelli}) are given by:
\begin{subequations} \label{eq:kin}
\begin{align}
\dot{\eta}
& =
\begin{bmatrix}
\dot{\eta}_1 \\
\dot{\eta}_2 \\
\end{bmatrix}
= \mathfrak{J}(\eta_2)
\begin{bmatrix}
\nu_1 \\
\nu_2 \\
\end{bmatrix}, \\
\mathfrak{J}(\eta_2) & \coloneqq
\begin{bmatrix}
\mathfrak{J}_1(\eta_2) & 0_{3 \times 3} \\
0_{3 \times 3} & \mathfrak{J}_{2}(\eta_2) \\
\end{bmatrix}, \\
\mathfrak{J}_1(\eta_2) & \coloneqq
\begin{bmatrix}
c_{\theta} c_{\psi} & s_{\phi} s_{\theta} c_{\psi}-s_{\psi} c_{\phi} &
s_{\theta} c_{\phi} c_{\psi}+s_{\phi} s_{\psi} \\
s_{\psi} c_{\theta} & s_{\phi} s_{\theta} s_{\psi}+c_{\phi} c_{\psi} &
s_{\theta} s_{\psi} c_{\phi}-s_{\phi} c_{\psi} \\
-s_{\theta} & s_{\phi} c_{\theta} & c_{\phi} c_{\theta} \\
\end{bmatrix}, \\
\mathfrak{J}_2(\eta_2) & \coloneqq
\begin{bmatrix}
1 & \tfrac{s_{\phi} s_{\theta}}{c_{\theta}} & \tfrac{c_{\phi} s_{\theta}}{c_{\theta}} \\
0 & c_{\phi} & -s_{\phi} \\
0 & \tfrac{s_{\phi}}{c_{\theta}} & \tfrac{c_{\phi}}{c_{\theta}} \\
\end{bmatrix},
\end{align}
\end{subequations}
where $\eta_{1} \coloneqq \left[ x, y, z \right]^{\tau} \in \mathbb{R}^3$, $\eta_{2} \coloneqq \left[\phi, \theta, \psi \right]^\top \in \mathbb{R}^3 $ denote the position vector and the orientation vector of the frame $\Sigma_B$ relative to the frame $\Sigma_I$, respectively; $ \nu_{1}$, $\nu_{2} \in \mathbb{R}^3 $ denote the linear and the angular velocity of the frame $\Sigma_B$ with respect to $\Sigma_I$ respectively; $\mathfrak{J}(\eta_2) \in \mathbb{R}^{6 \times 6}$ stands for the Jacobian matrix transforming the velocities from $\Sigma_B$ to $\Sigma_I$; $\mathfrak{J}_{1}(\eta_2)$, $\mathfrak{J}_2(\eta_2) \in \mathbb{R}^{3 \times 3}$ are the corresponding parts of the Jacobian related to position and orientation, respectively; The notation $s_{\varsigma}$ and $c_{\varsigma}$ stand for the trigonometric functions $\sin(\varsigma)$ and $\cos(\varsigma)$ of an angle $\varsigma \in \mathbb{R}$, respectively.
\begin{figure}
\caption{An AUV Equipped with a n DoF manipulator}
\label{fig:uvms_frames}
\end{figure}
\noindent Denote by $$\mathfrak{q} \coloneqq \left[\eta_1^\top, \eta_2^\top, q^\top \right]^\top\in\mathbb{R}^{6+n},$$ the pose configuration vector of the UVMS. Let $\mathfrak{p}$, $\mathfrak{o} \in \mathbb{R}^3$ be the position and orientation vectors of the end-effector with reference to the frame $\Sigma_I$, respectively. The vectors $\mathfrak{p}$, $\mathfrak{o}$ depend on the pose $\mathfrak{q}$ and they can be obtained by the the following homogeneous transformation:
\begin{equation}
\mathfrak{T}(\mathfrak{q}) \coloneqq
\begin{bmatrix}
R_E^{I}(\mathfrak{q}) & \mathfrak{p}(\mathfrak{q}) \\
0_{1 \times 3} & 1 \\
\end{bmatrix}
=
T_B^I T_0^B T_1^0 \cdots T_n^{n-1} T_E^n, \label{eq:forw_kinematics}
\end{equation}
where: $T^j_i$ is the homogeneous transformation matrix describing the position and orientation of frame $\Sigma_i$ with reference to the frame $\Sigma_j$ with $i$, $j \in \{1,\dots, n, I, 0, B, E\}$. The end-effector linear velocity $\dot{\mathfrak{p}} \in \mathbb{R}^3$ and the time derivative or Euler angles $\dot{\mathfrak{o}} \in \mathbb{R}^3$ are related to the body-fixed velocities $\nu_1$, $\nu_2$ and $\dot{q}$ with the following \emph{kinematics model:}
\begin{equation} \label{eq:kinematics}
\dot{\chi}
=
J(\mathfrak{q}) \zeta,
\end{equation}
where $$\chi \coloneqq [\mathfrak{p}^\top, \mathfrak{o}^\top]^\top \in \mathbb{R}^{6}, \ \ \zeta \coloneqq \left[ \nu_1^\top, \nu_2^\top, \dot{q}^\top \right]^\top \in \mathbb{R}^{{6+n}},$$ is the body-fixed system velocity vector. The Jacobian transformations matrices $$J(\mathfrak{q}) \in \mathbb{R}^{6 \times (6+n)}, \ \ J_{\rm pos}(\mathfrak{q}) \in \mathbb{R}^{3 \times (6 +n)}, \ \ J_{\rm{or}}(\mathfrak{q}) \in \mathbb{R}^{3 \times (6 +n)},$$ are respectively defined by:
\begin{align*}
J(\mathfrak{q}) &\coloneqq \begin{bmatrix}
J_{\rm pos}(\mathfrak{q}) \\
J_{\rm or}(\mathfrak{q}) \\
\end{bmatrix}, \\
J_{\rm pos}(\mathfrak{q}) &\coloneqq
\left[\mathfrak{J}_1(\eta_2)~\vline~- \mathfrak{J}_1(\eta_2) \mathcal{S}(p_{ee})~\vline~R_0^{I} J_{e,1} \right], \\
J_{\rm or}(\mathfrak{q}) &\coloneqq
\left[0_{3 \times 3}~\vline~\mathfrak{J}_2(\mathfrak{o}) R_B^E~\vline~\mathfrak{J}_2(\mathfrak{o}) R_0^{E} J_{e,2} \right].
\end{align*}
In the latter, the vector $p_{ee} \in \mathbb{R}^{3}$ is the local position of the end-effector with reference to the frame $\Sigma_B$; the matrices $J_{e,1}$, $J_{e,2} \in \mathbb{R}^{3 \times n}$ represent the manipulator Jacobian matrices with respect to the frame $\Sigma_0$; and $\mathcal{S}(\cdot)$ the skew-symmetric matrix as given in Section \ref{sec:notation_preliminaries}. For the aforementioned transformations we refer to \cite{sciavicco2012modelling}.
\subsection{Dynamic Model}
When the end-effector of the robotic system is in contact with the environment, the force at the tip of the manipulator acts on the whole system according to the following uncertain nonlinear dynamics:
\begin{align} \label{eq:dynamics}
\dot{\zeta} = f(\chi, \zeta)+ \mathfrak{u} + d(\mathfrak{q}, \zeta, t),
\end{align}
where:
\begin{align}
& \hspace{-4mm} f(\chi, \zeta) \coloneqq \notag \\
& \hspace{-4mm} - M(\mathfrak{q})^{-1} \Big\{ C(\zeta, \mathfrak{q}) \zeta + D(\zeta, \mathfrak{q}) \zeta +g(\mathfrak{q}) + J^{\top}(\mathfrak{q}) \mathfrak{F}(\chi) \Big\}, \hspace{-4mm} \label{eq:func_h}
\end{align}
where $M(\mathfrak{q}) \in \mathbb{R}^{({6+n}) \times ({6+n})}$ is the inertia matrix for which it holds that: $z^\top M(\mathfrak{q}) z > 0$, $\forall z \in \mathbb{R}^{6+n}$; $C(\zeta, \mathfrak{q}) \in \mathbb{R}^{({6+n}) \times ({6+n})}$ is the matrix of Coriolis and centripetal terms; $D(\zeta, \mathfrak{q}) \in \mathbb{R}^{({6+n}) \times ({6+n})}$ is the matrix of dissipative effects; $d(\mathfrak{q}, \zeta, t) \in \mathbb{R}^{6+n}$ is a vector that models the external disturbances, uncertainties and unmodeled dynamics of the system; $g(\mathfrak{q}) \in \mathbb{R}^{({6+n})}$ is the vector of gravity and buoyancy effects; $\mathfrak{u} \in \mathbb{R}^{6+n}$ denotes the vector of the propulsion forces and moments acting on the vehicle in the frame $\Sigma_{B}$ as well as the joint torques; $\mathfrak{F}(\chi) \in \mathbb{R}^{6}$ is the vector of interaction forces and torques exerted by the end-effector towards the environment expressed in $\Sigma_I$.
In this paper, an interaction between the end-effector and a frictionless, elastically compliant surface is assumed. Then, according to \cite{siciliano_force_control}, the vector of interaction forces and torques that is exerted by the end-effector can be written as:
\begin{equation} \label{eq:force_F}
\mathfrak{F}(\chi) \coloneqq K (\chi - \chi_{\scriptscriptstyle \rm eq}),
\end{equation}
where $K \in \mathbb{R}^{6 \times 6}$, $K > 0$ stands for the stiffness matrix, which represents elastic coefficient of the environment, and $\chi_{\scriptscriptstyle \rm eq} \in \mathbb{R}^{6}$ is the given constant vector of the equilibrium position/orientation of the undeformed environment.
We also consider that the UVMS is in the presence of state and input constraints given by $\mathfrak{q} \in \mathcal{Q}$, $\zeta \in \mathcal{Z}$, $u \in \mathcal{U}$,
where $\mathcal{Q} \subseteq \mathbb{R}^{6+n}$, $\mathcal{Z} \subseteq \mathbb{R}^{6+n}$ and $\mathcal{U} \subseteq \mathbb{R}^{6+n}$ are \emph{connected sets containing the origin}. For certain technical reasons that will be presented thereafter, the constraints imposed to the configuration states $\mathfrak{q}$ are given by:
\begin{align} \label{eq:set_Q}
\mathcal{Q} \coloneqq \Big\{ \mathfrak{q} \in \mathbb{R}^{6+n} : & \ \lambda_{\scriptscriptstyle \min}\left[ \tfrac{J^{+}(\mathfrak{q})+J^{+}(\mathfrak{q})^\top}{2} \right] \ge \underline{J}, \notag \\
& \ \ \|J(\mathfrak{q})\|_2 \le \overline{J}, \ \|\dot{J}(\mathfrak{q})\|_{2} \le \widetilde{J} \Big\},
\end{align}
where $J^{+}(\mathfrak{q}) \coloneqq J(\mathfrak{q}) J(\mathfrak{q})^\top$ and $\underline{J}$, $\overline{J}$, $\widetilde{J} > 0$. According to \eqref{eq:forw_kinematics}, the constraints $\mathfrak{q} \in \mathcal{Q}$ impose also constraints on the vector $\chi \in \mathcal{X} \subseteq \mathbb{R}^{6}$, where the set $\mathcal{X}$ can be computed by the transformation $\mathfrak{T}(\mathfrak{q})$, as given in \eqref{eq:forw_kinematics}. Note also that the function $f$ given in \eqref{eq:func_h} is continuously differentiable in the set $\mathcal{Q} \times \mathcal{X} \times \mathcal{Z}$. Furthermore, assume bounded disturbances $d \in \mathcal{D}$ where:
$\mathcal{D}$ $\coloneqq \big\{d \in \mathbb{R}^{6+n}:$ $\|d(\mathfrak{q}, \zeta, t)\|_{2} \le \widetilde{d}$, $\forall (\mathfrak{q}$, $\zeta) \in \mathcal{Q}$ $\times \mathcal{Z}\big\}$, where $\widetilde{d} > 0$.
For the kinematics/dynamics \eqref{eq:kinematics},\eqref{eq:dynamics}, define the corresponding \emph{nominal kinematics/dynamics} by:
\begin{subequations}
\begin{align}
\dot{\overline{\chi}} & = J(\overline{\mathfrak{q}}) \overline{\zeta}, \label{eq:nom_kinematics} \\
\dot{\overline{\zeta}} & = f(\overline{\chi}, \overline{\zeta})+ \overline{u}, \label{eq:nom_dynamics}
\end{align}
\end{subequations}
where $d(\cdot) \equiv 0$, $\overline{\mathfrak{q}} \in \mathcal{Q}$, $\overline{\chi} \in \mathcal{X}$, $\overline{\zeta} \in \mathcal{Z}$ and $\overline{u} \in \mathcal{U}$. Define the stack vector $\overline{\xi} \coloneqq [\overline{\chi}, \overline{\zeta}]^\top \in \mathbb{R}^{12+n}$ and consider the linear nominal system $\dot{\overline{\xi}} = A \overline{\xi} + B \overline{u}, \ \ A \in \mathbb{R}^{(12+n) \times (12+n)}, \ \ B \in \mathbb{R}^{(12+n) \times (6+n)}$,
which is the outcome of the Jacobian linearization of the nominal dynamics \eqref{eq:nom_kinematics},\eqref{eq:nom_dynamics} around the equilibrium point $\xi = 0$. Due to the dimension of the control input ($6+n > 6$), the stabilization of the state $\overline{\chi}$ to the desired state $\chi_{\scriptscriptstyle \rm des}$ can be achieved. Therefore, the linear system is stabilizable.
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\subsection{Problem Statement}
\begin{problem} \label{problem}
Consider a UVMS composed of an AUV and an attached manipulator with $n$ DoF, which is in contact with a surface of a compliant environment. The UVMS is governed by the kinematics and dynamics models given in \eqref{eq:kinematics} and \eqref{eq:dynamics}, respectively. The system is in the presence of state and input constraints as well as bounded disturbances which are respectively given by:
\begin{align} \label{eq:constr}
\mathfrak{q} \in \mathcal{Q}, \ \chi \in \mathcal{X}, \ \zeta \in \mathcal{Z}, \ \mathfrak{u} \in \mathcal{U}, \ d \in \mathcal{D}.
\end{align}
Given a vector $\mathfrak{F}_{\scriptscriptstyle \rm des} \in \mathbb{R}^{6}$ that satisfies \eqref{eq:force_F} and stands for the desired force/torque vector that the end-effector is required to exert towards a surface of the environment, design a \emph{feedback control law} $\mathfrak{u} \coloneqq \kappa(\chi, \zeta)$ such that $\lim\limits_{t \to \infty} \|\mathfrak{F}(\chi(t))-\mathfrak{F}_{\scriptscriptstyle \rm des}\|_{2} \to 0$, while all the constraints given in \eqref{eq:constr} are satisfied.
\end{problem}
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\section{Main Results} \label{sec:main_results}
In this section, we propose a novel feedback control law that solves Problem \ref{problem} in a systematic way. Due to the fact that it is required to design a feedback control law that guarantees the minimization of the term $\|\mathfrak{F}(t)-\mathfrak{F}_{\scriptscriptstyle \rm des}\|_{2}$, as $t \to \infty$, under state and input constraints given by \eqref{eq:constr}, we utilize a Nonlinear Model Predictive Control (NMPC) framework \cite{michalska_1993, frank_1998_quasi_infinite, mayne_2000_nmpc}. Furthermore, since the UVMS is under the presence of disturbances/uncertainties $d \in \mathcal{D}$, we provide a robust analysis, the so-called tube-based robust NMPC approach \cite{yu_2013_tube, alex_IJRNC_2018}. In particular, first, the error states and the corresponding transformed constraints sets are defined in Section \ref{sec:error_constr}. Then, the proposed feedback control law consists of two parts: an on-line control law which is the outcome of a solution to a Finite Horizon Optimal Control Problem (FHOCP) for the nominal system dynamics (see Section \ref{sec:optimal_contol}); and a state feedback law which is designed off-line and guarantees that the real system trajectories always lie within a hyper-tube centered along the nominal trajectories (see \ref{sec:state_feedback_law}).
\subsection{Errors and Constraints} \label{sec:error_constr}
According to \eqref{eq:force_F}, for the error between the actual $\mathfrak{F}$ and the desired $\mathfrak{F}_{\scriptscriptstyle \rm des}$ forces/torques exerted from the end-effector to the surface it holds that: $\mathfrak{F}-\mathfrak{F}_{\scriptscriptstyle \rm des}$ $= K (\chi - \chi_{\scriptscriptstyle \rm eq})$ $-K (\chi_{\scriptscriptstyle \rm des}$ $- \chi_{\scriptscriptstyle \rm eq})$ $= K (\chi-\chi_{\scriptscriptstyle \rm des})$,
where $\chi_{\scriptscriptstyle \rm des} \coloneqq K^{-1} \mathfrak{F}_{\scriptscriptstyle \rm des} + \chi_{\scriptscriptstyle \rm eq} \in \mathbb{R}^{6}$. The latter implies that if we design a feedback control law $u = \kappa(\chi, \zeta)$ which guarantees that $\lim\limits_{t \to \infty} \|\chi(t)-\chi_{\scriptscriptstyle \rm des}\|_{2} \to 0$, while all the constraints given in \eqref{eq:constr} are satisfied, Problem \ref{problem} will have been solved.
Define the error state $e \coloneqq \chi-\chi_{\scriptscriptstyle \rm des} \in \mathbb{R}^{6}$. Then, the \emph{uncertain error kinematics/dynamics} are given by:
\begin{subequations}
\begin{align}
\dot{e} & = J(\mathfrak{q}) \zeta, \label{eq:unsrt_error_kin} \\
\dot{\zeta} & = f(e+\chi_{\scriptscriptstyle \rm des}, \zeta)+ \mathfrak{u} + d(\mathfrak{q}, \zeta, t), \label{eq:unsrt_error_dyn}
\end{align}
\end{subequations}
and the corresponding \emph{nominal error kinematics/dynamics} by:
\begin{subequations}
\begin{align}
\dot{\overline{e}} & = J(\overline{\mathfrak{q}}) \overline{\zeta}, \label{eq:nom_error_kin} \\
\dot{\overline{\zeta}} & = f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \overline{\zeta})+ \overline{u}, \label{eq:nom_error_dyn}
\end{align}
\end{subequations}
In order to translate the constraints for the state $\chi \in \mathcal{X}$ to constraints that are dictated regarding the error $e$, the constraints set $\mathcal{E} \coloneqq \{e \in \mathbb{R}^{6}: e \in \mathcal{X} \oplus (-\chi_{\scriptscriptstyle \rm des}) \}$ is introduced.
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\subsection{Feedback Control Design} \label{sec:state_feedback_law}
\noindent Consider the feedback law:
\begin{equation} \label{eq:control_law_u}
\mathfrak{u} \coloneqq \overline{u}(\overline{e}, \overline{\zeta}) + \kappa(e, \zeta, \overline{e}, \overline{\zeta}),
\end{equation}
which consists of a nominal control law $\overline{u}(\overline{e}, \overline{\zeta}) \in \mathcal{U}$ and a state feedback law $\kappa(\cdot)$. The control action $\overline{u}(\overline{e}, \overline{\zeta})$ will be the outcome of a FHOCP for the nominal kinematics/dynamics \eqref{eq:nom_error_kin},\eqref{eq:nom_error_dyn} which is solved on-line at each sampling time. The state feedback law $\kappa(\cdot)$ is used to guarantee that the real trajectories $e(t)$, $\zeta(t)$, which are the solution to \eqref{eq:unsrt_error_kin},\eqref{eq:unsrt_error_dyn}, always remain within a bounded hyper-tube centered along the nominal trajectories $\overline{e}(t)$,~$\overline{\zeta}(t)$ which are~the~solution~ to~\eqref{eq:nom_error_kin},\eqref{eq:nom_error_dyn}.
Define by $\mathfrak{e} \coloneqq e - \overline{e} \in \mathbb{R}^{6}$ and $\mathfrak{z} \coloneqq \zeta - \overline{\zeta} \in \mathbb{R}^{6+n}$ the deviation between the real states of the uncertain system \eqref{eq:unsrt_error_kin},\eqref{eq:unsrt_error_dyn} and the states of the nominal system \eqref{eq:nom_error_kin},\eqref{eq:nom_error_dyn}, respectively, with $\mathfrak{e}(0) = \mathfrak{z}(0) = 0$. It will be proved hereafter that the trajectories $\mathfrak{e}(t)$, $\mathfrak{z}(t)$ remain invariant in compact sets. The dynamics of the states $\mathfrak{e}$, $\mathfrak{z}$ are written as:
\begin{subequations}
\begin{align}
\dot{\mathfrak{e}} & = \mathfrak{b}(\chi, \overline{\chi}, \zeta) + J(\overline{\mathfrak{q}}) \mathfrak{z}, \label{eq:frak_e} \\
\dot{\mathfrak{z}} & = \mathfrak{l}(e, \overline{e}, \zeta, \overline{\zeta})+(\mathfrak{u}-\overline{u}) + d(\mathfrak{q}, \zeta, t), \label{eq:frak_z}
\end{align}
\end{subequations}
where the functions $\mathfrak{b}$, $\mathfrak{l}$ are defined by: $\mathfrak{b}(\chi, \overline{\chi}, \zeta) \coloneqq \mathfrak{c}(\chi, \zeta)-\mathfrak{c}(\overline{\chi}, \zeta)$, $\mathfrak{l}(e, \overline{e}, \zeta, \overline{\zeta}) \coloneqq f(e+\chi_{\scriptscriptstyle \rm des}, \zeta)-f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \overline{\zeta})$, with $\mathfrak{c}(\chi, \zeta) \coloneqq J(\mathfrak{q}) \zeta$. Since the aforementioned functions are continuously differentiable, the following hold:
\begin{align*}
\|\mathfrak{b}(\cdot)\|_2 & = \|\mathfrak{c}(\chi, \zeta)-\mathfrak{c}(\overline{\chi}, \zeta)\|_2 \le L_{\scriptscriptstyle \mathfrak{c}} \|\chi-\overline{\chi}\|_2 = L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2, \\
\|\mathfrak{l}(\cdot)\|_{2} & \le \|f(e+\chi_{\scriptscriptstyle \rm des}, \zeta)- f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \zeta)\|_{2} \notag \\
&\hspace{12mm}+\|f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \zeta)- f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \overline{\zeta})\|_2 \notag \\
& \le L_1\|e-\overline{e}\|_{2} + L_2 \|\zeta-\overline{\zeta}\|_{2} \le L \left( \|\mathfrak{e}\|_{2} + \|\mathfrak{z}\|_{2} \right).
\end{align*}
The constant $L_{\scriptscriptstyle \mathfrak{c}}$ stands for the Lipschitz constant of function $\mathfrak{c}$ with respect to the variable $\chi$; $L_1$, $L_2$ stand for the Lipschitz constants of function $h$ with respect to the variables $\chi$ and $\zeta$, respectively, and $L \coloneqq \max\{L_1, L_2\}$.
\begin{lemma} \label{lemma:tube}
The state feedback law designed by:
\begin{equation} \label{eq:kappa_law}
\kappa(e, \overline{e}, \zeta, \overline{\zeta}) \coloneqq - k (e-\overline{e})-k \sigma J(\overline{q})^\top (\zeta -\overline{\zeta}),
\end{equation}
where $k$, $\sigma > 0$ are chosen such that the following hold:
\begin{subequations}
\begin{align}
\underline{\sigma} & > 0,\ \ \sigma \coloneqq \frac{L_{\scriptscriptstyle \mathfrak{c}}+\underline{\sigma}}{\underline{J}},
\ \ \rho > \tfrac{\Lambda_1}{4 \underline{\sigma}}, \ \ k > \rho \Lambda_1 + \Lambda_2, \label{eq:sigma_under_sigma} \\
\Lambda_1 & \coloneqq \left[L + \overline{J}+ \sigma \left( L_{\scriptscriptstyle \mathfrak{c}} + \widetilde{J}\right) \right], \Lambda_2 \coloneqq \left(L + \sigma \overline{J}^2\right), \label{eq:Lambda_1}
\end{align}
\end{subequations}
renders the sets:
\begin{subequations}
\begin{align}
\Omega_1 & \coloneqq \left\{\mathfrak{e} \in \mathbb{R}^{6} : \|\mathfrak{e}\|_2 \le \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}} \right\}, \label{eq:omega_1} \\
\Omega_2 & \coloneqq \left\{\mathfrak{z} \in \mathbb{R}^{6+n} : \|\mathfrak{z}\|_2 \le \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}} \right\}, \label{eq:omega_2}
\end{align}
\end{subequations}
RCI sets for the error dynamics \eqref{eq:frak_e}, \eqref{eq:frak_z}, according to Definition \ref{def:RPI_set}. The constants $\alpha_1$, $\alpha_2 > 0$ are defined by:
\begin{equation}
\alpha_1 \coloneqq \underline{\sigma}- \tfrac{\Lambda_1}{4 \rho}, \ \alpha_2 \coloneqq k-\rho \Lambda_1-\Lambda_2. \label{eq:a_1_a_2}
\end{equation}
\end{lemma}
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\begin{lemma} \label{lemma:tube}
The state feedback law designed by:
\begin{equation} \label{eq:kappa_law}
\kappa(e, \overline{e}, \zeta, \overline{\zeta}) \coloneqq - k (e-\overline{e})-k \sigma J(\overline{q})^\top (\zeta -\overline{\zeta}),
\end{equation}
where $k$, $\sigma > 0$ are chosen such that the following hold:
\begin{subequations}
\begin{align}
\underline{\sigma} & > 0,\ \ \sigma \coloneqq \frac{L_{\scriptscriptstyle \mathfrak{c}}+\underline{\sigma}}{\underline{J}},
\ \ \rho > \tfrac{\Lambda_1}{4 \underline{\sigma}}, \ \ k > \rho \Lambda_1 + \Lambda_2, \label{eq:sigma_under_sigma} \\
\Lambda_1 & \coloneqq \left[L + \overline{J}+ \sigma \left( L_{\scriptscriptstyle \mathfrak{c}} + \widetilde{J}\right) \right], \Lambda_2 \coloneqq \left(L + \sigma \overline{J}^2\right), \label{eq:Lambda_1}
\end{align}
\end{subequations}
renders the sets:
\begin{subequations}
\begin{align}
\Omega_1 & \coloneqq \left\{\mathfrak{e} \in \mathbb{R}^{6} : \|\mathfrak{e}\|_2 \le \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}} \right\}, \label{eq:omega_1} \\
\Omega_2 & \coloneqq \left\{\mathfrak{z} \in \mathbb{R}^{6+n} : \|\mathfrak{z}\|_2 \le \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}} \right\}, \label{eq:omega_2}
\end{align}
\end{subequations}
RCI sets for the error dynamics \eqref{eq:frak_e}, \eqref{eq:frak_z}, according to Definition \ref{def:RPI_set}. The constants $\alpha_1$, $\alpha_2 > 0$ are defined by:
\begin{equation}
\alpha_1 \coloneqq \underline{\sigma}- \tfrac{\Lambda_1}{4 \rho}, \ \alpha_2 \coloneqq k-\rho \Lambda_1-\Lambda_2. \label{eq:a_1_a_2}
\end{equation}
\end{lemma}
\noindent \textbf{Proof :} A backstepping control methodology will be used \cite{krstic1995nonlinear}. The state $\mathfrak{z}$ in \eqref{eq:frak_e} can be seen as virtual input to be designed such that the Lyapunov function $\mathfrak{L}_1(\mathfrak{e}) \coloneqq \frac{1}{2} \|\mathfrak{e}\|^2_2$ for the system \eqref{eq:frak_e} is always decreasing. The time derivative of $\mathfrak{L}_1$ along the trajectories of system \eqref{eq:frak_e} is given by:
\begin{align}
\dot{\mathfrak{L}}(\mathfrak{e}) & = \mathfrak{e}^\top J(\overline{\mathfrak{q}}) \mathfrak{z} + \mathfrak{e}^\top \mathfrak{b}(\cdot) \le \mathfrak{e}^\top J(\overline{\mathfrak{q}}) \mathfrak{z} + L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2^2. \label{eq:lyap1}
\end{align}
Design the virtual control input as $\mathfrak{z} \equiv - \sigma J(\overline{\mathfrak{q}})^\top \mathfrak{e}$, with $\underline{J}$, $\sigma$ as given in \eqref{eq:set_Q}, \eqref{eq:sigma_under_sigma}, respectively. Then, by employing \eqref{eq:set_Q}, \eqref{eq:lyap1} becomes:
\begin{align}
\dot{\mathfrak{L}}(\mathfrak{e}) & \le - \sigma \mathfrak{e}^\top J^{+}(\overline{\mathfrak{q}}) \mathfrak{e} + L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2^2 \notag \\
& \le - \sigma \lambda_{\min} \left[\tfrac{J^{+}(\overline{\mathfrak{q}})+J^{+}(\overline{\mathfrak{q}})^\top}{2}\right] \|\mathfrak{e}\|_{2}^2 + L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2^2 \notag \\
& \le - \sigma \underline{J} \|\mathfrak{e}\|_{2}^2 + L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2^2 = - \underline{\sigma} \|\mathfrak{e}\|_{2}^{2}. \label{eq:lyap11}
\end{align}
Define the backstepping auxiliary error state $\mathfrak{r} \coloneqq \mathfrak{z}+\sigma J(\overline{\mathfrak{q}})^\top \mathfrak{e} \in \mathbb{R}^{6+n}$ and the the stack vector $\mathfrak{y} \coloneqq [\mathfrak{e}^\top, \mathfrak{r}^\top]^\top \in \mathbb{R}^{12+n}$. Consider the Lyapunov function $\mathfrak{L}(\mathfrak{y}) = \tfrac{1}{2}\|\mathfrak{y}\|^2$. Its time derivative along the trajectories of the system \eqref{eq:frak_e},\eqref{eq:frak_z} is given by:
\begin{align}
\dot{\mathfrak{L}}(\mathfrak{y}) & = \mathfrak{e}^\top \dot{\mathfrak{e}}+\mathfrak{r}^\top \Big[\dot{\mathfrak{z}}+\sigma J(\overline{\mathfrak{q}})^\top \dot{\mathfrak{e}}+\sigma \dot{J}(\overline{\mathfrak{q}})^\top \mathfrak{e}\Big] \notag \\
&\hspace{-5mm} = \left[\mathfrak{e} + \sigma J(\overline{\mathfrak{q}}) \mathfrak{r} \right]^\top \dot{\mathfrak{e}} + \mathfrak{r}^\top \dot{\mathfrak{z}} +\sigma \mathfrak{r}^\top \dot{J}(\overline{\mathfrak{q}})^\top \mathfrak{e} = - \sigma \mathfrak{e}^\top J^{+}(\overline{\mathfrak{q}}) \mathfrak{e} \notag \\
&\hspace{2mm} + \mathfrak{e}^\top \mathfrak{b}(\cdot) +\sigma \mathfrak{r}^\top J(\overline{\mathfrak{q}})^\top \mathfrak{b}(\cdot) + \mathfrak{e}^\top J(\overline{\mathfrak{q}}) \mathfrak{r} +\sigma \mathfrak{r}^\top J^{+}(\overline{\mathfrak{q}}) \mathfrak{r} \notag \\
&\hspace{2mm} +\sigma \mathfrak{r}^\top \dot{J}(\overline{\mathfrak{q}})^\top \mathfrak{e} + \mathfrak{r}^\top \mathfrak{l}(\cdot) + \mathfrak{r}^\top (\mathfrak{u}-\overline{u}) + \mathfrak{r}^\top d(\cdot). \label{eq:lyap2}
\end{align}
By invoking \eqref{eq:lyap11} as well as the following:
\begin{align*}
\sigma \mathfrak{r}^\top J(\overline{\mathfrak{q}})^\top \mathfrak{b}(\cdot) & \le \sigma \|\mathfrak{r}\|_{2} \|J(\overline{\mathfrak{q}})\|_{2} \|\mathfrak{b}(\cdot)\|_{2} \le \sigma L_{\scriptscriptstyle \mathfrak{c}} \overline{J} \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2}, \\
\mathfrak{e}^\top J(\overline{\mathfrak{q}}) \mathfrak{r} & \le \|\mathfrak{e}\|_{2} \|J(\overline{\mathfrak{q}})\|_2 \|\mathfrak{r}\|_{2} \le \overline{J} \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2}, \\
\sigma \mathfrak{r}^\top J^{+}(\overline{\mathfrak{q}}) \mathfrak{r} & \le \sigma \|\mathfrak{r}\|_{2}^{2} \|J^{+}(\overline{\mathfrak{q}})\|_{2} \le \sigma \|\mathfrak{r}\|_{2}^{2} \|J(\overline{\mathfrak{q}})\|_{2} \big\|J^{\top}(\overline{\mathfrak{q}}) \big\|_{2} \\
& \le \sigma \overline{J}^2 \|\mathfrak{r}\|_{2}^{2}, \\
\sigma \mathfrak{r}^\top \dot{J}(\overline{\mathfrak{q}})^\top \mathfrak{e} & \le \sigma \|\mathfrak{e}\|_{2} \|\dot{J}(\overline{\mathfrak{q}})\|_{2} \|\mathfrak{r}\|_{2} \le \sigma \widetilde{J} \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2} \\
\mathfrak{r}^\top \mathfrak{l}(\cdot) & \le L \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2} + L \|\mathfrak{r}\|_{2}^{2}, \\
\mathfrak{r}^\top d(\cdot) & \le \|\mathfrak{r}\|_{2} \|d(\cdot)\|_{2} \le \|\mathfrak{y}\|_{2} \widetilde{d},
\end{align*}
\eqref{eq:lyap2} becomes:
\begin{align}
\dot{\mathfrak{L}}(\mathfrak{y}) & \le - \underline{\sigma} \|\mathfrak{e}\|_{2}^{2} + \Lambda_1 \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2} \notag \\
&\hspace{16mm} + \Lambda_2 \|\mathfrak{r}\|^{2}_{2} + \mathfrak{r}^\top (\mathfrak{u}-\overline{u}) + \|\mathfrak{y}\|_{2} \widetilde{d}. \label{eq:lyap3}
\end{align}
with $\Lambda_1$, $\Lambda_2$ given in \eqref{eq:Lambda_1}. By using Lemma \ref{lemma:basic_ineq} for $n = P = 1$, we get $\|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2}$ $\le \tfrac{1}{4 \rho} \|\mathfrak{e}\|_{2}^{2}$ $+ \rho \|\mathfrak{r}\|_{2}^{2}$,
with $\rho$ designed so that \eqref{eq:sigma_under_sigma} holds. Combining the latter with \eqref{eq:lyap3} it yields:
\begin{align*}
\dot{\mathfrak{L}}(\mathfrak{y}) & \le - \left(\underline{\sigma}- \tfrac{\Lambda_1}{4 \rho} \right) \|\mathfrak{e}\|_{2}^{2} + \big(\rho \Lambda_1 + \Lambda_2 \big) \|\mathfrak{r}\|^{2}_{2} \notag \\
&\hspace{30mm} + \mathfrak{r}^\top (\mathfrak{u}-\overline{u}) + \|\mathfrak{y}\|_{2} \widetilde{d}.
\end{align*}
By designing $\mathfrak{u} - \overline{u} = -k \mathfrak{r} = -k \mathfrak{e}-k \sigma J(\overline{\mathfrak{q}})^{\top} \mathfrak{z}$, which is compatible with \eqref{eq:control_law_u} and the same as in \eqref{eq:kappa_law}, we have:
\begin{align*}
\dot{\mathfrak{L}}(\mathfrak{y}) & \le - \left(\underline{\sigma}- \tfrac{\Lambda_1}{4 \rho} \right) \|\mathfrak{e}\|_{2}^{2} - \big(k -\rho \Lambda_1 - \Lambda_2 \big) \|\mathfrak{r}\|^{2}_{2} + \|\mathfrak{y}\|_{2} \widetilde{d} \\
& \le - \min\{\alpha_1, \alpha_2\}\|\mathfrak{y}\|_2^2 + \|\mathfrak{y}\|_2 \widetilde{d} \\
& = -\|\mathfrak{y}\|_2 \big[ \min\{\alpha_1, \alpha_2\}\|\mathfrak{y}\|_2 - \widetilde{d} \big],
\end{align*}
as $\alpha_1$ and $\alpha_2$ given in \eqref{eq:a_1_a_2}. Thus, $\dot{\mathfrak{L}}(\mathfrak{y}) < 0$, when $\|\mathfrak{y}\|_2 > \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}}$. Taking the latter into consideration and the fact that $\mathfrak{y}(0)$, we have that $\|\mathfrak{y}(t)\| \le \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}}$, $\forall t \ge 0$. Moreover, the following inequalities hold:
\begin{align*}
\|\mathfrak{e}\|_2 &\le \| \mathfrak{y} \|_2 \Rightarrow \|\mathfrak{e}(t)\|_2 \le \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}}, \forall t \ge 0, \\
\Big| \|\mathfrak{e}\|_2- \big\|J^\top \mathfrak{z} \big\|_2 \Big| &\le \big\|\mathfrak{e}+J^\top \mathfrak{z} \big\|_2 = \|\mathfrak{z}\|_2 \le \|\mathfrak{y}\|_2 \\
\Rightarrow \|\mathfrak{z}(t)\|_2 &\le \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}}, \forall t \ge 0. \hspace{34mm} \square
\end{align*}
\begin{remark}
According to Lemma \ref{lemma:tube}, the volume of the tube which is centered along the nominal trajectories $\overline{e}(t)$, $\overline{\zeta}(t)$, that are solution of system \eqref{eq:nom_error_kin},\eqref{eq:nom_error_dyn}, depends on the parameters $\widetilde{d}$, $\overline{J}$, $\underline{J}$, $\widetilde{J}$, $L$ and $L_{\scriptscriptstyle \mathfrak c}$. By tuning the parameters $\rho$ and $k$ from \eqref{eq:sigma_under_sigma} appropriately, the volume of the tube can be adjusted.
\end{remark}
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\subsection{On-line Optimal Control} \label{sec:optimal_contol}
Consider a sequence of sampling times $\{t_k\}$, $k \in \mathbb{N}$, with a constant sampling period $0 < h < T$, where $T$ is a prediction horizon such that $t_{k+1} \coloneqq t_{k} + h$, $\forall k \in \mathbb{N}$. At each sampling time $t_k$, a FHOCP is solved as follows:
\begin{subequations}
\begin{align}
&\hspace{-7mm}\min\limits_{\overline{u}(\cdot)} \left\{ \|\overline{\xi}(t_k+T)\|^2_{\scriptscriptstyle P} \hspace{-1mm} + \hspace{-2mm}\int_{t_k}^{t_k+T} \hspace{-1mm}\Big[ \|\overline{\xi}(\mathfrak{s})\|^2_{\scriptscriptstyle Q} +\|\overline{u}(\mathfrak{s})\|^2_{\scriptscriptstyle R} \Big] d\mathfrak{s} \right\} \hspace{0mm} \label{eq:mpc_cost_function} \hspace{-7mm}\\
&\hspace{-6mm}\text{subject to:} \notag \\
&\hspace{-3mm} \dot{\overline{\xi}}(\mathfrak{s}) = g(\overline{\xi}(\mathfrak{s}), \overline{u}(\mathfrak{s})), \ \ \overline{\xi}(t_k) = \xi(t_k), \label{eq:diff_mpc} \\
&\hspace{-3mm} \overline{\xi}(\mathfrak{s}) \in \overline{\mathcal{E}} \times \overline{\mathcal{Z}}, \ \ \overline{u}(\mathfrak{s}) \in \overline{\mathcal{U}}, \ \ \forall \mathfrak{s} \in [t_k,t_k+T], \label{eq:mpc_constrained_set} \\
&\hspace{-3mm} \overline{\xi}(t_k+T)\in \mathcal{F}, \label{eq:mpc_terminal_set}
\end{align}
\end{subequations}
where $\xi \hspace{-1mm}\coloneqq\hspace{-1mm}[e^\top,\zeta^\top]^\top \hspace{-2mm}\in \mathbb{R}^{12+n}$, $g(\xi,u)\hspace{-1mm}$ $\coloneqq\hspace{-1mm}\begin{bmatrix} J(\mathfrak{q}) \zeta \\ f(e+\chi_{\scriptscriptstyle \rm des}, \zeta)+u \end{bmatrix}$; $Q$, $P \in \mathbb{R}^{(12+n) \times (12+n)}$ and $R \in \mathbb{R}^{(6+n) \times (6+n)}$ are positive definite gain matrices to be appropriately tuned. We will explain hereafter the sets $\overline{\mathcal{E}}$, $\overline{\mathcal{V}}$, $\overline{\mathcal{U}}$ and $\mathcal{F}$.
In order to guarantee that while the FHOCP \eqref{eq:mpc_cost_function}-\eqref{eq:mpc_terminal_set} is solved for the nominal dynamics \eqref{eq:nom_error_kin}-\eqref{eq:nom_error_dyn}, the real states $e$, $\zeta$ and control input $\mathfrak{u}$ satisfy the corresponding state $\mathcal{E}$, $\mathcal{Z}$ and input constraints $\mathcal{U}$, respectively, the following modification is performed: $\overline{\mathcal{E}} \coloneqq \mathcal{E} \ominus \Omega_1, \ \ \overline{\mathcal{Z}} \coloneqq \mathcal{Z} \ominus \Omega_2, \ \ \overline{\mathcal{U}} \coloneqq \mathcal{U} \ominus \left[ \Lambda \circ \overline{\Omega} \right]$,
with $\Lambda \coloneqq {\rm diag} \{-k I_6, -k \sigma \overline{J} I_{6+n}\} \in \mathbb{R}^{(12+n) \times (12+n)}$, $\overline{\Omega} \coloneqq \Omega_1 \times \Omega_2$, the operators $\ominus$, $\circ$ as defined in Section \ref{sec:notation_preliminaries}, and $\Omega_1$, $\Omega_2$ as given in \eqref{eq:omega_1}, \eqref{eq:omega_2}, respectively. Intuitively, the sets $\mathcal{E}$, $\mathcal{Z}$ and $\mathcal{U}$ are tightened accordingly, in order to guarantee that while the nominal states $\overline{e}$, $\overline{\zeta}$ and the nominal control input $\overline{u}$ are calculated, the corresponding real states $e$, $\zeta$ and real control input $\mathfrak{u}$ satisfy the state and input constraints $\mathcal{E}$, $\mathcal{Z}$ and $\mathcal{U}$, respectively. This constitutes a standard constraints set modification technique adopted in tube-based NMPC frameworks (for more details see \cite{yu_2013_tube}). Define the \emph{terminal set} by:
\begin{align} \label{eq:terminal_set_F}
\mathcal{F} \coloneqq \big\{\overline{\xi} \in \overline{\mathcal{E}} \times \overline{\mathcal{Z}} : \|\overline{\xi}\|_{\scriptscriptstyle P} \le \epsilon \big\}, \ \ \epsilon > 0,
\end{align}
which is used to enforce the stability of the system \cite{frank_1998_quasi_infinite}. In particular, due to the fact that the linearized nominal dynamics $\dot{\overline{\xi}} = A \overline{\xi} + B \overline{u}$ are stabilizable, it can be proven that (see \cite[Lemma 1, p. 4]{frank_1998_quasi_infinite}) there exists a \emph{local controller} $u_{\scriptscriptstyle \rm loc} \coloneqq \mathfrak{K} \overline{\xi} \in \overline{\mathcal{U}}$, $\mathfrak{K} \in \mathbb{R}^{(6+n) \times (6+n)}$, $\mathfrak{K} > 0$ which guarantees that: $\tfrac{d}{dt}\left(\|\overline{\xi}\|^2_{\scriptscriptstyle P}\right) \le -\|\overline{\xi}\|^2_{\scriptscriptstyle \widetilde{Q}}$, $\forall \overline{\xi} \in \mathcal{F}$, with $\widetilde{Q} \coloneqq Q+\mathfrak{K}^\top R$.
\begin{theorem} \label{theorem_main}
Suppose also that the FHOCP \eqref{eq:mpc_cost_function}-\eqref{eq:mpc_terminal_set} is feasible at time $t = 0$. Then, the feedback control law \eqref{eq:control_law_u} applied to the system \eqref{eq:unsrt_error_kin}-\eqref{eq:unsrt_error_dyn} guarantees that there exists a time $\mathfrak{t}$ such that $\forall t \ge \mathfrak{t}$ it holds that:
\begin{subequations}
\begin{align}
\hspace{0mm} \|\chi(t)-\chi_{\scriptstyle \rm des}\|_{\scriptscriptstyle 2} & \le \tfrac{\epsilon}{\sqrt{\lambda_{\scriptscriptstyle \min}(P)}} + \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}}, \label{eq:theom_ineq_1} \\
\hspace{0mm} \|\zeta(t)\|_{\scriptscriptstyle 2} & \le \tfrac{\epsilon}{\sqrt{\lambda_{\scriptscriptstyle \min}(P)}} + \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}}. \label{eq:theom_ineq_2}
\end{align}
\end{subequations}
\end{theorem}
\begin{proof}
The proof of the theorem consists of two parts:
\noindent \textbf{Feasibility Analysis}: It can be shown that recursive feasibility is established and it implies subsequent feasibility. The proof of this part is similar to the feasibility proof of \cite[Theorem 2, Sec. 4, p. 12]{alex_IJRNC_2018}, and it is omitted here due to space constraints.
\noindent \textbf{Convergence Analysis}: Recall that $e = \chi-\chi_{\scriptscriptstyle \rm des}$, $\mathfrak{e} = e-\overline{e}$ and $\mathfrak{z} = \zeta-\overline{\zeta}$. Then, we get $\|\chi(t)-\chi_{\scriptstyle \rm des}\|_{\scriptscriptstyle 2}$ $\le \|\overline{e}(t)\|_{\scriptscriptstyle 2} + \|\mathfrak{e}(t)\|_{\scriptscriptstyle 2}$, $\|\zeta(t)\|_{\scriptscriptstyle 2}$ $\le \|\overline{\zeta}(t)\|_{\scriptscriptstyle 2} + \|\mathfrak{z}(t)\|_{\scriptscriptstyle 2}$, which, by using the fact that $\|\overline{e}\|$, $\|\overline{\zeta}\| \le \|\overline{\xi}\|_{2}$ as well as the bounds from \eqref{eq:omega_1}, \eqref{eq:omega_2} the latter inequalities become:
\begin{subequations}
\begin{align}
\|\chi(t)-\chi_{\scriptstyle \rm des}\|_{\scriptscriptstyle 2} & \le \|\overline{\xi}(t)\|_{\scriptscriptstyle 2} + \tfrac{ \widetilde{d}}{\min\{\alpha_1, \alpha_2\}}, \label{eq:conv_1}\\
\|\zeta(t)\|_{\scriptscriptstyle 2} & \le \|\overline{\xi}(t)\|_{\scriptscriptstyle 2} + \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}}, \forall t \ge 0. \label{eq:conv_2}
\end{align}
\end{subequations}
The nominal state $\overline{\xi}$ is controlled by the nominal control action $\overline{u} \in \overline{\mathcal{U}}$ which is the outcome of the solution to the FHOCP \eqref{eq:mpc_cost_function}-\eqref{eq:mpc_terminal_set} for the nominal dynamics \eqref{eq:nom_error_kin}-\eqref{eq:nom_error_dyn}. Hence, by invoking previous NMPC stability results found in \cite{frank_1998_quasi_infinite}, the state $\overline{\xi}(t)$ is driven to terminal set $\mathcal{F}$, given in \eqref{eq:terminal_set_F}, in finite time, and it remains there for all times. Thus, there exist a finite time $\mathfrak{t}$ such that $\overline{\xi}(t) \in \mathcal{F}$, $\forall t \ge \mathfrak{t}$. From \eqref{eq:terminal_set_F}, the latter implies that: $\|\overline{\xi}(t)\|_{\scriptscriptstyle P} \le \epsilon, \forall t \ge \mathfrak{t} \Rightarrow \|\overline{\xi}(t)\|_{\scriptscriptstyle 2} \le \tfrac{\epsilon}{\sqrt{\lambda_{\scriptscriptstyle \min}(P)}}, \forall t \ge \mathfrak{t}.$ The latter implication combined by \eqref{eq:conv_1}-\eqref{eq:conv_2} leads to the conclusion of the proof.
\end{proof}
\begin{figure}
\caption{ The GIRONA-UVMS composed of Girona500 AUV and ARM 5E Micro manipulator \cite{cieslak2015autonomous}
\label{fig:girona}
\end{figure}
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\section{Simulation Results} \label{sec:simulation_results}
For a simulation scenario, consider the Girona 500 AUV depicted in Fig. \ref{fig:girona} equipped with an ARM 5E Micro manipulator from \cite{cieslak2015autonomous}. The manipulator consists of $n = 4$ revolute joints with limits: $-0.52 \le q_1 \le 1.46$, $\-0.1471 \le q_2 \le 1.3114$, $-1.297 \le q_3 \le 0.73$ and $-3.14 \le q_4 \le 3.14$. The end-effector is in ready-to-grasp mode with initial state: $\chi(0) = [\mathfrak{p}(0)^\top, \mathfrak{o}(0)^\top]^\top =[-1.0, 1.3, -1.0, 0.0, -\tfrac{\pi}{8}, \tfrac{\pi}{12}]^\top$. The stiffness matrix is $K = I_{6}$ with $\chi_{\scriptscriptstyle \rm eq} = 0$ which results to $\mathfrak{F}_{\scriptscriptstyle \rm des} = \chi_{\scriptscriptstyle \rm des} = [\mathfrak{p}_{\scriptscriptstyle \rm des}^\top, \mathfrak{o}_{\scriptscriptstyle \rm des}^\top ]^\top = [0, 0, 0, \tfrac{\pi}{3}, \tfrac{\pi}{10}, 0]^\top$.
According to \eqref{eq:forw_kinematics}, the transformation matrices which lead to the forward kinematics are given by:
\begin{align*}
T_B^I & =
\begin{bmatrix}
\mathfrak{J}_1(\eta_2) & \eta_1 \\
0_{1 \times 3} & 1 \\
\end{bmatrix}, \ \ T_0^B =
\begin{bmatrix}
I_{3 \times 3} & \left[ 0.53, 0, 0.36 \right]^\top \\
0_{1 \times 3} & 1 \\
\end{bmatrix},
\end{align*}
and $T_{i}^{i-1}$, $i =1,\dots,4$ are given by the Denavit-Hantenberg parameters which can be calculated from Table \ref{table:DH_parameters}. By imposing the constraints $-\pi \le \phi$, $\psi \le \pi$ and $-\tfrac{\pi}{2}+\epsilon \le \theta \le \tfrac{\pi}{2} - \epsilon$, $\epsilon = 0.1$, according to \eqref{eq:set_Q} we get $\underline{J} = 0.5095$ and $L_{\scriptscriptstyle \mathfrak{c}} = 2 \sqrt{2}$. For simplified calculations, we apply the methodology of this paper by considering disturbance in the following disturbed kinematic model: $\dot{\chi} = J(\mathfrak{q}) \zeta + w(\mathfrak{q}, t)$,
with $w(\cdot) = 0.2 \sin(t) I_6$ $\Rightarrow \|w(\cdot)\|_{2} \le 0.2 = \widetilde{w}$, in which the vector $\zeta$ stands for the virtual control input to be designed such that $\lim_{t \to \infty} \|\chi(t) - \chi_{\scriptscriptstyle \rm des}\| \to 0$. The input constraints are set to $\|\nu_1\|_2 \le 2$, $\|\nu_2\|_2 \le 2$ and $\|\dot{q}\|_2 \le 2$. Then, by using \eqref{eq:frak_e} and \eqref{eq:lyap1} and designing the control gain $\sigma = 3.084$, the resulting RCI is $\Omega = \left\{\mathfrak{e} \in \mathbb{R}^{6} : \|\mathfrak{e}\|_{2} \le \tfrac{\widetilde{w}}{\sigma \underline{J} + L_{\scriptscriptstyle \mathfrak{c}}} = 0.3 \right\}$.
\begin{table}[t!]
\begin{center}
\begin{tabular}{|C{0.4cm}||C{1.2cm}||C{1.2cm}||C{1.2cm}||C{1.2cm}|}
\hline
& $d_i (m)$ & $q_i$ & $a_i (m)$ & $\alpha_i (\text{rad})$ \\
\hline \hline
$1$ & $0$ & $q_1$ & $0.1$ & $-\frac{\pi}{2}$ \\
\hline
$2$ & $0$ & $q_2$ & $0.26$ & $0$ \\
\hline
$3$ & $0$ & $q_3$ & $0.09$ & $\frac{\pi}{2}$ \\
\hline
$4$ & $0.29$ & $q_4$ & $0$ & $0$ \\
\hline \hline
E & \multicolumn{4}{c|}{$\text{Rot}(y,-\frac{\pi}{2})$} \\
\hline
\end{tabular}
\end{center}
\caption{Denavit-Hantenberg Parameters of the ARM 5E Micro}
\label{table:DH_parameters}
\end{table}
The simulation time is $6 \sec$. The optimization horizon and the sampling time are set to $T = 0.7 \sec$ and $h = 0.1 \sec$, respectively. The NMPC gains are set to $Q = P = 0.5 I_{6}$ and $R = 0.5 I_{10}$. Fig. \ref{fig:error} shows the evolution of the real and the nominal position errors of the end-effector. the corresponding real and nominal orientation errors are depicted in Fig. \ref{fig:error2}. Finally, the control inputs are presented in Fig. \ref{fig:inputs}. It can be observed that the desired task is performed while all the state/input constraints are satisfied.
\section{Conclusions and Future Research} \label{sec:conclusions}
This paper addresses the problem of force/torque control of UVMS under state/input constraints as well as external uncertainties/disturbances. In particular, we have proposed a tube-based robust NMPC framework that incorporates the aforementioned constraints in a novel way. Future efforts will be devoted towards extending the current framework under multi-UVMS which interact with each other through a common object in order to perform a collaborative manipulation task.
\begin{figure}
\caption{The evolution of the real position errors of the end-effector $\mathfrak{p}
\label{fig:error}
\end{figure}
\begin{figure}
\caption{The evolution of the real orientation errors $\mathfrak{o}
\label{fig:error2}
\end{figure}
\begin{figure}
\caption{The virtual control input signals $\|\nu_1(t)\|_{2}
\label{fig:inputs}
\end{figure}
\end{document}
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\begin{document}
\title{Large Sumsets from Medium-Sized Subsets}
\author{B\'ela Bollob\'as \and Imre Leader \and Marius Tiba}
\address{Department of Pure Mathematics and Mathematical Statistics,
Wilberforce Road,
Cambridge, CB3 0WA, UK, and Department of Mathematical Sciences,
University of Memphis, Memphis, TN 38152, USA}\email{[email protected]}
\address{Department of Pure Mathematics and Mathematical Statistics,
Wilberforce Road, Cambridge, CB3 0WA, UK}\email{[email protected]}
\address{IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil}\email{[email protected]}
\thanks{The first author was partially supported by NSF grant DMS-1855745}
\begin{abstract}
The classical Cauchy--Davenport inequality gives a lower bound for the size of
the sum of two subsets of ${\mathbb Z}_p$, where $p$ is a prime. Our main aim in this paper is to
prove a considerable strengthening of this inequality, where we take only a small number of points from each of
the two subsets when forming the sum. One of our results is that
there is an absolute constant $c>0$ such that if $A$ and $B$ are subsets of ${\mathbb Z}_p$ with
$|A|=|B|=n\le p/3$ then there are subsets $A'\subset A$ and
$B'\subset B$ with $|A'|=|B'|\le c \sqrt{n}$ such that $|A'+B'|\ge 2n-1$. In fact, we show that one
may take any sizes one likes: as long as $c_1$ and $c_2$ satisfy $c_1c_2 \ge cn$ then we may choose
$|A'|=c_1$ and $|B'|=c_2$. We prove related
results for general abelian groups.
\end{abstract}
\maketitle
\section{Introduction}
The Cauchy--Davenport theorem~\cite{Cauchy, Dav1, Dav2} states that if $p$ is a prime and $A$ and $B$ are non-empty
subsets of ${\mathbb Z}_p$ with $|A|+|B|\le p+1$ then $|A+B|\ge |A|+|B|-1$. Intervals show that this bound is best possible.
Over the years, this classical result
was followed by a host of important contributions about sums of subsets of groups, including other abelian groups such as
${\mathbb Z}$ itself. For these
contributions, see, among others,
Mann~\cite{Mann1, Mann2}, Kneser~\cite{Knes},
Erd\H{o}s and Heilbronn~\cite{ErdHeil}, Freiman~\cite{Frei-59, Frei-62, Frei-book, Frei-87}, Pl\"unnecke~\cite{Plu-70},
Ruzsa~\cite{Ruz-89}, Dias da Silva and Hamidoune~\cite{DiHa}, Alon, Nathanson and Ruzsa~\cite{AlNaRu},
Shao~\cite{shao}, Stanchescu~\cite{Stan}, Breuillard, Green and Tao~\cite{BGT-doubling}, as well as the books of
Nathanson~\cite{Nathbook}, Tao and Vu~\cite{taobook} and Grynkiewicz~\cite{grynkiewicz-2}.
Recently, a new direction of research was started in ~\cite{BLT}: can we get similar bounds for the size of the sum
if $A+B$ is replaced by $A+B'$, where $B'$ is a {\em small} subset of $B$? Among other results, it was proved that if
$A$ and $B$ are finite non-empty subsets of ${\mathbb Z}$ with $|A| \ge |B|$ then $B'$ can be taken to be really small:
there are {\em three} elements $b_1, b_2, b_3 \in B$ such that $|A+\{b_1,b_2,b_3\}| \ge |A|+|B|-1$. For
${\mathbb Z}_p$, it was shown that if
$A, B \subset {\mathbb Z}_p$ with $|A|=|B|=n\le p/3$ then $B$ has a subset $B'$ with at most $c$ elements
such that $|A+B'|\ge 2n-1$, where $c$ is an absolute constant. (Here again $p$ is prime, and for the rest of this paper $p$ will always denote a prime.)
Our aim in this paper is to prove that actually one can replace {\em both} $A$ and $B$ by appropriate small subsets.
In the result just mentioned, the product of the sizes of our two subsets in the sum, namely $A$ and $B'$, is $cn$,
and trivially we cannot ever get a sum of size linear in $n$ without the product of the sizes of the two sets being
linear in $n$. But, remarkably, one can indeed always choose subsets $A'$ of $A$ and $B'$ of $B$, of any desired
sizes, as long as the product of these sizes is linear in $n$.
The result mentioned in the Abstract has both sizes being a constant times $\sqrt{n}$, and the sets $A$ and $B$
themselves have size bounded away from $p/2$. The general form of our result is as follows.
\begin{theorem}\label{all_thm1}
For all $\alpha, \beta>0$ there exists $c>0$ such that the following holds.
Let $A$ and $B$ be non-empty subsets of $\mathbb{Z}_p$ with
$\alpha |B| \leq |A| \leq \alpha^{-1} |B|$ and $|A|+|B| \leq (1-\beta)p$.
Then, for any integers $1 \leq c_1 \leq |A|$ and $1 \leq c_2 \leq |B|$ such that $c_1c_2 \geq c\max(|A|,|B|)$,
there exist subsets $A' \subset A$ and $B' \subset B$, of sizes $c_1$ and $c_2$ such that
$|A'+B'| \geq |A|+|B|-1$.
\end{theorem}
It is rather surprising that this holds with no other restrictions on the values of $c_1$ and $c_2$, whatever the
form of the sets $A$ and $B$. We remark that the constant $c$ does have to depend on $\beta$: if our two sets
are allowed to have sizes whose sum approaches $p$ then taking random subsets shows that $c$ has to grow. Also, it is
easy to see that $c$ depends on $\alpha$. For example, if $B$ is far larger than $A$ then trivially taking $c$ points
from $B$ to be summed with all of $A$ will yield a sumset that is too small.
Our main tool is a similar result that is valid in general abelian groups. The reader will note that it is `worse'
than the above result in that there is an error term, but it is also far `better' because the lower bound comes from
the sum of two actual sets (the sets $A^*$ and $B^*$ below) rather than merely from the lower bound that comes from the
sum of their sizes (or, more precisely, the general Kneser lower bound that generalises the Cauchy--Davenport theorem).
\begin{theorem}\label{implication_shao1-prov} For all $K$ and $\varepsilon >0$ there exists $c$ such that the
following holds.
Let $A$ and $B$ be finite non-empty subsets of an abelian group, and let $1 \leq c_1 \leq |A|$ and
$1 \leq c_2 \leq |B|$ be integers satisfying $c_1c_2 \geq c \max(|A|,|B|)$. Then there are subsets $A^* \subset A$
and $B^* \subset B$, with $|A^*|\ge (1-\varepsilon)|A|$ and
$|B^*|\ge (1-\varepsilon)|B|$, such that
if we select points $a_1, \hdots, a_{c_1}$ and $b_1,\hdots,b_{c_2}$ uniformly at random from $A^*$ and
$B^*$ then, writing $A'$ for $\{a_1, \hdots, a_{c_1}\}$ and $B'$ for $\{b_1, \hdots, b_{c_2}\}$, we have
$${\mathbb E}|A'+ B'|
\geq \min\big( (1-\varepsilon)|A^*+ B^*|,\ K |A^*|,\ K |B^*| \big).$$
\end{theorem}
We remark that the terms $K |A^*|$ and $K |B^*|$ are only present to deal with unimportant cases: the key term is
$(1-\varepsilon)|A^*+ B^*|$. Thus the result is informally somehow saying that, in terms of sumsets, $A^*$ and $B^*$ may really be approximated by very small subsets of themselves (and indeed most subsets will do).
Interestingly, while this theorem is for general abelian groups, Theorem~\ref{all_thm1} is only about $\mathbb{Z}_p$. The passage
between these does require quite a lot of work.
The plan of the paper is as follows. In Section 2 we give various prerequisites that we shall need. Then in Section
3 we prove Theorem~\ref{implication_shao1-prov}, and also provide the consequence of it in $\mathbb{Z}_p$ that we use in
the proof of Theorem~\ref{all_thm1}.
In Section 4 we prove Theorem~\ref{all_thm1},
and the last section, Section 5, contains open problems.
Our notation is standard.
Sometimes we write `$x \mod d$' as shorthand for the infinite arithmetic progression
$\{ y \in \mathbb{Z}: y \equiv x \mod d \}$, and refer to it as a {\em fibre} mod $d$.
When $S$ is a subset of $\mathbb{Z}$ we often write $S^x$ for the intersection of this fibre with
$S$ -- when the value of $d$ is clear. (We sometimes write $S^x$ as $S^x_d$ when we want to stress the value of $d$.)
Thus $S^x=S\cap \pi^{-1}(x)$, where $\pi = \pi_d$ denotes
the natural projection from $\mathbb{Z}$ to $\mathbb{Z}_d$.
When we write a probability or an expectation over a finite set, we always assume that the elements
of the set are being sampled uniformly. Thus, for example,
for a finite set $X$ we denote the expectation and probability when we sample uniformly over
all $x \in X$ by ${\mathbb E}_{x\in X} \text{ and } \Prob_{x\in X}$.
We also often sample uniformly over all $c$-sets of a given set $X$. In most of those cases,
we could instead sample $c$ elements uniformly and independently, but the notation would tend to get
unwieldy, and this is why we use the sampling over all $c$-sets instead.
Before we turn to the next section, let us draw attention to the superficial similarity of
our problems to a beautiful result of Ellenberg~\cite{Ell}. Given a prime $p$ and a positive
integer
$d$, let $f(p^d)$ be the smallest integer such that for any sets $S, T \subset {\mathbb Z}_p^d$
there are subsets $S' \subset S$ and $T'\subset T$ satisfying $(S+T')\cup (S'+T)=S+T$
and $|S'|+|T'| \le f(p^d)$. Ellenberg proved that $f(p^d) \le (cp)^d$, where $c<1$ is an absolute
constant.
\section{Prerequisites}
In this section we collect together the various prerequisites that we will need. Each of these may be treated
as a `black box': knowledge of their proofs will not be required.
The first of the three theorems we shall need is due to Shao~\cite{shao}, and concerns
restricted sums. Let $A$ and $B$ be subsets of an abelian group, and let
$\Gamma \subset A \times B$. The {\em $\Gamma$-restricted sum} of $A$ and $B$ is
$A+_{\Gamma}B=\{a+b: \ a\in A,\ b\in B, (a,b) \in \Gamma \}$. Here is the result of Shao.
\begin{theorem}\label{thm_shao}
For all $\varepsilon, K>0$ there exists $\delta>0$ such that the following holds. Let $G$ be an
abelian group and let $N \in \mathbb{N}$. Let $A,B \subset G$
be two subsets with $|A|, |B| \geq N$. Let $\Gamma \subset A \times B$ be a subset with
$|\Gamma|\geq (1-\delta)|A||B|$. If $|A+_{\Gamma}B| \leq KN$, then there
exist $A_0 \subset A$ and $B_0 \subset B$ such that
\[
|A_0| \geq (1-\varepsilon)|A|
\text{ and } |B_0|\geq (1-\varepsilon)|B| \text{ and } |A_0+B_0| \leq |A+_{\Gamma}B|+\varepsilon N.
\]
\end{theorem}
The second theorem is an easy corollary of a theorem of Grynkiewicz~\cite{grynkiewicz-2}.
\begin{theorem}\label{Freiman_Zp}
Given $\beta, \gamma >0$ there is an $\varepsilon>0$ such that the following holds.
Let $A$ and $B$ be subsets of $\mathbb{Z}_p$. Suppose that
$2\leq \min(|A|,|B|)\text{ and }|A|+|B| \leq (1-\beta)p$
and $|A+B| \leq |A|+|B|-1 + \varepsilon \min(|A|,|B|)$.
Then there are arithmetic progressions $P$ and $Q$ with the same common difference that contain $A$ and $B$ and satisfy $|P \Delta A|\leq \gamma \min(|A|,|B|) \text{ and } |Q \Delta B|\leq \gamma \min(|A|,|B|)$.
\end{theorem}
The last theorem we need is a somewhat technical result from \cite{BLT}. It gives a strengthening of the result from \cite{BLT}
mentioned above about sums in $\mathbb{Z}_p$, when the sets $A$ and $B$ `relate nicely' to intervals.
\begin{theorem}\label{CD_technical}
For all $\beta>0$ there exists $\gamma>0$ such that for every $\alpha>0$ there is a value of $c$ for which the
following
holds. Let $A$ and $B$ be subsets of $\mathbb{Z}_p$ and let $I= [p_l, p_r]$ and $J= [q_l, q_r]$
be intervals in $\mathbb{Z}_p$ satisfying $\alpha |J| \leq |I| \leq \alpha^{-1} |J|$,\ $|I|+|J|\leq (1-\beta)p$,
$\max(|A\Delta I|, |B\Delta J|) \leq \gamma \min (|I|, |J|)$\ and\ $\{q_l, q_r\} \subset B \subset J$.
Then there is a family $\mathcal{F}\subset B^{(c)}$,\ depending only on $I$, $J$ and $B$ (but not on $A$),\ such that
\[
\E_{B'\in \mathcal{F}}|A+B'|\geq |A|+|J|-1 \geq |A|+|B|-1.
\]
\end{theorem}
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\section{Proof of Theorem 2}
In this section we prove our main result on general abelian groups, Theorem~\ref{implication_shao1-prov}.
We start by giving a brief overview of the proof. Although this paper is self-contained, we mention that the
reader who is familiar with \cite{BLT} will see that this proof is similar in spirit to the proof of Theorem 10 in
that paper.
We will repeatedly apply Theorem~\ref{thm_shao} in order to construct a decreasing sequence
of $s + 1$ pairs of sets $(A, B) = (A_0, B_0), (A_1, B_1), \dots , (A_s, B_s)$,
satisfying $A_i \subset A_{i-1}$ and $B_i
\subset B_{i-1}$ and $|A_i| >
(1 -
\varepsilon / s)^i |A|$ and $|B_i| > (1 - \varepsilon / s)^i |B|$.
Having constructed $A_i$ and $B_i$ we divide the elements of $A_i+B_i$ into the
set $P_i$ of `popular' ones (those hit at least $\alpha \min(|A_i|,|B_i|)$ times) and
the set $U_i$ of unpopular ones. And we let $\Gamma$ be the pairs summing to
popular elements.
We first deal with the situation when $|\Gamma|$ is much smaller than $|A_i| |B_i|$ or
$|P_i|$ is much larger than $K \min(|A_i|,|B_i|)$. In both cases a simple
computational check shows that the pair $(A_i, B_i)$ has the desired
properties. If we are not in this situation then we can apply Thm~\ref{thm_shao} to $A_i, B_i$ and $\Gamma$ to construct the sets
$A_{i+1}$, $B_{i+1}$. These satisfy $|A_{i+1} +
B_{i+1}| < |P_i| + (\varepsilon / s) \min(|A_{i+1}|, |B_{i+1}|)$.
We then deal with the case when the process continues for at least $s$
steps. In this case we have $|A_{i+1} + B_{i+1}| - (\varepsilon / s)
\min(|A_{i+1}|, |B_ {i+1}|) < |P_i| < |A_i+B_i|$, where the second inequality is
clear and the first is by Theorem~\ref{thm_shao}.
As $|P_0| < 10 K \min (|A_0|, |B_0|)$ and $|A_s + B_s| > |B_s| > |B_0|/2$, we
deduce that there exists $i$ such that $|P_i|$ is about
then easy to check that the pair of sets $(A_i, B_i)$ has the desired
property. Indeed, if $c$ is large enough (depending on $\alpha$),
then $|A'+B'|$ is about $|P_i|$ as each point in $P_i$ is hit with high probability. So
we conclude that $|A'+B'|$ is about $|A_i+B_i|$.
We now turn to the proof itself.
\begin{proof}[Proof of Theorem \ref{implication_shao1-prov}]
Fix $\varepsilon>0$ and $K>0$, where we assume that $\varepsilon$ is sufficiently small and
$K$ is sufficiently large.
Pick $s=\lfloor \frac{50K}{\varepsilon} \rfloor$. Let
$\delta$ be given by Theorem~\ref{thm_shao} with parameters $\frac{\varepsilon}{s}$ and $10K$.
Also pick $\alpha = \delta/16 K$. Finally, pick $c\geq \max(2^{10}K/ \delta, 2^{10} |\log(\epsilon)|/ \alpha)$. We may assume
that $|A| \geq |B|$.
We shall examine a process in which we repeatedly apply Theorem \ref{thm_shao} in order to construct a
decreasing sequence of $s+1$ pairs of sets $(A,B)=(A_0,B_0),(A_1,B_1), \hdots ,(A_s,B_s),$
satisfying $A_i \subset A_{i-1} \text{ and } B_i \subset B_{i-1}$ and
$|A_i|\geq (1-\varepsilon/s)^i |A|$ and $|B_i|\geq (1-\varepsilon/s)^i |B| $.
Fix $i<s$ and assume that the pair of sets $(A_i,B_i)$ has already been
constructed. We shall either stop the process at step $i$ or construct the pair of
sets $(A_{i+1},B_{i+1})$.
Let $A_i+B_i=C_i^+\sqcup C_i^-$ be the partition into `popular' and `unpopular' elements given by
$ C_i^+=\{c \in A_i+B_i: \ |(c-A_i)\cap B_i| \geq \alpha |B_i|\}$, and
$C_i^-=\{c \in A_i+B_i: \ |(c-A_i)\cap B_i| < \alpha |B_i|\}$.
Also, let the partition $A_i \times B_i=\Gamma_i\sqcup \Gamma_i^c$ be
given by $\Gamma_i=\{(a,b) \in A_i \times B_i: \ a+b \in C_i^+ \} \subset A_i \times B_i$, and $\Gamma_i^c=\{(a,b) \in A_i
\times B_i : \ a+b \in C_i^- \} \subset A_i \times B_i$,
so that $A_i+_{\Gamma_i}B_i=C_i^+$ and $A_i+_{\Gamma^c_i}B_i=C_i^-$.
Finally, for
each $x \in A_i+B_i$ set
$$
A_i^x=(x-B_i)\cap A_i \text{ and } B_i^x=(x-A_i)\cap B_i \text{ such that } A_i^x=x-B_i^x,$$
$$
r_i(x)=|A_i^x|=|B_i^x|=|\{(a,b)\in A_i \times B_i: \
x=a+b\}|,
$$
so that $ \sum_x r_i(x)=|A_i||B_i|.$
We stop this process `early', at step $i$, if $$|\Gamma_i|<(1-\delta)|A_i||B_i| \text{ or } |A_i+_{\Gamma_i}B_i|> 10K \min(|A_i|,|B_i|).$$
Otherwise, we apply Theorem~\ref{thm_shao} with parameters
$\varepsilon/s, 10K$ to the pair of sets $(A_i,B_i)$. Thus we produce a pair of sets $(A_{i+1},B_{i+1})$, satisfying
$A_{i+1} \subset A_i$, $B_{i+1} \subset B_i$, \
$|A_{i+1}| \geq (1-\varepsilon/s)|A_i|$, \ $|B_{i+1}|\geq (1-\varepsilon/s)|B_i|$
and $|A_{i+1}+B_{i+1}| \leq |A_i+_{\Gamma_i}B_i|+\frac{\varepsilon}{s} \min(|A_i|,|B_i|)$.
We shall analyse separately the cases in which the process continues until the end and in
which the process stops before that.
Before we begin, we need one easy estimate. Suppose the process continues until step $j$. If we
choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have
\begin{equation}\label{unification1}
\begin{split}
{\mathbb E}|A_j'+B_j'|
&= \sum_x \Prob(x \in A_j'+B_j')\\
&\geq \sum_{x} \sum_{\substack{ X\subset A_j^x, Y\subset B_j^x \\ X=x-Y \\ |X|=|Y|> \frac{n_Br_j(x)}{2|B_j|} }}
\mathbb{P} (B_j' \cap B_j^x = Y \text{ and } A_j' \cap X \neq \emptyset)\\
&= \sum_{x} \sum_{\substack{ X\subset A_j^x, Y\subset B_j^x \\ X=x-Y \\ |X|=|Y|> \frac{n_Br_j(x)}{2|B_j|} }}
\mathbb{P} (B_j' \cap B_j^x= Y ) \mathbb{P}(A_j' \cap X \neq \emptyset)\\
&= \sum_{x} \Prob(|B_j'\cap B_j^x| > \frac{n_Br_j(x)}{2|B_j|}) \min_{\substack{X \subset A_j^x \\ |X| >
\frac{n_Br_j(x)}{2|B_j|}}}\Prob(|A_j' \cap X| > 0)\\
&\geq \sum_{x} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|}))).
\end{split}
\end{equation}
Here, the last inequality follows from Chernoff's inequality (see for example Corollary 1.9 in \cite{taobook}) and the fact that $|X| > \frac{n_Br_j(x)}{2|B_j|} $ is equivalent to $|X| \geq \max(1, \frac{n_Br_j(x)}{2|B_j|})$.
{\bf Claim A.}
{\em
Suppose the process
stops early, say at step $j<s$. Then the pair of sets $(A_j,B_j)$ has the desired properties.
}
\begin{proof}
\noindent
\textbf{Case 1:} Consider first the
case when $|C^+_j|=|A_j+_{\Gamma_j}B_j|> 10K \min(|A_j|,|B_j|).$ For $x\in C_j^+$,
by construction we have that $r_j(x)=|(x-A_j)\cap B_j|\geq \alpha
|B_j|.$ If we choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have
\begin{eqnarray*}
{\mathbb E}|A_j'+B_j'|
&\geq& \sum_{x} (1-\exp(-\frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|}))) \\
&\geq& \sum_{x}(1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2
\geq \sum_{x \in C^+_j} (1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2\\
&\geq& \sum_{x \in C^+_j}(1- \exp(-2^{-4} \alpha c))^2
\geq |C_j^+|/2
\geq 5K \min(|A_j|,|B_j|).
\end{eqnarray*}
Here the first inequality follows from \eqref{unification1}; the second from the hypothesis $n_An_B \geq c|A|\geq c|A_j|$
which, in particular, gives $n_B \geq c|B_j|$; the fourth from the construction as $r_j(x) \geq \alpha |B_j|$
for $x \in C_j^+$; the fifth from the hypothesis $c \geq 2^{10}/ \alpha$; and the last inequality follows from
the assumption on the size of $C^+_j$.
\noindent
\textbf{Case 2:} Consider now the case when $|\Gamma_j|<(1-\delta)|A_j||B_j|.$ By construction,
$\sum_{x\in C_j^-}r_j(x) \geq \delta|A_j||B_j|.$ Moreover, for
$x\in C_j^-$ we have $r_j(x)=|(x-A_j)\cap B_j|\leq \alpha
|B_j|.$
If we choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have the following sequence of inequalities. To make the formulae less cluttered, we define $D_j^-=\{x: n_B r_j(x)\le 2|B_j|\}$ and $D_j^+=\{x: n_B r_j(x)> 2|B_j|\}$.
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\end{split}
\end{equation}
Here, the last inequality follows from Chernoff's inequality (see for example Corollary 1.9 in \cite{taobook}) and the fact that $|X| > \frac{n_Br_j(x)}{2|B_j|} $ is equivalent to $|X| \geq \max(1, \frac{n_Br_j(x)}{2|B_j|})$.
{\bf Claim A.}
{\em
Suppose the process
stops early, say at step $j<s$. Then the pair of sets $(A_j,B_j)$ has the desired properties.
}
\begin{proof}
\noindent
\textbf{Case 1:} Consider first the
case when $|C^+_j|=|A_j+_{\Gamma_j}B_j|> 10K \min(|A_j|,|B_j|).$ For $x\in C_j^+$,
by construction we have that $r_j(x)=|(x-A_j)\cap B_j|\geq \alpha
|B_j|.$ If we choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have
\begin{eqnarray*}
{\mathbb E}|A_j'+B_j'|
&\geq& \sum_{x} (1-\exp(-\frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|}))) \\
&\geq& \sum_{x}(1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2
\geq \sum_{x \in C^+_j} (1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2\\
&\geq& \sum_{x \in C^+_j}(1- \exp(-2^{-4} \alpha c))^2
\geq |C_j^+|/2
\geq 5K \min(|A_j|,|B_j|).
\end{eqnarray*}
Here the first inequality follows from \eqref{unification1}; the second from the hypothesis $n_An_B \geq c|A|\geq c|A_j|$
which, in particular, gives $n_B \geq c|B_j|$; the fourth from the construction as $r_j(x) \geq \alpha |B_j|$
for $x \in C_j^+$; the fifth from the hypothesis $c \geq 2^{10}/ \alpha$; and the last inequality follows from
the assumption on the size of $C^+_j$.
\noindent
\textbf{Case 2:} Consider now the case when $|\Gamma_j|<(1-\delta)|A_j||B_j|.$ By construction,
$\sum_{x\in C_j^-}r_j(x) \geq \delta|A_j||B_j|.$ Moreover, for
$x\in C_j^-$ we have $r_j(x)=|(x-A_j)\cap B_j|\leq \alpha
|B_j|.$
If we choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have the following sequence of inequalities. To make the formulae less cluttered, we define $D_j^-=\{x: n_B r_j(x)\le 2|B_j|\}$ and $D_j^+=\{x: n_B r_j(x)> 2|B_j|\}$.
\begin{eqnarray*}
{\mathbb E}|A_j'+B_j'|\hspace{-8pt}
&\geq& \hspace{-8pt} \sum_{x} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|}))) \\
&\geq& \hspace{-12pt} \sum_{x \in D_j^-} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} ))
(1-\exp(- \frac{n_A}{4|A_j|})) + \sum_{x\in D_j^+} 2^{-1}(1-\exp(- \frac{n_An_Br_j(x)}{8|A_j||B_j|}))\\
&\geq& \hspace{-12pt} \sum_{x\in D_j^-} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} ))
(1-\exp(- \frac{n_A}{4|A_j|})) + \sum_{x\in D_j^+} 2^{-1}(1-\exp(- \frac{8Kr_j(x)}{\delta|B_j|}))\\
&\geq& \hspace{-19pt} \sum_{x \in C_j^-\cap D_j^-} \frac{n_Br_j(x)}{32|B_j|} \frac{n_A}{8|A_j|}
+ \sum_{x \in C_j^- \cap D_j^+} \frac{Kr_j(x)}{\delta |B_j|}
\geq \sum_{x \in C_j^-} \frac{K r_j(x)}{\delta |B_j|} \geq K|A_j|.
\end{eqnarray*}
Here, the first inequality follows from \eqref{unification1}, the second inequality follows by splitting
into two cases according to how $\frac{n_B r(x)}{2|B_j|}$ compares to 1, the third inequality follows
from the hypothesis $n_An_B \geq c|A| \geq c|A_j|$ and $c \geq 2^{10}K/\delta$, the fourth inequality
follows from the two facts that $1-\exp(-t) \geq t/2$ for $0 \leq t \leq 1/2 $ and $\frac{8Kr_j(x)}{\delta |B_j|}
\leq \frac{8K \alpha}{\delta} <1/2 $ for $x \in C_j^-$, the fifth inequality follows again from the
hypothesis $n_An_B \geq c |A| \geq c |A_j| $ and $c\geq 2^{10} K/ \delta $, and the last inequality
follows from the original assumption that $\sum_{x \in C_j^-} r_j(x) \geq \delta |A_j||B_j|$.
We conclude the pair of sets $(|A_j|,|B_j|)$ has the desired properties, so Claim A is proved.
\end{proof}
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We now turn to the case when the process does not stop early.
{\bf Claim B.} {\em
Suppose that the process continues until the terminal step $s$.
Then there is an index $j\leq s$ such that the pair of sets $(A_j,B_j)$ has the desired
properties.}
\begin{proof}
Note that $|A_{1}+B_{1}| \leq |A_0+_{\Gamma_0}B_0|+\frac{\varepsilon}{s} \min(|A_0|,|B_0|) \leq 11K \min(|A_0|,|B_0|).$
Moreover, $11K \min(|A|,|B|)= 11K \min(|A_0|,|B_0|)\geq |A_{1}+B_{1}| \geq \hdots \geq |A_{s}+B_{s}| \geq 0.$ Therefore
$|A_{j+1}+B_{j+1}|\geq |A_{j}+B_{j}|-\frac{11K}{s-1}\min(|A|,|B|)$ for some index $j$,
$1 \leq j \leq s$. We shall
show that the pair $(A_j,B_j)$ has the desired properties.
Indeed, by construction,
\[
|A_{j}+_{\Gamma_j}B_{j}|+\frac{\varepsilon}{s}
\min(|A|,|B|) \geq |A_{j}+_{\Gamma_j}B_{j}|+\frac{\varepsilon}{s}
\min(|A_j|,|B_j|) \geq |A_{j+1}+B_{j+1}|.
\]
It follows that
\[
|A_j+_{\Gamma_j}B_j|\geq |A_{j}+B_{j}|- \big(\frac{11K}{s-1}+\frac{\varepsilon}{s} \big)\min(|A|,|B|)
\geq \big(1- \frac{20K}{s} \big)|A_{j}+B_{j}|.
\]
If we choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have
\begin{eqnarray*}
{\mathbb E}|A_j'+B_j'|
&\geq& \sum_{x} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|})))\\
&\geq& \sum_{x}(1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2
\geq \sum_{x \in C^+_j} (1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2\\
&\geq& \sum_{x \in C^+_j}(1- \exp(-2^{-4} \alpha c))^2
\geq (1- \exp(-2^{-4} \alpha c))^2 |A_j+_{\Gamma_j}B_j|\\
&\geq& (1- \exp(-2^{-4} \alpha c))^2 (1-\frac{20 K}{s}) |A_j+B_j|
\geq (1-\epsilon) |A_j+B_j|.
\end{eqnarray*}
Here the first inequality follows from \eqref{unification1}; the second from the hypothesis
$n_An_B \geq c|A|\geq c|A_j|$ which, in particular, gives $n_B \geq c|B_j|$; the fourth inequality
holds by the construction, as $r_j(x) \geq \alpha |B_j|$ for $x \in C_j^+$; the fifth inequality follows from the
hypothesis $c \geq 2^{10}/ \alpha |\log(\epsilon)|$; and the last inequality follows from the assumption on the size of $C^+_j$.
Thus the pair $(|A_j|,|B_j|)$ has the desired properties, so Claim B is proved.
\end{proof}
This concludes the proof of Theorem~\ref{implication_shao1-prov}: whether the process stops early
or does not, the pair of sets $(A_j,B_j)$ has the desired
properties.
\end{proof}
When we come to proving Theorem~\ref{all_thm1}, we shall need the following consequence of this.
\begin{theorem}\label{all_thm2}
For all $\beta, \gamma>0$ there exists $\epsilon>0$ such that for all $\alpha>0$ there is a value of $c$
for which the following holds. Let $A$ and $B
$ be subsets of $\mathbb{Z}_p$ and let $1 \leq c_1 \leq |A|$ and $1 \leq c_2 \leq |B|$ be integers such
that $c_1c_2 \geq c\max(|A|,|B|)$. Suppose that
\begin{equation}\label{all_eq2}
2 \leq \min(|A|,|B|), \alpha |B| \leq |A| \leq \alpha^{-1} |B| \text{ and } |A|+|B| \leq (1-\beta)p
\end{equation}
and
\begin{equation}\label{all_eq3}
\E_{A' \in A^{(c_1)},\ B' \in B^{(c_2)}} |A'+B'| \leq |A|+|B|-1 +\eps \min(|A|,|B|).
\end{equation}
Then there exist arithmetic progressions $P$ and $Q$ with the same common difference and
\begin{equation*}
\max(|A\Delta P|, |B \Delta Q|) \leq \gamma \min(|A|,|B|).
\end{equation*}
\end{theorem}
\begin{proof}
We start by fixing some parameters. Fix $2^{-10}>\beta>\gamma>\alpha>0$. Let $\varepsilon_1$ be the output of Theorem~\ref{Freiman_Zp} with
input $\beta,2^{-11}\gamma$. Let
$\varepsilon=\min(2^{-2}\varepsilon_1,2^{-6}\gamma)$. Let $\mu=2^{-12}\min(\alpha \varepsilon, \alpha \gamma)$.
Finally, let $c$ be the output of Theorem~\ref{implication_shao1-prov} with input $(4/\alpha,\mu)$.
By Theorem~\ref{implication_shao1-prov}, there are subsets $A^*\subset A$ and $B^*\subset B$ with
\begin{equation}\label{eq22.1*}
|A^*|\geq \lceil (1-\mu)|A|\rceil \text{ and } |B^*|\geq \lceil (1-\mu)|B|\rceil
\end{equation}
such that
\begin{equation}\label{eq22.2*}
\E_{A' \in A^{(c_1)}, \ B'\in B^{(c_2)}}| A'+B'| \geq \min \bigg((1-\mu)|A^*+B^*|\text{, } \frac{4}{\alpha}|A|\text{, }
\frac{4}{\alpha}|B|\bigg).
\end{equation}
By \eqref{all_eq2} we have
\begin{equation}\label{eq22.25*}
\min\bigg(\frac{4}{\alpha}|A|,\frac{4}{\alpha}|B|\bigg) \geq 4\max(|A|,|B|) >
|A|+|B|-1+\varepsilon\min(|A|,|B|),
\end{equation}
and by \eqref{all_eq2} and \eqref{eq22.1*} we find that
\begin{equation}\label{eq22.27*}
2\leq \min\big(|A^*|,|B^*|\big), \ \ \ \frac{\alpha}{2}
|B^*| \leq |A^*| \leq \frac{2}{\alpha}|B^*| \ \ \ {\rm and} \ \ \ |A^*|+|B^*| \leq (1-\beta)p.
\end{equation}
Hence \eqref{all_eq3}, \eqref{eq22.2*} and \eqref{eq22.25*} imply
\begin{equation*}
|A|+|B|-1+\varepsilon\min(|A|,|B|) \geq (1-\mu)|A^*+B^*|.
\end{equation*}
Combining this with \eqref{eq22.1*} and \eqref{eq22.27*}, we find that
\begin{equation}\label{eq22.3*}
\begin{split}
|A^*+B^*| &\leq (1-\mu)^{-1}(|A|+|B|-1+\varepsilon\min(|A|,|B|)) \\
&\leq (1-\mu)^{-2}|A^*|+(1-\mu)^{-2}|B^*|-1+2\varepsilon \min(|A^*|, |B^*|)\\
&\leq |A^*|+|B^*|-1+8\mu\max(|A^*|,|B^*|)+ 2\varepsilon\min(|A^*|,|B^*|)\\
&\leq |A^*|+|B^*|-1+4\varepsilon\min(|A^*|,|B^*|).
\end{split}
\end{equation}
Recalling Theorem~\ref{Freiman_Zp} and \eqref{eq22.27*}, we see that there
exist arithmetic progressions $P$ and $Q$ with
the same common difference such that
\begin{equation*}
|A^*\Delta P| \leq \frac{\gamma}{2^{11}}\min(|A^*|,|B^*|) \text{ and } |B^*\Delta Q| \leq \frac{\gamma}{2^{11}}\min(|A^*|,|B^*|)
\end{equation*}
The conclusion now follows from this and \eqref{eq22.1*}:
\begin{eqnarray*}
|A\Delta P|&\leq& |A^*\Delta P|+|A\Delta A^*| \leq \frac{\gamma}{2^{11}}\min(|A^*|,|B^*|)+\mu|A|\\
&\leq& \frac{\gamma}{2^{11}}\min(|A|,|B|)+\mu|A|
\leq \frac{\gamma}{2^{10}}\min(|A|,|B|).
\end{eqnarray*}
and
\begin{eqnarray*}
|B\Delta Q|&\leq& |B^*\Delta Q|+|B\Delta B^*|
\leq \frac{\gamma}{2^{11}}\min(|A^*|,|B^*|)+\mu|B|\\
&\leq& \frac{\gamma}{2^{11}}\min(|A|,|B|)+\mu|B|
\leq \frac{\gamma}{2^{10}}\min(|A|,|B|).
\end{eqnarray*}
This completes the proof of Theorem~\ref{all_thm2}.
\end{proof}
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\section{Proof of Theorem 1}
We start with a sketch of how the proof of Theorem~\ref{all_thm1} will proceed.
By Theorem~\ref{all_thm2}, we may assume that $A$ and $B$ are close to
intervals $I$ and $J$. Some delicate analysis around the endpoints of $I$ and
$J$, where we may slightly alter these intervals, will allow us to reduce to
the case where $A$ and $B$ are actually contained in $I$ and $J$. Thus there is
no `wraparound', and so we may as well be working in ${\mathbb Z}$ instead of ${\mathbb Z}_p$.
Assume for simplicity that $I = J$ has size $n$, that $A$ and $B$ are
contained in $I$ and $J$ and have size at least say $(1- 1/1000) n$, and that
$c_1 = c_2= c \sqrt{n}$ for some large constant $c$.
Fix $d$ to be about $\sqrt n$, and as usual write $X^y$ for $X \cap (y \mod d)$. Assume $A^0$ is the
largest of the fibres $A^y$. Then clearly $2|A^0| > |B^y|$ for all $y$. By
Theorem~\ref{CD_technical}, if $B'^y$ is a bounded set of $r$ random
points in $B^y$, chosen according to some distribution independent of $A$, we
have that $|A^0 + B'^y| \geq |A^0| + |B^y|-1$.
If we let $B'= \cup_y B'^y$, we have $|A^0 + B'| \geq |A^0| |\pi(B)| +|B| -
| \pi(B)|$ and $A^0 + B' \subset \pi^{-1} \pi(B)$.
Now pick a random fibre $y \mod d$ and let $A' = A^0 \cup A^y$.
We want to show that
for each fibre $z \mod d$ we have ${\mathbb E} |(A'^y+B')^z| \geq n/d$.
Indeed, once we have shown this, then combining the two inequalities and summing
over all $z$ not in $\pi(B)$ gives
${\mathbb E} |A'+B'| \geq |A^0| |\pi(B)| +|B| - |\pi(B)| + (d- |\pi(B)|) n/d \geq n/d
| \pi(B)| +|B| - |\pi(B)| + (d- |\pi(B)|) n/d \geq n+|B| - d$.
So fix sets $A'$ and $B'$ which satisfy this bound. To finish from
here we just note that by adding $d$ extra points in $A'$ and $d$ extra points in
$B'$ we can guarantee $|A'+B'| \geq |A|+|B|-1$. We also note $|A'| = |B'| =
O(\sqrt n)$.
To show ${\mathbb E} |(A'^y+B')^z| \geq n/d$, we proceed as follows. Note that the
proportion of fibres $y \mod d$ such that $A^y$ and $B^y$ have size at least
$9n/10d$ is at least $9/10$. Therefore, for a fixed fibre $z \mod d$, with
probability at least $8/10$ both $A^y$ and $B^{z-y}$ have size at least $9n/10d$.
Conditioned on this event, it follows that $2|A^y| \geq |B^{z-y}|$. Using
Theorem~\ref{CD_technical} again, we obtain $|A^y+B'^{z-y}|
\geq |A^y|+|B^{z-y}|-1 \geq 18n/10d - 1$.
Hence
\[
{\mathbb E} |(A'^y+B')^z| \geq (8/10) (18n/10d - 1) \geq n/d.
\]
We now start to work towards the proof of Theorem~\ref{all_thm1}. We collect together in advance
some results that we shall need. The first of these results will be applied when we already know that our
sets are close to intervals.
\begin{theorem}\label{all_thm3}
There exists $\gamma>0$ such that for all $\alpha>0$ there exists $c$ for which the following holds. Let $X$ and $Y$ be
subsets of two intervals $I$ and $J$ of $\mathbb{Z}_p$, and let $1 \leq c_1 \leq |X|$ and $1 \leq c_2 \leq |Y|$ be
integers such that $c_1c_2 \geq c\max(|X|,|Y|)$. Suppose that $\alpha |J| \leq |I| \leq \alpha^{-1}|J|$, \ $|I| +|J| \leq p$
and $\max(|I\setminus X|, |J \setminus Y| ) \leq \gamma \min(|I|, |J|)$. Then there exist
$X' \in X^{(c_1)}$ and $Y' \in Y^{(c_2)}$ such that $|X'+Y'| \geq |X|+|Y|-1$.
\end{theorem}
\begin{proof}
Since $|I| +|J| \leq p$, we may assume that the ambient space is $\mathbb{Z}$ rather than $\mathbb{Z}_p$.
Let $\gamma'$ and $k$ be the outputs of Theorem~\ref{CD_technical} with input $\alpha/2$ (since we are now in
$\mathbb{Z}$, there is no $\beta$, or more formally we are applying Theorem~\ref{CD_technical} inside $\mathbb{Z}_q$ for
some much larger $q$ with say $\beta=1/2$).
By increasing $\gamma'$ if
necessary we may assume that
$k \geq 100 /\gamma'$. Set $\gamma=\gamma'/100$ and
let $t=\lceil\log_{2/3}(1-(1+\alpha \gamma'/100)^{-1}) \rceil$, and put $c=2^5t(k+1)$. We may
assume by symmetry that
$c_1 \geq c_2$. Let $d= \lfloor c_2(k+1)^{-1}\rfloor$. Note that the hypothesis forces $c_2 \geq c \geq 2(k+1)$, which
ensures that $d$ is positive. \\
The definition of $d$ and the inequality $k \ge 100/\gamma'$ imply
\begin{equation}\label{03.005}
\min(|I|,|J|) \geq \min(|X|,|Y|) \geq c_2 \geq (k+1)d \geq 100d/ \gamma'.
\end{equation}
Given a set $Z$, recall that we write $Z^x_d$ for $Z \cap (x \text{ mod } d)$, the points of $Z$ in a fibre. Since $I$ and $J$ are intervals, for every $x \in \mathbb{Z}_d$ we have
\begin{equation*}\label{03.025}
|I|/d+1 \geq |I^x_d| \geq |I|/d-1 \text{ and } |J|/d+1 \geq |J^x_d| \geq |J|/d-1.
\end{equation*}
Combining the last two inequalities, for every $x \in \mathbb{Z}_d$ we have
\begin{equation}\label{03.027}
|I^x_d| \geq |I|/d- (\gamma'/100 d) \min(|I|,|J|) \text{ and } |J^x_d| \geq |J|/d-(\gamma'/100 d) \min(|I|,|J|)
\end{equation}
and
\begin{equation}\label{03.028}
|I^x_d| \leq |I|/d+ (\gamma'/100 d) \min(|I|,|J|) \text{ and } |J^x_d| \leq |J|/d+(\gamma'/100 d) \min(|I|
,|J|).
\end{equation}
\noindent
We may assume (by taking a translate of $X$, if necessary) that
$|X^0_{d}| = \max_{x \in \mathbb{Z}_{d}}|X^x_{d}|$. Then
\begin{equation*}\label{03.01}
|X^0_{d}| \geq |X|/d \geq |I|/d- (\gamma/ d) \min(|I|,|J|) \geq |I|/d- (\gamma'/3 d) \min(|I|,|J|).
\end{equation*}
Now define the sets $E_X, E_Y \subset \mathbb{Z}_d$ by
\[
E_X=\{x \in \mathbb{Z}_d \text{ : } |X^x_{d}| \geq (|I|/d)- (\gamma'/3 d) \min(|I|,|J|) \}
\]
and
\[
E_Y= \{x \in \mathbb{Z}_d \text{ : } |Y^x_{d}| \geq (|J|/d)- (\gamma'/3 d) \min(|I|,|J|) \}.
\]
For all $x\in E_X$ and $y \in E_Y$, noting that $X^x_d \subset I^x_d$ and $Y^y_d \subset J^y_d$, we see by \eqref{03.027} and
\eqref{03.028} that
\[
\max(|X^x_d \Delta I^x_d|, |Y^x_d \Delta J^y_d|) \leq (1/3+1/100) (\gamma'/d) \min(|I|,|J|) \leq \gamma'
\min(|I^x_d|, |J^y_d|).
\]
Since $(\alpha/2)|J^y_d| \leq |I^x_d| \leq (2/\alpha)|J^y_d|$, by Theorem~\ref{CD_technical}, there is a family $\mathcal{F}^y_d$ of subsets of $Y^y_d$ of size $k$, depending only on $Y^y_d$ (not on $X^x_d$), such that
\begin{equation}\label{03_06}
\E_{Z\in \mathcal{F}^y_d} |X^x_d+Z| \geq |X^x_d|+|Y^y_d|-1.
\end{equation}
Now construct a family $ \mathcal{F} = \{ \cup_{y \in E_Y} F^y_d \text{ : } F^y_d \in \mathcal{F}^y_d \}$,
and note that each set $F \in \mathcal{F}$ satisfies $|F| \leq |E_Y| k \leq dk \leq c_2-d$.
Define sets $ E_x' \subset E_X$ and $E_Y' \subset E_Y$ by
\[
E_X'=\{x \in \mathbb{Z}_d \text{ : } |X^x_{d}| \geq (|I|/d)- (\gamma'/10 d) \min(|I|,|J|) \}
\]
and
\[
E_Y'= \{x \in \mathbb{Z}_d \text{ : } |Y^x_{d}| \geq (|J|/d)- (\gamma'/10 d) \min(|I|,|J|) \}.
\]
By Markov's inequality,
\begin{equation*}
\mathbb{P}(E_X') \geq 1-\frac{\gamma}{(1/10-1/100)\gamma'}
\geq \frac{2}{3} \ \ \text{and} \ \ \mathbb{P}(E_Y') \geq 1- \frac{\gamma}{(
1/10-1/100)\gamma'} \geq \frac{2}{3}.
\end{equation*}
Simple calculations using \eqref{03.027} and \eqref{03.028} now show that for all $x' \in
E_X'\subset E_X, y' \in E_Y'\subset E_Y, x \in \mathbb{Z}_d$ and $y \in \mathbb{Z}_d \setminus E_y $ we have
\begin{eqnarray*}
|X^{x'}_d|+|Y^{y'}_d| &\geq& |I|/d +|J|/d -(\gamma'/5 d) \min(|I|,|J|)\\
&\geq& [|I|/d + (\gamma'/100 d) \min(|I|,|J|)] + [|J|/d- (\gamma'/3d) \min(|I|,|J|)]\\
&+& (1/3-1/5-1/100 )(\gamma'/d) \min(|I|,|J|)\\
&\geq& |X^x_d|+|Y^y_d| +(\gamma'/10 d) \min(|I|,|J|)\\
&\geq& |X^x_d|+|Y^y_d| +(\alpha \gamma'/10 d) \max(|I|,|J|)
\geq (1+\alpha \gamma'/100) (|X^x_d|+|Y^y_d|).
\end{eqnarray*}
\noindent
Now consider the family
$ \mathcal{G} = \{ X^0_d\cup_{i=1}^t X^{x_i}_d \text{ : } x_i \in E_X'\}$.
By \eqref{03.028}, every set $G \in \mathcal{G}$ satisfies
\begin{eqnarray*}
|G| &\leq& (t+1)(|I|/d+ (\gamma'/100 d) \min(|I|,|J|)) \leq 8td^{-1}|X|\\
&\leq& 2^{4}t(k+1)c_2^{-1}|X| \leq 2^{-1}cc_2^{-1}|X| \leq 2^{-1}c_1\leq c_1-d.
\end{eqnarray*}
The last ingredient needed to complete the proof of Theorem~\ref{all_thm3} is the following lemma.
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Now define the sets $E_X, E_Y \subset \mathbb{Z}_d$ by
\[
E_X=\{x \in \mathbb{Z}_d \text{ : } |X^x_{d}| \geq (|I|/d)- (\gamma'/3 d) \min(|I|,|J|) \}
\]
and
\[
E_Y= \{x \in \mathbb{Z}_d \text{ : } |Y^x_{d}| \geq (|J|/d)- (\gamma'/3 d) \min(|I|,|J|) \}.
\]
For all $x\in E_X$ and $y \in E_Y$, noting that $X^x_d \subset I^x_d$ and $Y^y_d \subset J^y_d$, we see by \eqref{03.027} and
\eqref{03.028} that
\[
\max(|X^x_d \Delta I^x_d|, |Y^x_d \Delta J^y_d|) \leq (1/3+1/100) (\gamma'/d) \min(|I|,|J|) \leq \gamma'
\min(|I^x_d|, |J^y_d|).
\]
Since $(\alpha/2)|J^y_d| \leq |I^x_d| \leq (2/\alpha)|J^y_d|$, by Theorem~\ref{CD_technical}, there is a family $\mathcal{F}^y_d$ of subsets of $Y^y_d$ of size $k$, depending only on $Y^y_d$ (not on $X^x_d$), such that
\begin{equation}\label{03_06}
\E_{Z\in \mathcal{F}^y_d} |X^x_d+Z| \geq |X^x_d|+|Y^y_d|-1.
\end{equation}
Now construct a family $ \mathcal{F} = \{ \cup_{y \in E_Y} F^y_d \text{ : } F^y_d \in \mathcal{F}^y_d \}$,
and note that each set $F \in \mathcal{F}$ satisfies $|F| \leq |E_Y| k \leq dk \leq c_2-d$.
Define sets $ E_x' \subset E_X$ and $E_Y' \subset E_Y$ by
\[
E_X'=\{x \in \mathbb{Z}_d \text{ : } |X^x_{d}| \geq (|I|/d)- (\gamma'/10 d) \min(|I|,|J|) \}
\]
and
\[
E_Y'= \{x \in \mathbb{Z}_d \text{ : } |Y^x_{d}| \geq (|J|/d)- (\gamma'/10 d) \min(|I|,|J|) \}.
\]
By Markov's inequality,
\begin{equation*}
\mathbb{P}(E_X') \geq 1-\frac{\gamma}{(1/10-1/100)\gamma'}
\geq \frac{2}{3} \ \ \text{and} \ \ \mathbb{P}(E_Y') \geq 1- \frac{\gamma}{(
1/10-1/100)\gamma'} \geq \frac{2}{3}.
\end{equation*}
Simple calculations using \eqref{03.027} and \eqref{03.028} now show that for all $x' \in
E_X'\subset E_X, y' \in E_Y'\subset E_Y, x \in \mathbb{Z}_d$ and $y \in \mathbb{Z}_d \setminus E_y $ we have
\begin{eqnarray*}
|X^{x'}_d|+|Y^{y'}_d| &\geq& |I|/d +|J|/d -(\gamma'/5 d) \min(|I|,|J|)\\
&\geq& [|I|/d + (\gamma'/100 d) \min(|I|,|J|)] + [|J|/d- (\gamma'/3d) \min(|I|,|J|)]\\
&+& (1/3-1/5-1/100 )(\gamma'/d) \min(|I|,|J|)\\
&\geq& |X^x_d|+|Y^y_d| +(\gamma'/10 d) \min(|I|,|J|)\\
&\geq& |X^x_d|+|Y^y_d| +(\alpha \gamma'/10 d) \max(|I|,|J|)
\geq (1+\alpha \gamma'/100) (|X^x_d|+|Y^y_d|).
\end{eqnarray*}
\noindent
Now consider the family
$ \mathcal{G} = \{ X^0_d\cup_{i=1}^t X^{x_i}_d \text{ : } x_i \in E_X'\}$.
By \eqref{03.028}, every set $G \in \mathcal{G}$ satisfies
\begin{eqnarray*}
|G| &\leq& (t+1)(|I|/d+ (\gamma'/100 d) \min(|I|,|J|)) \leq 8td^{-1}|X|\\
&\leq& 2^{4}t(k+1)c_2^{-1}|X| \leq 2^{-1}cc_2^{-1}|X| \leq 2^{-1}c_1\leq c_1-d.
\end{eqnarray*}
The last ingredient needed to complete the proof of Theorem~\ref{all_thm3} is the following lemma.
\begin{lemma}\label{F-and-G}
The families ${\mathcal F}$ and ${\mathcal G}$ are such that
$$\E_{X' \in \mathcal{G},\ Y'\in \mathcal{F} } |X'+Y'| \geq |X|+|Y|-d.$$
\end{lemma}
\begin{proof}
By the linearity of expectation, it is enough to show that for all $z \in \mathbb{Z}_d$ we have
$$\E_{X' \in \mathcal{G}, \ Y'\in \mathcal{F} } |(X'+Y')^z_d| \geq |X^z_d|+|Y^z_d|-1.$$
First assume that $z \in E_Y$. Then, using \eqref{03_06}, we get
\begin{equation*}
\E_{X' \in \mathcal{G},\ Y'\in \mathcal{F} } |(X'+Y')^z_d| \geq \E_{Z\in \mathcal{F}^z_d } |(X^0_d+Z)^z_d|
= \E_{Z\in \mathcal{F}^z_d } |X^0_d+Z|
\geq |X^0_d|+|Y^z_d|-1 \geq |X^z_d|+|Y^z_d|-1.
\end{equation*}
Now assume instead that $z \not \in E_Y$. Then, using \eqref{03_06}, ${\mathbb P}(E_X')\ge 2/3$, and our bound on $|X^{x'}_d|+|Y^{y'}_d|$, we obtain
\begin{eqnarray*}
&&\E_{X' \in \mathcal{G}, \ Y'\in \mathcal{F} } |(X'+Y')^z_d| \geq
\E_{\substack{x_1, \hdots, x_t \in E_X' \\ Y'\in \mathcal{F} }} |((\cup_iX^{x_i}_d)+Y' )^z_d|\\
&& \hspace{30pt}\geq \E_{\substack{x_1, \hdots, x_t \in E_X' \\ Y'\in \mathcal{F} }}
\bigg( |((\cup_iX^{x_i}_d)+Y' )^z_d| \text{ : } \exists i \text{ such that } z-x_i \in E_Y' \bigg) \\
&& \hspace{130pt} \times\Prob_{x_1, \hdots, x_t \in E_X'} \bigg(\exists i \text{ such that } z-x_i \in E_Y' \bigg)\\
&&\hspace{30pt}\geq \E_{\substack{x \in E_X' \cap (z-E_Y') \\ Y'\in \mathcal{F} }} |(X^{x}_d+Y' )^z_d| (1-(2/3)^t)
\geq \E_{\substack{x \in E_X' \cap (z-E_Y') \\ Z\in \mathcal{F}^{z-x}_d }} |X^{x}_d+Z | (1-(2/3)^t)\\
&&\hspace{30pt} \geq (|X^x_d|+|Y^{z-d}_d|-1) (1-(2/3)^t)
\geq \bigg((1+\alpha \gamma'/100) (|X^0_d|+|Y^z_d|)-1\bigg)(1-(2/3)^t)\\
&&\hspace{30pt}\geq (1+100^{-1}\alpha \gamma') (|X^0_d|+|Y^z_d|)(1-(2/3)^t)-1
\geq |X^0_d|+|Y^z_d|-1 \geq |X^z_d|+|Y^z_d|-1,
\end{eqnarray*}
proving Lemma~\ref{F-and-G}.
\end{proof}
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To complete the proof of Theorem~\ref{all_thm3}, note that if
$X' \in X^{(c_1-d)}$ and $Y' \in Y^{(c_2-d)}$ satisfy
$|X'+Y'| \geq |X|+|Y|-d$, then
there exist sets $X'' \in X^{(c_1)}$ and $Y'' \in Y^{(c_2)}$ such that
$|X''+Y''| \geq |X|+|Y|-1.$
\end{proof}
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Having proved Theorem~\ref{all_thm3}, we turn to `improving the setup by changing the ends of the
intervals', the step we mentioned in our sketch of the proof at the start of the section.
\begin{lemma}\label{all_lem4}
For all $\beta$ and $\gamma$ with $2^{-20}> \beta >2^{20} \gamma >0$ the following holds. Let $A$ and $B$ be
subsets of $\mathbb{Z}_p$ and
let $I$ and $J$ be intervals of $\mathbb{Z}_p$ such that
$|A|+|B| \leq (1-\beta)p$ and $\max(|A\Delta I|, |B\Delta J|) \leq \gamma \min(|A|, |B|)$.
Then in $\mathbb{Z}_p$ there are two sets of three consecutive intervals, $I_1, I_2, I_3$ and $J_1, J_2, J_3$, with
\[
\lfloor (\beta/8) p \rfloor \le \min(|I_1|, |I_3|,|J_1|,|J_3|)\le \max(|I_1|, |I_3|,|J_1|,|J_3|) \leq (\beta/4) p
\]
such that, setting $A_i=A\cap I_i$ and $B_i=B \cap J_i$, we have $|A_1|=|B_1|$, $|A_3|=|B_3|$, and
\begin{equation}\label{all_eq7}
\max(|A\Delta I_2|, |B\Delta J_2|) \leq 2^{10}\gamma \min(|A|, |B|).
\end{equation}
Moreover, for $i \in \{1,3\}$ and for any two arithmetic progressions $P$ and $Q$ we have
\begin{equation*}
\max(|A_i \Delta P|, |B_i \Delta Q |) \geq |A_i|/2^{10}= |B_i|/2^{10}.
\end{equation*}
\end{lemma}
\begin{proof}
Let $I_2'$ and $J_2'$ be maximal intervals in $\mathbb{Z}_p$ satisfying \eqref{all_eq7}: note that such intervals do exist by
hypothesis. By our bounds on $|A|+|B|$, $|A\Delta I|$ and $|B\Delta J|$, we have
\begin{equation*}\label{eqp_001}
|I_2'|+|J_2'| \leq |A|+|B|+ 2 \gamma \min(|A|,|B|) \leq (1-\beta +2\gamma) p \leq (1- (\beta/2)) p.
\end{equation*}
Thus we can construct intervals $I_1', I_3'$ and $J_1', J_3'$ such
that $I_1',I_2',I_3'$ and $J_1', J_2', J_3'$ are two families of three consecutive
intervals of $\mathbb{Z}_p$ with
\begin{equation}\label{eqp_002}
|I_1'|=|I_3'|=|J_1'|=|J_3'|= \lfloor (\beta/8) p \rfloor.
\end{equation}
Note that, by the maximality of $I_2'$ and $J_2'$, for any intervals $I_1''', I_3'''$ and $J_1''', J_3'''$
such that $I_1''', I_2', I_3'''$ and $J_1''', J_2', J_3'''$ are familes of consecutive
intervals, if we set $A_i'''=A \cap I_i'''$ and $B_i'''= B \cap J_i'''$, then we have
\begin{equation}\label{eqp_003}
|I_i'''| > 2 |A_i'''| \text{ and } |J_i'''| > 2 |B_i'''|.
\end{equation}
Let $A_i'=A \cap I_i'$ and $B_i'=B\cap J_i'$, and assume that
$|A_1'| \geq |B_1'| \text{ and } |A_3'| \geq |B_3'|$.
(The three other cases are analogous.)
Note that by \eqref{all_eq7} we have
\begin{equation}\label{eqp_0030}
\max(|A_1'|, |A_3'|, |I_2' \setminus A_2'|) \leq \gamma \min(|A|, |B|) \leq 2\gamma \min( |A_2'|, |B_2'|) .
\end{equation}
Now consider subintervals $I_1''$ and $I_3''$ at the ends of interval $I_2'$ that are minimal subject
to the following two properties:
setting $A_i''=A \cap I_i''$, we have
\begin{equation}\label{eqp_004}
|A_1''| \geq 14|A_1'| \text{ and } |A_3''|\geq 14|A_3'|
\end{equation}
and there exist intervals $P_1''$ and $P_3''$ (contained inside $I_1''$ and $I_3''$ respectively) such that
\begin{equation}\label{eqp_005}
|P_1'' \Delta A_1''| \leq |A_1''|/14 \text{ and } |P_3'' \Delta A_3''|\leq |A_3''|/14.
\end{equation}
(Here we insist that if $A_1'=\emptyset$ or $A_3'=\emptyset$ then $I_1''=P_1''=A_1''=
\emptyset$ or $I_3''=P_3''=A_3''=\emptyset$, respectively.)
Note that such intervals $I_1''$ and $I_3''$ do exist, as the intervals $I_1''=P_1''=I_3''=P_3''=I_2'$ have the desired properties by \eqref{eqp_0030}.
We now show that
\begin{equation}\label{eqp_006}
|I_1''|+|I_3''| \leq 2^6\gamma \min(|A_2'|, |B_2'|)<|I_2'|.
\end{equation}
The right-hand inequality is immediate, since $A_2' \subset I_2'$ and $\gamma < 2^{-6}$. For the left-hand
inequality, suppose for a contradiction that say $ |I_1''| > 2^5 \gamma \min(|A_2'|,|B_2'|)$. Consider first the case when $2 \gamma \min(|A_2'|,|B_2'|) <1
$. By \eqref{eqp_0030} we have $A_1'=A_3'=\emptyset$, and hence we obtain $I_1''=I_3''=\emptyset$, which is a contradiction.
Consider now the case when $2 \gamma \min(|A_2'|,|B_2'|) \geq 1$. In this case we consider the proper subinterval $I_1'''$ at the
end of $I_1''$ with $|I_1'''|=\lfloor 2^5\gamma \min(|A_2'|,|B_2'|) \rfloor
\geq 30 \gamma \min(|A_2'|,|B_2'|)$. Let $A_1'''=I_1''' \cap A$. By \eqref{eqp_0030}, we
have $| I_1''' \setminus A_1'''| \leq |I_2' \setminus A_2'| \leq 2 \gamma \min(|A_2'|,|B_2'|)$. In particular,
we have $|A_1'''| \geq 28 \gamma \min(|A_2'|, |B_2'|) $, which implies $|I_1''' \setminus A_1'''| \leq |A_1'''|/14$. But by \eqref{eqp_0030},
we also have $|A_1'| \leq 2\gamma \min(|A_2'|, |B_2'|)$, which implies $|A_1'''| \geq 14|A_1'|$.
Therefore $I_1'''$ is an interval strictly smaller than $I_1''$ with the desired properties, giving a contradiction. This proves
inequality \eqref{eqp_006}.
By \eqref{eqp_006}, the intervals $I_1''$ and $I_3''$ induce a partition $I_2'=I_1''\sqcup I_2 \sqcup I_3''$ into consecutive
intervals. Moreover, by \eqref{eqp_006} and \eqref{all_eq7} we get
\begin{equation}\label{eqp_007}
|A\Delta I_2| \leq |A\Delta I_2'| + |I_1''|+|I_3''| \leq 2^7 \gamma \min(|A_2'|,|B_2'|).
\end{equation}
Note also that by \eqref{all_eq7} we have
\begin{equation}\label{eqp_0031}
|J_2'\setminus B_2'| \leq \gamma \min(|A|,|B|) \leq 2 \gamma \min(|A_2'|,|B_2'|).
\end{equation}
Now consider subintervals $J_1''$ and $J_3''$ at the ends of $J_2'$ such that with $B_i''=B\cap J_i''$ we
have
\begin{equation}\label{eqp_0032}
|B_1'|+|B_1''|=|A_1'|+|A_1''| \text{ and } |B_3'|+|B_3''|=|A_3'|+|A_3''|
\end{equation}
Note that there are such intervals, since for $i \in \{1,3\}$ both $|B_i'| \leq |A_i'| \leq |A_i'|+|A_i''|$
and $|B_i'|+|B_2'| \geq |B_2'| \geq |A_i'|+|I_i''| \geq |A_i'|+|A_i''|$ hold. We now show that
\begin{equation}\label{eqp_0040}
|J_1''|+|J_3''| \leq 2^8\gamma \min(|A_2'|, |B_2'|)<|J_2'|.
\end{equation}
Assume for a contradiction that $|J_1''| \geq 2^7 \gamma \min(|A_2'|, |B_2'|)$. On the one hand, by \eqref{eqp_0031}
we deduce $|B_1''| \geq |J_1''|- |J_2' \setminus B_2'| \geq 126 \gamma \min(|A_2'|,|B_2'|)$. On the other hand,
by \eqref{eqp_0030} we have $|A_1'| \leq 2 \gamma
\min(|A_2'|, |B_2'|)$ and by \eqref{eqp_006} we have $|A_1''| \leq |I_1''| \leq 2^6 \gamma \min(|A_2'|, |B_2'|)$.
Thus we
obtain $|B_1''| > |A_1'|+|A_1''|$, which gives the desired contradiction. This proves inequality \eqref{eqp_0040}.
By \eqref{eqp_0040}, the intervals $J_1''$ and $J_3''$ induce a partition $J_2'=J_1''\sqcup J_2 \sqcup J_3''$ into
consecutive intervals. Moreover, by \eqref{eqp_0040} and \eqref{all_eq7} we have
\begin{equation}\label{eqp_008}
|B\Delta J_2| \leq |B\Delta J_2'| + |J_1''|+|J_3''| \leq 2^9 \gamma \min(|A_2'|,|B_2'|).
\end{equation}
For $i \in \{1,3\} $, set $I_i=I_i' \sqcup I_i''$, $J_i= I_i'\sqcup I_i''$, and note that $I_1, I_2, I_3$ and $J_1, J_2, J_3$ are consecutive intervals. Moreover, by \eqref{eqp_002} we have
\begin{equation}\label{eqp_009}
\min(|I_1|, |I_3|, |J_1|, |J_3|) \geq \lfloor (\beta/8) p \rfloor.
\end{equation}
In addition, by \eqref{eqp_002}, \eqref{eqp_006} and \eqref{eqp_0031}, we also have
\begin{equation}\label{eqp_0010}
\max(|I_1|, |I_3|, |J_1|, |J_3|) \leq (\beta/4) p .
\end{equation}
For $i \in \{1,2,3\}$ let $A_i=A\cap I_i$ and $B_i = B \cap J_i$. By \eqref{eqp_0032} we have
\begin{equation}\label{eqp_010}
|A_1|=|B_1| \text{ and } |A_3|=|B_3|.
\end{equation}
It remains to show that for any arithmetic progressions $P_1$ and $P_3$ we have
\begin{equation}\label{eqp_011}
|A_1 \Delta P_1| \geq 2^{-10}|A_1| \text{ and } |A_3 \Delta P_3| \geq 2^{-10}|A_3|.
\end{equation}
Assume for a contradiction that
\begin{equation}\label{eqp_012}
|A_1 \Delta P_1| < 2^{-10}|A_1|,
\end{equation}
which in particular means that
\begin{equation}\label{eqp_0125}
A_1\neq \emptyset \text{ i.e. } A_1' \neq \emptyset.
\end{equation}
Note that \eqref{eqp_004}, \eqref{eqp_005} and \eqref{eqp_012} imply
\begin{eqnarray*}
|P_1'' \Delta P_1| &&\leq |A_1 \Delta P_1| + |P_1'' \Delta A_1|
\leq |A_1 \Delta P| + |A_1'| + |P_1'' \Delta A_1''| \\
&&\leq |A_1|/2^{10} + |A_1''|/14+|A_1''|/14 \leq (1/2^9+1/7) |A_1''| \le |P_1''|/4.
\end{eqnarray*}
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Now consider subintervals $J_1''$ and $J_3''$ at the ends of $J_2'$ such that with $B_i''=B\cap J_i''$ we
have
\begin{equation}\label{eqp_0032}
|B_1'|+|B_1''|=|A_1'|+|A_1''| \text{ and } |B_3'|+|B_3''|=|A_3'|+|A_3''|
\end{equation}
Note that there are such intervals, since for $i \in \{1,3\}$ both $|B_i'| \leq |A_i'| \leq |A_i'|+|A_i''|$
and $|B_i'|+|B_2'| \geq |B_2'| \geq |A_i'|+|I_i''| \geq |A_i'|+|A_i''|$ hold. We now show that
\begin{equation}\label{eqp_0040}
|J_1''|+|J_3''| \leq 2^8\gamma \min(|A_2'|, |B_2'|)<|J_2'|.
\end{equation}
Assume for a contradiction that $|J_1''| \geq 2^7 \gamma \min(|A_2'|, |B_2'|)$. On the one hand, by \eqref{eqp_0031}
we deduce $|B_1''| \geq |J_1''|- |J_2' \setminus B_2'| \geq 126 \gamma \min(|A_2'|,|B_2'|)$. On the other hand,
by \eqref{eqp_0030} we have $|A_1'| \leq 2 \gamma
\min(|A_2'|, |B_2'|)$ and by \eqref{eqp_006} we have $|A_1''| \leq |I_1''| \leq 2^6 \gamma \min(|A_2'|, |B_2'|)$.
Thus we
obtain $|B_1''| > |A_1'|+|A_1''|$, which gives the desired contradiction. This proves inequality \eqref{eqp_0040}.
By \eqref{eqp_0040}, the intervals $J_1''$ and $J_3''$ induce a partition $J_2'=J_1''\sqcup J_2 \sqcup J_3''$ into
consecutive intervals. Moreover, by \eqref{eqp_0040} and \eqref{all_eq7} we have
\begin{equation}\label{eqp_008}
|B\Delta J_2| \leq |B\Delta J_2'| + |J_1''|+|J_3''| \leq 2^9 \gamma \min(|A_2'|,|B_2'|).
\end{equation}
For $i \in \{1,3\} $, set $I_i=I_i' \sqcup I_i''$, $J_i= I_i'\sqcup I_i''$, and note that $I_1, I_2, I_3$ and $J_1, J_2, J_3$ are consecutive intervals. Moreover, by \eqref{eqp_002} we have
\begin{equation}\label{eqp_009}
\min(|I_1|, |I_3|, |J_1|, |J_3|) \geq \lfloor (\beta/8) p \rfloor.
\end{equation}
In addition, by \eqref{eqp_002}, \eqref{eqp_006} and \eqref{eqp_0031}, we also have
\begin{equation}\label{eqp_0010}
\max(|I_1|, |I_3|, |J_1|, |J_3|) \leq (\beta/4) p .
\end{equation}
For $i \in \{1,2,3\}$ let $A_i=A\cap I_i$ and $B_i = B \cap J_i$. By \eqref{eqp_0032} we have
\begin{equation}\label{eqp_010}
|A_1|=|B_1| \text{ and } |A_3|=|B_3|.
\end{equation}
It remains to show that for any arithmetic progressions $P_1$ and $P_3$ we have
\begin{equation}\label{eqp_011}
|A_1 \Delta P_1| \geq 2^{-10}|A_1| \text{ and } |A_3 \Delta P_3| \geq 2^{-10}|A_3|.
\end{equation}
Assume for a contradiction that
\begin{equation}\label{eqp_012}
|A_1 \Delta P_1| < 2^{-10}|A_1|,
\end{equation}
which in particular means that
\begin{equation}\label{eqp_0125}
A_1\neq \emptyset \text{ i.e. } A_1' \neq \emptyset.
\end{equation}
Note that \eqref{eqp_004}, \eqref{eqp_005} and \eqref{eqp_012} imply
\begin{eqnarray*}
|P_1'' \Delta P_1| &&\leq |A_1 \Delta P_1| + |P_1'' \Delta A_1|
\leq |A_1 \Delta P| + |A_1'| + |P_1'' \Delta A_1''| \\
&&\leq |A_1|/2^{10} + |A_1''|/14+|A_1''|/14 \leq (1/2^9+1/7) |A_1''| \le |P_1''|/4.
\end{eqnarray*}
Recall that, when $A_1' \neq \emptyset$, $P_1''$ is a subinterval of $I_1''$ with $p/8 \geq |P_1''| \geq 8$
by \eqref{eqp_004}, \eqref{eqp_005} and \eqref{eqp_006}. It follows that $P_1$ is an interval intersecting the
interval $I_1''$ of size $p/4 \geq |P_1| \geq 4$. By \eqref{eqp_0010} we also
deduce $|P_1|+|I_1| \leq p/2$, which implies
that $P_1 \cap I_1$ is an interval. By replacing $P_1$ with $P_1\cap I_1$, we may assume that $P_1$ is a
subinterval of $I_1$.
We distinguish two cases: recalling that $I_1''$ and $I_3''$ were chosen to be minimal subject
to \eqref{eqp_004} and \eqref{eqp_005}, we ask whether the lower bound on the size of $A_1''$ in \eqref{eqp_004} is attained or not.
\textbf{Case A. $|A_1''|=14|A_1'|$.}\\
In this case, by \eqref{eqp_003}, we have
\begin{equation*}\label{eqp_015}
|(P_1 \cap I_1')\Delta A_1'| \geq |A_1'|.
\end{equation*}
So we obtain
\begin{equation}\label{eqp_016}
|A_1\Delta P_1| \geq |(P_1 \cap I_1')\Delta A_1'| \geq |A_1'| = |A_1|/15.
\end{equation}
\textbf{Case B. $|A_1''|>14|A_1'|\geq 14$.}\\
In this case, by the minimality of $I_1''$, if we let $x$ be the last point inside $A_1''$, then by \eqref{eqp_005} we deduce
\begin{equation*}\label{eqp_018}
|(P_1 \cap I_1'') \Delta (A_1'' \setminus \{x\})| > |A_1'' \setminus \{x\}| /14 .
\end{equation*}
This is equivalent to
\begin{equation*}\label{eqp_019}
|(P_1 \cap I_1'') \Delta (A_1'' \setminus \{x\})| \geq \min(2, |A_1'' \setminus \{x\}| /14) ,
\end{equation*}
and hence
\begin{equation*}\label{eqp_020}
|(P_1 \cap I_1'') \Delta A_1''| \geq \min(1, (|A_1''|-1)/14-1) \geq |A_1''|/28.
\end{equation*}
Thus we obtain
\begin{equation}\label{eqp_021}
|A_1\Delta P_1| \geq |(P_1 \cap I_1'')\Delta A_1''| \geq |A_1''|/28 \geq |A_1|/56.
\end{equation}
Inequalities \eqref{eqp_016} and \eqref{eqp_021} imply that, in either case,
$|A_1 \Delta P_1| \geq |A_1|/56$ , contradicting \eqref{eqp_012} and so proving \eqref{eqp_011}.
The proof of Lemma~\ref{all_lem4} is now complete, thanks to \eqref{eqp_009}, \eqref{eqp_0010} \eqref{eqp_010},
\eqref{eqp_007}, \eqref{eqp_008} and \eqref{eqp_011}.
\end{proof}
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Our next lemma is a simple fact about the `stickout' of sumsets from a set of fixed size:
roughly speaking, it says that if $Y$ is much larger than $X$ then the sum of $X$ with a random few points of
$Y$ is expected to be
much larger than $2|X|$.
\begin{lemma}\label{all_lem7}
Let $X$, $Y$ and $Z$ be subsets of $\mathbb{Z}_p$ and let $1 \leq c_1 \leq |X|$ and $1 \leq c_2 \leq |Y|$ be
integers such that $c_1c_2 \geq 16 |X| $. Suppose that $8|X|, 8|Z| \leq |Y|<p/2$.
Then there exist
$X' \in X^{(c_1)}$ and $Y' \in Y^{(c_2)}$ such that $|(X'+Y')\setminus Z| \geq 2|X|$.
\end{lemma}
\begin{proof}
Suppose the assertion is false, i.e. $\max_{X' \in X^{(c_1)}, Y' \in Y^{(c_2)}} |(X'+Y')\setminus Z| < 2|X|$ which, in particular, implies that $\max_{X' \in X^{(c_1)}, Y' \in Y^{(c_2/2)}} |(X'+Y')\setminus Z| < 2|X|$. Let $X'=\{x_1, \hdots, x_{c_1}\}$ and $Y'=\{y_1, \hdots, y_{c_2/2}\}$ be elements of $X^{(c_1)}$ and $Y^{(c_2/2)}$ chosen uniformly at random. Then ${\mathbb E} |(X'+Y')\setminus Z|$ is bounded from below as follows:
\begin{eqnarray*}
&& \ \ \ \sum_{i,j} {\mathbb E}|\{x_i+y_j\}\, \setminus \, [Z \cup (\{x_1, \hdots, x_{i-1}, x_{i+1}, \hdots, x_{c_1}\} + \{y_1, \hdots, y_{j-1}, y_{j+1}, \hdots, y_{c_2}\} ) ] |\\
&&= \sum_{i,j} \Prob | x_i+y_j \not \in Z \cup (\{x_1, \hdots, x_{i-1}, x_{i+1}, \hdots, x_{c_1}\} + \{y_1, \hdots, y_{j-1}, y_{j+1}, \hdots, y_{c_2}\} ) |\\
&&\geq \sum_{i,j} 1-\max_{ X' \in X^{(c_1)}, Y'\in Y^{(c_2/2)}}\frac{|Z \cup (X'+Y')|}{|Y|-(c_2/2)}\\
&&\geq \sum_{i,j} 1- \frac{|Z|+2|X|}{|Y|/2} \geq \frac{c_1c_2}{2} (1-\frac{3}{4}) \geq 2|X|,
\end{eqnarray*}
so $ {\mathbb E} |(X'+Y')\setminus Z| \geq 2|X|$, completing the proof.
\end{proof}
As an immediate corollary we have the following, obtained by applying the previous lemma inductively on $k$
(increasing $Z$ at each stage).
\begin{corollary}\label{cor_new000}
Let $X_1, \hdots, X_k$, $Y_1, \hdots, Y_k$ and $Z$ be subsets of $\mathbb{Z}_p$ and
let $1 \leq c_1^i \leq |X_i|$ and $1 \leq c_2^i \leq |Y_i|$ be integers such that
for all $i$ we have $c_1^ic_2^i\geq 16|X_i|$. Suppose
that for all $i$ we have $16|X_i|, 16|Z|\leq |Y_i| <p/2$. Then there exist
$X'_i \in (X_i)^{(c_1^i)}$ and $Y'_i \in (Y_i)^{(c_2^i)}$ (for each $i$) such that
\[
\ \ \ \ \ \ |\cup_i(X'_i+Y'_i)\setminus Z|
\geq \min(16^{-1}|Y_1|, \hdots, 16^{-1}|Y_k|, 2\sum_i|X_i|). \ \ \ \ \ \ \square
\]
\end{corollary}
\noindent
The final ingredient we need is a somewhat cumbersome result about partitions into intervals.
\begin{lemma}\label{all_lem6}
For all $1/2^{10}> \alpha, \beta >0$ the following holds. Consider partitions $\mathbb{Z}_p= I_0 \sqcup I_1
\sqcup I_2 \sqcup I_3 =J_0 \sqcup J_1 \sqcup J_2 \sqcup J_3$ into consecutive intervals such that
$|I_2|+|J_2| \leq (1-\beta/2)p$, $(\alpha/2)|J_2| \leq |I_2| \leq (2/\alpha) |J_2|$,
$\min(|I_2|, |J_2|) \ge 24/\beta$ and
\[
\lfloor (\beta/8) p \rfloor \le \min(|I_1|,|I_3|,|J_1|,|J_3|) \le \max(|I_1|, |I_3|,|J_1|,|J_3|) \le (\beta/4) p.
\]
Then there are four families of subsets of $\mathbb{Z}_p$, each of size $k \leq 100/ \alpha \beta$,
\[
\mathcal{I}_0=\{I^1_0, \hdots, I^k_0\}, \ \ \mathcal{I}_2=\{I^1_2, \hdots, I^k_2\}, \ \
\mathcal{J}_0=\{J_0^1, \hdots,J_0^k\}, \ \ \mathcal{J}_2=\{J_2^1, \hdots, J_2^k\}
\]
such that $\cup_i I_0^i = I_0, \cup_i J_0^i = J_0\ \text{ and }\ \cup_i I_2^i \subset I_2, \cup_i J_2^i \subset J_2$.
Furthermore, for every $1 \leq i \leq k$ we have \ $|I_2^i|=|J_2^i|= \lfloor (\beta/24) \min(|I_2|, |J_2|) \rfloor$
and $(I_0^i+J_2^i) \cap (I_2+J_2) = (J_0^i + I_2^i) \cap (I_2+J_2) = \emptyset$.
\end{lemma}
\begin{proof}
We construct the families of intervals $\mathcal{I}_0$ and $\mathcal{J}_2$. The construction of the
$\mathcal{J}_0$ and $\mathcal{I}_2$ is identical.
By the conditions above, for every $x \in I_0$ there is a
subinterval $J_2'$ of $J_2$ of size $\lfloor (\beta/8) \min(|I_2|,|J_2|) \rfloor$ such that
$(x+J_2') \cap (I_2+J_2) =\emptyset.$
Let $\mathcal{J}_2$ be a maximal collection of disjoint intervals of size $\lfloor (\beta/24) \min(|I_2|,|J_2|) \rfloor$
contained inside interval $J_2$.
First note that
\begin{equation*}
|\mathcal{J}_2| \leq |J_2|/\lfloor (\beta/24) \min(|I_2|,|J_2|) \rfloor \leq (48/ \beta)
(2/ \alpha) = 100/ (\alpha \beta).
\end{equation*}
Secondly, note that for any subinterval $J_2'$ of $J_2$ of size $\lfloor (\beta/8) \min(|I_2|,|J_2|) \rfloor$
there exists $J_2'' \in \mathcal{J}_2$
such that $J_2'' \subset J_2'$. Therefore, for any point $x \in I_0$ there exists $J_2^x \in \mathcal{J}_2$ such that
$$(x+J_2^x) \cap (I_2+J_2) =\emptyset.$$
Let $\mathcal{J}_2=\{J_2^1, \hdots, J_2^k\}$. For each $1 \leq i \leq k$
set $I_0^i=\{x \in I_0 \text{ : } J_2^x=J_2^i\}$, and finally put
$\mathcal{I}_0=\{I_0^1, \hdots, I_0^k\}$.
These families $\mathcal{I}_0$ and $\mathcal{J}_2$ have the desired properties.
\end{proof}
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We are now ready to prove Theorem~\ref{all_thm1}.
\begin{proof}[Proof of Theorem~\ref{all_thm1}]
Fix $\alpha, \beta>0$ and assume $\alpha, \beta<2^{-30}$. Let $\gamma'$ and $c'$ be the output of
Theorem~\ref{all_thm3} with input $\alpha/2$. Let $\gamma=2^{-30}\min(\gamma',\beta)$.
Let $\epsilon>0$ and $c''>0$ be the
output of Theorem~\ref{all_thm2} with input $\alpha, \beta, \gamma$. Choose
$c=10^{20} \alpha ^{-2} \beta^{-2}\max(c',c'')$.
Let $A$ and $B$ be
subsets of $\mathbb{Z}_p$ and let $1 \leq c_1 \leq |A|$ and $1 \leq c_2 \leq |B|$ such that
$c_1 c_2 \geq c \max(|A|,|B|)$. This forces $\min(c_1,c_2) \geq c$, which, in particular, implies $\min(|A|,|B|) \geq c$. Let $c_1'=c_1 \alpha \beta/ 10^5 $
and $c_2'=c_2 \alpha \beta/ 10^5 $ and note that $c_1'c_2' \geq 10^{10}\max(c',c'') \max(|A|,|B|)$ and
$\min(c_1',c_2') \geq 10^{10}\max(c',c'')$. Observe that we are done unless
\[
\max_{A' \in A^{(c_1')}, B' \in B^{(c_2')}}|A'+B'| < |A|+|B|-1,
\]
so we may assume that this holds. But then we may apply Theorem~\ref{all_thm2} to deduce that
there are arithmetic progressions $I$ and $J$ with the same common difference such that
\begin{equation}\label{main_2}
\max(|A\Delta I|, |B \Delta J|) \leq \gamma \min(|I|,|J|).
\end{equation}
Furthermore, we may and shall assume that $I$ and $J$ are intervals.
But then Lemma~\ref{all_lem4} can be used to deduce that $\mathbb{Z}_p$ has disjoint partitions into
consecutive intervals, $\mathbb{Z}_p=I_0\sqcup I_1\sqcup I_2 \sqcup I_3= J_0\sqcup J_1\sqcup J_2 \sqcup J_3$,
that have the following properties. (Here and elsewhere, the notation $\sqcup$ indicates that we are
taking the union of disjoint sets.) On the one hand we have
\begin{equation}\label{main_3}
\lfloor (\beta/8) p \rfloor \le \min(|I_1|, |I_3|,|J_1|,|J_3|) \le \max(|I_1|, |I_3|,|J_1|,|J_3|) \leq (\beta/4) p
\end{equation}
On the other hand, writing $A_i=A\cap I_i$ and $B_i=B \cap J_i$, we have $|A_1|=|B_1|$, $|A_3|=|B_3|$, and
\begin{equation}\label{main_6}
\max(|A\Delta I_2|, |B\Delta J_2|) \leq 2^{10}\gamma \min(|A|, |B|).
\end{equation}
Moreover, if $i \in \{1,3\}$ and $P$ and $Q$ are arithmetic progressions, then
\begin{equation}\label{main_7}
\max(|A_i \Delta P|, |B_i \Delta Q |) \geq 2^{-10} |A_i|=2^{-10} |B_i|.
\end{equation}
In particular, if $i \in \{1,3\}$ we have $|A_i|=|B_i|=0$ or $|A_i|=|B_i| \geq 2$. Because $\min(c_1',c_2') \geq c''$ and $c_1'c_2' \geq c''|A_i||B_i|$, we either have $\min(c_1',c_2') \geq |A_i|=|B_i|$ or $\min(c_1',|A_i|) \min(c_2', |B_i|) \geq c''|A_i|=c''|B_i|$. By the
contrapositive of Theorem~\ref{all_thm2}, we deduce that
\begin{equation*}
\E_{A' \in A_i^{(c_1')}, \ B' \in B_i^{(c_2')}} |A'+B'| > |A_i|+|B_i|-1.
\end{equation*}
We further deduce that there are sets $A_1' \in A_{1}^{(c_1')},\ B_1' \in B_{1}^{(c_2')}, \
A_3' \in A_{3}^{(c_3')}$\ and \ $B_3' \in B_{3}^{(c_2')}$\ such that
\begin{equation}\label{main_008}
|A_1'+B_1'|\geq |A_1|+|B_1|\ \ \text{and} \ \ |A_3'+B_3'|\geq |A_3|+|B_3|.
\end{equation}
Choose a subset $Z$ of $\cup_{i \in \{1,3\}}(A_i'+B_i') $
with $|A_1|+|B_1|+|A_3|+|B_3|$ elements, which, by \eqref{main_6}, satisfies
\begin{equation}\label{ccc_000}
|Z|=|A_1|+|B_1|+|A_3|+|B_3|\le |A\Delta I_2|+ |B \Delta J_2| \leq 2^{11} \gamma \min(|A|,|B|).
\end{equation}
Now, by \eqref{main_6} we also have
\begin{equation*}
(1- 2^{10} \gamma)|A| \leq |I_2| \leq (1+2^{10}\gamma) |A| \ \ \ \text{and} \ \ \
(1- 2^{10} \gamma)|B| \leq |J_2| \leq (1+2^{10}\gamma) |B|,
\end{equation*}
and so
\begin{equation}\label{main_9}
|I_2|+|J_2| \leq (1-\beta)(1+2^{10}\gamma) p \leq (1-\beta/2)p
\end{equation}
and
\begin{equation}\label{main_10}
(\alpha/2) |I_2| \leq |J_2| \leq (2/ \alpha)|I_2|.
\end{equation}
Furthermore,
we deduce that
\begin{equation}\label{main_11}
\min(|I_2|,|J_2|) \geq 2^{-1}\min(|A|,|B|) \geq 2^{-1}c \ge 24/\beta.
\end{equation}
First, by construction, we have $A_2 \subset I_2 \text{ and } B_2 \subset J_2$.
Hence, recalling the relations \eqref{main_6}, \eqref{main_9} and \eqref{main_10}, and the fact that $\min(c_1', |A_2|) \min(c_2', |B_2|) \geq 4^{-1}c_1'c_2' \geq c'\max(|A_2'|,|B_2'|)$, we may apply Theorem~\ref{all_thm3}
with parameters $\alpha/2$, $\gamma'\geq 2^{30} \gamma$ and $c'$ to obtain the following: there exist $A_2' \in A_2^{(c_1')}$
and $B_2' \in B_2^{(c_2')}$ such that
\begin{equation}\label{ddd_000}
|A_2'+B_2'| \geq |A_2|+|B_2|-1.
\end{equation}
Second, by \eqref{main_3}, \eqref{main_9}, \eqref{main_10} and \eqref{main_11}, we may apply
Lemma~\ref{all_lem6} to find four families of subsets of $\mathbb{Z}_p$, each with
$k\le 100/ \alpha \beta$ sets,
\[
\mathcal{I}_0=\{I^1_0, \hdots, I^k_0\}, \mathcal{I}_2=\{I^1_2, \hdots, I^k_2\}, \mathcal{J}_0=\{J_0^1,
\hdots, J_0^k\}, \mathcal{J}_2=\{J_2^1, \hdots, J_2^k\},
\]
such that
\begin{equation}\label{main_14}
\cup_i I_0^i = I_0,\ \ \cup_i J_0^i = J_0 \ \ \ \text{ and }\ \ \ \cup_i I_2^i \subset I_2, \ \ \ \cup_i J_2^i \subset J_2
\end{equation}
and for every $1 \leq i \leq k$ and we have
\begin{equation}\label{main_15}
|I_2^i|=|J_2^i|= \lfloor (\beta/24) \min(|I_2|, |J_2|) \rfloor
\end{equation}
and
\begin{equation}\label{main_16}
(I_0^i+J_2^i) \cap (I_2+J_2) = (J_0^i + I_2^i) \cap (I_2+J_2) = \emptyset.
\end{equation}
Writing $A_j^i= A \cap I_j^i$ and $B_j^i=B \cap J_j^i$, by \eqref{main_6}, we have
\begin{equation*}\label{ccc_002}
\max \{|A_0^i|, |B_0^i|\} \leq 2^{10} \gamma \min(|A|,|B|) .
\end{equation*}
and by \eqref{main_6}, \eqref{main_11} and \eqref{main_15}, we have
\begin{equation}\label{ccc_001}
\big(\beta/100\big)\ \min(|A|,|B|) \leq \min \{ |A_2^i|, |B_2^i|\} \le \max \{ |A_2^i|, |B_2^i|\} \leq p/2.
\end{equation}
The last two inequalities and inequality \eqref{ccc_000} imply that
\begin{equation}\label{ccc_0025}
10^{10} \max \{|A_0^i|,\ |B_0^i|,\ |Z|\} \leq \min \{ |A_2^i|, |B_2^i|\} \le \max \{ |A_2^i|, |B_2^i|\} \leq p/2.
\end{equation}
As we have $\min(c_1',c_2') \geq 10^{10}$ and $c_1'c_2' \geq 10^{10} \max(|A|,|B|) \geq 10^{10}\max(|A_0^i|,|B_0^i|)$, we further deduce that
\begin{equation}\label{ccc_003}
\min(c_1',|A_0^i|) \min(c_2', |B_2^i|) \geq 16 |A_0^i| \ \ \text{and} \ \ \min(c_1',|B_0^i|) \min(c_2', |A_2^i|) \geq 16 |B_0^i|.
\end{equation}
By Corollary~\ref{cor_new000} together with \eqref{ccc_0025} and \eqref{ccc_003} we deduce that for $i \in [k]$ there exist $A'^i_0 \in (A_0^i)^{(c_1')}, B'^i_2 \in (B_2^i)^{(c_2')}, A'^i_2 \in (A_2^i)^{(c_1')}, B'^i_0 \in (B_0^i)^{(c_2')}$ such that
\begin{equation*}
|\cup_i(A'^i_0+B'^i_2)\cup_i(A'^i_2+B'^i_0)\setminus Z| \geq 16^{-1}\min(|A_2^1|, \hdots, |A_2^k|,|B_2^1|, \hdots, |B_2^k|, 32\sum_i|A^i_0|+|B^i_0|)
\end{equation*}
By \eqref{main_6} and \eqref{ccc_001}, we further deduce
\begin{equation}\label{ccc_005}
|\cup_i(A'^i_0+B'^i_2)\cup_i(A'^i_2+B'^i_0)\setminus Z| \geq |A_0|+|B_0|
\end{equation}
Finally, note that from \eqref{main_3} and \eqref{main_9} it follows that
\[
I_1+J_1, I_2+J_2 \text{ and } I_3+J_3
\]
are disjoint sets, which in particular implies that
$$A_1'+B_1', A_2'+B_2' \text{ and } A_3'+B_3'$$
are disjoint sets. Moreover, by \eqref{main_16}, it follows that
$$\cup_i(A'^i_0+B'^i_2)\cup_i (A'^i_2+B'^i_0) \text{ and } A_2'+B_2' $$
are disjoint sets.
Let $$A'= A_1'\cup A_2'\cup A_3' \cup_i A'^i_0 \cup_i A'^i_2 \text{ and } B'= B_1'\cup B_2'\cup B_3' \cup_i B'^i_0 \cup_i B'^i_2.$$
From \eqref{main_008}, \eqref{ccc_000}, \eqref{ddd_000} and \eqref{ccc_005} we conclude that
$$|A'+B'| \geq (|A_0|+|B_0|)+(|A_1|+|B_1|)+(|A_2|+|B_2|-1)+(|A_3|+|B_3|) = |A|+|B|-1$$
and that
\[
|A'| \leq (3+2\times 100 \alpha^{-1}\beta^{-1})c_1' \leq c_1 \text{ and }|B'| \leq (3+2\times 100 \alpha^{-1}\beta^{-1})c_2' \leq c_2.
\]
This concludes the proof of Theorem~\ref{all_thm1}.
\end{proof}
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\section{Open problems}
One very natural question to ask is as follows. Suppose that as usual we are choosing $c_1$ points of $A$ and $c_2$
points of $B$, where $|A|=|B|=n$ and $c_1c_2$ is a fixed multiple of $n$. Are there phenomena that may not occur in
the regime where say we are
choosing $c_1=n$ (in other words, we choose the whole of $A$) and $c_2$ bounded, but might possibly
always hold when both $c_1$ and $c_2$ are of order $\sqrt{n}$?
One example is the following.
{\bf Question 1. }{\em
Is there a constant $c$ such that the following is true? If $A$ and $B$ are non-empty subsets of
${\mathbb Z}_p$ with $|A|=|B|=n \leq (p+1)/2$ then there are subsets $A'\subset A$ and
$B'\subset B$ with $|A'|=|B'|\le c \sqrt{n}$ such that $|A'+B'|\ge 2n-1$.
}
As remarked in the Introduction, this is not true for $c_1=n$ and $c_2$ bounded, in other words for $A'=A$ and
$|B'|$ bounded, as may be seen by taking $A$ and $B$ to be random subsets of ${\mathbb Z}_p$ of size approaching
$p/2$. But it might conceivably be true when we force both $c_1$ and $c_2$ to be large.
In a similar vein, one might ask whether the case of both $c_1$ and $c_2$ being of order $\sqrt{n}$ is
in fact always the `best' case (where $A$ and $B$ are set of size $n$, say). Thus for Theorem~\ref{all_thm1}
we would be asking the following.
{\bf Question 2. }{\em
Let $c>0$ and $c_1=c_1(n)$ be such that whenever $A$ and $B$ are subsets of ${\mathbb Z}_p$ with
$|A|=|B|=n \leq p/3$ there exist subsets $A'\subset A$ and
$B'\subset B$, with $|A'| \leq c_1$ and $|B'| \leq cn/c_1$, such that $|A'+B'|\ge 2n-1$. Does it follow that
whenever $A$ and $B$ are subsets of ${\mathbb Z}_p$ with
$|A|=|B|=n \leq p/3$ there exist subsets $A'\subset A$ and
$B'\subset B$ with $|A'|=|B'|\le \sqrt{cn}$ such that $|A'+B'|\ge 2n-1$?
}
One could also ask what the `worst' case is: is it when $c_1=c$ and $c_2=n$? More generally, is there
`monotonicity' as $c_1$ varies from $c$ to $\sqrt{cn}$?
It would also be very interesting to obtain good bounds on the constants appearing in our various results.
For example, in Theorem~\ref{all_thm1}, what is the form of the dependence of $c$ on $\alpha$ and $\beta$?
Finally, we consider what happens for discrete versions of the Brunn--Minkowski inequality. Green and
Tao~\cite{GreenTao}
showed
that, given a dimension $k$ and a constant $\varepsilon >0$, there exists $t$ such that if $A$ is a subset of
${\mathbb Z}^k$ of size $n$ that is not
contained inside $t$ parallel hyperplanes (intuitively, $A$ `does not look lower-dimensional'),
then $|A+A| \geq (2^k - \varepsilon) n$. We wonder if the following might be true. Although this is a question
about $\mathbb{Z}$ rather than $\mathbb{Z}_p$, we feel that the methods in this paper are likely to be
relevant.
{\bf Question 3. }{\em
For a given dimension $k$, does there exist a constant $c$ such that the
following holds? For any $\varepsilon>0$ there exists $t$ such that if $A$ is a subset of ${\mathbb Z}^k$ of
size $n$ that is not contained in
$t$ parallel hyperplanes, then there exists a subset $A'$ of $A$ of size at most $c \sqrt{n}$ such
that $|A'+A'| \geq (2^k - \varepsilon) n$.
}
\end{document}
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\begin{document}
\newtheorem{definition}{Definition}[section]
\newtheorem{theorem}[definition]{Theorem}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{remark}[definition]{Remark}
\newtheorem{remarks}[definition]{Remarks}
\newtheorem{examples}[definition]{Examples}
\newtheorem{leeg}[definition]{}
\newtheorem{definitions}[definition]{Definitions}
\newtheorem{proposition}[definition]{Proposition}
\newtheorem{example}[definition]{Example}
\newtheorem{comments}[definition]{Some comments}
\newtheorem{corollary}[definition]{Corollary}
\def\Box{\Box}
\newtheorem{observation}[definition]{Observation}
\newtheorem{observations}[definition]{Observations}
\newtheorem{defsobs}[definition]{Definitions and Observations}
\newenvironment{prf}[1]{ \trivlist
\item[\hskip \labelsep{\it
#1.\hspace*{.3em}}]}{~\hspace{\fill}~$\Box$\endtrivlist}
\newenvironment{proof}{\begin{prf}{Proof}}{\end{prf}}
\title{Analytic $q$-difference equations}
\author{Marius van der Put \\
\footnotesize Department of Mathematics, University of Groningen,
P.O.Box 800,\\
\footnotesize 9700 AV Groningen,
The Netherlands, [email protected] }
\date{}
\maketitle
\noindent
\section*{Introduction}
A complex number $q$ with $0<|q|<1$ is fixed. By an analytic $q$-difference equation we mean
an equation which can be represented by a matrix equation $Y(z)=A(z)Y(qz)$ where $A(z)$
is an invertible $n\times n$-matrix with coefficients in the field $K=\mathbb{C}(\{z\})$ of the
convergent Laurent series and where $Y(z)$ is a vector of size $n$. The aim of this paper
is to give an overview of our present knowledge of these equations and their solutions.
Definitions and statements are presented in detail. Examples illustrate the main results. For proofs we refer to the cited literature.
For section 1 and part of the following sections, the reference is [P-S]. For the later sections the reference is [P-R]. The last section presents unpublished results. The theory of linear differential equations over
$K$ (see [P-S.2], especially Chapter 10) has many of the features presented in this survey.
The manuscript [R-S]
contains a concise overview of analytic $q$-difference equations and its main purpose is
to develop a theory of $q$-summation leading to, in our terminology, a description of a
universal difference Galois group. As the authors of [R-S] note, this is only partially achieved
and part of their work is still conjectural. However for a certain class of $q$-difference equations,
namely those having at most two slopes and such that the slopes are integral,
they have explicit results. An important part of the extensive literature on $q$-difference equations can be found in the papers cited here.
\section{Difference equations in general}
A difference field $F$ is a field provided with an automorphism $\phi$ of infinite order.
A scalar linear difference equation is an equation of the form
\[\phi ^n(y)+a_{n-1}\phi ^{n-1}(y)+\cdots +a_1\phi (y) +a_0y=0\,\]
with given $a_i\in F$ and, say, $a_0\neq 0$.
As in the case of linear differential equations, one can transform this equation into a matrix difference equation, i.e., an equation of the form $Y=A\phi (Y)$, where $A$ is a given invertible $n\times n$-matrix with coefficients in $F$ and $Y$ denotes a vector of length $n$. On the vector space $M=F^n$ one considers the operator
$\Phi : Y\mapsto A\phi (Y)$. The bijective map $\Phi :M\rightarrow M$ is additive and
$\Phi (f\cdot m)=\phi (f)\cdot \Phi (m)$ for $m\in M$ and $f\in F$. This leads to the following definition of a {\it difference module $M=(M,\Phi )$ over $F$}:\\
$M$ is a finite dimensional vector space over $F$ and $\Phi :M\rightarrow M$ is an additive, bijective map
satisfying $\Phi (f\cdot m)=\phi (f)\cdot \Phi (m)$ for $m\in M$ and $f\in F$. The equation, corresponding
to a difference module is $Y=\Phi (Y)$. In general this equation has few solutions in $M$ itself. Like ordinary
polynomial equations over fields, one has to construct extensions of $F$ in order to have sufficiently many
solutions.
We will now assume that $F$ has {\it characteristic zero} and that its field of constants
$C: =\{f\in F|\ \phi (f)=f\}$ is {\it algebraically closed}. A {\it Picard-Vessiot ring (or extension) $R$ for a difference module $M$}, say, represented by the matrix equation $Y=A\phi (Y)$ is defined by:\\
(i) $R$ is an $F$-algebra (commutative and with a $1$),\\
(ii) $R$ is provided with an automorphism $\phi$ extending $\phi$ on $F$,\\
(iii) $R$ has only trivial $\phi$-invariant ideals,\\
(iv) there exists an invertible matrix $U$ (called fundamental matrix) with coefficients in $R$ such that
$U=A\phi (U)$,\\
(v) $R$ is generated over $F$ by the coefficients of $U$ and $\frac{1}{\det U}$.\\
Property (iii) translates in terms of $M$ into $V:=\{a\in R\otimes _FM|\ \Phi (a)=a\}$ is a vector space over
$C$ and the natural map $R\otimes _CV\rightarrow R\otimes _FM$ is an isomorphisms.
One constructs $R$ as follows. Let $X$ denote a matrix $(X_{i,j})$ of indeterminates and put $D:=\det X$.
The $F$-algebra $R_0:=F[X,\frac{1}{D}]$ is provided with a $\phi$-action, extending the one on $F$, by
the formula $(\phi X_{i,j} )=A^{-1}(X_{i,j})$. Let $I\subset R_0$ denote an ideal maximal among the ideals
invariant under $\phi$. Then $R=R_0/I$ is a Picard-Vessiot ring for the given equation.
The basic results of difference Galois theory are:\\
(1) A Picard-Vessiot ring $R$ exists and is unique up to a, non unique, isomorphism. \\
(2) $R$ is reduced (i.e., has no nilpotent elements).\\
(3) The set constants of the ring of total fractions of $R$ is $C$.\\
(4) Let $G$ be the group of the $F$-linear automorphism of $R$, commuting with $\phi$.
The natural action of $G$ on $R\otimes _FM$ induces a faithful action of $G$ on $V$, the solution space.
The image of $G$ in ${\rm GL}(V)$ is a linear algebraic subgroup of the latter. This makes $G$ into a linear algebraic group over $C$.\\
(5) The action of $G$ on $Spec(R)$ makes the latter into an $G$-torsor over $F$. In other words, there exists
a finite extension $F^+\supset F$ and a $G$-equivariant isomorphism
$F^+\otimes _CC[G]\rightarrow F^+ \otimes _FR$, where $C[G]$ is the coordinate ring of $G$.\\
Most of the notions and `operations of linear algebra', such as morphisms, kernels, cokernels, direct sums
have an obvious equivalent for difference modules. Let $(M_1,\Phi _1),\ (M_2,\Phi _2)$ denote two difference modules. The tensor product of the two modules is defined as $M_1\otimes _FM_2$ with $\Phi$ given by
$\Phi (m_1\otimes m_2)=(\Phi _1m_1)\otimes (\Phi _2m_2)$. The internal hom of the two modules is
${\rm Hom}_F(M_1,M_2)$ with $\Phi$ defined by $(\Phi (L))(m_1)=\Phi _2^{-1}(L(\Phi _1m_1))$ for
$L\in {\rm Hom}_F(M_1,M_2)$ and $m_1\in M_1$.
This leads to another, more abstract but very useful, formulation of the above Picard-Vessiot theory, namely that of (neutral) Tannakian categories. The category of all difference modules over $F$ is a neutral Tannakian
category. We will return to this in section 6. For a specific difference field $F$ one can
say much more than the above formalism (analogous to the case of ordinary Galois theory for specific fields).
\section{First examples of $q$-difference equations}
This exposition is concerned with the difference field $K=\mathbb{C}(\{z\})$, i.e., the field of convergent Laurent
series over $\mathbb{C}$, provided with the automorphism $\phi$ given by $\phi (z)=qz$, where
$q$ is a fixed complex number satisfying $0<|q|<1$. In order to define $\phi$ on the algebraic closure
$K_\infty =\cup _{n\geq 1} K_n$, with $K_n:=\mathbb{C}(\{z^{1/n}\})$, of
$K$ we choose a $\tau \in \mathbb{C}$ with $\Im (\tau )>0$ such that $q=e^{2\pi i \tau}$. Define $q^\lambda$ for $\lambda \in \mathbb{Q}$ (or any $\lambda \in \mathbb{C}$) as $e^{2\pi i\lambda \tau}$. Then the action of $\phi$ on
$K_\infty$ is given by $\phi (z^\lambda)=q^\lambda z^\lambda $. The action of $\phi$ on
$\widehat{K}=\mathbb{C}((z))$, i.e., the field of the formal Laurent series, and on its algebraic closure $\widehat{K}_\infty =\cup _{n\geq 1}\widehat{K}_n$, with $\widehat{K}_n:=\mathbb{C}((z^{1/n}))$,
is defined in a similar way.
Some $q$-difference rings $F=(F,\phi )$ will be considered, namely $\mathbb{C}[z,z^{-1}]$ and
$O$, the ring of the holomorphic functions on $\mathbb{C}^*$. A difference module $M=(M,\Phi )$ over a $q$-difference ring $F$ will be a {\it free} $F$-module of finite rank provided with a bijective additive map
$\Phi :M\rightarrow M$ such that $\Phi (f\cdot m)=\phi (f)\cdot \Phi (m)$ for $f\in F,\ m\in M$.
\begin{examples} {\rm $\ $\\
(a) $Ke, \Phi (e)=ce,\ c\in \mathbb{ C}^*$. There exists a solution ($\neq 0$)
in $K$ $\Leftrightarrow \ c\in q^{\mathbb{Z}}$. Indeed, $\Phi (z^te)=cq^tz^te$. If $c\in q^{\mathbb{Z}}$, then
the Picard-Vessiot ring is $K$ itself and the difference Galois group is $\{1\}$.
There exists a solution ($\neq 0$) in $K_n$ $\Leftrightarrow$ $c\in q^{\frac{1}{n}\mathbb{Z}}$. If
$c\in q^{\frac{1}{n}\mathbb{Z}}$ with $n$ minimal, then $K_n$ is the Picard-Vessiot ring and the difference
Galois group is $\mu _n:=\{a\in \mathbb{C}|\ a^n=1\}$.
In the remaining case, the Picard-Vessiot ring is $K[X,X^{-1}]$ with the action of $\phi$ given by
$\phi (X)=c^{-1}X$. The difference Galois group is $\mathbb{G}_m:=\mathbb{C}^*$. It consists of the automorphisms $\sigma$ given by $\sigma (X)=aX$ (for any $a\in \mathbb{C}^*$). We will use the symbol
$e(c)$ for this $X$. It can be given an interpretation as multivalued function $z^b$ for a $b$ with $q^b=c^{-1}$.
\noindent (b) The difference module $U_n:=Ke_1+\cdots +Ke_n$ with $\Phi$ given by the matrix
\[\left[\begin{array}{cccc}1&1& &\\
&1&1 &\\
&&&1\\
&&& 1 \end{array}\right] \]
is called {\it unipotent of length $n$}. The Picard-Vessiot ring is $K[X]$, with $\phi (X )=X +1$.
The difference Galois group is $\mathbb{G}_a:=\mathbb{C}$. The elements $\sigma$ of this group have the form $\sigma (X )=X +a$ (any $a\in \mathbb{C}$). We will use the symbol $\ell$ for this $X$. It has an interpretation as multivalued function $\frac{\log z}{2\pi i \tau}$. For $n=2$ one easily verifies that the solution space has $\mathbb{C}$-basis $\{e_1-\ell e_2,\ e_2\}$.
$\Box$ }\end{examples}
A difference module $M$ over $K$ is called {\it regular singular} if $M$ has a basis $e_1,\dots ,e_m$ such that
$\mathbb{C}\{z\}e_1+\cdots +\mathbb{C}\{z\}e_m$ is invariant under $\Phi$ and $\Phi ^{-1}$.
\begin{theorem} The following are equivalent\\
{\rm (i)} $M$ regular singular.\\
{\rm (ii)} $M=K\otimes _{\mathbb{C}}W$, $\dim _{\mathbb{C}}W < \infty$ and
$\Phi (f\otimes w)=\phi (f)\otimes A(w)$ for some $A\in {\rm GL}(W)$. Moreover, there is a unique $A$ such that
every eigenvalue $c$ satisfies $|q|<|c|\leq 1$.\\
{\rm (iii)} $M$ is obtained by $\oplus ,\otimes $ from the examples in {\rm 2.1}.
\end{theorem}
The difference module $(Ke,\ \Phi e=(-z)e)$ is the basic example of an {\it irregular singular} difference module. Its Picard-Vessiot
ring is $K[X,X^{-1}]$ with $\phi (X)=(-z)^{-1}X$. The difference Galois group is $\mathbb{G}_m$. We will
use the symbol $e(-z)$ for this $X$. It has the interpretation $ \Theta (z):=\sum _{n\in \bf Z}q^{n(n-1)/2}(-z)^n$
because of the well known formula $(-z)\Theta (qz)=\Theta (z)$.
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\section{Towards a classification of modules}
We present here the `classics' by G.D.~Birkhof, P.E.~Guenther, C.R.~Adams et al. and `modern' work
by J.-P.~Ramis, Ch.~Zhang, J.~Sauloy, A.~Duval, M.F.~Singer, M.~van der Put, M.~Reversat et al.,
concerning $q$-difference equations over $K$.
Let $K[\Phi ,\Phi ^{-1}]$ be the skew ring of difference operators. The elements of this ring are the finite
formal sums $\sum _{n\in \mathbb{Z}}a_n\Phi ^n$ and the multiplication is given by the rule $\Phi \cdot f
=\phi (f)\Phi$. This ring is (left and right) Euclidean. Any difference module $(M,\Phi _M)$ can be seen as
a left $K[\Phi ,\Phi ^{-1}]$-module where the action of $\Phi$ on $M$ is just $\Phi _M$. This left module is
{\it cyclic} (i.e., generated by one element) and therefore $M\cong K[\Phi ,\Phi ^{-1}]/K[\Phi ,\Phi ^{-1}]L$ for
some $L=\Phi ^m+a_{m-1}\Phi ^{m-1}+\cdots +a_1\Phi +a_0$ with $a_0\neq 0$.
The usual discrete valuation $v$ on $K$ is given by $v(0)=+\infty$ and for $a\in K^*$, $v(a)$ is the order of
$a$. Using the values $v(a_i)$ one defines the {\it Newton polygon} of $L$, as in the case of an ordinary
polynomial in $K[T]$. This Newton polygon depends on $M$ only. The slopes of the Newton polygon are in
$\mathbb{Q}$. A difference module $M$ is called {\it pure} if there is only one slope.
\begin{examples} {\rm
$M$ is regular singular if and only if $M$ is pure of slope 0.
\noindent
Further $(Ke,\ \Phi e=c(-z)^te)$ with $c\in \mathbb{C}^*,\ t\in \mathbb{Z}$ is pure of slope $t$. }
$\Box$\end{examples}
\begin{theorem}[Adams, Birkhoff, Guenther, Ramis, Sauloy] $\ $\\
$M$ has a unique tower of submodules
$0=M_0\subset M_1\subset \cdots \subset M_r=M$ such that every $M_i/M_{i-1}$
is pure of slope $\lambda _i$ and $\lambda _1<\cdots <\lambda _r$.
\end{theorem}
This filtration is called the {\it slope filtration} of $M$ and one defines the {\it graded module} $gr(M)$ of $M$
by $gr(M)=\oplus _i M_i/M_{i-1}$. In proving Theorem 3.2, one observes that the above operator $L$ has a unique factorization $L_1\cdot L_2\cdots L_r$ with each $L_i\in \widehat{K}[\Phi ,\Phi ^{-1}]$ monic and having only one slope $\lambda _i$. The main step is to show that these $L_i$ are actually convergent, i.e., belong
to $K[\Phi ,\Phi ^{-1}]$. This factorization of $L$ induces the slope filtration.
We note that for obtaining convergence, it is essential that $\lambda _1<\cdots <\lambda _r$. For any order
of the slopes $\lambda _i$ there is a similar factorization of $L$ in the ring $\widehat{K}[\Phi ,\Phi ^{-1}]$.
This has as consequence that $\widehat{K}\otimes _KM$ is in fact equal to the direct sum
$\oplus _i(\widehat{K}\otimes _K M_i/M_{i-1})=\widehat{K}\otimes _Kgr(M)$. In the next section we will study the moduli spaces that describe the modules $M$ with a fixed graded module $gr(M)$. A difference module
$M$ over $K$ is called {\it split} if it is isomorphic to its graded module $gr(M)$.
Here we continue with the classification of pure
modules over $K$. We note that by Theorem 3.2 any irreducible module over $K$ is pure.
\begin{definition} The module $E(cz^{t/n})$.\\{\rm
Given are the data $t/n,\ n\geq 1,\ (t,n)=1, \ c\in \mathbb{C}^*, |q|^{1/n}<|c|\leq 1$. They define
a difference module over $K_n$ of dimension 1, namely $(K_ne, \ \Phi e=cz^{t/n}e)$. This object,
considered as difference module over $K$, has dimension $n$ over $K$ and is called
$E(cz^{t/n})$.}
$\Box$\end{definition}
\begin{theorem} $E(cz^{t/n})$ is pure, irreducible and has slope $t/n$. Further,
$E(c_1z^{t/n})\cong E(c_2z^{t/n})$ if and only if $ c_1^n=c_2^n $. Moreover,
every irreducible $M$ over $K$ is isomorphic to some $E(cz^{t/n})$.
\end{theorem}
A difference module is called {\it indecomposable} if it is not the direct sum of two proper submodules.
We note that an indecomposable module over $K$ need not be pure.
\begin{theorem} The indecomposable pure modules over $K$ are
$E(cz^{t/n})\otimes U_m$ with $|q|^{1/n}<|c|\leq 1$. Further, the triple $(t/n, c^n,m)$ is unique.
\end{theorem}
\begin{definition} Global lattices. \\{\rm
A global lattice $\Lambda$ for a $q$-difference module $M$ over $K$ is a finitely generated
$\mathbb{C}[z,z^{-1}]$-submodule of $M$ (and hence free), invariant under $\Phi$ and $ \Phi ^{-1}$, such that
the natural map $K\otimes _{\mathbb{C}[z,z^{-1}]}\Lambda \rightarrow M$ is an
isomorphism. }
$\Box$\end{definition}
We will see later that {\it any} difference module $M$ has a unique global lattice. This means that the $q$-difference equation, defined locally at $z=0$, is equivalent to an equation on all of $\mathbb{P}^1$ with
at most poles at $z=0$ and $z=\infty$. From the explicit description of the indecomposable pure modules over $K$ it is not hard to deduce the following.
\begin{corollary} Every pure indecomposable difference module $M$ over $K$ has a unique global lattice.
This lattice, {\em denoted by} $M_{global}$, is a difference module over $\mathbb{C}[z,z^{-1}]$. The same holds for {\em split} difference modules over $K$.\\
Moreover, any morphism $f:M\rightarrow N$ between split difference modules over $K$ maps
$M_{global}$ into $N_{global}$. \end{corollary}
Before going on, we remark that the {\it classification of the difference modules over $\widehat{K}$} is remarkably simple. With the same methods used in the proof of Theorem 3.5, one shows that
every pure indecomposable difference module over $\widehat{K}$ has the form
$\widehat{K}\otimes _K(E(cz^{t/n})\otimes U_m)$ (again with unique $(t/n,c^n,m)$). Finally, as remarked before, any difference module over $\widehat{K}$ is a direct sum of pure modules over $\widehat{K}$.\\
It is well known that the elliptic curve $E_q:={\mathbb{C}}^*/q^{\mathbb{Z}}$, which we like to call {\it the Tate curve}, plays an important role for $q$-difference equations. With the help of 3.5, 3.6 and 3.7 one can deduce the following
rather striking result.
\begin{theorem} There is an additive, faithful functor $V$ from the category of the split difference modules over $K$ to the category of the vector bundles on $E_q$. It has the properties:\\
{\rm (i)} $V$ induces a bijection between the (isomorphy classes of) indecomposable modules over
$K$ and the (isomorphy classes of) indecomposable vector bundles on $E_q$.\\
{\rm (ii)} $V$ induces a bijection between (isomorphy classes of) objects. \\
{\rm (iii)} $V$ respects the constructions of linear algebra, i.e., tensor products, exterior powers etc.
\end{theorem}
\begin{proof} We will {\it sketch} a proof. \\
(1). We recall that $O$ denotes the algebra of the holomorphic functions on
${\mathbb C}^*$ and that a difference module $M$ over $O$ is a left module
over the ring $O[\Phi ,\Phi ^{-1}]$, free of some rank $m<\infty $ over
$O$. Further $pr:{\mathbb C}^*\rightarrow E_q:={\mathbb C}^*/q^{\mathbb Z}$
denotes the canonical map. One associates
to $M$ the vector bundle $v(M)$ of rank $m$ on $E_q$ given by
$v(M)(U)=\{f\in O(pr ^{-1}U)\otimes _OM|\ \Phi (f)=f\}$,
where, for any open $W\subset {\mathbb C}^*$, $O(W)$ is the algebra of the
holomorphic functions on $W$.
On the other hand, let a vector bundle $\mathcal M$ of rank
$m$ on $E_q$ be given. Then ${\mathcal N}:=pr^*{\mathcal M}$ is a vector
bundle on ${\mathbb C}^*$ provided with a natural isomorphism
$\sigma _q^*{\mathcal N}\rightarrow {\mathcal N}$, where $\sigma _q$ is the
map $\sigma _q(z)=qz$. One knows that ${\mathcal N}$ is in fact a free
vector bundle of rank $m$ on ${\mathbb C}^*$. Therefore, $M$, the collection
of the global sections of $\mathcal N$, is a free $O$-module
of rank $m$ provided with an invertible action $\Phi$ satisfying
$\Phi (fm)=\phi (f)\Phi (m)$ for $f\in O$ and $m\in M$.
It is easily verified that the above describes an equivalence $v$ of tensor categories.\\
\noindent (2). One associates to any split difference module $M$ over $K$, its global lattice $M_{global}$
and the $q$-difference module $O\otimes _{\mathbb{C}[z,z^{-1}]}M_{global}$ over $O$. The latter
induces by (1) a vector bundle on $E_q$, which we call $V(M)$. For a morphism $f:M\rightarrow N$ between split modules one has $f:M_{global}\rightarrow N_{global}$.Therefore $f$ induces a morphism $V(f):V(M)\rightarrow V(N)$. Thus we found the additive, faithful functor $V$. Clearly $V$ respects the constructions of linear algebra.\\
\noindent (3). As (ii) is an immediate consequence of (i), we are left with proving (i). The indecomposable module $M:=E(cz^{t/n})\otimes U_m$ is producing a vector bundle $V(M)$ of rank $nm$ and degree $tm$.
One can show that $V(M)$ is indecomposable and that $V(M)$ is irreducible if $m=1$. Further one can verify
that non isomorphic indecomposable $M, N$ produce non isomorphism vector bundles
$V(M), V(N)$. It is somewhat more complicated to show that every indecomposable vector bundle is isomorphic to $V(M)$ for a suitable indecomposable $M$. This last step can be avoided by an inspection of Atiyah's paper where the classification of the indecomposable vector bundles on $E_q$ is explicitly
given. \end{proof}
\begin{corollary} Let $B$ be a split module over $K$, then\\
{\rm (i)} ${\rm ker}(\Phi -1,O\otimes B_{global})\cong H^0(E_q,V(B))$.\\
{\rm (ii)} ${\rm coker}(\Phi -1,O\otimes B_{global})\cong H^1(E_q,V(B))$.\\
{\rm (iii)} The two canonical maps ${\rm coker}(\Phi -1,B_{global})
\rightarrow {\rm coker}(\Phi -1,B)$ and ${\rm coker}(\Phi -1,B_{global})
\rightarrow {\rm coker}(\Phi -1,O\otimes B_{global})$ are isomorphisms.
\end{corollary}
\begin{examples}{\rm Consider the difference module
$(B=Ke,\ \Phi e=(-z)^te)$ with $t\in \mathbb{Z}$. The line bundle
$V(B)$ is equal to $O_{E_q}(t\cdot [1])$, where $1$ denotes the neutral element of $E_q$.
For $t\geq 1$, ${\rm coker}(\Phi -1,O\otimes B_{global})=0$ and
${\rm ker}(\Phi -1,O\otimes B_{global})$ is the $t$-dimensional vector space with basis
$\{ \Theta (\zeta z)^te| \zeta ^t=1\}$.
For $t=0$, ${\rm ker}(\Phi -1,O)=\mathbb{C}1$ and
${\rm coker}(\Phi -1,O)$ has dimension 1. Explicitly, $O$ is the vector space of the everywhere
convergent Laurent series and has the formula
$(\Phi -1)(\sum _{n\in \mathbb{Z}}a_nz^n)
=\sum _{n\in \mathbb{Z}}(q^n-1)a_nz^n$. }
$\Box$ \end{examples}
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\section{Moduli spaces for difference modules}
Fix a split difference module $S=P_1\oplus \cdots \oplus P_r$ over $K$, where $P_i$ is pure
with slope $\lambda _i$ and $\lambda _1<\cdots <\lambda _r$. The problem is to classify
the difference modules $M$ over $K$ such that $gr(M)$ is isomorphic to $S$. The collection
of the isomorphy classes is a rather ugly object due to the fact that $S$ has many automorphisms.
A better way to formulate the problem is to consider pairs $(M,f)$ consisting
of a difference module $M$ and an isomorphism $f:gr(M)\rightarrow S$. Two pairs
$(M_1,f_1), (M_2,f_2)$ are called {\it equivalent} if there exists an isomorphism
$g:M_1\rightarrow M_2$ such that the induced graded isomorphism $gr(g):gr(M_1)\rightarrow
gr(M_2)$ has the property $f_1=f_2\circ gr(g)$. Let $Equiv(S)$ denote the set of equivalence classes.
This formulation allows us to define a covariant functor $\mathcal F$ from the category of finitely generated $\mathbb{C}$-algebras $R$ (i.e., $R$ is commutative and has a $1$) to the category of sets. For a finitely generated $\mathbb{C}$-algebra $R$ one considers
$K_R:=R\otimes _{\mathbb{C}}K$ and one can define the notion of difference module over
$K_R$. The set ${\mathcal F}(R)$ is the set of equivalence classes of pairs
$(M,f)$ with $M$ a difference module over $K_R$ and $f$ is a $K_R$-isomorphism $f:gr(M)\rightarrow K_R\otimes S$. Equivalence of two pairs is defined as above.
We note that ${\mathcal F}(\mathbb{C})$ is precisely $Equiv(S)$.
One can see the functor $\mathcal F$ as a contravariant functor on the category of affine
$\mathbb{C}$-schemes (of finite type).
\begin{theorem} The contravariant functor $\mathcal F$ on affine $\mathbb{C}$-schemes is representable. In fact, the affine space $\mathbb{A}_{\mathbb{C}}^N$ with
$N=\sum _{i<j}(\lambda _j-\lambda _i)\dim P_i\cdot \dim P_j$, represents $\mathcal F$.
In other words, the covariant
functor $\mathcal F$ is represented by a certain universal $q$-difference module $M$ over
$\mathbb{C}[X_1,\dots ,X_N]$. Further $Equiv(S)$ identifies with $\mathbb{C}^N$.
\end{theorem}
For the case of integer slopes $\lambda _i$, the above result is announced by
J.-P.~ Ramis and J.~Sauloy. The general case is treated in [P-R].
One can normalize the representing space $\mathbb{A}^N_{\mathbb{C}}$ by letting
$0$ correspond to the class of the pair $(S,id_S)$. The vector space structure of
$\mathbb{A}^N_{\mathbb{C}}$ has an interpretation for $s=2$, namely as the vector space
${\rm coker} (\Phi -1,{\rm Hom}(P_2,P_1))$. For $s>2$, the functor $\mathcal F$ and its representing space
still have a weaker structure, namely that of an iterated torsor.
We illustrate Theorem 4.1 by the following {\it basic example}:\\
$S=P_1\oplus P_2$, where $P_1=(Ke_1,\ \Phi e_1=e_1)$ and $P_2=(Ke_2,\ \Phi e_2=(-z)^te_2)$ with $t>0$. The moduli space is $\mathbb{A}^t_{\mathbb{C}}$ and the universal
family above this moduli space is
\[K[x_0,\dots ,x_{t-1}]e_1+K[x_0,\dots ,x_{t-1}]e_2,\ \Phi e_1=e_1,\]
\[ \Phi e_2=(-z)^te_2+(x_0+x_1z+\cdots +x_{t-1}z^{t-1})e_1\ . \]
Surprisingly enough, Theorem 4.1 (for the case of integer slopes) and this example are already present in the work of Birkhoff of Guenther.
\begin{corollary} Every difference module $M$ over $K$ has a unique global lattice.
This lattice will be called, as before, $M_{global}$.
Moreover, every morphism $f:M\rightarrow N$ satisfies $f(M_{global})\subset
N_{global}$. In particular, one can extend the functor $V$ of Theorem {\rm 3.8} to the category
of all $q$-difference modules over $K$.
\end{corollary}
This corollary can be deduced from the Theorem 4.1 and Corollary 3.9.
\section{Difference Galois groups}
In the last section a complete, however complicated, classification of the difference modules over $K$ is given. Using this classification we will be able to give a complete description of the
difference Galois groups. The difference Galois group of a module $M$ will be denoted by
$Gal(M)$.We start with the easiest case and build up to the general case.\\
\noindent (1) {\it Regular singular modules}. \\
We recall that a regular singular module has the form $M=K\otimes _{\mathbb{C}}W$ and
$\Phi (f\otimes w)=\phi (f)\otimes A(w)$ with $A\in {\rm GL}(W)$. We normalize $A$ such that
the eigenvalues of $A$ have absolute values in $(|q|,1]\subset \mathbb{R}$.
Let $L\subset \mathbb{C}^*/q^{\mathbb{Z}}=E_q$ be the group generated by the images of the
eigenvalues of $A$. Then:
\[Gal(M)={\rm Hom}(L,\mathbb {C}^*)\;(\times {\mathbb{C}}). \]
Write $L=L_{free}\oplus L_{torsion}$ with the first summand a free $\mathbb{Z}$-module of
rank $g\geq 0$ and where the second term is a finite commutative group. Then
${\rm Hom}(L,\mathbb{C}^*)$ is a product of $\mathbb{G}_m^g$ with a finite commutative group generated by at most two elements. If $A$ is semi-simple, then this is $Gal(M)$.
If $A$ is not semi-simple, then the term $\mathbb{G}_a=\mathbb{C}$ is also present. \\
\noindent (2) {\it Irreducible modules}.\\
We recall that $M=E(cz^{t/n})$. For $n=1$ and $t\neq 0$, the group $Gal(M)$ is
$\mathbb{G}_m=\mathbb{C}^*$. For $n>1$ one can describe $Gal(M)$ be an exact sequence
$1\rightarrow \mathbb{G}_m\rightarrow Gal(M)\rightarrow (\mathbb{Z}/n\mathbb{Z})^2\rightarrow 0$. The group $Gal(M)$ is not commutative and is not a semi-direct product of
$\mathbb{G}_m$ and $(\mathbb{Z}/n\mathbb{Z})^2$.\\
\noindent (3) {\it Indecomposable modules}.\\
We may suppose $M=E(cz^{t/n})\otimes U_m$ with $t/n\neq 0$ and $m>1$. Then
$Gal(M)=Gal(E(cz^{t/n})\times \mathbb{G}_a$.\\
\noindent (4) {\it Split modules}.\\
A split module $M$ is a direct sum of pure modules $M_i$. An explicit combination of the difference Galois groups of the $M_i$ (described above) yields $Gal(M)$.\\
\noindent (5) {\it The general case}.\\
We recall that $M$ has a slope filtration
$0=M_0\subset M_1\subset \cdots \subset M_r=M$ with $P_i:=M_i/M_{i-1}$ pure
and $gr(M)=S:=P_1\oplus \cdots \oplus P_r$. Let $\xi$ in the moduli space, introduced in section 4, represent $M$. Then there exists an exact sequence
\[1\rightarrow U_\xi \rightarrow Gal(M)\rightarrow Gal(S)\rightarrow 1\ ,\]
with $U_\xi$ a unipotent group, explicitly determined by $\xi$. This sequence is in fact a semi-direct product and the action, by conjugation, of $Gal(S)$ on $U$ is again explicit.\\
\begin{remarks} $\ $\\ {\rm
(1) The above description of $Gal(M)$ implies that $Gal(M)^o$ is a solvable group. This is in contrast with the differential Galois groups that occur for differential equations over $K$.\\
\noindent (2) {\it Difference modules over $\widehat{K}$}.
For a difference module $N$ over $\widehat{K}$, there exists a unique split difference
module $M$ over $K$ such that $N\cong \widehat{K}\otimes _KM$. Further, $N$ and $M$
have the same difference Galois group. \\
\noindent (3)
Let $M$ be a difference module over $K$. The step from $M$ to its global lattice $M_{global}$
is probably not algorithmic since it involves a computation with arbitrary complex numbers.
However, the classification in sections 3 and 4, assuming the knowledge of $M_{global}$,
is algebraic and can be shown to be algorithmic. The computation of $Gal(M)$
(and of the Picard-Vessiot ring for $M$), on the basis of the classification, is algorithmic as well.
We note that for {\it linear differential equations over $K$}, the existence of a theoretical algorithm is proven by E. Hrushovski (2001).
In that case no explicit algorithm is known. }
$\Box$ \end{remarks}
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\section{Universal Picard-Vessiot rings and \\
universal difference Galois groups}
We start by explaining some notions and constructions (see also [P-S2]).
An {\it affine group scheme $G$ over $\mathbb{C}$} is given by a
$\mathbb{C}$-algebra $A$ provided with the structure of a Hopf-algebra. The latter is defined by a triple $(m,e,i)$ of
$\mathbb{C}$-algebra morphisms:\\
(a) $m:A\rightarrow A\otimes _\mathbb{C}A$ (the co-multiplication)\\
(b) $e:A\rightarrow \mathbb{C}$ with $e(1)=1$ (the co-unit element),\\
(c) $i:A\rightarrow A$ is an isomorphism (the co-inverse).\\
Put $G:=Spec(A)$. The induced map $m^*:G\times G\rightarrow G$
is the multiplication, further $e^*:Spec(\mathbb{C})\rightarrow G$ is the unit element of $G$ and the induced map $i^*:G\rightarrow G$
is the map $g\mapsto g^{-1}$. The usual rules for a group, expressed
in $m^*,e^*,i^*$, are translated into rules for $m,e,i$. These rules
define a Hopf-algebra.
If $A$ is finitely generated over $\mathbb{C}$, then $G$ is an ordinary linear algebraic group over $\mathbb{C}$. In general, $A$ is the direct limit (in fact
a filtered union) of finitely generated sub-Hopf-algebras. This means that $G$ is the projective limit of linear algebraic groups.
A {\it representation of an affine group scheme $G$ over $\mathbb{C}$} is a morphism of affine group schemes $G\rightarrow {\rm GL}(W)$,
where $W$ is a finite dimensional vector space over $\mathbb{C}$.
Morphisms between representations are defined in the obvious way
and thus we can talk about the category $Repr_G$ of all representations of $G$. In this category one can perform all `constructions of linear algebra', e.g., kernels, co-kernels, direct sums,
tensor products, duals, and the rules that one knows from linear algebra are valid. We note that an equivalence between
$Repr_{G_1}$ and $Repr_{G_2}$, preserving the constructions of
linear algebra, comes from an isomorphism $G_1\rightarrow G_2$
of affine group schemes (this is Tannaka's theorem).
We adopt here the following rather trivial definition of {\it neutral Tannakian category}, namely it is a category $T$ having all
constructions and rules of linear algebra and is, for these structures,
equivalent to $Repr_G$ for a suitable affine group scheme $G$.
Of course there is an {\it intrinsic} definition of neutral Tannakian category. Using that, one can show that $\Delta _K$, {\it the category
of all differential modules over $K$}, is a Tannakian category.
The affine group scheme $G$ such that $\Delta _K$ is isomorphic to
$Repr_G$, is called the {\it universal difference Galois group} for
the category $\Delta_K$. It is this group that we want to describe.
A full subcategory $\Delta$ of $\Delta _K$ (i.e., for any objects $A,B$ of $\Delta$, one has ${\rm Hom}_{\Delta}(A,B)=
{\rm Hom}_{\Delta _K}(A,B)$ ), closed under the operations of linear algebra, is again a neutral Tannakian category.
Consider an object $M$ of $\Delta _K$. Write $\{\{M\}\}$ for the
full subcategory of $\Delta _K$ generated by $M$ and all constructions of linear algebra applied to $M$. Then $\{\{M\}\}$
is isomorphic to some $Repr_H$. Moreover, there is a Picard-Vessiot
ring $PVR(M)$ attached to $M$. It is a {\it general result} that $H$ can be identified with the linear algebraic group consisting of the
$K$-automorphisms of $PVR(M)$ that commute with $\phi$. In other
words, $H$ can be identified with the difference Galois group of
$M$.
The above holds for any full subcategory $\Delta$ of $\Delta _K$,
closed under the operations of linear algebra. There is a Picard-Vessiot ring $PVR(\Delta )$ for $\Delta$, namely the direct limit of the $PVR(M)$ for all objects of $\Delta$. Moreover the affine group scheme $H$, such that $Repr_H$ is equivalent to $\Delta$, identifies with the group of the $K$-linear automorphism of $PVR(\Delta )$,
commuting with $\phi$. An explicit description of the universal
Picard-Vessiot ring for $\Delta _K$ is what we are aiming to produce.
We build up a description of the universal Picard-Vessiot ring and the universal Galois group of $\Delta _K$, by considering suitable
subcategories of $\Delta _K$.\\
\noindent (1) {\it $\Delta _{rs}$, the category of the regular singular
difference modules over $K$}.\\
$PVR(\Delta _{rs})=K[\{e(c)\}_{c\in {\mathbb{C}}^*},\ell ]$ with rules:
\[e(c_1c_2)=e(c_1)\cdot e(c_2),\ e(q)=z^{-1} \mbox{ and }
\phi (e(c))=c^{-1}e(c),\ \phi (\ell )=1+\ell \]
One identifies $z^\lambda =e(q^{-\lambda} )$ for $\lambda \in {\bf Q}$ and thus $PVR(\Delta _{rs})$ contains the {\it algebraic closure} $K_\infty$ of $K$. We can therefore rewrite $PVR(\Delta _{rs})$
as $K_\infty [\{e(c)\},\ell ]$ with the additional relations
$z^\lambda =e(q^{-\lambda })$ for all $\lambda \in \mathbb{Q}$. Further,
the difference Galois group $G_{rs}$ is ${\rm Hom}({\mathbb{C}}^*/q^{\mathbb{Z}},{\mathbb{C}}^*)\times {\mathbb{C}}$.\\
A similar description holds for regular difference modules over
$\widehat{K}$. Let $\widehat{K}_\infty$ denote the {\it algebraic closure} of
$\widehat{K}$. Then the universal Picard-Vessiot ring is
$\widehat{K}[\{e(c)\},\ell ]$ and the universal difference Galois group
coincides with the above group $G_{rs}$.\\
{\it Comments}. This description follows from the observation that
$PVR(\Delta _{rs})$ is generated by the solutions for the modules in Examples 2.1. The given expression for $G_{rs}$ has to be interpreted
as an affine group scheme. The description follows from section 5,
part (1).\\
\noindent (2) {\it $\Delta _{split}$, the category of the split difference
modules over $K$}.
\[PVR(\Delta _{split})=K_\infty [\{e(c)\}_{c\in {\mathbb{C}}^*},\ell ,\{e(z^\lambda )\}_{\lambda \in {\mathbb{Q}}}]\]
with additional rules
$ e(z^{\lambda +\mu})=e(z^\lambda )\cdot e(z^\mu),\
\phi (e(z^\lambda ))=z^{-\lambda} \cdot e(z^\lambda ) $.\\
Let the corresponding universal difference Galois group be denoted by $G_{split}$. From the inclusion $\Delta _{rs}\subset \Delta _{split}$ one obtains an exact sequence of affine group schemes
\[1\rightarrow {\rm Hom}({\mathbb{Q}},{\mathbb{C}}^*)\rightarrow
G_{split}\rightarrow G_{rs}\rightarrow 1\ .\]
The group scheme $G_{split}$ is {\it not} a semi-direct product and
${\rm Hom}({\mathbb{Q}},{\mathbb{C}}^*)$ lies in the center of
$G_{split}$.\\
{\it Comments}. The new universal Picard-Vessiot ring is generated over the one of (1) by solutions for the modules $E(cz^{t/n})$. This explains the terms $e(z^\lambda )$. The exact sequence and its features follow from an explicit calculation of the automorphism
of $PVR(\Delta _{split})$. We note the contrast with the differential case! Finally, the descriptions for the universal Picard-Vessiot ring
and the universal difference Galois group for $\Delta _{\widehat{K}}$,
the category of all difference modules over $\widehat{K}$, is rather
similar.\\
\noindent (3) {\it $\Delta _K$, this is the most interesting and the most complicated case}.\\
What can be proved at present is the following:\\
(a) $PVR(\Delta _K)=\mathcal{D}[\{e(c)\},\ell ,\{e(z^\lambda )\}]$ and the latter is a subalgebra of the explicit universal Picard-Vessiot ring
$PVR(\Delta _{\widehat{K}})=
\widehat{K}_\infty [\{e(c)\},\ell, \{e(z^\lambda )\}]$.
Further $\mathcal{D}$ is the $K_\infty $-subalgebra of
$\widehat{K}_\infty $ consisting of the elements $f\in \widehat{K}_\infty$ satisfying a scalar $q$-differential equation over $K_\infty$.\\
(b) The $K$-algebra $\mathcal D$ is generated over $K_\infty$ by the solutions in $\widehat{K}_\infty $ of all equations of the form
\[ (c_1z^{-\lambda _1}\phi -1)^{m_1}\cdots (c_rz^{-\lambda _r}\phi -1)^{m_r}f=z^\mu \ ,\]
where $ 0<\lambda _1<\cdots <\lambda _r,\ r\geq 1,\ m_1,\dots ,m_r\geq 1,\ \mu \in \mathbb{Q}$.\\
(c) The universal difference group $G$ admits an exact sequence (in
fact is a canonical semi-direct product)
$1\rightarrow N\rightarrow G\rightarrow G_{split}\rightarrow 1$ ,
where $N$ is a (connected) unipotent group scheme. \\
(d) The (pro)-Lie algebra $Lie(N)$ of $N$ consists of the
$K_\infty$-linear
derivations $D$ of $PVR(\Delta _K)$ commuting with $\phi$ and
zero on the elements $e(c), \ell , e(z^\lambda )$. We note that any
such $D$ is determined by its restriction to $\mathcal D$.
\begin{remarks} $\ $\\ {\rm
(1) A standard example for (b) is $f=\sum _{n\geq 1}q^{-n(n+1)/2}z^n$, the only solution of $(z^{-1}\phi -1)f=1$ in $\widehat{K}_\infty $. \\
(2) In [R-S] it is suggested, in analogy with the differential case, that
$Lie(N)$ is a nilpotent completion of a free Lie algebra with a set of free generators derived from the analytic tool of $q$-summation.
}
$\Box$ \end{remarks}
The main obstruction for the determination of $Lie(N)$ is the absence of an explicit description of the algebra $\mathcal D$. Now we present
an {\it intermediate Tannakian category} $\Delta _{2,K}$. It is the Tannakian subcategory of $\Delta _K$ generated by the difference modules having at most two slopes. For this category one has
$PVR(\Delta _{2,K})=\mathcal{D}_2[\{e(c)\},\ell ,\{e(z^\lambda )\}]$
for a certain $K_\infty$-subalgebra
$\mathcal{D}_2 \subset \mathcal{D} \subset \widehat{K}_\infty$ and
a universal difference Galois group $G_2$ which is the semi-direct
product of $G_{split}$ and a (connected) unipotent group scheme
$N_2$. The latter is a quotient of $N$ and the pro-Lie algebra
$Lie(N_2)$ is a quotient of $Lie (N)$.
Any $D\in Lie(N_2)$ is a $K_\infty$-linear derivation
$D:\mathcal{D}_2\rightarrow PVR(\Delta _{2,K})$, commuting
with $\phi$.
\begin{definition} The elements $f_{m,c,\mu}$.
\\ {\rm For $\mu \in \mathbb{Q}$ with $\mu >0$, $c\in \mathbb{C}^*$
with $|q^\mu |<|c|\leq 1$ and $m\geq 1$, the unique solution in
$\widehat{K}_\infty$ of the equation $(c^{-1}z^{-\mu }\phi -1)^my=1$
is called $f_{m,c,\mu}$. }
$\Box$ \end{definition}
\begin{theorem} ${\mathcal D}_2$ is generated over $K_\infty$ by the
elements $f_{m,c,\mu}$. These elements are algebraically independent over $K_\infty$. \end{theorem}
We note that $\phi (f_{m,c,\mu})=cz^\mu (f_{m,c,\mu}+f_{m-1,c,\mu})$,
where we use the notation $f_{0,c,\mu}=1$ for all $c,\mu$. Thus the action of $\phi$ on $\mathcal{D}_2$ is explicit. An element
$D\in Lie(N_2)$ is a $K_\infty$-derivation $\mathcal{D}_2\rightarrow
PVR(\Delta _{2,K})$, commuting with $\phi$. Since the $f_{m,c,\mu}$
are free generators of $\mathcal{D}_2$, the values $D(f_{m,c,\mu})$
have the only restriction that $\phi (D(f_{m,c,\mu}))=
cz^\mu (D(f_{m,c,\mu})+D(f_{m-1,c,\mu}))$.
Choose for every $\mu ,c$ as above, a sequence of complex
numbers\\
$a_0(\mu ,c), a_1(\mu ,c), a_2(\mu ,c),\dots $ . Define $D$ by the formula $D(f_{m,c,\mu}):=$
\[(a_0(\mu ,c){\ell \choose m-1}+a_1(\mu ,c) {\ell \choose m-2}+\cdots + a_{m-1}(\mu ,c){\ell \choose 0})\cdot e(c^{-1})e(z^{-\mu} )\ .\]
One can verify that $D$ commutes with the action of $\phi$ and thus $D$ defines an element of $Lie(N_2)$. Moreover, every element of $Lie(N_2)$ has this form.
One observes that $Lie(N_2)$ is commutative. Now we propose {\it topological generators} for the pro-Lie algebra $Lie(N_2)$ by considering the elements $D_{\mu ,c,n}$ with
$\mu >0,\ |q^\mu |<|c|\leq 1,\ n\geq 0$ defined by the sequences
$\{ a_k(\mu ',c') \}$ with $a_k(\mu ',c')=
\delta _{\mu ,\mu '}\delta _{c,c'}\delta _{k,n}$.
It is not difficult to verify that any element $\xi \in PVR(\Delta _{2,K})$, invariant under $G_{split}$ and satisfying $D _{\mu ,c,n}\xi =0$ for all
$\mu ,c,n$ , lies in $K$. This implies that $N_2$ is connected and that its pro-Lie-algebra is actually topologically generated by
$\{D_{\mu ,c,n}\}$. We {\it conjecture} that $Lie(N_2)$ is actually
$Lie(N)_{ab}:=Lie(N)/[Lie(N),Lie(N)]$.
In [R-S] a set of free topological generators for the pro-Lie-algebra
$Lie(N)$ is proposed. It seems that the restriction of this set
to the quotient $Lie(N_2)$ has a translation into our set $\{D_{\mu ,c, n}\}$. \\
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\begin{remarks} $\ $\\ {\rm
(1) A standard example for (b) is $f=\sum _{n\geq 1}q^{-n(n+1)/2}z^n$, the only solution of $(z^{-1}\phi -1)f=1$ in $\widehat{K}_\infty $. \\
(2) In [R-S] it is suggested, in analogy with the differential case, that
$Lie(N)$ is a nilpotent completion of a free Lie algebra with a set of free generators derived from the analytic tool of $q$-summation.
}
$\Box$ \end{remarks}
The main obstruction for the determination of $Lie(N)$ is the absence of an explicit description of the algebra $\mathcal D$. Now we present
an {\it intermediate Tannakian category} $\Delta _{2,K}$. It is the Tannakian subcategory of $\Delta _K$ generated by the difference modules having at most two slopes. For this category one has
$PVR(\Delta _{2,K})=\mathcal{D}_2[\{e(c)\},\ell ,\{e(z^\lambda )\}]$
for a certain $K_\infty$-subalgebra
$\mathcal{D}_2 \subset \mathcal{D} \subset \widehat{K}_\infty$ and
a universal difference Galois group $G_2$ which is the semi-direct
product of $G_{split}$ and a (connected) unipotent group scheme
$N_2$. The latter is a quotient of $N$ and the pro-Lie algebra
$Lie(N_2)$ is a quotient of $Lie (N)$.
Any $D\in Lie(N_2)$ is a $K_\infty$-linear derivation
$D:\mathcal{D}_2\rightarrow PVR(\Delta _{2,K})$, commuting
with $\phi$.
\begin{definition} The elements $f_{m,c,\mu}$.
\\ {\rm For $\mu \in \mathbb{Q}$ with $\mu >0$, $c\in \mathbb{C}^*$
with $|q^\mu |<|c|\leq 1$ and $m\geq 1$, the unique solution in
$\widehat{K}_\infty$ of the equation $(c^{-1}z^{-\mu }\phi -1)^my=1$
is called $f_{m,c,\mu}$. }
$\Box$ \end{definition}
\begin{theorem} ${\mathcal D}_2$ is generated over $K_\infty$ by the
elements $f_{m,c,\mu}$. These elements are algebraically independent over $K_\infty$. \end{theorem}
We note that $\phi (f_{m,c,\mu})=cz^\mu (f_{m,c,\mu}+f_{m-1,c,\mu})$,
where we use the notation $f_{0,c,\mu}=1$ for all $c,\mu$. Thus the action of $\phi$ on $\mathcal{D}_2$ is explicit. An element
$D\in Lie(N_2)$ is a $K_\infty$-derivation $\mathcal{D}_2\rightarrow
PVR(\Delta _{2,K})$, commuting with $\phi$. Since the $f_{m,c,\mu}$
are free generators of $\mathcal{D}_2$, the values $D(f_{m,c,\mu})$
have the only restriction that $\phi (D(f_{m,c,\mu}))=
cz^\mu (D(f_{m,c,\mu})+D(f_{m-1,c,\mu}))$.
Choose for every $\mu ,c$ as above, a sequence of complex
numbers\\
$a_0(\mu ,c), a_1(\mu ,c), a_2(\mu ,c),\dots $ . Define $D$ by the formula $D(f_{m,c,\mu}):=$
\[(a_0(\mu ,c){\ell \choose m-1}+a_1(\mu ,c) {\ell \choose m-2}+\cdots + a_{m-1}(\mu ,c){\ell \choose 0})\cdot e(c^{-1})e(z^{-\mu} )\ .\]
One can verify that $D$ commutes with the action of $\phi$ and thus $D$ defines an element of $Lie(N_2)$. Moreover, every element of $Lie(N_2)$ has this form.
One observes that $Lie(N_2)$ is commutative. Now we propose {\it topological generators} for the pro-Lie algebra $Lie(N_2)$ by considering the elements $D_{\mu ,c,n}$ with
$\mu >0,\ |q^\mu |<|c|\leq 1,\ n\geq 0$ defined by the sequences
$\{ a_k(\mu ',c') \}$ with $a_k(\mu ',c')=
\delta _{\mu ,\mu '}\delta _{c,c'}\delta _{k,n}$.
It is not difficult to verify that any element $\xi \in PVR(\Delta _{2,K})$, invariant under $G_{split}$ and satisfying $D _{\mu ,c,n}\xi =0$ for all
$\mu ,c,n$ , lies in $K$. This implies that $N_2$ is connected and that its pro-Lie-algebra is actually topologically generated by
$\{D_{\mu ,c,n}\}$. We {\it conjecture} that $Lie(N_2)$ is actually
$Lie(N)_{ab}:=Lie(N)/[Lie(N),Lie(N)]$.
In [R-S] a set of free topological generators for the pro-Lie-algebra
$Lie(N)$ is proposed. It seems that the restriction of this set
to the quotient $Lie(N_2)$ has a translation into our set $\{D_{\mu ,c, n}\}$. \\
{\bf References}\\
\noindent [P-R] M. van der Put and M. Reversat - {\it Galois theory of $q$-difference equations} -
Ann. Fac. Sci. de Toulouse, vol XVI, no 2, p. 1-54, 2007\\
\noindent [P-S] M. van der Put and M.F. Singer - {\it Galois theory of difference equations} -
Lecture Notes in Mathematics, 1666, Springer Verlag, 1997\\
\noindent [P-S.2] M. van der Put and M.F. Singer - {\it Galois theory of linear differential equations} -
Grundlehren der mathematische Wissenschaften, 328, Springer Verlag, 2003\\
\noindent [R-S] J.-P. Ramis and J. Sauloy -{\it The $q$-analogue of the wild fundamental group} (I) -
arXiv:math.QA/0611521 v1 17 Nov 2006\\
\end{document}
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\begin{document}
\title{Product-free sets in the free semigroup}
\author{Imre Leader}
\address{Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3\thinspace0WB, UK}
\email{[email protected]}
\author{Shoham Letzter}
\address{ETH Institute for Theoretical Studies, 8092 Zurich, Switzerland}
\email{[email protected]}
\author{Bhargav Narayanan}
\address{Department of Mathematics, Rutgers University, Piscataway NJ 08854, USA}
\email{[email protected]}
\author{Mark Walters}
\address{School of Mathematical Sciences, Queen Mary, University of London, London E1\thinspace4NS, UK}
\email{[email protected]}
\date{6 December 2018}
\subjclass[2010]{Primary 20M05; Secondary 05D05}
\begin{abstract}
In this paper, we study product-free subsets of the free semigroup over a finite alphabet $\mathscr{A}$. We prove that the maximum density of a product-free subset of the free semigroup over $\mathscr{A}$, with respect to the natural measure that assigns a weight of $|\mathscr{A}|^{-n}$ to each word of length $n$, is precisely $1/2$.
\end{abstract}
\maketitle
\section {Introduction}
A subset $S$ of a semigroup is said to be \emph{product-free} if there do not exist $x,y,z \in S$ (not necessarily distinct) such that $x\bigcdot y = z$; it is customary to call $S$ \emph{sum-free} when the underlying semigroup is abelian.
It is a well known fact (and an easy exercise) that any sum-free subset of the integers has upper density at most $1/2$. Sum-free subsets of the integers, and of abelian groups in general, have been studied by very many researchers over the last fifty years. For example, from the work of Green and Ruzsa~\citep{green}, there is now a complete picture of how large a sum-free set we can find in any finite abelian group. We refer the reader to the surveys of Tao and Vu~\citep{tao-s} and Kedlaya~\citep{ked-s} for more information on these questions.
Product-free subsets of finite non-abelian groups were first investigated by Babai and S\'os~\citep{babai}. Following foundational work by Gowers~\citep{gowers} demonstrating so-called `product-mixing' phenomena in groups with no low-dimensional representations, there has been a great deal of recent work in the non-abelian setting; for instance, in a recent breakthrough, Eberhard~\citep{eber} determined how large a product-free subset of the alternating group can be.
In light of these developments, it is natural to ask what one can say about product-free sets in infinite non-abelian structures, a setting in which our knowledge is a bit more limited. Perhaps the first natural place to look among infinite non-abelian structures is among those that are free, so here, we shall investigate how large product-free subsets of the free semigroup can be.
\section{Our results}
Let $\mathscr{A}$ be a finite set. We write $\mathcal{F} = \mathcal{F}_\mathscr{A}$ for the free semigroup over $\mathscr{A}$; in other words, $\mathcal{F}$ is the set of all finite words over the alphabet $\mathscr{A}$ equipped with the associative operation of concatenation. While we state and prove our results for finite alphabets of all possible sizes for the sake of completeness, the reader will lose nothing by supposing that $\mathscr{A}$ is a two-element set in what follows; indeed, this case captures all the difficulties inherent in the questions we study.
Recall that a set $S \subset \mathcal{F}$ is \emph{product-free} if, writing $\bigcdot$ for the operation of concatenation, there do not exist words $x,y,z \in S$ (not necessarily distinct) such that $x\bigcdot y = z$. There is an obvious example of a `large' subset of $\mathcal{F}$ that is product-free: when $\mathscr{A} = \{ a, b\}$ for instance, the set of words which contain an odd number of occurrences of the symbol $a$ (or $b$, for that matter) is easily seen to be a product-free set that contains, roughly, half the words from $\mathcal{F}$.
Our aim in this paper is to prove that these sets are, in a precise sense, the largest product-free subsets of $\mathcal{F}$. We remark in passing that there are several other product-free sets that are `equally large': for any nonempty subset $\Gamma \subset \mathscr{A}$, the \emph{odd-occurrence set} $\mathcal{O}_\Gamma \subset \mathcal{F}$ generated by $\Gamma$, namely the set of words in which the total number of occurrences of symbols from $\Gamma$ is odd, is easily seen to be a product-free set; in the case where $\mathscr{A} = \{ a, b\}$, our earlier example corresponds to taking $\Gamma = \{a\}$, and taking $\Gamma = \{a, b\}$ gives us the set of all words of odd length, for example.
To formally state our results, we need a way to measure the size of a set $S\subset \mathcal{F}$. For an integer $n\in \mathbb{N}$, the \emph{layer} $\mathcal{F}(n) \subset \mathcal{F}$ is the set of words of length $n$, and the \emph{ball} $\mathcal{F}_{\le}(n) \subset \mathcal{F}$ is the set of words of length at most $n$. As a first attempt, one might define the density of a set $S \subset \mathcal{F}$ via its densities in balls, namely as the quantity
\[\limsup_{n \to \infty}\frac{|S \cap \mathcal{F}_{\le}(n)|}{|\mathcal{F}_{\le}(n)|}.\]
However, a little thought should convince the reader that the counting measure is somewhat ill-suited for our purposes. Indeed, when $|\mathscr{A}| > 1$, almost all the words in $\mathcal{F}_{\le}(n)$ are long since $|\mathcal{F}(n)| \ge |\mathcal{F}_{\le}(n)|/2$. Consequently, we may find product-free sets that are intuitively small, and yet have density arbitrarily close to $1$ in the above sense; for example, for any sufficiently large $c \in \mathbb{N}$, the set
\[\bigcup_{n\ge c}(\mathcal{F}_{\le 2^n + c} \setminus \mathcal{F}_{\le 2^n})\]
is product-free and has density at least $1-1/c$ in the above sense, provided $|\mathscr{A}| > 1$.
A more natural approach is to assign a weight of $|\mathscr{A}|^{-n}$ to each word of $\mathcal{F}(n)$, thereby ensuring that the layers $\mathcal{F}(n)$ have the same total weight for all $n\in\mathbb{N}$. To this end, for a subset $S \subset \mathcal{F}$ and an integer $n \in \mathbb{N}$, we define \emph{the density of $S$ in the layer $\mathcal{F}(n)$} by $d_S(n) = |S \cap \mathcal{F}(n)|/|\mathcal{F}(n)|$. With this definition in place, most standard notions of density may now be carried over: we define the \emph{upper asymptotic density} of $S$ by
\[ \bar d(S) = \limsup_{n \to \infty} \frac{ \sum_{i=1}^{n}d_S(i)}{n},\]
and the \emph{upper Banach density} of $S$ by
\[ d^*(S) = \limsup_{n-m \to \infty} \frac{ \sum_{i=m}^{n}d_S(i)}{n-m+1}.\]
Of course, the latter is a weaker notion of density than the former; indeed, it is clear that $\bar d(S) \le d^*(S)$ for any $S \subset \mathcal{F}$.
It is easy to see that any odd-occurrence set has both an upper asymptotic density and an upper Banach density of $1/2$. Our aim in this note is to show that product-free sets cannot be any larger; our main result is as follows.
\begin{theorem}\label{main-res}
Let $\mathscr{A}$ be a finite set. If $S \subset \mathcal{F}_\mathscr{A}$ is product-free, then $d^*(S) \le 1/2$.
\end{theorem}
\begin{comment}
We shall also show the only maximally dense product-free sets are subsets of odd-occurrence sets.
\begin{theorem}\label{unique}
Let $\mathscr{A}$ be a finite set. If $S \subset \mathcal{F}_\mathscr{A}$ is product-free and $d^*(S) = 1/2$, then $S \subset \mathcal{O}_\Gamma$ for some nonempty subset $\Gamma \subset \mathscr{A}$.
\end{theorem}
\end{comment}
Let us mention that product-free sets in cancellative semigroups have been studied by {\L}uczak and Schoen~\citep{semigr}; while their results are sharp for such semigroups in general, these results do not give us any effective bounds on the size of a product-free subset of $\mathcal{F}$.
Before we turn to the proof of Theorem~\ref{main-res}, it is worth pointing out that there is a simple argument that allows us to bound the upper asymptotic density of a product-free subset of $\mathcal{F}$ away from $1$. Indeed, suppose that $S\subset\mathcal{F}$ is product-free. We then have
\[d_S(m)d_S(n) + d_S(m+n) \le 1\]
for any $m,n\in\mathbb{N}$ since the sets $S \cap \mathcal{F}(m+n)$ and $(S \cap \mathcal{F}(m))\bigcdot (S \cap \mathcal{F}(n))$ must be disjoint. Now, consider the set of integers $n \in \mathbb{N}$ for which $d_S(n) > \phi$, where $\phi = (\sqrt 5 - 1)/2 \approx 0.618$ is the unique positive solution to the equation $x^2 + x = 1$. It follows from the inequality above that this set of integers must be sum-free. It is now easy to see that $\bar d(S) \le (1 + \phi)/2 \approx 0.809$.
We shall have to work somewhat harder to prove Theorem~\ref{main-res}, which improves this bound of $(1 + \phi)/2$ for the upper asymptotic density to the optimal bound of $1/2$ for the upper Banach density. The proof of Theorem~\ref{main-res} is given in Section~\ref{sec-proof}. We conclude this note with a discussion of some open problems in Section~\ref{sec-conc}.
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\section{Proof of the main result}\label{sec-proof}
We begin by fixing our finite alphabet $\mathscr{A}$. In the sequel, $\mathcal{F}$ will always mean $\mathcal{F}_\mathscr{A}$, the free semigroup over this fixed alphabet $\mathscr{A}$.
It will be helpful to establish some notation. For a pair of words $x, w \in \mathcal{F}$, we say that $x$ is a \emph{prefix} of $w$ if $w = x\bigcdot y$ for some $y \in \mathcal{F}$, and that $x$ is a \emph{suffix} of $w$ if $w = y\bigcdot x$ for some $y \in \mathcal{F}$. For a pair of sets $S_1, S_2 \subset \mathcal{F}$, we write $S_1 \bigcdot S_2$ for their (Minkowski) product; in other words,
\[S_1 \bigcdot S_2 = \{ w_1 \bigcdot w_2 : w_1 \in S_1, w_2 \in S_2\}.\]
For a set $S\subset \mathcal{F}$ and an integer $n \in \mathbb{N}$, we set $S(n) = S \cap \mathcal{F}(n)$. One of the key ideas in the proof of Theorem~\ref{main-res} is the following definition. For any sequence of positive integers $ \ell_1 < \ell_2 <\dots <\ell_k <n$, we define
\[ S(n; \ell_1, \ell_2,\dots,\ell_k) = \mathopen{}\mathclose\bgroup\originalleft\{ w \in S(n): w \text{ has no prefix in } S(\ell_1) \cup S(\ell_2) \cup \dots \cup S(\ell_k) \aftergroup\egroup\originalright \}; \]
in other words,
\[ S(n; \ell_1, \ell_2,\dots,\ell_k) = S(n) \setminus \mathopen{}\mathclose\bgroup\originalleft( \bigcup_{i=1}^{k}S(\ell_i) \bigcdot \mathcal{F}(n-\ell_i) \aftergroup\egroup\originalright).
\]
Let us note, for any $S \subset \mathcal{F}$, that the sets $S(n;m)$ and $S(m) \bigcdot \mathcal{F}(n-m)$ are disjoint for any pair of positive integers $m < n$. Recall that $d_S(n) = |S(n)||\mathcal{F}(n)|^{-1}$; we analogously define
\[d_S(n; \ell_1, \ell_2,\dots,\ell_k) = \frac{|S(n; \ell_1, \ell_2,\dots,\ell_k)|}{|\mathcal{F}(n)|}.\]
When the set $S$ in question is clear, we write $d(n)$ and $d(n; \ell_1, \ell_2,\dots,\ell_k)$ for $d_S(n)$ and $d_S(n; \ell_1, \ell_2,\dots,\ell_k)$, respectively. Recall that for any product-free set $S \subset \mathcal{F}$ and any $m,n \in \mathbb{N}$, we have
\[d(m)d(n) + d(m+n) \le 1.\]
We start by proving a generalisation of this fact.
\begin{proposition}\label{doublecount}
If $S \subset \mathcal{F}$ is product-free, then for any sequence of positive integers $\ell_1 < \ell_2 <\dots <\ell_k <n$, we have
\begin{align*}
& d(\ell_1)d(n-\ell_1) + d(\ell_2;\ell_1)d(n-\ell_2) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})d(n-\ell_k) + d(n) \\
\le \,\, & d(\ell_1) + d(\ell_2;\ell_1) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) + d(n; \ell_1, \ell_2, \dots, \ell_k) \le 1.
\end{align*}
\end{proposition}
\begin{proof}
First, consider the products
\[ S(\ell_1) \bigcdot S(n-\ell_1), S(\ell_2;\ell_1)\bigcdot S(n-\ell_2), \dots, S(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) \bigcdot S(n-\ell_k).\]
These subsets of $\mathcal{F}(n)$ are by definition disjoint. Let $L'$ be the union of these $k$ sets. Since $S$ is product-free, $L'$ and $S(n)$ are disjoint as well. Let $L = L' \cup S(n)$; clearly, the density of $L$ in $\mathcal{F}(n)$ is
\[d(\ell_1)d(n-\ell_1) + d(\ell_2;\ell_1)d(n-\ell_2) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})d(n-\ell_k) + d(n).
\]
Next, consider the Minkowski products
\[
S(\ell_1) \bigcdot \mathcal{F}(n-\ell_1),\, S(\ell_2;\ell_1)\bigcdot \mathcal{F}(n-\ell_2),\, \dots,\, S(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) \bigcdot \mathcal{F}(n-\ell_k).
\]
These subsets of $\mathcal{F}(n)$ are again disjoint by definition; let $R'$ denote their union. Note that $R'$ and $S(n; \ell_1, \ell_2, \dots, \ell_k)$ are disjoint. Let $R = R' \cup S(n; \ell_1, \ell_2, \dots, \ell_k)$; it is easy to see that the density of $R$ in $\mathcal{F}(n)$ is
\[d(\ell_1) + d(\ell_2;\ell_1) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) + d(n; \ell_1, \ell_2, \dots, \ell_k)\]
and that this quantity is therefore at most $1$.
To finish the proof, it suffices to show that
\[L' \cup S(n) = L \subset R = R' \cup S(n; \ell_1, \ell_2, \dots, \ell_k).\]
It is easy to see that $L' \subset R'$. Therefore, it is sufficient to show that $S(n)$ is a subset of $R' \cup S(n; \ell_1, \ell_2, \dots, \ell_k)$. To see this, note that any word from $S(n)$ which has a prefix in $S(\ell_1) \cup S(\ell_2) \cup \dots \cup S(\ell_k)$ is also contained in $R'$. In other words, $S(n) \setminus S(n; \ell_1, \ell_2, \dots, \ell_k) \subset R'$; the result follows.
\end{proof}
With the above observation in hand, we are now ready to prove Theorem~\ref{main-res}.
\begin{proof}[Proof of Theorem~\ref{main-res}]
We prove by contradiction that the upper Banach density of a product-free set is at most $1/2$.
Suppose that $S \subset \mathcal{F}$ is product-free and that $d^*(S) > 1/2 + \eps$ for some $\eps > 0$. We then claim that we may find an increasing sequence of positive integers $(\ell_k)_{k \in \mathbb{N}}$ such that
\[ d(\ell_1) + d(\ell_2; \ell_1) + \dots + d(\ell_k; \ell_1, \ell_2, \dots, \ell_{k-1}) \ge \frac{1}{2} + \frac{1}{4} + \dots + \frac{1}{2^k} = 1 - \frac{1}{2^k}\]
for each $k \in \mathbb{N}$.
We construct this sequence inductively. Since $d^*(S) > 1/2$, it is clear that we may find $\ell_1 \in \mathbb{N}$ such that $d(\ell_1) \ge 1/2$. Having found $\ell_1< \ell_2< \dots< \ell_k$ as required, we choose $\ell_{k+1}$ as follows. Since $d^*(S) > 1/2 + \eps$, there exist arbitrarily long intervals $I \subset \mathbb{N}$ that satisfy
\[
\frac{\sum_{n \in I}d(n)}{|I|} > \frac{1}{2} + \eps.
\]
Choose such an interval $I$ whose length is sufficiently larger than $\ell_k$; we may assume, by passing to a sub-interval if necessary, that $\min I > \ell_k$. We claim that it is possible to choose $\ell_{k+1}$ from $I$; in other words, we claim that there exists an $n \in I$ such that
\begin{multline*}
d(\ell_1) + d(\ell_2; \ell_1) + \dots + d(\ell_k; \ell_1, \ell_2, \dots, \ell_{k-1}) + d(n; \ell_1, \ell_2, \dots, \ell_{k}) \ge 1 - \frac{1}{2^{k+1}}.
\end{multline*}
We prove this claim by contradiction. Suppose that there is no such $n \in I$. Then, by Proposition~\ref{doublecount}, we have
\begin{align*}
& d(\ell_1)d(n-\ell_1) + d(\ell_2;\ell_1)d(n-\ell_2) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})d(n-\ell_k) + d(n) \\
\le \,\, & d(\ell_1) + d(\ell_2;\ell_1) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) + d(n; \ell_1, \ell_2, \dots, \ell_k) < 1-\frac{1}{2^{k+1}}
\end{align*}
for each $n \in I$. By summing the above inequality over all $n \in I$, we get
\[
\sum_{n \in I'}d(n)\mathopen{}\mathclose\bgroup\originalleft(1+d(\ell_1)+ d(\ell_2;\ell_1) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})\aftergroup\egroup\originalright) < |I|\mathopen{}\mathclose\bgroup\originalleft( 1 -\frac{1}{2^{k+1}}\aftergroup\egroup\originalright),
\]
where $I'\subset I$ is the set of $n \in I$ with $n + \ell_k < \max I$. This implies, by the inductive hypothesis, that
\[\sum_{n \in I'}d(n)\mathopen{}\mathclose\bgroup\originalleft(2 - \frac{1}{2^k}\aftergroup\egroup\originalright) < |I|\mathopen{}\mathclose\bgroup\originalleft(1-\frac{1}{2^{k+1}}\aftergroup\egroup\originalright),\]
or equivalently, $\sum_{n \in I'}d(n) < |I|/2$. Therefore, we have
\[ \sum_{n \in I}d(n) \le \sum_{n \in I'}d(n) + \ell_k + 1 < \frac{|I|}{2} + \ell_k + 1,\]
which contradicts the fact that $\sum_{n \in I}d(n) > |I|/2 + \eps|I|$, provided $|I| > (\ell_k+1)/\eps$.
We now finish the proof of the proposition by showing that the existence of this sequence $(\ell_k)_{k \in \mathbb{N}}$ contradicts our initial assumption that $d^*(S) > 1/2 + \eps$. Fix a $k\in \mathbb{N}$ large enough to ensure that
\[\frac{2^k}{2^{k+1} - 1} < \frac{1 + \eps}{2}\]
and consider any interval $I \subset \mathbb{N}$ with $|I| > 4(\ell_k + 1) / \eps$. We know from Proposition~\ref{doublecount} that
\[ d(\ell_1)d(n-\ell_1) + d(\ell_2;\ell_1)d(n-\ell_2) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})d(n-\ell_k) + d(n) \le 1\]
for each $n \in \mathbb{N}$ with $n > \ell_k$; summing this inequality over such $n \in I$, we get
\[\sum_{n \in I'} d(n)\mathopen{}\mathclose\bgroup\originalleft(2 - \frac{1}{2^k}\aftergroup\egroup\originalright) \le |I|, \]
where $I'$ is the set of $n \in I$ with $n > \ell_k$ and $n + \ell_k < \max I$. Therefore,
\[ \frac{\sum_{n \in I} d(n)}{|I|} \le \frac{2^k}{(2^{k+1} - 1)} + \frac{2(\ell_k + 1)}{|I|} < \frac{1}{2} + \eps, \]
which is a contradiction; this proves the claimed upper bound in Theorem~\ref{main-res}.
\end{proof}
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\section{Conclusion}\label{sec-conc}
A common line of enquiry in the study of product-free sets is to ask for `asymmetric' versions of results bounding the upper density of product-free sets. In this spirit, it is natural to ask whether an analogue of Theorem~\ref{main-res} continues to hold when one wishes to solve the equation $x \bigcdot y = z$ with $x$, $y$ and $z$ in specified subsets of $\mathcal{F}$. More precisely, if $X, Y, Z \subset \mathcal{F}$ are such that there are no solutions to $x \bigcdot y = z$ with $x \in X$, $y \in Y$ and $z\in Z$, one might ask if one of $X$, $Y$ or $Z$ has an upper asymptotic density of at most $1/2$. However, it is not hard to construct for any $\eps>0$, three sets $X, Y, Z \subset \mathcal{F}$, each of upper asymptotic density at least $\phi - \eps$, where $\phi = (\sqrt 5 - 1)/2$, such that there are no solutions to $x \bigcdot y = z$ with $x \in X$, $y \in Y$ and $z\in Z$. Indeed, pick a suitably large $n \in\mathbb{N}$ and choose any set $W \subset \mathcal{F}(n)$ such that $||W|/|\mathcal{F}(n)| - \phi| < \eps/3$. Now take $X$ to be the set of all words with a prefix in $W$, $Y$ to be the set of all words with a suffix in $W$, and $Z$ to be the set $\mathcal{F} \setminus (X \bigcdot Y)$. Clearly, there are no solutions to $x \bigcdot y = z$ with $x \in X$, $y \in Y$ and $z\in Z$; it is also not hard to check that each of $X$, $Y$ and $Z$ has an upper asymptotic density at least $\phi - \eps$.
Next, it would be interesting to understand what product-free sets of maximal density look like. As we saw earlier, several non-isomorphic extremal constructions are furnished by the family of odd-occurrence sets. We suspect that these might be the only constructions of maximal density, and make the following conjecture.
\begin{conjecture}\label{unique}
Let $\mathscr{A}$ be a finite set. If $S \subset \mathcal{F}_\mathscr{A}$ is product-free and $d^*(S) = 1/2$, then $S \subset \mathcal{O}_\Gamma$ for some nonempty subset $\Gamma \subset \mathscr{A}$.
\end{conjecture}
Finally, another natural direction is to study product-free subsets of the \emph{free group $\mathbf{F}_\mathscr{A}$} over a finite alphabet $\mathscr{A}$. Similarly to the situation in this paper, the most natural measure to consider in the case of the free group $\mathbf{F}_\mathscr{A}$ would be the one that assigns a weight of $|\mathscr{A}|(|\mathscr{A}|-1)^{-(n-1)}$ to each irreducible word of length $n$. The different notions of density defined here for the free semigroup then have analogous definitions in the free group, and we believe that an analogue of Theorem~\ref{main-res} should hold in the free group as well; concretely, we conjecture the following.
\begin{conjecture}\label{freegrp}
For any finite alphabet $\mathscr{A}$, no product-free subset of the free group $\mathbf{F}_\mathscr{A}$ has upper Banach density exceeding $1/2$.
\end{conjecture}
Note that, in the proof of Theorem~\ref{main-res}, we rely crucially on the fact that there is exactly one way to write a word of length $m+n$ as the concatenation of a word of length $m$ with a word of length $n$; of course, we lose this property when working with free groups, so we believe that some new ideas will be required to understand product-free sets in free groups.
\end{document}
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\begin{equation}gin{document}
\title{Homogenization of dislocation dynamics}
\alphauthor{Ahmad El Hajj, Hassan Ibrahim and R\'egis Monneau}
\alphaddress{CERMICS, ENPC, 6 \& 8 avenue Blaise
Pascal, Cit\'e Descartes, Champs sur Marne, 77455 Marne-la-Vall\'ee
Cedex 2, France}
\epsilonnd{eqnarray}d{[email protected], [email protected], [email protected]}
\begin{equation}gin{abstract}
In this paper we consider the dynamics of dislocations with the same
Burgers vector, contained in the same glide plane, and moving in a material
with periodic obstacles. We study two cases: i) the
particular case of parallel straight dislocations and ii) the general case of curved
dislocations. In each case, we perform rigorously the homogenization of the
dynamics and predict the corresponding effective macroscopic elasto-visco-plastic flow rule.
\epsilonnd{abstract}
{\bf S}E{Introduction}
In the recent years, an important effort has been done, both to improve the
methods to compute discrete dislocation dynamics (see for instance the book
of Bulatov and Cai
\cite{BC} and the references therein) and also to connect them
to continuum models of plasticity in crystalline solids (see for instance
Fivel et al. \cite{FTRC} and more recently Hoc et
al. \cite{HDK}). Although continuum models of
dislocations are known since the 50's (see Kr\"{o}ner \cite{K,K2}), the
dynamics has been taken into account only recently : see Groma et
al. \cite{GB,GCZ} in 2D (and their mathematical studies in
\cite{FE,IJM}), Hochrainer et al. \cite{HZG}, and Monneau \cite{M} in 3D. The goal
of our work is to present, on a particular
example, a rigorous justification of a continuum model with densities of
dislocations bridging the
gap with dislocation dynamics at the microscale. Indeed for a
very special geometry, we are able to deduce by homogenization, the
macroscopic elasto-visco-plastic flow rule relating the plastic strain
velocity to the shear stress. The full technical details are presented
in \cite{FIM}.
{\bf S}E{Homogenization of straight dislocations}
In this section, we consider the case of parallel straight edge dislocations
with the same Burgers vector $\mbox{\bf b}=be_x$ with $b>0$, where
$(e_x,e_y,e_z)$ is an orthonormal basis with corresponding coordinates
$(x,y,z)$. All these dislocation lines are
assumed to be contained in the same glide plane $(x,y)$ and to move in
this plane.
\sigmaubsection{The microscopic model for straight dislocations}
Because of our assumptions, for every integer $i\inftyn \mathbb{Z}$, we can
simply describe the position of the $i$-th dislocation
by its real abscissa that we call $x_i(t)$ where $t$ is the time. We want
to take into account the interactions of each dislocation with other
defects in the crystal, that constitute obstacles
to their motion. Those obstacles can be for instance other pinned
dislocations or precipitates. In order to simplify the analysis, we will
assume that these obstacles are periodically distributed, of spatial period
$\lambdaambda$. In our model, those obstacles will be simply modeled by a smooth
periodic potential $V^{per}$ satisfying
$V^{per}(x+\lambdaambda)=V^{per}(x)$. Then the energy of the system is the sum
of two contributions: the interactions of each dislocations with the
periodic potential and the sum of the two-body interactions between
dislocations associated to a pair potential $V$.
The energy of a set of
dislocations is then
given by
$$E=\sigmaum_{i} V^{per}(x_i) + \sigmaum_{i<j} V(x_i-x_j)\quad
\mbox{with}\quad V(x)=-\begin{eqnarray}r{\mu} b\lambdan |x| \quad \mbox{and}\quad
\begin{eqnarray}r{\mu}=\frac{\mu }{2\partiali (1-\nu)}$$
where the constants $\mu$ and $\nu$ are respectively the shear modulus
and the Poisson ratio. Remark that
the force $-V'(x)$ is then the usual Peach-Koehler force created at the
point $x$ by an edge dislocation positioned at the origin.
We then consider the fully overdamped dynamics, where the velocity is
proportional to the force, i.e.
\begin{equation}gin{equation}\lambdaabel{eq::1}
B\frac{dx_i}{dt}=-\nabla_{x_i} E + \tau_{ext}
\epsilonnd{equation}
where $B$ is the viscous drag coefficient and the force is
on the right hand side. The first contribution to the force is a term deriving from the
energy and $\tau_{ext}$ is a real exterior applied shear stress, that can be
seen as a driving force of the system.
A natural question is then: what is the macroscopic behavior of this system ?
In order to answer this question (which is done in Theorem \ref{th::1}),
we have to introduce the plastic strain. To each dislocation is associated a three-dimensional
displacement in the crystal, whose plastic strain is localized in the glide
plane $z=0$ and is equal to $\gamma \deltaelta_0(z)$ where $\deltaelta_0$ is the
Dirac mass. For instance, for a dislocation $x_i$, the intensity
$\gamma$ (that we continue to call plastic strain) is equal to $-b
H(x-x_i)$ where the Heaviside
function $H(x)$ is equal to $1$ for positive $x$ and zero otherwise. Here
the sign defining the plastic strain is such that the quantity
$\gamma$ increases when $x_i$ increases. Then the total plastic strain can
be written as
$$\gamma(x,t)=-b \sigmaum_{i} H(x-x_i(t)).$$
\sigmaubsection{The normalization procedure}
We are now interested in the behavior of the system at a macroscopic scale
$\Lambdaambda$ such that $\Lambdaambda >> \lambdaambda =\omegal b$ where $\omegal >1$ is a fixed
ratio. Then we introduce several dimensionless quantities. We call
$\begin{eqnarray}r{x}$ and $\omegat$ the normalized
spatial and time coordinates at the macroscopic level, and introduce a
parameter $\varepsilon$ and the associated
normalized macroscopic plastic strain
${\gamma}^\varepsilon$ such that
\begin{equation}gin{equation}\lambdaabel{eq::0}
\begin{eqnarray}r{x} = \frac{x}{\Lambdaambda}, \quad \omegat =
\frac{\begin{eqnarray}r{\mu}}{B} \frac{t}{\Lambdaambda},\quad \varepsilon = \frac{b}{\Lambdaambda} \quad \mbox{and}\quad
\deltaisplaystyle{{\gamma}^\varepsilon(\omegax,\omegat)=
\frac{1}{\Lambdaambda}\gamma(x,t)} \quad \mbox{with}\quad {\gamma}^\varepsilon(\omegax,0)=\varepsilon\lambdaeft[\frac{1}{\varepsilon}\gamma_0(\omegax)\right]
\epsilonnd{equation}
where $\lambdaeft[\cdot\right]$ is the floor function, $\gamma_0$ is a given function and $B\Lambdaambda/\begin{eqnarray}r{\mu}$ is a
typical macroscopic time deduced from equation (\ref{eq::1}).
Remark that $\varepsilon$ can be very small in our application (for instance $\varepsilon \sigmaimeq 10^{-6}$ if
$b\sigmaimeq 10^{-9}m$ and $\Lambdaambda\sigmaimeq 10^{-3}m$).
We expect that the macroscopic behavior of the model is well described by
{\inftyt the limit macroscopic plastic strain $\gamma^0(\omegax,\omegat)$ of $\gamma^\varepsilon(\omegax,\omegat)$ as
$\varepsilon$ goes to zero}.
\sigmaubsection{Heuristics for the macroscopic stress field}
In this subsection, we want to give heuristic expressions of the normalized
dislocation density and the macroscopic stress field, in terms of the limit
macroscopic plastic strain.
We remark that the gradient of the map $x\mapsto -\gamma^\varepsilon
(x/\Lambdaambda,\omegat) /\varepsilon$
is a sum of Dirac masses, and then the number of dislocations in a large segment
of length ${\cal D}elta x$ is formally given by $-\inftynt_{0}^{{\cal D}elta x}
\frac{1}{\varepsilon \Lambdaambda } \frac{\partialartial \gamma^\varepsilon}{\partialartial \omegax}(x/\Lambdaambda,\omegat)\
dx$. This shows at least formally that the dislocation density can be
estimated as $\rho(x,t)=-\frac{1}{\varepsilon\Lambdaambda} \frac{\partialartial \gamma^0}{\partialartial
\omegax}(\omegax,\omegat)$.
Then the total stress on the right hand side of (\ref{eq::1}) can be formally
described at the macroscopic scale by
\begin{equation}gin{equation}\lambdaabel{eq::2}
\deltaisplaystyle{\tau = \tau_{ext} + \tau_{sc} \quad \mbox{with}\quad
\tau_{sc}(\omegax,\omegat)= -\begin{eqnarray}r{\mu} \inftynt_{-\inftynfty}^{+\inftynfty}
\frac{d\omegax'}{\omegax-\omegax'} \frac{\partialartial \gamma^0}{\partialartial \omegax}(\omegax',\omegat)}
\epsilonnd{equation}
where we take the principal value in the integral defining the
self-consistent field $\tau_{sc}$. This expression can be deduced from the
equation $\tau_{sc}(\omegax,\omegat)= -(V'\sigmatar_x \rho)(x,t)$, where $\sigmatar_x$ denotes the
convolution with respect to the variable $x$.
Remark also that the expression (\ref{eq::2}) of
$\tau_{sc}$
is known to be the resolved shear stress created by the normalized
dislocation density
\begin{equation}gin{equation}\lambdaabel{eq::3}
\rho^0=- \frac{\partialartial \gamma^0}{\partialartial \omegax}
\epsilonnd{equation}
where for instance $\rho^0=1/\begin{eqnarray}r{\lambdaambda}$ when there is one dislocation by spatial period $\lambdaambda$.
In particular, we see that $\tau_{sc}$ keeps the memory of the long range interactions
between dislocations.
\sigmaubsection{The homogenization result}\lambdaabel{s1.3}
We expect that the effective equation satisfied by the limit
$\gamma^0$ can be written
\begin{equation}gin{equation}\lambdaabel{eq::5}
\lambdaeft\{\begin{equation}gin{array}{l}
\deltaisplaystyle{\frac{\partialartial \gamma^0}{\partialartial \omegat} = f(\rho^0, \tau), \quad
\mbox{for all}\quad \omegax\inftyn\mathbb{R},\quad \omegat\inftyn (0,+\inftynfty)},\\
\\
\gamma^0(\omegax,0)=\gamma_0(\omegax) \quad \mbox{for all}\quad \omegax\inftyn \mathbb{R}
\epsilonnd{array}\right.
\epsilonnd{equation}
where $\rho^0$ is given in (\ref{eq::3}) and $\tau$ in (\ref{eq::2}).
Then our main result is:
\begin{equation}gin{theo}\lambdaabel{th::1}{\bf (Homogenization of straight dislocations)}\\
Assume that the initial data $\gamma_0$ is non-decreasing and satisfies
$|\gamma_0|+ |\gamma_0'|+ |\gamma_0''| \lambdae C$ for some constant $C$.
Then for any $C^2$ periodic potential $V^{per}$, there exists
a continuous function $f: \mathbb{R}^2 \to \mathbb{R}$ such that $\tau \mapsto
f(\rho^0,\tau)$ is nondecreasing. And there exists a unique viscosity
solution $\gamma^0$ of the equation (\ref{eq::5}).\\
Moreover, under the assumptions and notation of this
section, there exists a unique solution $\gamma^\varepsilon$ associated to
the dynamics (\ref{eq::1}) with initial
data given in (\ref{eq::0}), and $\gamma^\varepsilon$ converges to $\gamma^0$ locally
uniformly on $\mathbb{R} \times [0,+\inftynfty )$.
\epsilonnd{theo}
This result is proven rigorously in \cite{FIM} in the mathematical framework of
viscosity solutions (see for instance Crandall, Ishii, Lions \cite{CIL} for an introduction to
this theory). We explain in the next section how we compute the
function $f$, which keeps the memory of the short range interactions
between the dislocations and the periodic potential $V^{per}$.
\sigmaubsection{Computation of $f$ using Orowan's law}\lambdaabel{ssf}
In this subsection, we briefly explain (without any justifications) how to
compute the function $f$. We refer the reader to \cite{FIM} for the proofs
of those results.\\
\noindent {\inftyt Case A:} $V^{per}\epsilonquiv 0$.\\
In this special case, we can show that
\begin{equation}gin{equation}\lambdaabel{eq::6}
f(\rho^0,\tau)=\rho^0 \begin{eqnarray}r{v} \quad \mbox{with}\quad \begin{eqnarray}r{v}=\frac{\tau}{\begin{eqnarray}r{\mu}}
\epsilonnd{equation}
which is nothing else than the normalized Orowan's law giving, in a
dimensionless form, the
plastic strain velocity as the product of the normalized dislocation
density $\rho^0$ and the normalized mean velocity $\begin{eqnarray}r{v}$ of the
dislocations.\\
\noindent {\inftyt Case B:} General periodic potential $V^{per}$.\\
In that case, the function $f$ can
be computed using the following two steps.\\
\noindent {\underline{Step 1}.}\\
For $i\inftyn\mathbb{Z}$, we look for solutions to (\ref{eq::1}) of the following
special form
$$x_i(t)=b\cdot h\lambdaeft(\frac{vt}{b} + \frac{i}{\rho^0}\right), \quad \mbox{with}\quad
h(a+\begin{eqnarray}r{\lambdaambda})=\begin{eqnarray}r{\lambdaambda}+h(a) \quad \mbox{for all}\quad a\inftyn\mathbb{R}$$
for some constant $v$ and for a function $h$ which is called a {\inftyt hull
function}. Both $v$ and $h$ have to be determined.
Because of the convexity of the two-body potential $V$ outside the origin,
it is possible to show that the constant $v$ exists and is unique. Moreover
this constant $v$ can be interpreted as the mean velocity of each dislocation.\\
\noindent {\underline{Step 2}.}\\
We simply define $f(\rho^0,\tau_{ext})$ using the normalized Orowan's law as in
(\ref{eq::6}), but with the normalized velocity
$\begin{eqnarray}r{v}$ replaced by the constant $\begin{eqnarray}r{v}=\frac{B}{\begin{eqnarray}r{\mu}}v$.
\sigmaubsection{Numerical computation of $f$}
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\sigmaubsection{The homogenization result}\lambdaabel{s1.3}
We expect that the effective equation satisfied by the limit
$\gamma^0$ can be written
\begin{equation}gin{equation}\lambdaabel{eq::5}
\lambdaeft\{\begin{equation}gin{array}{l}
\deltaisplaystyle{\frac{\partialartial \gamma^0}{\partialartial \omegat} = f(\rho^0, \tau), \quad
\mbox{for all}\quad \omegax\inftyn\mathbb{R},\quad \omegat\inftyn (0,+\inftynfty)},\\
\\
\gamma^0(\omegax,0)=\gamma_0(\omegax) \quad \mbox{for all}\quad \omegax\inftyn \mathbb{R}
\epsilonnd{array}\right.
\epsilonnd{equation}
where $\rho^0$ is given in (\ref{eq::3}) and $\tau$ in (\ref{eq::2}).
Then our main result is:
\begin{equation}gin{theo}\lambdaabel{th::1}{\bf (Homogenization of straight dislocations)}\\
Assume that the initial data $\gamma_0$ is non-decreasing and satisfies
$|\gamma_0|+ |\gamma_0'|+ |\gamma_0''| \lambdae C$ for some constant $C$.
Then for any $C^2$ periodic potential $V^{per}$, there exists
a continuous function $f: \mathbb{R}^2 \to \mathbb{R}$ such that $\tau \mapsto
f(\rho^0,\tau)$ is nondecreasing. And there exists a unique viscosity
solution $\gamma^0$ of the equation (\ref{eq::5}).\\
Moreover, under the assumptions and notation of this
section, there exists a unique solution $\gamma^\varepsilon$ associated to
the dynamics (\ref{eq::1}) with initial
data given in (\ref{eq::0}), and $\gamma^\varepsilon$ converges to $\gamma^0$ locally
uniformly on $\mathbb{R} \times [0,+\inftynfty )$.
\epsilonnd{theo}
This result is proven rigorously in \cite{FIM} in the mathematical framework of
viscosity solutions (see for instance Crandall, Ishii, Lions \cite{CIL} for an introduction to
this theory). We explain in the next section how we compute the
function $f$, which keeps the memory of the short range interactions
between the dislocations and the periodic potential $V^{per}$.
\sigmaubsection{Computation of $f$ using Orowan's law}\lambdaabel{ssf}
In this subsection, we briefly explain (without any justifications) how to
compute the function $f$. We refer the reader to \cite{FIM} for the proofs
of those results.\\
\noindent {\inftyt Case A:} $V^{per}\epsilonquiv 0$.\\
In this special case, we can show that
\begin{equation}gin{equation}\lambdaabel{eq::6}
f(\rho^0,\tau)=\rho^0 \begin{eqnarray}r{v} \quad \mbox{with}\quad \begin{eqnarray}r{v}=\frac{\tau}{\begin{eqnarray}r{\mu}}
\epsilonnd{equation}
which is nothing else than the normalized Orowan's law giving, in a
dimensionless form, the
plastic strain velocity as the product of the normalized dislocation
density $\rho^0$ and the normalized mean velocity $\begin{eqnarray}r{v}$ of the
dislocations.\\
\noindent {\inftyt Case B:} General periodic potential $V^{per}$.\\
In that case, the function $f$ can
be computed using the following two steps.\\
\noindent {\underline{Step 1}.}\\
For $i\inftyn\mathbb{Z}$, we look for solutions to (\ref{eq::1}) of the following
special form
$$x_i(t)=b\cdot h\lambdaeft(\frac{vt}{b} + \frac{i}{\rho^0}\right), \quad \mbox{with}\quad
h(a+\begin{eqnarray}r{\lambdaambda})=\begin{eqnarray}r{\lambdaambda}+h(a) \quad \mbox{for all}\quad a\inftyn\mathbb{R}$$
for some constant $v$ and for a function $h$ which is called a {\inftyt hull
function}. Both $v$ and $h$ have to be determined.
Because of the convexity of the two-body potential $V$ outside the origin,
it is possible to show that the constant $v$ exists and is unique. Moreover
this constant $v$ can be interpreted as the mean velocity of each dislocation.\\
\noindent {\underline{Step 2}.}\\
We simply define $f(\rho^0,\tau_{ext})$ using the normalized Orowan's law as in
(\ref{eq::6}), but with the normalized velocity
$\begin{eqnarray}r{v}$ replaced by the constant $\begin{eqnarray}r{v}=\frac{B}{\begin{eqnarray}r{\mu}}v$.
\sigmaubsection{Numerical computation of $f$}
We present numerical simulations for the computation of the function
$f$. We work with dimensionless quantities:
$\lambdaambda=1=\begin{eqnarray}r{\lambdaambda}=b=B=\begin{eqnarray}r{\mu}$.
We put initially $N$ dislocations in
an interval of length $l=10$ which is repeated periodically. Therefore this
interval contains $l$ times the period of
the periodic potential that we choose equal to $V^{per}(x)=\frac{A}{2\partiali}\sigmain(2\partiali x)$
with $A=3$. We discretize the ODE system (\ref{eq::1}), using an
explicit Euler scheme with a time step ${\cal D}elta t =0.01$. We compute
numerically the mean velocity $v$ of the dislocations after a final time
$T=1000$. We then set $f=\rho^0 v$ with $\rho^0=N/l$. We do the computation
with $N=1,...,200$ and $0\lambdae \tau_{ext}\lambdae 9$ with ${\cal D}elta \tau_{ext} =
\frac{9}{200}$. Remark that
we can restrict our computation for positive $\tau_{ext}$, because we have
$f(\rho^0,-\tau_{ext})=-f(\rho^0,\tau_{ext})$, from the symmetry of the
potential $V^{per}$ in our problem. The
level sets of the function $f$ are represented on Figure \ref{F1}. In
order to have a better view of the set where $f=0$, this set is
conventionally represented in Figure \ref{F1} with artificial negative
values of $f$. We remark that this figure shows in particular a
collective behavior of the dislocations: higher is the density of
dislocations, then easier the dislocations move above the obstacles.
Figure \ref{F2} shows the map $\tau_{ext}\mapsto f(\rho^0,\tau_{ext})$ for
$\rho^0=N/l$ with $N=1,10,20$. We see in particular that for $\tau_{ext}$ under a threshold
(that depends on the dislocation density $\rho^0$) the function $f$ vanishes.
\begin{equation}gin{figure}[!h]
\begin{equation}gin{minipage}[b]{.46\lambdainewidth}
\centering\epsilonpsfig{figure=Photos68.eps,width=\lambdainewidth}
\caption{Level sets of the effective $f(N/l,\tau_{ext})$ with $N$ on
abscissas and $\tau_{ext}$ on ordinates\lambdaabel{F1}}
\epsilonnd{minipage}
\begin{equation}gin{minipage}[b]{.46\lambdainewidth}
\centering\epsilonpsfig{figure=Photos32new.eps ,width=\lambdainewidth}
\caption{For $N=1,10,20$, graph of the map $\tau_{ext}\mapsto f(N/l,\tau_{ext})$\\\lambdaabel{F2}}
\epsilonnd{minipage}
\epsilonnd{figure}
{\bf S}E{Homogenization of curved dislocations}
In this section, we very briefly generalize the previous analysis to the
case of curved dislocations all contained in the same plane $(x,y)$ with the
same Burgers vector $\mbox{\bf b}=be_x$ with $b>0$.
\sigmaubsection{The microscopic model for curved dislocations}
For $i\inftyn\mathbb{Z}$, the motion of the $i$-th dislocation curve $\Gammaamma_i(t)$ at the point
$X\inftyn\mathbb{R}^2$ is given by its
normal velocity ${\mathcal V}$ defined by
\begin{equation}gin{equation}\lambdaabel{eq::1bis}
B\cdot{\mathcal V}(X,t)= \tau^{per}(X) + \sigmaum_{j} F_{j}(X,t)
\epsilonnd{equation}
where $F_{j}(X,t)$ is the resolved Peach-Koehler force created by the
dislocation $\Gammaamma_j(t)$ at the point $X$. Here $\tau^{per}$ is a smooth
periodic function satisfying
$\tau^{per}(X +\lambdaambda k)=\tau^{per}(X)$ for all $k\inftyn\mathbb{Z}^2$, which
represents the periodic obstacles to the motion of the dislocations and can also
include the exterior applied stress. To give the expression of this
force, it is convenient
to introduce a continuous function $\tilde{\gamma}(X,t)$
such that each dislocation curve $\Gammaamma_j(t)$ can be seen as the level set
$\tilde{\gamma}(X,t)=jb$ (when this level set is non-degenerated). Then a
good approximation is given by
$$\deltaisplaystyle{F_{j}(X,t)=\frac12 \inftynt_{\mathbb{R}^2} dZ\ J(X-Z)\ \mbox{sign}(\tilde{\gamma}(Z,t)-jb)}$$
where, in the integral, the sign function takes values $-1,0,1$. Here the
kernel $J$ is smooth and satisfies for a cut-off radius $R=\begin{eqnarray}r{R}b$ with $\begin{eqnarray}r{R}>1$ fixed:
$$J(-X)=J(X)\ge 0, \quad \mbox{and}\quad
\deltaisplaystyle{J(X)=J_{\inftynfty}
(X):=\frac{1}{|X|^3}g\lambdaeft(\frac{X}{|X|}\right) \quad \mbox{for}\quad
|X|>R>0}$$
where for isotropic elasticity with $X=(x,y)$, we have
$g\lambdaeft(\frac{X}{|X|}\right)=\frac{\mu b}{4\partiali}\lambdaeft\{\frac{x^2(2\begin{equation}ta -1) +
y^2(2-\begin{equation}ta)}{x^2+y^2}\right\}$ with $\begin{equation}ta=\frac{1}{1-\nu}$. Remark that
this formula allows to describe with the same formalism edge, screw and
mixed dislocations (see for instance \cite{AHLM}).
We also define the plastic strain $\gamma$ as
$$\gamma= b \lambdaeft[
\frac{\tilde{\gamma}}{b}\right]$$
where we recall that $\lambdaeft[\cdot\right]$ is the
floor function. Then we proceed as in the previous section and define
\begin{equation}gin{equation}\lambdaabel{eq::0bis}
\begin{eqnarray}r{X}=\frac{X}{\Lambdaambda},\quad
\begin{eqnarray}r{t}=\frac{\mu}{B}\frac{t}{\Lambdaambda},\quad
\varepsilon=\frac{b}{\Lambdaambda},\quad \mbox{and}\quad
\gamma^{\varepsilon}(\begin{eqnarray}r{X},\omegat)=\frac{1}{\Lambdaambda}\gamma(X,t),\quad \mbox{with}\quad
\gamma^{\varepsilon}(\begin{eqnarray}r{X},0)=\varepsilon
\lambdaeft[\frac{1}{\varepsilon}\gamma_0(\omegaX)\right].
\epsilonnd{equation}
\sigmaubsection{The homogenization result}
We expect that the effective equation satisfied by the limit
$\gamma^0$ of $\gamma^\varepsilon$ can be written
\begin{equation}gin{equation}\lambdaabel{eq::5bis}
\lambdaeft\{\begin{equation}gin{array}{l}
\deltaisplaystyle{\frac{\partialartial \gamma^0}{\partialartial \omegat} = f(-\nabla \gamma^0, \tau_{sc}), \quad
\mbox{for all}\quad \omegaX\inftyn\mathbb{R}^2,\quad \omegat\inftyn (0,+\inftynfty)},\\
\\
\gamma^0(\omegaX,0)=\gamma_0(\omegaX) \quad \mbox{for all}\quad \omegaX\inftyn \mathbb{R}^2
\epsilonnd{array}\right.
\epsilonnd{equation}
with
$$\tau_{sc}(\omegaX,\omegat)=\inftynt_{\mathbb{R}^2} dZ\ J_\inftynfty (\omegaX-Z) \gamma^0(Z,\omegat)$$
where we take the principal value of the integral. Remark that this
expression of $\tau_{sc}$ is consistent with the one given in (\ref{eq::2})
in the special case where $\gamma^{0}(\begin{eqnarray}r{x}, \begin{eqnarray}r{y}, \begin{eqnarray}r{t})$ is
independent of $\begin{eqnarray}r{y}$.
Then we have
\begin{equation}gin{theo}\lambdaabel{th::1bis}{\bf (Homogenization of curved dislocations)}\\
Assume that the initial data satisfies $|\gamma_0|+ |\nabla \gamma_0|+
|D^2 \gamma_0|\lambdae C$ for some constant $C$.
Then for any $C^2$ periodic function $\tau^{per}$, there exists
a continuous function $f: \mathbb{R}^2\times \mathbb{R} \to \mathbb{R}$ such that $\tau \mapsto
f(\cdot,\tau)$ is nondecreasing. And there exists a unique viscosity
solution $\gamma^0$ of the equation (\ref{eq::5bis}).\\
Moreover, under the assumptions and notation of this
section, there exists a unique solution $\gamma^\varepsilon$ associated to
the dynamics (\ref{eq::1bis}) with initial
data given in (\ref{eq::0bis}), and $\gamma^\varepsilon$ converges to $\gamma^0$ locally
uniformly on $\mathbb{R}^2 \times [0,+\inftynfty )$.
\epsilonnd{theo}
{\bf S}E{Conclusion}
The main result of our work is the justification of the
elasto-visco-plastic flow rule by the homogenization of the
dynamics of dislocations with the same
Burgers vector, moving in the same glide plane with periodic obstacles.
Even if this geometry is very particular, this is, up
to our knowledge, the first rigorous result in this direction. We also
explained how to compute the flow rule, and presented numerical results.
The proof of the homogenization for straight dislocations uses strongly the local convexity of the
two-body potential $V$ (which is equivalent to the non-negativity of the
kernel $J$ in the case of curved dislocations).
Remark that for the same dynamics, it is
possible to find non-convex potentials $V$, for which there is no
homogenization. For a general geometry, there is in general no hope to find any
convexity argument to justify homogenization. On the contrary, it seems
reasonable to think that homogenization could arise in general, if we
assume moreover that the dynamics is modified by the addition of a small
random noise. But this is still an open problem to investigate.
\noindent {\bf Acknowledgements}\\
This work was supported by the contract ANR MICA (2006-2009).
\noindent {\bf References}
\begin{equation}gin{thebibliography}{99}
\bibitem{BC}
{Bulatov V V and Cai W},
{\inftyt Oxford University Press}, (2006).
\bibitem{FTRC}
{Fivel M, Tabourot L, Rauch E and Canova G R},
{\inftyt J. Phys.} IV, {\bf 8} (1998), 151-158.
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\sigmaubsection{The homogenization result}
We expect that the effective equation satisfied by the limit
$\gamma^0$ of $\gamma^\varepsilon$ can be written
\begin{equation}gin{equation}\lambdaabel{eq::5bis}
\lambdaeft\{\begin{equation}gin{array}{l}
\deltaisplaystyle{\frac{\partialartial \gamma^0}{\partialartial \omegat} = f(-\nabla \gamma^0, \tau_{sc}), \quad
\mbox{for all}\quad \omegaX\inftyn\mathbb{R}^2,\quad \omegat\inftyn (0,+\inftynfty)},\\
\\
\gamma^0(\omegaX,0)=\gamma_0(\omegaX) \quad \mbox{for all}\quad \omegaX\inftyn \mathbb{R}^2
\epsilonnd{array}\right.
\epsilonnd{equation}
with
$$\tau_{sc}(\omegaX,\omegat)=\inftynt_{\mathbb{R}^2} dZ\ J_\inftynfty (\omegaX-Z) \gamma^0(Z,\omegat)$$
where we take the principal value of the integral. Remark that this
expression of $\tau_{sc}$ is consistent with the one given in (\ref{eq::2})
in the special case where $\gamma^{0}(\begin{eqnarray}r{x}, \begin{eqnarray}r{y}, \begin{eqnarray}r{t})$ is
independent of $\begin{eqnarray}r{y}$.
Then we have
\begin{equation}gin{theo}\lambdaabel{th::1bis}{\bf (Homogenization of curved dislocations)}\\
Assume that the initial data satisfies $|\gamma_0|+ |\nabla \gamma_0|+
|D^2 \gamma_0|\lambdae C$ for some constant $C$.
Then for any $C^2$ periodic function $\tau^{per}$, there exists
a continuous function $f: \mathbb{R}^2\times \mathbb{R} \to \mathbb{R}$ such that $\tau \mapsto
f(\cdot,\tau)$ is nondecreasing. And there exists a unique viscosity
solution $\gamma^0$ of the equation (\ref{eq::5bis}).\\
Moreover, under the assumptions and notation of this
section, there exists a unique solution $\gamma^\varepsilon$ associated to
the dynamics (\ref{eq::1bis}) with initial
data given in (\ref{eq::0bis}), and $\gamma^\varepsilon$ converges to $\gamma^0$ locally
uniformly on $\mathbb{R}^2 \times [0,+\inftynfty )$.
\epsilonnd{theo}
{\bf S}E{Conclusion}
The main result of our work is the justification of the
elasto-visco-plastic flow rule by the homogenization of the
dynamics of dislocations with the same
Burgers vector, moving in the same glide plane with periodic obstacles.
Even if this geometry is very particular, this is, up
to our knowledge, the first rigorous result in this direction. We also
explained how to compute the flow rule, and presented numerical results.
The proof of the homogenization for straight dislocations uses strongly the local convexity of the
two-body potential $V$ (which is equivalent to the non-negativity of the
kernel $J$ in the case of curved dislocations).
Remark that for the same dynamics, it is
possible to find non-convex potentials $V$, for which there is no
homogenization. For a general geometry, there is in general no hope to find any
convexity argument to justify homogenization. On the contrary, it seems
reasonable to think that homogenization could arise in general, if we
assume moreover that the dynamics is modified by the addition of a small
random noise. But this is still an open problem to investigate.
\noindent {\bf Acknowledgements}\\
This work was supported by the contract ANR MICA (2006-2009).
\noindent {\bf References}
\begin{equation}gin{thebibliography}{99}
\bibitem{BC}
{Bulatov V V and Cai W},
{\inftyt Oxford University Press}, (2006).
\bibitem{FTRC}
{Fivel M, Tabourot L, Rauch E and Canova G R},
{\inftyt J. Phys.} IV, {\bf 8} (1998), 151-158.
\bibitem{HDK}
{Hoc T, Devincre B and Kubin L P},
{\inftyt In C. et al. Gundlach, editor, Riso National Laboratory, Denmark} (2004), 43-59.
\bibitem{K}
{Kr\"{o}ner E},
{\inftyt Erg. Angew. Math.} {\bf 5} (1958), 1-179, Berlin: Springer.
\bibitem{K2}
{Kr\"{o}ner E},
{\inftyt Int. J. Solids and Structures} {\bf 38} (2001), 1115-1134.
\bibitem{GB}
{Groma I and Balogh P},
{\inftyt Mat. Sci. Eng.} A {\bf 234-236} (1997), 249-252.
\bibitem{GCZ}
{Groma I, Cikor F F and Zaiser M},
{\inftyt Acta Mater.} {\bf 51} (2003), 1271-1281.
\bibitem{FE}
{El Hajj A and Forcadel N},
{\inftyt Math. Comp.} {\bf 77} (2008), 789-812.
\bibitem{IJM}
{Ibrahim H, Jazar M and Monneau R},
{\inftyt C. R. Acad. Sci. Paris}, Ser I {\bf 346} (2008) 945-950.
\bibitem{HZG}
{Hochrainer T, Zaiser M and Gumbsch P},
{\inftyt Philosophical Magazine} {\bf 87} (8 \& 9) (2007), 1261-1282.
\bibitem{M}
{R. Monneau},
{\inftyt Interfaces Free Bound.} {\bf 9} (2007), 383-409.
\bibitem{FIM}
{Forcadel N, Imbert C and Monneau},
{\inftyt Discrete Contin. Dyn. Syst.} A {\bf 23} (3), to appear (March
2009), and HAL: hal-00140545 (12-27-2007).
\bibitem{CIL}
{Crandall M G, Ishii H and Lions P -L},
{\inftyt Bull. Amer. Math. Soc.} (N.S.) {\bf 27} (1992), 1-67.
\bibitem{AHLM}
{Alvarez O, Hoch P, Le Bouar Y and Monneau R},
{\inftyt Arch. Ration. Mech. Anal.} {\bf 181} (3) (2006), 449-504.
\epsilonnd{thebibliography}
\epsilonnd{document}
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\betaegin{document}
\betaegin{abstract} In this paper we construct and study the actions of certain deformations of the Lie
algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the
relative symmetric powers ${\cal C}^{[\betaullet]}$ and the relative Jacobian ${\cal J}$ of a family of curves
${\cal C}/S$. As one of the applications, we show that
in the case of a single curve $C$ this action induces a ${\Bbb Z}$-form of a Lefschetz $\operatorname{sl}_2$-action on the Chow groups of $C^{[N]}$. Another application gives a new grading on the ring ${\Bbb C}H_0(J)$
of $0$-cycles on the Jacobian $J$ of $C$ (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes
in ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ and in ${\Bbb C}H^*({\cal J})$ and prove for them analogs of the properties established
in the case of the Jacobian of a single curve by Beauville in \cite{Bmain}. We show that the our
algebras of operators preserve the subrings of tautological cycles and act on them via some explicit differential operators.
\operatorname{e}nd{abstract}
\title{Algebraic cycles on the relative symmetric powers and on the
relative Jacobian of a family of curves. I}
\betaigskip
\centerline{\sc Introduction}
\betaigskip
Let ${\cal C}/S$ be a family of smooth projective curves over a smooth quasiprojective base $S$,
and let ${\cal C}^{[N]}$ denote the $N$th relative symmetric power of ${\cal C}$ over $S$.
In this paper we construct and study the natural action of a certain modification of the Lie algebra of differential operators on the line on the direct sum of the Chow groups
$${\Bbb C}H^*({\cal C}^{[\betaullet]}):=\betaigoplus_{N\ge 0}{\Bbb C}H^*({\cal C}^{[N]}),$$
where ${\cal C}^{[0]}=S$ (and on the similar direct sum of cohomology).
We also construct a related action of another algebra on ${\Bbb C}H^*({\cal J})$, where ${\cal J}/S$
is the corresponding relative Jacobian. These constructions are motivated by their potential use in the study of the Chow rings of Jacobians of curves, and in particular, in the study of the tautological subrings (see \cite{Bmain}, \cite{P-univ}, \cite{P-lie}). It was observed that the subalgebra generated by the standard cycles in the Jacobian of a curve depends in an interesting way on the corresponding point in the moduli space. Therefore, it is important to develop the corresponding calculus in the relative situation. On the other hand,
our main construction is reminiscent of the well known construction of the Heisenberg action on
cohomology of Hilbert schemes of surfaces (see \cite{Nak}, \cite{G}), and one can hope that there might
be a direct link between the two actions in the case of a family of curves lying on a surface.
Let us describe our main construction.
Let ${\cal D}={\Bbb Z}[t,\frac{d}{dt}]$ be the algebra of differential operators on the line.
Adjoin an independent variable $h$ and consider the subalgebra
${\cal D}_h\subset {\cal D}\otimesimes{\Bbb Z}[h]$ generated over ${\Bbb Z}[h]$ by $t$ and by $h\frac{d}{dt}$.
We view it as a Lie algebra with the commutator
$$[D_1,D_2]_h=(D_1D_2-D_2D_1)/h.$$
Note that ${\cal D}_h$ is a deformation of the Lie algebra $\widetilde{{\cal HV}}={\Bbb Z}[t,p]$ of polynomial
Hamiltonians on the plane (equipped with the standard Poisson bracket).
Now for any supercommutative ring $A$ and an even element ${\betaf a}_0\in A$ we define the Lie superalgebra
$${\cal D}(A,{\betaf a}_0):={\cal D}_h\otimesimes_{{\Bbb Z}[h]} A,$$
where the homomorphism ${\Bbb Z}[h]\to A$ sends $h$ to ${\betaf a}_0$.
The (super)bracket (resp., ${\Bbb Z}/2{\Bbb Z}$-grading) on ${\cal D}(A,{\betaf a}_0)$ is induced by the bracket on ${\cal D}_h$ and
the product on $A$ (resp., by the ${\Bbb Z}/2{\Bbb Z}$-grading on $A$). More explicitly,
${\cal D}(A,{\betaf a}_0)$ is generated as an abelian group by the elements
$${\betaf P}_{m,k}(a):=t^m(h\frac{d}{dt})^k\otimes a, \ \ a\in A.$$
The supercommutator is given by the formula
\betaegin{equation}\lambdabel{main-com-rel}
[{\betaf P}_{m,k}(a),{\betaf P}_{m',k'}(a')]= \sum_{i\ge 1}(-1)^{i-1}i!\cdotot
\left({k\choose i}{m'\choose i}-{m\choose i}{k'\choose i}\right)
{\betaf P}_{m+m'-i,k+k'-i}(a\cdotot a'\cdotot {\betaf a}_0^{i-1}).
\operatorname{e}nd{equation}
If $R\subset A$ is a subring then we can talk about an $R$-linear action of ${\cal D}(A,{\betaf a}_0)$ on an $R$-module (viewing ${\cal D}(A,{\betaf a}_0)$ as a Lie superalgebra over $R$). Our main construction gives a natural ${\Bbb C}H^*(S)$-linear (resp., $H^*(S)$-linear) action of ${\cal D}({\Bbb C}H^*({\cal C}),K)$ (resp.,
${\cal D}(H^*({\cal C}),cl(K))$) on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ (resp., $\betaigoplus H^*({\cal C}^{[N]})$),
where $K\in{\Bbb C}H^1({\cal C})$ is the relative canonical class,
$cl(K)\in H^2({\cal C})$ is the corresponding cohomology class (see Theorem \ref{action-thm} below).
For every integers $N\ge m\ge 0$ let us consider the morphism
$$s_{m,N}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[N]},$$
sending a point $(p,D)\in {\cal C}_s\tildemes {\cal C}_s^{[N-m]}$ to $mp+D$, where
${\cal C}_s\subset {\cal C}$ is the fiber of our family over $s\in S$. Here
we identify points of ${\cal C}_s^{[N]}$ with effective divisors of degree $N$ on ${\cal C}_s$.
Note that in the case $m=0$ this map is just the projection to ${\cal C}^{[N]}$.
For a cycle $a\in{\Bbb C}H^*({\cal C})$ and integers $m\ge 0$, $k\ge 0$, we consider the operator
$P_{m,k}(a)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ defined by the formula
\betaegin{equation}\lambdabel{main-oper-eq}
P_{m,k}(a)(x)=s_{m,N-k+m,*}(p_1^*a\cdotot s_{k,N}^*x),
\operatorname{e}nd{equation}
where $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
In the case $a=[{\cal C}]$ we will simply write $P_{m,k}({\cal C})$.
Note that in the case $N<k$ we have $P_{m,k}(a)(x)=0$.
If $a\in{\Bbb C}H^i({\cal C})$ then
$P_{m,k}(a)$ sends ${\Bbb C}H^p({\cal C}^{[N]})$ to ${\Bbb C}H^{p+i+m-1}({\cal C}^{[N-k+m]})$.
\betaegin{thm}\lambdabel{action-thm}
(a) The map ${\betaf P}_{m,k}(a)\mapsto P_{m,k}(a)$ defines a ${\Bbb C}H^*(S)$-linear action of
${\cal D}({\Bbb C}H^*({\cal C}),K)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$,
where $K\in{\Bbb C}H^1({\cal C})$ is the relative canonical class.
More precisely, the commutation relations \operatorname{e}qref{main-com-rel} for the operators $(P_{m,k}(a))$
(with ${\betaf a}_0=K$) hold on the level of relative correspondences over
$S$, i.e., they correspond to certain equalities in ${\Bbb C}H^*({\cal C}^{[\betaullet]}\tildemes_S {\cal C}^{[\betaullet]})$.
\noindent
(b) If we work over ${\Bbb C}$, the same construction defines an action of ${\cal D}(H^*({\cal C},{\Bbb Z}),cl(K))$ on
$H^*({\cal C}^{[\betaullet]})=\betaigoplus_N H^*({\cal C}^{[N]},{\Bbb Z})$.
\operatorname{e}nd{thm}
In the case of a trivial family ${\cal C}=C\tildemes S$ the above relations can be rewritten in a simpler form
(due to the fact that ${\cal D}_h/h^2$ becomes a trivial deformation of $\widetilde{{\cal HV}}$).
Recall that the commutator in the Lie algebra $\widetilde{{\cal HV}}={\Bbb Z}[x,p]$ of polynomial Hamiltonians on the plane is given by
$$\{ x^mp^k, x^{m'}p^{k'} \}=(km'-mk') x^{m+m'-1}p^{k+k'-1}.$$
\betaegin{cor}\lambdabel{triv-base-cor}
Let $C$ be a smooth projective curve over a field $k$.
Choose a theta characteristic $\chi\in{\Bbb C}H^1(C)$ (so that $2\chi=K$) and set
$$L_{m,k}(a)=P_{m,k}(a)-mk P_{m-1,k-1}(p_1^*\chi\cdotot a)$$
for $k\ge 0$, $m\ge 0$, $a\in{\Bbb C}H^*(C\tildemes S)$. Then one has the following relations:
$$[L_{m,k}(a),L_{m',k'}(a')]=(km'-mk')L_{m+m'-1,k+k'-1}(a\cdotot a').$$
In other words, the map $x^mp^k\otimes a\mapsto L_{m,k}(a)$ defines an action of
$\widetilde{{\cal HV}}\otimes{\Bbb C}H^*(C\tildemes S)$ on ${\Bbb C}H^*(C^{[\betaullet]}\tildemes S)$.
Similarly, we can define an action of $\widetilde{{\cal HV}}\otimes H^*(C\tildemes S)$ on $H^*(C^{[\betaullet]}\tildemes S)$.
We have $L_{0,0}([C\tildemes S])=0$, so the operators $(L_{m,k}([C\tildemes S]))$ define the action
of ${\cal HV}=\widetilde{{\cal HV}}/{\Bbb Z}$ on ${\Bbb C}H^*(C^{[\betaullet]}\tildemes S)$.
\operatorname{e}nd{cor}
\betaegin{rem} The operators $P_{n,0}(a)$ (resp., $P_{0,n}(a)$) for $n\ge 0$
are defined by the correspondences that are similar to those defining
Nakajima's operators $q_n(a)$ (resp., $q_{-n}(a)$) for the Hilbert schemes of points on a surface,
where we use the notation of \cite{Lehn}. It is somewhat surprising that in the curve case
the Lie superalgebra generated by these operators is more complicated
(the relations for $q_n(a)$ are simply those of the Heisenberg superalgebra).
\operatorname{e}nd{rem}
Looking at the simplest of the above operators (such as $P_{m,1}(C)$ and $P_{0,1}([p])$, where
$p\in C$ is a point) in the case $S=\operatorname{Spec}(k)$ we will derive the following result.
\betaegin{thm}\lambdabel{curve-thm}
Let $C$ be a smooth projective curve of genus $g\ge 1$ over an algebraically closed field $k$ and let
$J$ be the Jacobian of $C$. Fix a point $p_0\in C(k)$ and consider the embedding $\iota:C\to J$
associated with $p_0$, so that $\iota(p_0)=0\in J(k)$.
Let us denote by $I_C\subset{\Bbb C}H_0(J)$ the subgroup of classes represented by
$0$-cycles of degree $0$ supported on $\iota(C)$.
\noindent
(i) One has a direct sum decomposition
$${\Bbb C}H_0(J)={\Bbb Z}\cdotot [0]\oplus I_C\oplus I_C^{*2}\oplus\ldots\oplus I_C^{*g},$$
where $I_C^{*n}$ denotes the $n$th Pontryagin power of $I_C$.
The associated filtration $(\betaigoplus_{i\ge n}I_C^{*i})$ coincides with the
standard filtration $(I^{*n})$, where $I\subset{\Bbb C}H_0(J)$ is the subgroup of cycles of degree zero.
\noindent
(ii) There exists a family of derivations $(\delta_m)_{m\ge 1}$ of the graded algebra
$${\Bbb C}H_0(J)={\Bbb Z}\oplus \betaigoplus_{n=1}^g I_C^{*n}\sigmameq\betaigoplus_{n=0}^g I^{*n}/I^{*(n+1)},$$
where the multiplication is given by the Pontryagin product, such that for every $x\in I_C$ one has
$$\delta_m(x)=\sum_{i=0}^{m-1}(-1)^i{m\choose i}[m-i]_*x\in I_C^{*m}.$$
Equivalently, $\delta_m|_{I_C}$ can be characterized by the property
$$\delta_m([\iota(p)]-[0])=([\iota(p)]-[0])^{*m} \text{ for all }p\in C(k).$$
These derivations satisfy the commutation relations
$$[\delta_m,\delta_{m'}]=(m'-m)\delta_{m+m'-1},$$
i.e., they define an action of the Lie algebra of polynomial
vector fields on the line vanishing at the origin by $t^m\frac{d}{dt}\mapsto\delta_m$.
\operatorname{e}nd{thm}
We will show (see Remark 2 in the end of section \ref{first-sec})
that for a general curve of genus $\ge 3$ the decomposition in Theorem \ref{curve-thm}(i)
is different from
the decomposition defined by Beauville (see \cite{B1}, p.254; \cite{B2}, Prop.~4).
Theorems \ref{action-thm} and \ref{curve-thm} will be proved in section \ref{first-sec}.
In sections \ref{Jac-sec} and \ref{div-sec}, that are somewhat more technical,
we reprove (and generalize to the relative case) some known results using our algebra of operators.
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\betaegin{cor}\lambdabel{triv-base-cor}
Let $C$ be a smooth projective curve over a field $k$.
Choose a theta characteristic $\chi\in{\Bbb C}H^1(C)$ (so that $2\chi=K$) and set
$$L_{m,k}(a)=P_{m,k}(a)-mk P_{m-1,k-1}(p_1^*\chi\cdotot a)$$
for $k\ge 0$, $m\ge 0$, $a\in{\Bbb C}H^*(C\tildemes S)$. Then one has the following relations:
$$[L_{m,k}(a),L_{m',k'}(a')]=(km'-mk')L_{m+m'-1,k+k'-1}(a\cdotot a').$$
In other words, the map $x^mp^k\otimes a\mapsto L_{m,k}(a)$ defines an action of
$\widetilde{{\cal HV}}\otimes{\Bbb C}H^*(C\tildemes S)$ on ${\Bbb C}H^*(C^{[\betaullet]}\tildemes S)$.
Similarly, we can define an action of $\widetilde{{\cal HV}}\otimes H^*(C\tildemes S)$ on $H^*(C^{[\betaullet]}\tildemes S)$.
We have $L_{0,0}([C\tildemes S])=0$, so the operators $(L_{m,k}([C\tildemes S]))$ define the action
of ${\cal HV}=\widetilde{{\cal HV}}/{\Bbb Z}$ on ${\Bbb C}H^*(C^{[\betaullet]}\tildemes S)$.
\operatorname{e}nd{cor}
\betaegin{rem} The operators $P_{n,0}(a)$ (resp., $P_{0,n}(a)$) for $n\ge 0$
are defined by the correspondences that are similar to those defining
Nakajima's operators $q_n(a)$ (resp., $q_{-n}(a)$) for the Hilbert schemes of points on a surface,
where we use the notation of \cite{Lehn}. It is somewhat surprising that in the curve case
the Lie superalgebra generated by these operators is more complicated
(the relations for $q_n(a)$ are simply those of the Heisenberg superalgebra).
\operatorname{e}nd{rem}
Looking at the simplest of the above operators (such as $P_{m,1}(C)$ and $P_{0,1}([p])$, where
$p\in C$ is a point) in the case $S=\operatorname{Spec}(k)$ we will derive the following result.
\betaegin{thm}\lambdabel{curve-thm}
Let $C$ be a smooth projective curve of genus $g\ge 1$ over an algebraically closed field $k$ and let
$J$ be the Jacobian of $C$. Fix a point $p_0\in C(k)$ and consider the embedding $\iota:C\to J$
associated with $p_0$, so that $\iota(p_0)=0\in J(k)$.
Let us denote by $I_C\subset{\Bbb C}H_0(J)$ the subgroup of classes represented by
$0$-cycles of degree $0$ supported on $\iota(C)$.
\noindent
(i) One has a direct sum decomposition
$${\Bbb C}H_0(J)={\Bbb Z}\cdotot [0]\oplus I_C\oplus I_C^{*2}\oplus\ldots\oplus I_C^{*g},$$
where $I_C^{*n}$ denotes the $n$th Pontryagin power of $I_C$.
The associated filtration $(\betaigoplus_{i\ge n}I_C^{*i})$ coincides with the
standard filtration $(I^{*n})$, where $I\subset{\Bbb C}H_0(J)$ is the subgroup of cycles of degree zero.
\noindent
(ii) There exists a family of derivations $(\delta_m)_{m\ge 1}$ of the graded algebra
$${\Bbb C}H_0(J)={\Bbb Z}\oplus \betaigoplus_{n=1}^g I_C^{*n}\sigmameq\betaigoplus_{n=0}^g I^{*n}/I^{*(n+1)},$$
where the multiplication is given by the Pontryagin product, such that for every $x\in I_C$ one has
$$\delta_m(x)=\sum_{i=0}^{m-1}(-1)^i{m\choose i}[m-i]_*x\in I_C^{*m}.$$
Equivalently, $\delta_m|_{I_C}$ can be characterized by the property
$$\delta_m([\iota(p)]-[0])=([\iota(p)]-[0])^{*m} \text{ for all }p\in C(k).$$
These derivations satisfy the commutation relations
$$[\delta_m,\delta_{m'}]=(m'-m)\delta_{m+m'-1},$$
i.e., they define an action of the Lie algebra of polynomial
vector fields on the line vanishing at the origin by $t^m\frac{d}{dt}\mapsto\delta_m$.
\operatorname{e}nd{thm}
We will show (see Remark 2 in the end of section \ref{first-sec})
that for a general curve of genus $\ge 3$ the decomposition in Theorem \ref{curve-thm}(i)
is different from
the decomposition defined by Beauville (see \cite{B1}, p.254; \cite{B2}, Prop.~4).
Theorems \ref{action-thm} and \ref{curve-thm} will be proved in section \ref{first-sec}.
In sections \ref{Jac-sec} and \ref{div-sec}, that are somewhat more technical,
we reprove (and generalize to the relative case) some known results using our algebra of operators.
In section \ref{Jac-sec} we study the relation between our operators $(P_{m,k}(a))$
and the operators $(X_{n,k})_{n+k\ge 2}$ on ${\Bbb C}H(J)_{{\Bbb Q}}$,
where $J$ is the Jacobian of a curve $C$, constructed in \cite{P-lie}.
The latter family of operators satisfies the commutation relations of the
Lie subalgebra ${\cal HV}'\subset{\cal HV}$ spanned over ${\Bbb Z}$ by the elements $x^np^k$ with
$n+k\ge 2$ (it corresponds to Hamiltonian vector fields on the plane vanishing at the origin).
It is natural to ask how this action is related to the one given by Corollary \ref{triv-base-cor} (say,
in terms of the push-forward map ${\Bbb C}H^*(C^{[\betaullet]})_{{\Bbb Q}}\to{\Bbb C}H^*(J)_{{\Bbb Q}}$).
The relation turns out to be not quite straightforward. To work it out we
introduce another family of operators $(T_k(m,a))$,
acting both on ${\Bbb C}H^*(C^{[\betaullet]})$ and on ${\Bbb C}H^*(J)$, and
compatible with the push-forward map. In the case of ${\Bbb C}H^*(C^{[\betaullet]})$ we find an explicit expression of $T_k(m,a)$ in terms of the operators $(P_{k,m}(a))$ (see Proposition \ref{T-P-prop}).
On the other hand, in the
case of ${\Bbb C}H^*(J)_{{\Bbb Q}}$ we find that the operators $T_k(m,a)$ depend polynomially on $m$ and
the corresponding coefficients are closely related to the operators $X_{n,k}$ considered in
\cite{P-lie}. Similar computations work in the case of the relative Jacobian
${\cal J}/S$ of a family of curves ${\cal C}/S$. However, in the relative case the relations
between the operators $X_{n,k}$ get deformed in an interesting way: the corresponding
(quadratic) algebra of operators acting on ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$ is not a universal enveloping algebra
of a Lie algebra anymore (see \operatorname{e}qref{X-rel-eq}). We will study this algebra in detail elsewhere.
In section \ref{div-sec} we revisit algebraic Lefschetz $\operatorname{sl}_2$-actions for $C^{[N]}$
using our operators and study related questions of integrality.
The algebraic Lefschetz operators over ${\Bbb Q}$ are easily obtained from the action of
the algebra of (polynomial) differential operators in two variables ${\cal D}_{t,u,{\Bbb Q}}$
on ${\Bbb C}H^*({\cal C}^{[\betaullet]})_{{\Bbb Q}}$ generated by operators of the form $P_{10}(a)$ and $P_{01}(a')$
(see Corollary \ref{Heis-K-cor}).
After studying the divided powers of the operators $P_{n,0}({\cal C})$ and $P_{0,n}({\cal C})$ we construct an action of the divided powers subalgebra
${\Bbb Z}[t,u^{[\betaullet]},\partial_t^{[\betaullet]},\partial_u]\subset{\cal D}_{t,u,{\Bbb Q}}$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$. Using this action
we generalize to the relative case
the result of Collino \cite{Col2} on injectivity of the homomorphism
$i_{N*}$ (resp., surjectivity of $i_N^*$), where $i_N:{\cal C}^{[N-1]}\to{\cal C}^{[N]}$ is the embedding
associated with $p_0$.
The corresponding $\operatorname{sl}_2$-action given by the operators $e=t\partial_u$, $f=u\partial_t$ and
$h=t\partial_t-u\partial_u$ preserves the Chow groups of the individual symmetric powers.
In the case when $S$ is a point we show that in this way we get a Lefschetz
$\operatorname{sl}_2$-triple for $C^{[N]}$ (see Theorem \ref{Lefschetz-thm}).
Working with divided powers allows us to reprove the fact
(observed by del Ba\~{n}o in \cite{dB2})
that the hard Lefschetz isomorphism for $C^{[N]}$ holds over ${\Bbb Z}$ (see Corollary \ref{Lefschetz-cor}).
Finally, in section \ref{taut-sec} we define the groups of tautological classes in ${\Bbb C}H^*({\cal C}^{[\betaullet]})$
and in ${\Bbb C}H^*({\cal J})$. For tautological classes in ${\Bbb C}H^*({\cal J})$ we establish the properties similar
to those obtained in the case $S=\operatorname{Spec}(k)$ by Beauville in \cite{Bmain}. We also show that tautological
subspaces are preserved by the operators constructed in this paper and by push-forward (resp.,
pull-back) associated with the relative Albanese maps ${\cal C}^{[N]}\to{\cal J}$ (see Theorem \ref{taut-thm}).
As an application of
our techniques we relate the modified diagonal classes in ${\Bbb C}H^{k-1}(C^{[k]})$ introduced by
Gross and Schoen in \cite{GS} to the pull-backs of some tautological classes on $J$ (see
Corollary \ref{pull-back-cor}).
\noindent
{\it Notations and conventions}.
Throughout this paper we work with a family $\pi:{\cal C}\to S$ of smooth projective curves of genus $g$,
where $S$ is smooth quasiprojective over a field $k$ (when we mention cohomology we assume
that $k={\Bbb C}$). We denote by ${\cal J}/S$ the corresponding relative Jacobian.
In the case when $S$ is a point we denote ${\cal C}$ (resp., ${\cal J}$) simply by $C$ (resp., $J$).
We will often use a natural product operation on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ (resp., ${\Bbb C}H^*({\cal J})$)
called the {\it Pontryagin product}. It is defined using the map
$$\alpha_{m,n}:{\cal C}^{[m]}\tildemes_S {\cal C}^{[n]}\to {\cal C}^{[m+n]}:(D_1,D_2)\mapsto D_1+D_2.$$
by the formula
$$x*y=\alpha_{m,n,*}(p_1^*x\cdotot p_2^*y)$$
for $x\in{\Bbb C}H^*({\cal C}^{[m]})$ and $y\in{\Bbb C}H^*({\cal C}^{[n]})$ (where $p_1$ and $p_2$ are the projections).
It is easy to see that this operation makes ${\Bbb C}H^*({\cal C}^{\betaullet})$ into an associative commutative algebra over ${\Bbb C}H^*(S)={\Bbb C}H^*(C^{[0]})$. The Pontryagin product on ${\Bbb C}H^*({\cal J})$ is defined similarly
using the addition map ${\cal J}\tildemes_S{\cal J}\to{\cal J}$.
For every integer $m\in{\Bbb Z}$ we denote by $[m]:{\cal J}\to{\cal J}$ the corresponding map $\xi\mapsto m\xi$.
We usually fix a point $p_0\in{\cal C}(S)$ and consider the corresponding embedding
$\iota:{\cal C}\to{\cal J}$. We denote by ${\cal L}$ the biextension on ${\cal J}\tildemes_S{\cal J}$ corresponding to the autoduality of ${\cal J}$, normalized by the condition that ${\cal L}|_{{\cal C}\tildemes_S{\cal J}}\sigmameq{\cal P}_{{\cal C}}$, where
${\cal P}_{{\cal C}}$ is the Poincar\'e line bundle on ${\cal C}\tildemes_S{\cal J}$ trivialized over $p_0$ and over
the zero section of ${\cal J}$.
We often use results from Fulton's book \cite{Fulton}.
We usually consider Chow groups
only for nonsingular varieties and use the upper grading (by codimension).
For a cartesian diagram
\betaegin{diagram}
X' &\rTo{} &Y'\\
\dTo{} & &\dTo{}\\
X &\rTo{f} & Y
\operatorname{e}nd{diagram}
where $f$ is a locally complete intersection morphism,
we denote by $f^!:{\Bbb C}H^*(Y')\to{\Bbb C}H^*(X')$ the refined Gysin map defined in section 6.6 of
\cite{Fulton}.
When we talk
about $0$-cycles we mean cycles of dimension zero and use the notation ${\Bbb C}H_0$.
Also, on one occasion
in section \ref{Jac-sec} we also use Chow homology groups ${\Bbb C}H_*$ for a possibly singular scheme.
In the relative situation we view Chow groups of an $S$-scheme as a module over ${\Bbb C}H^*(S)$.
The analogs of our results for integral cohomology are based on the formalism developed in
\cite{FM} (that includes in particular Gysin maps $f_*:H^iX\to H^{i-2d}Y$ for proper locally complete
intersection morphisms $f:X\to Y$ such that $\dim X-\dim Y=d$).
The summation variables for which no range is given are supposed to be nonnegative integers.
The symbol $x^i$ for $i<0$ in algebraic formulas should be treated as zero. We denote divided powers
of a variable $x$ by $x^{[d]}=x^d/d!$.
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\section{Cycles on symmetric powers}\lambdabel{first-sec}
In this section we will prove Theorems \ref{action-thm} and \ref{curve-thm}.
We start with computing some intersection products.
We will need to work with the closed embedding
$$t_{m,N}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}\tildemes_S {\cal C}^{[N]}:(p,D)\mapsto(p,mp+D).$$
Note that its composition with the projection to the second factor is the
map $s_{m,N}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[N]}$ considered before.
We will denote by ${\cal D}_N\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ the relative universal divisor (the image of
$t_{1,N}$).
\betaegin{lem}\lambdabel{mult-lem2}
Let us consider the cartesian diagram
\betaegin{diagram}
{\Bbb P}i_{m,M,N} &\rTo{} & {\cal C}^{[M]}\tildemes_S {\cal C}^{[N]}\\
\dTo{} & &\dTo{\alpha_{M,N}}\\
{\cal C}\tildemes_S {\cal C}^{[M+N-m]} &\rTo{s_{m,M+N}}& {\cal C}^{[M+N]}
\operatorname{e}nd{diagram}
For every decomposition $m=k+l$ we have a natural closed embedding
$$q_{k,l}:{\cal C}\tildemes_S {\cal C}^{[M-k]}\tildemes_S {\cal C}^{[N-l]}\to {\Bbb P}i_{m,M,N}:
(x,D_1,D_2)\mapsto (x,D_1+D_2,D_1+kx,D_2+lx),$$
where we view ${\Bbb P}i_{m,M,N}$ as a subset of
${\cal C}\tildemes_S {\cal C}^{[M+N-m]}\tildemes_S{\cal C}^{[M]}\tildemes_S {\cal C}^{[N]}$.
Then one has the following identity in ${\Bbb C}H^*({\Bbb P}i_{m,M,N})$:
$$\alpha_{M,N}^![{\cal C}\tildemes_S {\cal C}^{[M+N-m]}]=
\sum_{k+l=m}{m\choose k} q_{k,l,*}[{\cal C}\tildemes_S {\cal C}^{[M-k]}\tildemes_S {\cal C}^{[N-l]}].$$
\operatorname{e}nd{lem}
\noindent {\it Proof} .
We have the following commutative diagram with cartesian squares
\betaegin{diagram}
{\Bbb P}i_{m,M,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[M]}\tildemes_S {\cal C}^{[N]} &\rTo{p_{23}} &
{\cal C}^{[M]}\tildemes_S {\cal C}^{[N]}\\
\dTo{} & & \dTo{\operatorname{id}\tildemes\alpha_{M,N}} & &\dTo{\alpha_{M,N}}\\
{\cal C}\tildemes_S {\cal C}^{[M+N-m]} &\rTo{t_{m,M+N}} & {\cal C}\tildemes_S {\cal C}^{[M+N]} &\rTo{p_2}& {\cal C}^{[M+N]}
\operatorname{e}nd{diagram}
Therefore, we have
$$\alpha_{M,N}^![{\cal C}\tildemes_S {\cal C}^{[M+N-m]}]=(\operatorname{id}\tildemes\alpha_{M,N})^![{\cal C}\tildemes_S {\cal C}^{[M+N-m]}].$$
In the case $m=1$ the image of $t_{1,M+N}$ is exactly the universal divisor ${\cal D}_{M+N}$.
By definition of the map $\alpha_{M,N}$, we have
\betaegin{equation}\lambdabel{div-pullback-eq}
(\operatorname{id}\tildemes\alpha_{M,N})^*[{\cal D}_{M+N}]=p_{12}^*[{\cal D}_M]+p_{13}^*[{\cal D}_N].
\operatorname{e}nd{equation}
This implies the required formula for $m=1$. The general case follows easily by induction in $m$.
\operatorname{e}d
\betaegin{lem}\lambdabel{diag-lem}
Consider the cartesian square
\betaegin{diagram}
\Sigma_{m,N} &\rTo{} & {\cal C}\\
\dTo{} & &\dTo{{\cal D}e_N}\\
{\cal C}\tildemes_S {\cal C}^{[N-m]} &\rTo{s_{m,N}} &{\cal C}^{[N]}
\operatorname{e}nd{diagram}
where ${\cal D}e_N:{\cal C}\to {\cal C}^{[N]}$ is the relative diagonal embedding.
Note that there is a natural isomorphism $\Sigma_{m,N}\sigmameq {\cal C}$ for $m>0$,
while $\Sigma_{0,N}\sigmameq {\cal C}\tildemes_S {\cal C}$.
Then we have
$${\cal D}e_N^!([{\cal C}\tildemes_S {\cal C}^{[N]})=[{\cal C}\tildemes_S {\cal C}]\in {\Bbb C}H^0({\cal C}\tildemes_S {\cal C}), \text{ and}$$
$${\cal D}e_N^!([{\cal C}\tildemes_S {\cal C}^{[N-m]}])=(-1)^{m-1}m!{N\choose m}K^{m-1}\in{\Bbb C}H^{m-1}({\cal C})$$
for $m\ge 1$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . The case $m=0$ is clear since $s_{0,N}$
is simply the projection ${\cal C}\tildemes_S {\cal C}^{[N]}\to {\cal C}^{[N]}$. In the case $m=1$
we should compute the intersection-product for the cartesian diagram
\betaegin{diagram}
{\cal C} &\rTo{{\cal D}e} & {\cal C}\tildemes_S {\cal C}\\
\dTo{} & &\dTo{\operatorname{id}\tildemes{\cal D}e_N}\\
{\cal C}\tildemes_S {\cal C}^{[N-1]} &\rTo{t_{1,N}} &{\cal C}\tildemes_S {\cal C}^{[N]}
\operatorname{e}nd{diagram}
In other words, we have to compute the intersection of the universal divisor
${\cal D}_N\subset {\cal C}\tildemes {\cal C}^{[N]}$ with ${\cal C}\tildemes_S{\cal D}e_N({\cal C})$. We need to check that the corresponding multiplicity with the diagonal
${\cal D}e({\cal C})\subset {\cal C}\tildemes_S {\cal C}$ is equal to $N$.
This is a local problem, so we can pick a local parameter $t$ along the fibers of ${\cal C}\to S$
and think of ${\cal D}_N$ as the set of pairs $(x,f(t))$, where $f$ is a unital polynomial of degree
$N$ in $t$ such that $f(x)=0$.
The diagonal embedding ${\cal D}e_N$ sends a point $y$ to the polynomial $(t-y)^N$. Hence, the restriction of the equation $f(x)=0$ will have form $(x-y)^N$, which gives multiplicity $N$ with the diagonal ${\cal D}e\subset {\cal C}\tildemes_S {\cal C}$.
The case of $m>1$ follows by induction: from the commutative diagram with cartesian squares
\betaegin{diagram}
{\cal C} &\rTo{\operatorname{id}} & {\cal C} &\rTo{\operatorname{id}} & {\cal C}\\
\dTo{} & &\dTo{\operatorname{id}\tildemes{\cal D}e_{N-m+1}} & &\dTo{{\cal D}e_N}\\
{\cal C}\tildemes_S {\cal C}^{[N-m]} &\rTo{t_{1,N-m+1}} &{\cal C}\tildemes_S {\cal C}^{[N-m+1]}&\rTo{s_{m-1,N}}&{\cal C}^{[N]}
\operatorname{e}nd{diagram}
we see that
$${\cal D}e_N^!([{\cal C}\tildemes_S {\cal C}^{[N-m]}])=(\operatorname{id}\tildemes{\cal D}e_{N-m+1})^!([{\cal C}\tildemes_S {\cal C}^{[N-m]}]).$$
Hence, the step of induction follows from the previous computation (for $N-m+1$ instead of $N$)
along with the formula ${\cal D}e^*([{\cal D}e({\cal C})])=K$.
\operatorname{e}d
\betaegin{lem}\lambdabel{inter-mult-lem}
Consider the cartesian square
\betaegin{diagram}
Z_{m,k,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-k]}\\
\dTo{} & & \dTo{s_{k,N}}\\
{\cal C}\tildemes_S {\cal C}^{[N-m]} &\rTo{s_{m,N}} & {\cal C}^{[N]}
\operatorname{e}nd{diagram}
We have natural closed embeddings
$$q^0:{\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]}\to Z_{m,k,N}: (x,x',D)\mapsto (x,D+kx',x',D+mx),$$
$$q^i:{\cal C}\tildemes_S {\cal C}^{[N-m-k+i]}\to Z_{m,k,N}: (x,D)\mapsto (x,D+(k-i)x,x,D+(m-i)x),$$
where $1\le i\le\min(m,k)$ (we view $Z_{m,k,N}$ as a subset of
${\cal C}\tildemes_S {\cal C}^{[N-m]}\tildemes_S{\cal C}\tildemes_S {\cal C}^{[N-k]}$).
Then we have the following formula for the intersection-product in the above diagram:
$$[{\cal C}\tildemes_S {\cal C}^{[N-m]}]\cdotot [{\cal C}\tildemes_S {\cal C}^{[N-k]}]=
[q^0({\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]})]+
\sum_{i\ge 1}(-1)^{i-1}i!{m\choose i}{k\choose i}q^i_*(K^{i-1}\tildemes [C^{[N-m-k+i]}]).$$
\operatorname{e}nd{lem}
\noindent {\it Proof} .
We can represent $s_{m,N}$ as the composition of ${\cal D}e_m\tildemes\operatorname{id}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to
{\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}$ followed by $\alpha_{m,N-m}:{\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[N]}$.
Therefore,
$$s_{m,N}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=({\cal D}e_m\tildemes\operatorname{id})^!\alpha_{m,N-m}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}].$$
Using Lemma \ref{mult-lem2} we obtain
$$s_{m,N}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=
\sum_{i+l=k}{k\choose i}z^{i,l}_*({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]}],$$
where for $i+l=k$ we consider a closed subset $Z^{i,l}\stackrel{z^{i,l}}{\hookrightarrow} Z=Z_{m,k,N}$
defined from the cartesian square
\betaegin{diagram}
Z^{i,l} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-m]}\\
\dTo{} & & \dTo{{\cal D}e_m\tildemes\operatorname{id}}\\
{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]} &\rTo{s^{i,l}} & {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}
\operatorname{e}nd{diagram}
with
$$s^{i,l}(x,D_1,D_2)=(D_1+ix,D_2+lx).$$
Note that $s^{i,l}$ factors into the composition of the map
$${\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]}\to {\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m]}$$
induced by $t_{l,N-m}$ (identical on the second factor), followed by
$$s_{i,m}\tildemes\operatorname{id}:{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}.$$
Now we can apply Lemma \ref{diag-lem}.
For $i=0$ we immediately get $Z^{0,k}\sigmameq {\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]}$ and
$$({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m-k]}]=
[{\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]}].$$
Similarly, for $i\ge 1$ we get
$Z^{i,k-i}=q^i({\cal C}\tildemes_S {\cal C}^{[N-m-k+i]})$ and
$$({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-k+i]}]=(-1)^{i-1}i!{m\choose i}
p_1^*K^{i-1}\cdotot [{\cal C}\tildemes_S {\cal C}^{[N-m-k+i]}].$$
\operatorname{e}d
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\noindent {\it Proof} .
We can represent $s_{m,N}$ as the composition of ${\cal D}e_m\tildemes\operatorname{id}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to
{\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}$ followed by $\alpha_{m,N-m}:{\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[N]}$.
Therefore,
$$s_{m,N}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=({\cal D}e_m\tildemes\operatorname{id})^!\alpha_{m,N-m}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}].$$
Using Lemma \ref{mult-lem2} we obtain
$$s_{m,N}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=
\sum_{i+l=k}{k\choose i}z^{i,l}_*({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]}],$$
where for $i+l=k$ we consider a closed subset $Z^{i,l}\stackrel{z^{i,l}}{\hookrightarrow} Z=Z_{m,k,N}$
defined from the cartesian square
\betaegin{diagram}
Z^{i,l} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-m]}\\
\dTo{} & & \dTo{{\cal D}e_m\tildemes\operatorname{id}}\\
{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]} &\rTo{s^{i,l}} & {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}
\operatorname{e}nd{diagram}
with
$$s^{i,l}(x,D_1,D_2)=(D_1+ix,D_2+lx).$$
Note that $s^{i,l}$ factors into the composition of the map
$${\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]}\to {\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m]}$$
induced by $t_{l,N-m}$ (identical on the second factor), followed by
$$s_{i,m}\tildemes\operatorname{id}:{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}.$$
Now we can apply Lemma \ref{diag-lem}.
For $i=0$ we immediately get $Z^{0,k}\sigmameq {\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]}$ and
$$({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m-k]}]=
[{\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]}].$$
Similarly, for $i\ge 1$ we get
$Z^{i,k-i}=q^i({\cal C}\tildemes_S {\cal C}^{[N-m-k+i]})$ and
$$({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-k+i]}]=(-1)^{i-1}i!{m\choose i}
p_1^*K^{i-1}\cdotot [{\cal C}\tildemes_S {\cal C}^{[N-m-k+i]}].$$
\operatorname{e}d
\noindent
{\it Proof of Theorem \ref{action-thm}.}
The operator $P_{k,m}(a)$ acting on ${\Bbb C}H^*({\cal C}^{[N]})$ is given by the relative correspondence
$f_{k,m*}(p_1^*(a))$, where
$$f_{k,m}:{\cal C}\tildemes_S{\cal C}^{[N-k]}\to{\cal C}^{[N]}\tildemes_S{\cal C}^{[N-k+m]}:(p,D)\mapsto(kp+D,mp+D).$$
Therefore, to compute the correspondence inducing
the composition $P_{k,m}(a)\circ P_{k',m'}(a')$ acting on ${\Bbb C}H^*(C^{[N]})$ we have to calculate
the push-forward to ${\cal C}^{[N]}\tildemes_S{\cal C}^{[N'-k+m]}$ of
the intersection product in the following diagram
\betaegin{diagram}
{\cal C}\tildemes_S {\cal C}^{[N-k']}\tildemes_S{\cal C}^{[N'-k+m]} &&&&{\cal C}^{[N]}\tildemes_S{\cal C}\tildemes_S{\cal C}^{[N'-k]}\\
&\rdTo{f_{k',m'}\tildemes\operatorname{id}}&&\ldTo{\operatorname{id}\tildemes f_{k,m}}\\
&&{\cal C}^{[N]}\tildemes_S{\cal C}^{[N']}\tildemes_S{\cal C}^{[N'-k+m]}
\operatorname{e}nd{diagram}
multiplied with the pullbacks of $b$ and $a$, where $N'=N-k'+m'$,
It is easy to see that this intersection-product is exactly the one computed in
Lemma \ref{inter-mult-lem} for $Z_{m',k,N'}\subsetset {\cal C}\tildemes_S{\cal C}^{[N-k']}\tildemes_S
{\cal C}\tildemes_S{\cal C}^{[N'-k]}$.
It follows that the above composition is given by the relative correspondence
$$(s_{k',N}\tildemes s_{m,N'-k+m})_*(w\cdotot p_1^*a'\cdotot p_3^*a)\in{\Bbb C}H^*({\cal C}^{[N]}\tildemes_S {\cal C}^{[N'-k+m]}),$$
where $p_1, p_3: Z_{m',k,N'}\to {\cal C}$ are the projections, and
$w\in{\Bbb C}H^*(Z_{m',k,N'})$ is the intersection-product computed in Lemma \ref{inter-mult-lem}.
When we substitute the formula for $w$ in the above equation and
subtract the similar expression for $P_{k',m'}(a')\circ P_{k,m}(a)$ we note that the first
terms (corresponding to the images of $q^0$) will cancel out due to the symmetry exchanging
the two factors ${\cal C}$. The remaining terms will give the required formula for the commutator.
In the case of cohomology we have to work with the supercommutator since switching the order
of $a$ and $a'$ will introduce the standard sign.
\operatorname{e}d
\betaegin{cor}\lambdabel{Heis-K-cor} The operators $(P_{1,0}(a))$ and $(P_{0,1}(a))$ on
${\Bbb C}H^*({\cal C}^{[\betaullet]})$ satisfy the following relations:
\betaegin{align*}
&[P_{1,0}(a),P_{1,0}(a')]=[P_{0,1}(a),P_{0,1}(a')]=0,\\
&[P_{0,1}(a),P_{1,0}(a')]=\lambdangle a,a'\rightarrowngle\cdotot\operatorname{id},
\operatorname{e}nd{align*}
where $\lambdangle a,a'\rightarrowngle=\pi_*(a\cdotot a')\in{\Bbb C}H^*(S)$
(recall that we view ${\Bbb C}H^*({\cal C}^{[N]})$ as a ${\Bbb C}H^*(S)$-module
using the product with the pull-back under the projection ${\cal C}^{[N]}\to S$).
In particular, if we are given a pair of divisor classes $\alpha,\beta\in{\Bbb C}H^1({\cal C})$ of nonzero relative degrees
$\deltag(\alpha)$ and $\deltag(\beta)$ then there is an action of the algebra
${\cal D}_{t,u,{\Bbb Q}}={\Bbb Q}[t,u,\partial_t,\partial_u]$ of differential operators in two variables on
${\Bbb C}H^*({\cal C}^{[\betaullet]})_{{\Bbb Q}}$ such that
\betaegin{align*}
&t\mapsto P_{1,0}(\alpha)-\frac{\lambdangle\alpha,\beta\rightarrowngle}{\deltag(\beta)}\cdotot P_{1,0}({\cal C}) \ \
(\text{Pontryagin product with }\alpha-\frac{\lambdangle\alpha,\beta\rightarrowngle}{\deltag(\beta)}\cdotot[{\cal C}])\\
&u\mapsto \frac{1}{\deltag(\beta)}P_{1,0}({\cal C}) \ \ (\text{Pontryagin product with } \frac{1}{\deltag(\beta)}[{\cal C}]),\\
&\partial_t\mapsto \frac{1}{\deltag(\alpha)}P_{0,1}({\cal C}),\\
&\partial_u\mapsto P_{0,1}(\beta).
\operatorname{e}nd{align*}
For example, for $g\neq 1$ we can take $\alpha=\beta=K$.
\operatorname{e}nd{cor}
\noindent {\it Proof} . This follows from the relations of Theorem \ref{action-thm} together with the identity
$P_{0,0}(a)=\pi_*(a)\cdotot\operatorname{id}$.
\operatorname{e}d
\betaegin{rem} In the case $S=\operatorname{Spec}({\Bbb C})$
the cohomology $H^*(C^{[\betaullet]},{\Bbb Q})$ can be identified with the
super-symmetric algebra of $H^*(C,{\Bbb Q})$. Then the operators
$P_{1,0}(a)$ and $P_{0,1}(a)$ for $a\in H^*(C,{\Bbb Q})$ are identified with the standard operators
on the super-symmetric algebra (products and contractions).
\operatorname{e}nd{rem}
\noindent
{\it Proof of Theorem \ref{curve-thm}.}
(i) Consider the Abel-Jacobi map $S:{\Bbb C}H_0(J)\to J(k): \sum m_i[a_i]\mapsto \sum m_ia_i$.
It is well known and easy to see that $S$ induces an isomorphism $I/I^{*2}\widetilde{\to} J(k)$
(see sec.0 of \cite{Bl}).
Since the composition ${\Bbb C}H_0(C)\stackrel{\iota_*}{\to}{\Bbb C}H_0(J)\stackrel{S}{\to} J(k)$ is the Abel-Jacobi
map for $C$ that induces an isomorphism of degree zero cycles with $J(k)$, we obtain
a decomposition
\betaegin{equation}\lambdabel{I-eq}
I=I_C\oplus I^{*2}.
\operatorname{e}nd{equation}
By taking the Pontryagin powers we immediately derive that
$$I^{*n}=I_C^{*n}+I^{*(n+1)}.$$
Since $I^{*(g+1)}=0$ by the result of Bloch (see \cite{Bl}), we deduce that
$${\Bbb C}H_0(J)={\Bbb Z}[0]+I_C+I_C^{*2}+\ldots+I_C^{*g}.$$
It remains to prove that this decomposition is direct, i.e., the summands are linearly independent.
The version of Roitman's theorem in
arbitrary characterstic proved by Milne~\cite{Milne} implies that $I^{*2}\subset\operatorname{ker}(S)$ has no torsion.
In view of \operatorname{e}qref{I-eq}, this shows that it is enough to prove our statement after tensoring with
${\Bbb Q}$. A more direct way of getting our decomposition over ${\Bbb Z}$ will be outlined in Remark 3 after
Corollary \ref{module-cor}.
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\betaegin{cor}\lambdabel{Heis-K-cor} The operators $(P_{1,0}(a))$ and $(P_{0,1}(a))$ on
${\Bbb C}H^*({\cal C}^{[\betaullet]})$ satisfy the following relations:
\betaegin{align*}
&[P_{1,0}(a),P_{1,0}(a')]=[P_{0,1}(a),P_{0,1}(a')]=0,\\
&[P_{0,1}(a),P_{1,0}(a')]=\lambdangle a,a'\rightarrowngle\cdotot\operatorname{id},
\operatorname{e}nd{align*}
where $\lambdangle a,a'\rightarrowngle=\pi_*(a\cdotot a')\in{\Bbb C}H^*(S)$
(recall that we view ${\Bbb C}H^*({\cal C}^{[N]})$ as a ${\Bbb C}H^*(S)$-module
using the product with the pull-back under the projection ${\cal C}^{[N]}\to S$).
In particular, if we are given a pair of divisor classes $\alpha,\beta\in{\Bbb C}H^1({\cal C})$ of nonzero relative degrees
$\deltag(\alpha)$ and $\deltag(\beta)$ then there is an action of the algebra
${\cal D}_{t,u,{\Bbb Q}}={\Bbb Q}[t,u,\partial_t,\partial_u]$ of differential operators in two variables on
${\Bbb C}H^*({\cal C}^{[\betaullet]})_{{\Bbb Q}}$ such that
\betaegin{align*}
&t\mapsto P_{1,0}(\alpha)-\frac{\lambdangle\alpha,\beta\rightarrowngle}{\deltag(\beta)}\cdotot P_{1,0}({\cal C}) \ \
(\text{Pontryagin product with }\alpha-\frac{\lambdangle\alpha,\beta\rightarrowngle}{\deltag(\beta)}\cdotot[{\cal C}])\\
&u\mapsto \frac{1}{\deltag(\beta)}P_{1,0}({\cal C}) \ \ (\text{Pontryagin product with } \frac{1}{\deltag(\beta)}[{\cal C}]),\\
&\partial_t\mapsto \frac{1}{\deltag(\alpha)}P_{0,1}({\cal C}),\\
&\partial_u\mapsto P_{0,1}(\beta).
\operatorname{e}nd{align*}
For example, for $g\neq 1$ we can take $\alpha=\beta=K$.
\operatorname{e}nd{cor}
\noindent {\it Proof} . This follows from the relations of Theorem \ref{action-thm} together with the identity
$P_{0,0}(a)=\pi_*(a)\cdotot\operatorname{id}$.
\operatorname{e}d
\betaegin{rem} In the case $S=\operatorname{Spec}({\Bbb C})$
the cohomology $H^*(C^{[\betaullet]},{\Bbb Q})$ can be identified with the
super-symmetric algebra of $H^*(C,{\Bbb Q})$. Then the operators
$P_{1,0}(a)$ and $P_{0,1}(a)$ for $a\in H^*(C,{\Bbb Q})$ are identified with the standard operators
on the super-symmetric algebra (products and contractions).
\operatorname{e}nd{rem}
\noindent
{\it Proof of Theorem \ref{curve-thm}.}
(i) Consider the Abel-Jacobi map $S:{\Bbb C}H_0(J)\to J(k): \sum m_i[a_i]\mapsto \sum m_ia_i$.
It is well known and easy to see that $S$ induces an isomorphism $I/I^{*2}\widetilde{\to} J(k)$
(see sec.0 of \cite{Bl}).
Since the composition ${\Bbb C}H_0(C)\stackrel{\iota_*}{\to}{\Bbb C}H_0(J)\stackrel{S}{\to} J(k)$ is the Abel-Jacobi
map for $C$ that induces an isomorphism of degree zero cycles with $J(k)$, we obtain
a decomposition
\betaegin{equation}\lambdabel{I-eq}
I=I_C\oplus I^{*2}.
\operatorname{e}nd{equation}
By taking the Pontryagin powers we immediately derive that
$$I^{*n}=I_C^{*n}+I^{*(n+1)}.$$
Since $I^{*(g+1)}=0$ by the result of Bloch (see \cite{Bl}), we deduce that
$${\Bbb C}H_0(J)={\Bbb Z}[0]+I_C+I_C^{*2}+\ldots+I_C^{*g}.$$
It remains to prove that this decomposition is direct, i.e., the summands are linearly independent.
The version of Roitman's theorem in
arbitrary characterstic proved by Milne~\cite{Milne} implies that $I^{*2}\subset\operatorname{ker}(S)$ has no torsion.
In view of \operatorname{e}qref{I-eq}, this shows that it is enough to prove our statement after tensoring with
${\Bbb Q}$. A more direct way of getting our decomposition over ${\Bbb Z}$ will be outlined in Remark 3 after
Corollary \ref{module-cor}.
Let us consider the morphisms $C^{[N]}\to J$ induced by $\iota:C\to J$.
It is easy to see that
the induced push-forward map $\sigma_*:{\Bbb C}H_0(C^{[\betaullet]})\to{\Bbb C}H_0(J)$ is compatible with the
Pontryagin products. Also, since $\iota(p_0)=0\in J(k)$, we have $\sigma_*(x*[p_0])=\sigma_*(x)$ for
any $x\in{\Bbb C}H_0(C^{[\betaullet]})$. Let $A_0(C)\subset{\Bbb C}H_0(C)$ denote the subgroup of classes of degree zero. Note that $\iota_*(A_0(C))=I_C$ and the push-forward
map $\sigma_*:{\Bbb C}H_0(C^{[g]})\to{\Bbb C}H_0(J)$ is an isomorphism (since $C^{[g]}\to J$ is birational).
Therefore, it suffices to establish the direct sum decomposition
\betaegin{equation}\lambdabel{AC-dec-eq}
{\Bbb C}H_0(C^{[g]})_{{\Bbb Q}}=
{\Bbb Q}\cdotot [p_0]^{*g}\oplus A_0(C)_{{\Bbb Q}}*[p_0]^{*(g-1)}\oplus \ldots \oplus A_0(C)_{{\Bbb Q}}^{*(g-1)} *[p_0]\oplus
A_0(C)_{{\Bbb Q}}^{*g},
\operatorname{e}nd{equation}
where the Pontryagin products are taken in ${\Bbb C}H_0(C^{[\betaullet]})_{{\Bbb Q}}$.
To this end we will use the action of the algebra of differential operators ${\cal D}_{t,{\Bbb Q}}$ in one variable
on ${\Bbb C}H^*(C^{[\betaullet]})_{{\Bbb Q}}$ given by $t\mapsto P_{1,0}([p_0])$,
$\frac{d}{dt}\mapsto P_{0,1}(C)$ (see Corollary \ref{Heis-K-cor}). Note that this
action preserves the subspace of $0$-cycles ${\Bbb C}H_0(C^{[\betaullet]})_{{\Bbb Q}}$.
Since $\frac{d}{dt}$ acts locally nilpotently, we have a natural isomorphism of ${\cal D}_{t,{\Bbb Q}}$-modules
\betaegin{equation}\lambdabel{D-mod-eq}
{\Bbb C}H_0(C^{[\betaullet]})_{{\Bbb Q}}\sigmameq K_{{\Bbb Q}}[t],
\operatorname{e}nd{equation}
where $K=\operatorname{ker}(P_{0,1}(C))\cap{\Bbb C}H_0(C^{[\betaullet]})$.
Furthermore, this isomorphism is compatible with gradings, where the grading on the right-hand side
is induced by the grading of $K$ and the rule $\deltag(t)=1$.
Next, we observe that for every $a\in{\Bbb C}H_0(C)$ we have the relation
$$[P_{0,1}(C),P_{1,0}(a)]=\deltag(a)\cdotot\operatorname{id}.$$
Hence, $P_{0,1}(C)$ commutes with the Pontryagin product with any $0$-cycle of degree zero on $C$.
Thus, we obtain
$$A_0(C)^{*n}\subset K_n=\operatorname{ker}(P_{0,1}(C))\cap{\Bbb C}H_0(C^{[n]}) \text{ for }n\ge 1.$$
On the other hand, the algebra ${\Bbb C}H_0(C^{\betaullet})_{{\Bbb Q}}$ is generated over ${\Bbb Q}={\Bbb C}H_0(C^{[0]})_{{\Bbb Q}}$
by ${\Bbb C}H_0(C)_{{\Bbb Q}}=A_0(C)_{{\Bbb Q}}\oplus{\Bbb Q}\cdotot [p_0]$.
Therefore, the natural map
$$\betaigoplus_{n\ge 0}A_0(C)_{{\Bbb Q}}^{*n}[t]\to{\Bbb C}H_0(C^{[\betaullet]})_{{\Bbb Q}}$$
is an isomorphism (where we set $A_0(C)_{{\Bbb Q}}^{*0}={\Bbb Q}$).
Looking at the grading component of degree $g$ we get the decomposition \operatorname{e}qref{AC-dec-eq}.
\noindent
(ii) Let us set $A={\Bbb Z}\oplus A_0(C)\oplus A_0(C)^{*2}\oplus\ldots\oplus A_0(C)^{*g}$ (from (i)
we know that this algebra is isomorphic to ${\Bbb C}H_0(J)$) and consider
the natural homomorphism of algebras
\betaegin{equation}\lambdabel{A-hom-eq}
A[t]\to {\Bbb C}H_0(C^{[\betaullet]})
\operatorname{e}nd{equation}
as in part (i). We claim that it is an isomorphism. Indeed, it is surjective, since ${\Bbb C}H_0(C^{[\betaullet]})$
is generated by $[p_0]$ and by $A_0(C)$ as an algebra over ${\Bbb Z}$. Also, from part (i) we know that
\operatorname{e}qref{A-hom-eq} is injective modulo torsion. It remains to show that it is injective
on the torsion subgroup. But the torsion in $A$ is contained in $A_0(C)$ (by Roitman's theorem), so the statement boils down to the fact that the natural map $A_0(C)t^N\to {\Bbb C}H_0(C^{[N]})$ is an embedding.
But this follows from the fact that its composition with the Abel-Jacobi map to $J(k)$ is an isomorphism.
Note that as in part (i) we could have avoided referring to Roitman's theorem and used
the divided powers instead (see Remark 3 after Corollary \ref{module-cor}).
Thus, we can view $(P_{m,1}(C))$ as operators on $A[t]$. For example, $P_{0,1}(C)$ acts by
$\frac{d}{dt}$.
For $m\ge 1$ and $a\in{\Bbb C}H_0(C)$ we have the relation
$$[P_{m,1}(C),P_{1,0}(a)]=P_{m,0}(a).$$
This implies that for every point $p\in C(k)$ one has
$$P_{m,1}(C)([p]*x)=[p]*P_{m,1}(C)(x)+[p]^{*m}*x.$$
Since the classes $[p]$ generate our algebra, this implies that $P_{m,1}(C)$ is a derivation
of $A[t]$ characterized by
$$P_{m,1}(C)([p])=[p]^{*m} \text{ for all }p\in C(k).$$
Setting $x_p=[p]-[p_0]\in A_0(C)$ we derive that $P_{m,1}(C)(t)=t^m$ and
$P_{m,1}(C)(x_p)=(x_p+t)^m-t^m$. Now let us define $\delta_m$ as the following composition
$$A\to A[t]\stackrel{P_{m,1}(C)}{\rightarrow} A[t]\to A,$$
where the first map is the natural embedding and the last map is the evaluation at $t=0$.
Then $\delta_m$ is a derivation of $A$ with the property $\delta_m(x_p)=x_p^m$ for all $p\in C(k)$.
The commutation relations for $\delta_m$ are easily checked on the generators $x_p$.
The formula for $\delta_m|_{I_C}$ follows from the simple identity
$$([a]-[0])^{*m}=\sum_{i=0}^{m-1}(-1)^i{m\choose i}[m-i]_*([a]-[0])$$
in ${\Bbb C}H_0(J)$, where $m\ge 1$, $a\in J(k)$.
\operatorname{e}d
\betaegin{rems} 1. Note that $\delta_1$ is just the grading derivation: it is equal to $n\operatorname{id}$ on the
grading component of degree $n$.
It is easy to see that under the identification \operatorname{e}qref{A-hom-eq} the operators
$P_{m,1}(C)$ are given by
$$P_{m,1}(C)=\delta_m+mt\delta_{m-1}+{m\choose 2}t^2\delta_{m-2}+\ldots+mt^{m-1}\delta_1+t^m\frac{d}{dt},$$
where $\delta_m$ are extended to operators on $A[t]$ commuting with $t$.
\noindent
2. It is natural to compare our decomposition of ${\Bbb C}H_0(J)={\Bbb C}H^g(J)$ (tensored with ${\Bbb Q}$)
with the Beauville's decomposition
$${\Bbb C}H^g(J)_{{\Bbb Q}}=\betaigoplus_{s=0}^g{\Bbb C}H^g_s(J),$$
where ${\Bbb C}H^g_s(J)\subset{\Bbb C}H^g(J)_{{\Bbb Q}}$ is characterized by the condition $x\in{\Bbb C}H^g_s(J)$ if and only
if $[m]_*x=m^s x$ for all $m\in{\Bbb Z}$. The corresponding filtrations
$$\betaigoplus_{s\ge n}{\Bbb C}H^g_s(J)=I^{*n}=\betaigoplus_{s\ge n}I_C^*s$$
are the same (see \cite{B2}). However, the decompositions themselves are different.
Indeed, if they were the same we would have
$(I_C)_{{\Bbb Q}}\subset{\Bbb C}H^g_1(J)$ which would imply that $[2]_*([\iota(p)]-[0])=2[\iota(p)]-2[0]$ in
${\Bbb C}H^g(J)_{{\Bbb Q}}$ for all $p\in C$. But this would mean that $([\iota(p)]-[0])^{*2}=0$ in ${\Bbb C}H^g(J)$, hence
$(p,p)-(p,p_0)-(p_0,p)+(p_0,p_0)$
is a torsion class in ${\Bbb C}H_0(C\tildemes C)$ for all $p$, which is known not to be the case for a general
curve of genus $g\ge 3$ (see \cite{BV}, Prop. 3.2).
\operatorname{e}nd{rems}
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\section{Connection with cycles on the relative Jacobian}
\lambdabel{Jac-sec}
Assume that our family $\pi:{\cal C}\to S$ is equipped with a section
$p_0:S\to{\cal C}$, and let $\sigma_N:{\cal C}^{[N]}\to {\cal J}$
denote the corresponding map to the relative Jacobian of ${\cal C}$
sending a divisor $D\in{\cal C}_s^{[N]}$ to the class of the line bundle ${\cal O}_{{\cal C}_s}(D-Np_0)$.
We normalize the relative Poincar\'e line bundle ${\cal P}_{{\cal C}}$ on ${\cal C}\tildemes_S {\cal J}$ so
that its pull-backs under $p_0\tildemes\operatorname{id}_{{\cal J}}$ and under $\operatorname{id}_{{\cal C}}\tildemes e$ are trivial, where
$e:S\to{\cal J}$ is the zero section. Then we have the following equality in ${\Bbb C}H^1({\cal C}\tildemes_S{\cal C}^{[N]})$:
\betaegin{equation}\lambdabel{pull-back-P-b-eq}
(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})=[{\cal D}_N]-Np_1^*[p_0]-p_2^*([{\cal R}_N])-N\psi,
\operatorname{e}nd{equation}
where ${\cal D}_N\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ is
the universal divisor,
$${\cal R}_N=(p_0\tildemes\operatorname{id})^{-1}({\cal D}_N)=s_{1,N-1}(p_0\tildemes \operatorname{id})({\cal C}^{[N-1]})\subset {\cal C}^{[N]}$$
is the divisor in ${\cal C}^{[N]}$ associated with $p_0$, and
$$\psi=p_0^*K\in{\Bbb C}H^1(S)$$
(we view $\psi$ as a divisor class on any $S$-scheme via the pull-back).
We are going to introduce a new family of operators on ${\Bbb C}H^*({\cal J})$ and on
${\Bbb C}H^*({\cal C}^{[\betaullet]})$ that are compatible with respect to the push-forward map
$\sigma_*:{\Bbb C}H^*({\cal C}^{[\betaullet]})\to{\Bbb C}H^*({\cal J})$ (that has $\sigma_{N*}$ as components).
It is convenient to consider a more general setup.
Let ${\cal X}_{\betaullet}=(\sigma_N:{\cal X}_N\to{\cal J})_{N\in{\Bbb Z}}$ be a family of proper ${\cal J}$-schemes equipped with
a collection of morphisms
$$s_N:{\cal C}\tildemes_S{\cal X}_{N-1}\to {\cal X}_N,$$
where $N\in{\Bbb Z}$, such that
\noindent
(i) the diagram
\betaegin{equation}
\betaegin{diagram}
{\cal C}\tildemes_S{\cal X}_{N-1} & \rTo{s_N} & {\cal X}_N\\
\dTo{\operatorname{id}\tildemes\sigma_{N-1}} & &\dTo{\sigma_N}\\
{\cal C}\tildemes_S{\cal J} &\rTo{s_N}& {\cal J}
\operatorname{e}nd{diagram}
\operatorname{e}nd{equation}
is commutative, where the lower horizontal arrow is induced
by the map $\iota=\sigma_1:{\cal C}\to{\cal J}$ and by the group law on the Jacobian;
\noindent
(ii) for each $N$ the map ${\cal C}^m\tildemes_S{\cal X}_{N-m}\to{\cal X}_N$
obtained from $(s_N)$ by iteration (where ${\cal C}^m$ is the $m$th cartesian power of ${\cal C}/S$),
factors through a map
${\cal C}^{[m]}\tildemes_S{\cal X}_{N-m}\to{\cal X}_N$.
Two main examples of the above situations are: ${\cal X}_N={\cal C}^{[N]}$ for $N\ge 0$
(where $\sigma_N:{\cal C}^{[N]}\to{\cal J}$ are associated with a point $p_0\in{\cal C}(S)$, and ${\cal X}_N=\operatorname{e}mptyset$ for
$N<0$) and ${\cal X}_N={\cal J}$ for all $N\in{\Bbb Z}$.
Another (singular) example is obtained by taking ${\cal X}_N$
to be the image of the map ${\cal C}^{[N]}\to{\cal J}$.
Restricting the above maps ${\cal C}^m\tildemes_S{\cal X}_{N-n}\to{\cal X}_N$ to the diagonal in ${\cal C}^m$
we get morphisms
$$s_{m,N}:{\cal C}\tildemes_S{\cal X}_{N-m}\to{\cal X}_N.$$
Now, let us define the operator $T_k(m,a)$ on
${\Bbb C}H_*({\cal X}_{\betaullet})=\betaigoplus_{N\in{\Bbb Z}}{\Bbb C}H_*({\cal X}_N)$, where $k\ge 0$, $m\ge 0$,
$a\in{\Bbb C}H^*({\cal C})$, by the formula
$$T_k(m,a)(x)=s_{m,m+N,*}((\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})^k\cdotot p_1^*a\cdotot p_2^*x),$$
where $x\in{\Bbb C}H_*({\cal X}_N)$, $p_1$ and $p_2$ are projections from the product
${\cal C}\tildemes_S {\cal X}_N$ to its factors. In the case $a=[{\cal C}]$ we will simply write $T_k(m,{\cal C})$.
Note that from the projection formula we get
\betaegin{equation}\lambdabel{T-k-0-eq}
T_k(0,a)(x)=\sigma_N^*\tau_k(a)\cdotot x \ \text{ for }x\in{\Bbb C}H_*({\cal X}_N),
\operatorname{e}nd{equation}
where for $a\in{\Bbb C}H^*({\cal C})$ and $k\ge 0$ we set
\betaegin{equation}\lambdabel{tau-eq}
\tau_k(a)=p_{2*}(c_1({\cal P}_{{\cal C}})^k\cdotot p_1^*a)\in{\Bbb C}H^*({\cal J}).
\operatorname{e}nd{equation}
Also, if $(f:{\cal X}_N\to{\cal Y}_N)_{N\in{\Bbb Z}}$ is a morphism of two families as above then
it follows immediately from the definition that the above operators commute with the
push-forward map $f_*:{\Bbb C}H_*({\cal X}_{\betaullet})\to{\Bbb C}H_*({\cal Y}_{\betaullet})$, i.e.,
$$T_k(m,a)\circ f_*=f_*\circ T_k(m,a).$$
\betaegin{thm}\lambdabel{relations-thm}
One has the following relations between operators on ${\Bbb C}H_*({\cal X}_{\betaullet})$:
\betaegin{align*}
&\sum_{i\ge 0}\psi^i\cdotot\left({k\choose i}m^{\operatorname{pr}ime i}
T_{k-i}(m,a)T_{k'}(m',a')-{k'\choose i}m^i T_{k'-i}(m',a')T_k(m,a)\right)=\\
&\sum_{i\ge 1}(-1)^{i-1}
\left({k\choose i}m^{\operatorname{pr}ime i}-{k'\choose i}m^i\right)T_{k+k'-i}(m+m',a\cdotot a'\cdotot
(K+2[p_0(S)])^{i-1})+\\
&\psi^{k'-1}m^{k'}p_0^*(a')T_k(m,a)-\psi^{k-1}m^{\operatorname{pr}ime k}p_0^*(a)T_{k'}(m',a')+\\
&\delta_{k,0}\cdotot\sum_{i\ge 1}{k'\choose i}m^i\psi^{i-1}p_0^*(a)T_{k'-i}(m',a')
-\delta_{k',0}\cdotot\sum_{i\ge 1}{k\choose i}m^{\operatorname{pr}ime i}\psi^{i-1}p_0^*(a') T_{k-i}(m,a)
\operatorname{e}nd{align*}
where $a,a'\in{\Bbb C}H^*({\cal C})$, $k\ge 0$, $k'\ge 0$, $m\ge 0$, $m'\ge 0$.
\operatorname{e}nd{thm}
Let us set $\operatorname{e}ll_N=(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_C)\in{\Bbb C}H^1({\cal C}\tildemes_S {\cal X}_N)$.
Also let us denote $\mu=\operatorname{e}ll_1=(\operatorname{id}\tildemes\sigma_1)^*c_1({\cal P}_C)\in{\Bbb C}H^1({\cal C}\tildemes_S{\cal C})$.
Recall that ${\cal P}_C$ is the pull-back of the biextension ${\cal L}$ of ${\cal J}\tildemes_S{\cal J}$
under the embedding $(\iota\tildemes\operatorname{id}):{\cal C}\tildemes_S {\cal J}\to {\cal J}\tildemes_S {\cal J}$
corresponding to point $p_0$. This implies the following isomorphism in
$CH^1({\cal C}\tildemes_S{\cal C}\tildemes_S{\cal X}_{N-m})$:
\betaegin{equation}\lambdabel{biext-eq}
(\operatorname{id}_{{\cal C}}\tildemes s_{m,N})^*\operatorname{e}ll_N=m\cdotot p_{12}^*\mu+p_{13}^*\operatorname{e}ll_{N-m}.
\operatorname{e}nd{equation}
\betaegin{lem}\lambdabel{diag-lem2}
One has the following identity in ${\Bbb C}H^*({\cal C}\tildemes_S{\cal C})$ for $n\ge 1$:
$$\mu^n=(-\psi)^n+(-1)^n\psi^{n-1}\cdotot
\left((p_0\tildemes\operatorname{id})_*[{\cal C}]+(\operatorname{id}\tildemes p_0)_*[{\cal C}]\right)+(-1)^{n-1}
\sum_{i\ge 1}{n\choose i}\psi^{n-i}\cdotot{\cal D}e_*(K+2[p_0(S)])^{i-1}.$$
\operatorname{e}nd{lem}
\noindent {\it Proof} . In the case $N=1$ the equality \operatorname{e}qref{pull-back-P-b-eq} gives
$$\mu={\cal D}e_*[{\cal C}]-(p_0\tildemes\operatorname{id})_*[{\cal C}]-(\operatorname{id}\tildemes p_0)_*[{\cal C}]-\psi\cdotot [{\cal C}\tildemes_S{\cal C}],$$
where ${\cal D}e:{\cal C}\to{\cal C}\tildemes_S{\cal C}$ is the diagonal. Now the required identity is easily proved by induction in $n$ using the equalities
$${\cal D}e^*\left({\cal D}e_*[{\cal C}]-(p_0\tildemes\operatorname{id})_*[{\cal C}]-(\operatorname{id}\tildemes p_0)_*[{\cal C}]\right)=-(K+2[p_0(S)]),$$
$$\mu\cdotot (p_0\tildemes\operatorname{id})_*[{\cal C}]=\mu\cdotot (\operatorname{id}\tildemes p_0)_*[{\cal C}]=0.$$
\operatorname{e}d
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Also, if $(f:{\cal X}_N\to{\cal Y}_N)_{N\in{\Bbb Z}}$ is a morphism of two families as above then
it follows immediately from the definition that the above operators commute with the
push-forward map $f_*:{\Bbb C}H_*({\cal X}_{\betaullet})\to{\Bbb C}H_*({\cal Y}_{\betaullet})$, i.e.,
$$T_k(m,a)\circ f_*=f_*\circ T_k(m,a).$$
\betaegin{thm}\lambdabel{relations-thm}
One has the following relations between operators on ${\Bbb C}H_*({\cal X}_{\betaullet})$:
\betaegin{align*}
&\sum_{i\ge 0}\psi^i\cdotot\left({k\choose i}m^{\operatorname{pr}ime i}
T_{k-i}(m,a)T_{k'}(m',a')-{k'\choose i}m^i T_{k'-i}(m',a')T_k(m,a)\right)=\\
&\sum_{i\ge 1}(-1)^{i-1}
\left({k\choose i}m^{\operatorname{pr}ime i}-{k'\choose i}m^i\right)T_{k+k'-i}(m+m',a\cdotot a'\cdotot
(K+2[p_0(S)])^{i-1})+\\
&\psi^{k'-1}m^{k'}p_0^*(a')T_k(m,a)-\psi^{k-1}m^{\operatorname{pr}ime k}p_0^*(a)T_{k'}(m',a')+\\
&\delta_{k,0}\cdotot\sum_{i\ge 1}{k'\choose i}m^i\psi^{i-1}p_0^*(a)T_{k'-i}(m',a')
-\delta_{k',0}\cdotot\sum_{i\ge 1}{k\choose i}m^{\operatorname{pr}ime i}\psi^{i-1}p_0^*(a') T_{k-i}(m,a)
\operatorname{e}nd{align*}
where $a,a'\in{\Bbb C}H^*({\cal C})$, $k\ge 0$, $k'\ge 0$, $m\ge 0$, $m'\ge 0$.
\operatorname{e}nd{thm}
Let us set $\operatorname{e}ll_N=(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_C)\in{\Bbb C}H^1({\cal C}\tildemes_S {\cal X}_N)$.
Also let us denote $\mu=\operatorname{e}ll_1=(\operatorname{id}\tildemes\sigma_1)^*c_1({\cal P}_C)\in{\Bbb C}H^1({\cal C}\tildemes_S{\cal C})$.
Recall that ${\cal P}_C$ is the pull-back of the biextension ${\cal L}$ of ${\cal J}\tildemes_S{\cal J}$
under the embedding $(\iota\tildemes\operatorname{id}):{\cal C}\tildemes_S {\cal J}\to {\cal J}\tildemes_S {\cal J}$
corresponding to point $p_0$. This implies the following isomorphism in
$CH^1({\cal C}\tildemes_S{\cal C}\tildemes_S{\cal X}_{N-m})$:
\betaegin{equation}\lambdabel{biext-eq}
(\operatorname{id}_{{\cal C}}\tildemes s_{m,N})^*\operatorname{e}ll_N=m\cdotot p_{12}^*\mu+p_{13}^*\operatorname{e}ll_{N-m}.
\operatorname{e}nd{equation}
\betaegin{lem}\lambdabel{diag-lem2}
One has the following identity in ${\Bbb C}H^*({\cal C}\tildemes_S{\cal C})$ for $n\ge 1$:
$$\mu^n=(-\psi)^n+(-1)^n\psi^{n-1}\cdotot
\left((p_0\tildemes\operatorname{id})_*[{\cal C}]+(\operatorname{id}\tildemes p_0)_*[{\cal C}]\right)+(-1)^{n-1}
\sum_{i\ge 1}{n\choose i}\psi^{n-i}\cdotot{\cal D}e_*(K+2[p_0(S)])^{i-1}.$$
\operatorname{e}nd{lem}
\noindent {\it Proof} . In the case $N=1$ the equality \operatorname{e}qref{pull-back-P-b-eq} gives
$$\mu={\cal D}e_*[{\cal C}]-(p_0\tildemes\operatorname{id})_*[{\cal C}]-(\operatorname{id}\tildemes p_0)_*[{\cal C}]-\psi\cdotot [{\cal C}\tildemes_S{\cal C}],$$
where ${\cal D}e:{\cal C}\to{\cal C}\tildemes_S{\cal C}$ is the diagonal. Now the required identity is easily proved by induction in $n$ using the equalities
$${\cal D}e^*\left({\cal D}e_*[{\cal C}]-(p_0\tildemes\operatorname{id})_*[{\cal C}]-(\operatorname{id}\tildemes p_0)_*[{\cal C}]\right)=-(K+2[p_0(S)]),$$
$$\mu\cdotot (p_0\tildemes\operatorname{id})_*[{\cal C}]=\mu\cdotot (\operatorname{id}\tildemes p_0)_*[{\cal C}]=0.$$
\operatorname{e}d
\noindent
{\it Proof of Theorem \ref{relations-thm}.}
The composition $T_k(m,a)\circ T_{k'}(m',a')$ acting on ${\Bbb C}H_*({\cal X}_N)$ is given by
the operator
$$x\mapsto s_{m,m',*}\left((\operatorname{id}_{{\cal C}}\tildemes s_{m',m'+N})^*\operatorname{e}ll_{m'+N}^{k}
p_{23}^*\operatorname{e}ll_N^{k'}\cdotot p_1^*(a)\cdotot p_2^*(a')\cdotot p_3^*(x)\right),$$
where $p_i$ ($i=1,2,3$) are the projections from the product ${\cal C}\tildemes_S{\cal C}\tildemes_S{\cal X}_N$ to its factors, and $s_{m,m'}$ denotes the following composition
$$
\betaegin{diagram}
{\cal C}\tildemes_S{\cal C}\tildemes_S{\cal X}_N &\rTo{\operatorname{id}_{{\cal C}}\tildemes s_{m',m'+N}}&{\cal C}\tildemes_S{\cal X}_{m'+N}
&\rTo{s_{m,m+m'+N}}&{\cal X}_{m+m'+N}.
\operatorname{e}nd{diagram}
$$
From \operatorname{e}qref{biext-eq} we get
\betaegin{equation}\lambdabel{biext-pow-eq}
(\operatorname{id}_{{\cal C}}\tildemes s_{m',m'+N})^*\operatorname{e}ll_{m'+N}^k=\sum_i {k\choose i}m^{\operatorname{pr}ime i} p_{12}^*\mu^i
\cdotot p_{13}^*\operatorname{e}ll_N^{k-i}.
\operatorname{e}nd{equation}
Let us set
$$S_{k,k';m,m'}(a,a')(x)=s_{m,m',*}\left(p_{12}^*\operatorname{e}ll_N^k\cdotot p_{23}^*\operatorname{e}ll_N^{k'}\cdotot p_1^*(a)\cdotot p_2^*(a')\cdotot p_3^*(x)\right).$$
Then using Lemma \ref{diag-lem2} and \operatorname{e}qref{biext-pow-eq} we derive
\betaegin{align*}
&T_k(m,a)\circ T_{k'}(m',a')=
\sum_{n\ge 0}(-m'\psi)^n{k\choose n}S_{k-n,k';m,m'}(a,a')+\\
&\sum_{n\ge 1,i\ge 1}(-1)^{n-1}m^{\operatorname{pr}ime n}\psi^{n-i}{k\choose n}{n\choose i}
T_{k+k'-n}(m+m',aa'(K+2[p_0(S)])^{i-1})+\\
&(-1)^km^{\operatorname{pr}ime k}\psi^{k-1}p_0^*(a)\cdotot T_{k'}(m',a')+
\delta_{k',0}\sum_{n\ge 1}(-1)^nm^{\operatorname{pr}ime n}\psi^{n-1}{k\choose n}p_0^*(a')\cdotot T_{k-n}(m,a).
\operatorname{e}nd{align*}
From this we deduce that
\betaegin{align*}
&\sum_{p\ge 0}{k\choose p}(m'\psi)^p T_{k-p}(m,a)T_{k'}(m',a')=S_{k,k';m,m'}(a,a')+\\
&\sum_{i\ge 1}(-1)^{i-1}{k\choose i}m^{\operatorname{pr}ime i}T_{k+k'-i}(m+m',aa'(K+2[p_0(S)])^{i-1})
-m^{\operatorname{pr}ime k}\psi^{k-1}p_0^*(a)T_{k'}(m',a')\\
&-\delta_{k',0}\cdotot\sum_{i\ge 1}{k\choose i}m^{\operatorname{pr}ime i}\psi^{i-1}p_0^*(a')T_{k-i}(m,a).
\operatorname{e}nd{align*}
It remains to observe that condition (ii) imposed on $({\cal X}_N)$ implies that
$$S_{k,k';m,m'}(a,a')=S_{k',k;m',m}(a',a).$$
Expressing both sides of this equality in terms of the operators $(T_k(m,a))$ we get the required
relation.
\operatorname{e}d
\betaegin{rems} 1. In the case of ${\cal X}_N={\cal J}$ the definition of the operators $T_k(m,a)$
can be extended to the case of arbitrary $m\in{\Bbb Z}$, so that the relations of Theorem \ref{relations-thm}
still hold. Namely, we define the morphism $s_{m,N}$ for $m\in{\Bbb Z}$, so that it maps $(p,\xi)\in{\cal C}_s\tildemes{\cal J}_s$ to $m\iota(p)+\xi\in{\cal J}_s$.
\noindent
2. Similar relation holds if we replace Chow groups with cohomology
provided we insert the standard sign $(-1)^{\deltag(a)\deltag(a')}$ whenever $a'$ goes before $a$.
\operatorname{e}nd{rems}
We are mostly interested in two cases: $({\cal X}_N={\cal C}^{[N]})$ and $({\cal X}_N={\cal J})$.
First, let us consider the case $({\cal X}_N={\cal C}^{[N]})$. In this case we will deduce the relation
between the operators $(T_k(m,a))$ and $(P_{i,j}(a))$. We need one auxiliary result for this.
Let us denote by $Z_{m,N}\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ the image of the closed embedding $t_{m,N}$.
In other words, $Z_{m,N}$
consists of $(p,D)\in{\cal C}_s\tildemes{\cal C}_s^{[N]}$ such that $D-mp\ge 0$. Note that
${\cal D}_N:=Z_{1,N}$ is the universal divisor in ${\cal C}\tildemes_S {\cal C}^{[N]}$.
\betaegin{lem}\lambdabel{Z-lem}
One has the following equality in ${\Bbb C}H^m({\cal C}\tildemes_S {\cal C}^{[N]})$:
$$[Z_{m,N}]=[{\cal D}_N]\cdotot([{\cal D}_N]+p_1^*K)\cdotot\ldots\cdotot([{\cal D}_N]+(m-1)p_1^*K).$$
where $p_1:{\cal C}\tildemes_S {\cal C}^{[N]}\to {\cal C}$ is the projection.
Equivalently,
$$[{\cal D}_N]^m=\sum_{i=0}^m (-1)^{m-i}S(m,i)\cdotot p_1^*K^{m-i}\cdotot [Z_{i,N}],$$
where $S(m,i)=\frac{1}{i!}\sum_{j=0}^i (-1)^j{i\choose j}(i-j)^m$ are the Stirling numbers of
the second kind.
\operatorname{e}nd{lem}
\noindent {\it Proof} . In the case $m=1$ the equality is clear. The general case follows by induction in $m$ using the
identity
\betaegin{equation}\lambdabel{D-res-eq}
[{\cal D}_N]=t_{1,N+1}^*({\cal D}_{N+1}+p_1^*K)
\operatorname{e}nd{equation}
in ${\Bbb C}H^1({\cal C}\tildemes_S {\cal C}^{[N]})$. To obtain this identity one can start with the equality
\operatorname{e}qref{div-pullback-eq} (for $M=1$) in ${\Bbb C}H^1({\cal C}\tildemes_S{\cal C}\tildemes_S{\cal C}^{[N]})$ and then
apply the pull-back with respect to the diagonal embedding
${\cal D}e\tildemes\operatorname{id}:{\cal C}\tildemes_S{\cal C}^{[N]}\to{\cal C}\tildemes_S{\cal C}\tildemes_S{\cal C}^{[N]}$
(when the base is a point this was observed in Proposition 19.1 of \cite{P-av}).
\operatorname{e}d
\betaegin{prop}\lambdabel{T-P-prop}
One has the following equality of operators on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$:
\betaegin{align*}
&T_k(m,a)=(-1)^kP_{m,0}(a\cdotot [p_0])P_{1,1}([{\cal C}])^k\psi^{k-1}+\\
&\sum_{i+n+j=k}(-1)^{n+j}{k\choose j}S(i+n,i)
P_{i+m,i}(a\cdotot K^n)P_{1,1}([p_0]+\psi)^j.
\operatorname{e}nd{align*}
\operatorname{e}nd{prop}
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\betaegin{rems} 1. In the case of ${\cal X}_N={\cal J}$ the definition of the operators $T_k(m,a)$
can be extended to the case of arbitrary $m\in{\Bbb Z}$, so that the relations of Theorem \ref{relations-thm}
still hold. Namely, we define the morphism $s_{m,N}$ for $m\in{\Bbb Z}$, so that it maps $(p,\xi)\in{\cal C}_s\tildemes{\cal J}_s$ to $m\iota(p)+\xi\in{\cal J}_s$.
\noindent
2. Similar relation holds if we replace Chow groups with cohomology
provided we insert the standard sign $(-1)^{\deltag(a)\deltag(a')}$ whenever $a'$ goes before $a$.
\operatorname{e}nd{rems}
We are mostly interested in two cases: $({\cal X}_N={\cal C}^{[N]})$ and $({\cal X}_N={\cal J})$.
First, let us consider the case $({\cal X}_N={\cal C}^{[N]})$. In this case we will deduce the relation
between the operators $(T_k(m,a))$ and $(P_{i,j}(a))$. We need one auxiliary result for this.
Let us denote by $Z_{m,N}\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ the image of the closed embedding $t_{m,N}$.
In other words, $Z_{m,N}$
consists of $(p,D)\in{\cal C}_s\tildemes{\cal C}_s^{[N]}$ such that $D-mp\ge 0$. Note that
${\cal D}_N:=Z_{1,N}$ is the universal divisor in ${\cal C}\tildemes_S {\cal C}^{[N]}$.
\betaegin{lem}\lambdabel{Z-lem}
One has the following equality in ${\Bbb C}H^m({\cal C}\tildemes_S {\cal C}^{[N]})$:
$$[Z_{m,N}]=[{\cal D}_N]\cdotot([{\cal D}_N]+p_1^*K)\cdotot\ldots\cdotot([{\cal D}_N]+(m-1)p_1^*K).$$
where $p_1:{\cal C}\tildemes_S {\cal C}^{[N]}\to {\cal C}$ is the projection.
Equivalently,
$$[{\cal D}_N]^m=\sum_{i=0}^m (-1)^{m-i}S(m,i)\cdotot p_1^*K^{m-i}\cdotot [Z_{i,N}],$$
where $S(m,i)=\frac{1}{i!}\sum_{j=0}^i (-1)^j{i\choose j}(i-j)^m$ are the Stirling numbers of
the second kind.
\operatorname{e}nd{lem}
\noindent {\it Proof} . In the case $m=1$ the equality is clear. The general case follows by induction in $m$ using the
identity
\betaegin{equation}\lambdabel{D-res-eq}
[{\cal D}_N]=t_{1,N+1}^*({\cal D}_{N+1}+p_1^*K)
\operatorname{e}nd{equation}
in ${\Bbb C}H^1({\cal C}\tildemes_S {\cal C}^{[N]})$. To obtain this identity one can start with the equality
\operatorname{e}qref{div-pullback-eq} (for $M=1$) in ${\Bbb C}H^1({\cal C}\tildemes_S{\cal C}\tildemes_S{\cal C}^{[N]})$ and then
apply the pull-back with respect to the diagonal embedding
${\cal D}e\tildemes\operatorname{id}:{\cal C}\tildemes_S{\cal C}^{[N]}\to{\cal C}\tildemes_S{\cal C}\tildemes_S{\cal C}^{[N]}$
(when the base is a point this was observed in Proposition 19.1 of \cite{P-av}).
\operatorname{e}d
\betaegin{prop}\lambdabel{T-P-prop}
One has the following equality of operators on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$:
\betaegin{align*}
&T_k(m,a)=(-1)^kP_{m,0}(a\cdotot [p_0])P_{1,1}([{\cal C}])^k\psi^{k-1}+\\
&\sum_{i+n+j=k}(-1)^{n+j}{k\choose j}S(i+n,i)
P_{i+m,i}(a\cdotot K^n)P_{1,1}([p_0]+\psi)^j.
\operatorname{e}nd{align*}
\operatorname{e}nd{prop}
\noindent {\it Proof} . Since $[Z_{k,N}]=t_{k,N,*}([{\cal C}\tildemes_S {\cal C}^{[N-k]}])$,
we can rewrite $P_{k+m,k}(a)$ in the form similar to
that of $T_{k}(m,a)$:
\betaegin{equation}\lambdabel{P-k-m-Z-eq}
P_{k+m,k}(a)(x)=s_{m,N,*}([Z_{k,N}]\cdotot p_1^*a\cdotot p_2^*x),
\operatorname{e}nd{equation}
where $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
Let us use the following shorthand notation for divisors on ${\cal C}\tildemes_S{\cal C}^{[N]}$:
${\cal D}={\cal D}_N$, ${\cal R}=p_2^*{\cal R}_N$, $K=p_1^*K$,
$[p_0]=p_1^*[p_0]$. Then we can write \operatorname{e}qref{pull-back-P-b-eq}
as
$$(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})={\cal D}-{\cal R}-N[p_0]-N\psi.$$
Note that we have the following relations in ${\Bbb C}H^2({\cal C}\tildemes_S{\cal C}^{[N]})$:
$$({\cal D}-{\cal R})\cdotot[p_0]=0, \ \ [p_0]^2=-\psi\cdotot[p_0].$$
It follows that for $j\ge 1$ one has $({\cal D}-{\cal R}-N\psi)^i\cdotot [p_0]^j=N^i(-\psi)^{i+j-1}\cdotot [p_0]$,
so we derive
$$(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})^k=({\cal D}-{\cal R}-N([p_0]+\psi))^k=({\cal D}-{\cal R}-N\psi)^k+(-N)^k[p_0]\psi^{k-1}.
$$
Therefore, using Lemma \ref{Z-lem} we find
$$(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})^k=(-N)^k[p_0]\psi^{k-1}+
\sum_{i+n+j=k}(-1)^{n+j}{k\choose j}S(i+n,i)[Z_{i,N}]\cdotot K^n\cdotot ({\cal R}+N\psi)^j.$$
Taking into the account \operatorname{e}qref{P-k-m-Z-eq}, we get
\betaegin{align*}
&T_k(m,a)(x)=(-N)^kP_{m,0}(a\cdotot[p_0])(x)\cdotot\psi^{k-1}+\\
&\sum_{i+n+j=k}(-1)^{n+j}{k\choose j}S(i+n,i)
P_{i+m,i}(a\cdotot K^n)(({\cal R}+N\psi)^j\cdotot x),
\operatorname{e}nd{align*}
where $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
It remains to use the equalities
$$P_{1,1}([{\cal C}])(x)=Nx,$$
$$P_{1,1}([p_0])(x)={\cal R}\cdotot x$$
for $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
\operatorname{e}d
Now let us specialize to the case ${\cal X}_N={\cal J}$.
In this case we can relate the operators $T_k(m,a)$
to the operators considered in \cite{P-lie} (in the case $S=\operatorname{Spec}(k)$).
Following \cite{P-lie} let us define the operators $A_k(\alpha)$ on ${\Bbb C}H^*({\cal J})$ for $\alpha\in{\Bbb C}H^*({\cal J})$
and $k\ge 0$ by
$$A_k(\alpha)(x)=(p_1+p_2)_*(c_1({\cal L})^k\cdotot p_1^*\alpha\cdotot p_2^*x),$$
where $p_1$ and $p_2$ are projections from the product ${\cal J}\tildemes_S {\cal J}$ to its factors.
\betaegin{lem}\lambdabel{A-T-lem1}
For $a\in{\Bbb C}H^*({\cal C})$ one has
$A_k([m]_*\iota_*a)=m^k T_k(m,a)$ (with the convention that $0^0=1$),
where $[m]:{\cal J}\to {\cal J}:\xi\to m\xi$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . We have
\betaegin{align*}
&A_k([m]_*\iota_*a)=(mp_1+p_2)_*\left(([m]\tildemes\operatorname{id})^*c_1({\cal L})^k\cdotot p_1^*(\iota_*a)\cdotot p_2^*x\right)=\\
&m^k(mp_1+p_2)_*\left(c_1({\cal L})^k\cdotot (\iota\tildemes\operatorname{id}_{{\cal J}})_*(p_1^*a)\cdotot p_2^*x\right).
\operatorname{e}nd{align*}
Now the result follows immediately from the isomorphism ${\cal L}|_{{\cal C}\tildemes_S {\cal J}}\sigmameq{\cal P}_{{\cal C}}$.
\operatorname{e}d
Working with rational coefficients we can
consider the decomposition
$${\Bbb C}H^*({\cal J})_{{\Bbb Q}}=\betaigoplus_{i=0}^{2g}{\Bbb C}H^*({\cal J})_i,$$
where $[m]_*x=m^ix$ for $x\in{\Bbb C}H^*({\cal J})_i$ (see \cite{DM} Thm. 3.1).
It follows that for fixed $k$ and $\alpha\in{\Bbb C}H^*({\cal J})$ the operator valued function
$m\to A_k([m]_*\alpha)$ is a polynomial of degree $\le 2g$. By Lemma \ref{A-T-lem1} the same is true for
$T_k(m,a)$, where $a\in{\Bbb C}H^*({\cal C})$ (more precisely, it is a polynomial in $m$ of degree $\le 2g-k$). Therefore, we can write
$$T_k(m,a)=\sum_{n=0}^{2g-k} \frac{m^n}{n!} \widetilde{X}_{n,k}(a)$$
for some operators $\widetilde{X}_{n,k}(a)$ on ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$.
Then the relations of Theorem \ref{relations-thm}
are equivalent to the following relations for $(\widetilde{X}_{n,k}(a))$:
\betaegin{equation}\lambdabel{X-rel-eq}
\betaegin{array}{l}
\sum_{i\ge 0}\psi^i\cdotot i!\left({k\choose i}{n'\choose i} \widetilde{X}_{n,k-i}(a)\widetilde{X}_{n'-i,k'}(a')-
{k'\choose i}{n\choose i} \widetilde{X}_{n',k'-i}(a')\widetilde{X}_{n-i,k}(a)\right)=\\
\sum_{i\ge 1}(-1)^{i-1}i!\left({k\choose i}{n'\choose i}-
{k'\choose i}{n\choose i}\right) \widetilde{X}_{n+n'-i,k+k'-i}(aa'(K+2[p_0(S)])^{i-1})+\\
\delta_{n',0}p_0^*(a')\psi^{k'-1}k'!{n\choose k'}\widetilde{X}_{n-k',k}(a)-
\delta_{n,0}p_0^*(a)\psi^{k-1}k!{n'\choose k}\widetilde{X}_{n'-k,k'}(a')+\\
\delta_{k,0}p_0^*(a)\psi^{n-1}n!{k'\choose n}\widetilde{X}_{n',k'-n}(a')-
\delta_{k',0}p_0^*(a')\psi^{n'-1}n'!{k\choose n'}\widetilde{X}_{n,k-n'}(a).
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
We will denote $\widetilde{X}_{n,k}([{\cal C}])$ simply by $\widetilde{X}_{n,k}({\cal C})$.
In the case $S=\operatorname{Spec}(k)$ the above relations are essentially equivalent to those of
Theorem 2.6 of \cite{P-lie}. Recall that in \cite{P-lie} we showed that
the operators $\widetilde{X}_{n,k}(C)-nk\widetilde{X}_{n-1,k-1}(K/2+[p_0])$
satisfy the commutation relations of the Lie
algebra ${\cal HV}'$ and calculated their Fourier transform. We are going to
present a similar computation in the relative case (i.e., when $S$ is arbitrary).
We refer to \cite{DM} for the basic properties of the Fourier transform on cycles over
abelian schemes (originally introduced in \cite{M} and studied in \cite{B1} and \cite{B2}).
\betaegin{thm}\lambdabel{four-thm}
(i) The operators
\betaegin{equation}
e=\frac{1}{2}\widetilde{X}_{0,2}({\cal C}),\ \
f=-\frac{1}{2}\widetilde{X}_{2,0}({\cal C}),\ \
h=-\widetilde{X}_{1,1}({\cal C})+g\cdotot\operatorname{id}
\operatorname{e}nd{equation}
on ${\Bbb C}H({\cal J})_{{\Bbb Q}}$ define an action of the Lie algebra $\operatorname{sl}_2$.
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\betaegin{lem}\lambdabel{A-T-lem1}
For $a\in{\Bbb C}H^*({\cal C})$ one has
$A_k([m]_*\iota_*a)=m^k T_k(m,a)$ (with the convention that $0^0=1$),
where $[m]:{\cal J}\to {\cal J}:\xi\to m\xi$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . We have
\betaegin{align*}
&A_k([m]_*\iota_*a)=(mp_1+p_2)_*\left(([m]\tildemes\operatorname{id})^*c_1({\cal L})^k\cdotot p_1^*(\iota_*a)\cdotot p_2^*x\right)=\\
&m^k(mp_1+p_2)_*\left(c_1({\cal L})^k\cdotot (\iota\tildemes\operatorname{id}_{{\cal J}})_*(p_1^*a)\cdotot p_2^*x\right).
\operatorname{e}nd{align*}
Now the result follows immediately from the isomorphism ${\cal L}|_{{\cal C}\tildemes_S {\cal J}}\sigmameq{\cal P}_{{\cal C}}$.
\operatorname{e}d
Working with rational coefficients we can
consider the decomposition
$${\Bbb C}H^*({\cal J})_{{\Bbb Q}}=\betaigoplus_{i=0}^{2g}{\Bbb C}H^*({\cal J})_i,$$
where $[m]_*x=m^ix$ for $x\in{\Bbb C}H^*({\cal J})_i$ (see \cite{DM} Thm. 3.1).
It follows that for fixed $k$ and $\alpha\in{\Bbb C}H^*({\cal J})$ the operator valued function
$m\to A_k([m]_*\alpha)$ is a polynomial of degree $\le 2g$. By Lemma \ref{A-T-lem1} the same is true for
$T_k(m,a)$, where $a\in{\Bbb C}H^*({\cal C})$ (more precisely, it is a polynomial in $m$ of degree $\le 2g-k$). Therefore, we can write
$$T_k(m,a)=\sum_{n=0}^{2g-k} \frac{m^n}{n!} \widetilde{X}_{n,k}(a)$$
for some operators $\widetilde{X}_{n,k}(a)$ on ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$.
Then the relations of Theorem \ref{relations-thm}
are equivalent to the following relations for $(\widetilde{X}_{n,k}(a))$:
\betaegin{equation}\lambdabel{X-rel-eq}
\betaegin{array}{l}
\sum_{i\ge 0}\psi^i\cdotot i!\left({k\choose i}{n'\choose i} \widetilde{X}_{n,k-i}(a)\widetilde{X}_{n'-i,k'}(a')-
{k'\choose i}{n\choose i} \widetilde{X}_{n',k'-i}(a')\widetilde{X}_{n-i,k}(a)\right)=\\
\sum_{i\ge 1}(-1)^{i-1}i!\left({k\choose i}{n'\choose i}-
{k'\choose i}{n\choose i}\right) \widetilde{X}_{n+n'-i,k+k'-i}(aa'(K+2[p_0(S)])^{i-1})+\\
\delta_{n',0}p_0^*(a')\psi^{k'-1}k'!{n\choose k'}\widetilde{X}_{n-k',k}(a)-
\delta_{n,0}p_0^*(a)\psi^{k-1}k!{n'\choose k}\widetilde{X}_{n'-k,k'}(a')+\\
\delta_{k,0}p_0^*(a)\psi^{n-1}n!{k'\choose n}\widetilde{X}_{n',k'-n}(a')-
\delta_{k',0}p_0^*(a')\psi^{n'-1}n'!{k\choose n'}\widetilde{X}_{n,k-n'}(a).
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
We will denote $\widetilde{X}_{n,k}([{\cal C}])$ simply by $\widetilde{X}_{n,k}({\cal C})$.
In the case $S=\operatorname{Spec}(k)$ the above relations are essentially equivalent to those of
Theorem 2.6 of \cite{P-lie}. Recall that in \cite{P-lie} we showed that
the operators $\widetilde{X}_{n,k}(C)-nk\widetilde{X}_{n-1,k-1}(K/2+[p_0])$
satisfy the commutation relations of the Lie
algebra ${\cal HV}'$ and calculated their Fourier transform. We are going to
present a similar computation in the relative case (i.e., when $S$ is arbitrary).
We refer to \cite{DM} for the basic properties of the Fourier transform on cycles over
abelian schemes (originally introduced in \cite{M} and studied in \cite{B1} and \cite{B2}).
\betaegin{thm}\lambdabel{four-thm}
(i) The operators
\betaegin{equation}
e=\frac{1}{2}\widetilde{X}_{0,2}({\cal C}),\ \
f=-\frac{1}{2}\widetilde{X}_{2,0}({\cal C}),\ \
h=-\widetilde{X}_{1,1}({\cal C})+g\cdotot\operatorname{id}
\operatorname{e}nd{equation}
on ${\Bbb C}H({\cal J})_{{\Bbb Q}}$ define an action of the Lie algebra $\operatorname{sl}_2$.
\noindent (ii) Let us set for $a\in{\Bbb C}H^*({\cal C})$, $n\ge 0$, $k\ge 0$,
$$X_{n,k}(a)=\sum_{i\ge 0}(-1)^i i!{n\choose i}{k\choose i}\widetilde{X}_{n-i,k-i}(a\operatorname{e}ta^i),$$
where $\operatorname{e}ta:=K/2+[p_0(S)]+\psi/2\in{\Bbb C}H^1({\cal C})_{{\Bbb Q}}$.
Consider the Fourier transform defined by
$$F:{\Bbb C}H^*({\cal J})_{{\Bbb Q}}\to{\Bbb C}H^*({\cal J})_{{\Bbb Q}}:x\mapsto p_{2*}(\operatorname{e}xp(c_1({\cal L}))\cdotot p_1^*x).$$
Then one has
$$FX_{n,k}(a)F^{-1}=(-1)^kX_{k,n}(a).$$
$$[e,X_{n,k}(a)]=nX_{n-1,k+1}, \ \ [f,X_{n,k}(a)]=kX_{n+1,k-1}(a), \ \ [h,X_{n,k}(a)]=(k-n)X_{n,k}(a).$$
\operatorname{e}nd{thm}
\betaegin{rem} The $\operatorname{sl}_2$-action of Theorem \ref{four-thm} differs only by a sign from
the relative Lefschetz action associated with the relatively ample class $-\tau_2({\cal C})/2$ on ${\cal J}$
(see \cite{K}).
\operatorname{e}nd{rem}
We start with some preliminary statements.
\betaegin{lem}\lambdabel{four-lem}
One has $F\widetilde{X}_{n,0}(a)F^{-1}=\widetilde{X}_{0,n}(a)$ for every $a\in{\Bbb C}H^*({\cal C})$, $n\ge 0$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . By definition, $\widetilde{X}_{n,0}(a)/n!$ is the Pontryagin product with $a_n\in{\Bbb C}H^*({\cal J})_{{\Bbb Q}}$
where $[m]_*\iota_*a=\sum_{i\ge 0}m^ia_i$ for all $m\in{\Bbb Z}$. On the other hand,
$\widetilde{X}_{0,n}(a)=T_n(0,a)$
is the usual product with $\tau_n(a)$ (see \operatorname{e}qref{T-k-0-eq} and \operatorname{e}qref{tau-eq}).
Since $F(x*y)=F(x)\cdotot F(y)$, it remains to show that
\betaegin{equation}\lambdabel{F-a-eq}
F(a_n)=\frac{1}{n!}\tau_n(a)=p_{2*}(\frac{c_1({\cal L})^n}{n!}\cdotot p_1^*\iota_*a),
\operatorname{e}nd{equation}
where $p_1$ and $p_2$ are the projections of the product ${\cal J}\tildemes_S{\cal J}$ on its factors.
This fact is well known but we will give the proof since it is very short. We have
$$\sum_{i\ge 0}m^i F(a_i)=F([m]_*\iota_*a)=[m]^*F(\iota_*a)=[m]^*p_{2*}(\operatorname{e}xp(c_1({\cal L}))\cdotot
p_1^*\iota_*a)=p_{2*}((\operatorname{id}\tildemes[m])^*\operatorname{e}xp(c_1({\cal L}))\cdotot p_1^*\iota_*a).$$
Using the identity $(\operatorname{id}\tildemes [m])^*c_1({\cal L})=m c_1({\cal L})$, we see that this is equal
to $\sum_{i\ge 0}m^i\tau_i(a)/i!$. Now \operatorname{e}qref{F-a-eq} is obtained by equating the coefficients with $m^n$.
\operatorname{e}d
\betaegin{lem}\lambdabel{sl2-lem}
(i) One has $\widetilde{X}_{0,i}({\cal C})=\widetilde{X}_{i,0}({\cal C})=0$ for $i\le 1$.
\noindent (ii) One has
$$[e,\widetilde{X}_{n,k}(a)]=n\widetilde{X}_{n-1,k+1}(a)-n(n-1)\widetilde{X}_{n-2,k}(a\cdotot\operatorname{e}ta),$$
$$[f,\widetilde{X}_{n,k}(a)]=k\widetilde{X}_{n+1,k-1}(a)-k(k-1)\widetilde{X}_{n,k-2}(a\cdotot\operatorname{e}ta).$$
\operatorname{e}nd{lem}
\noindent {\it Proof} . (i) The operator $\widetilde{X}_{0,0}({\cal C})=T_0(0,{\cal C})$ is the product with the pull-back of
$\pi_*[{\cal C}]=0$, where $\pi:{\cal C}\to S$ is the projection. On the other hand, the operator
$\widetilde{X}_{0,1}({\cal C})=T_1(0,{\cal C})$ is the product with $\tau_1({\cal C})=p_{2*}(c_1({\cal P}_{{\cal C}}))$,
where $p_2:{\cal C}\tildemes_S{\cal J}\to{\cal J}$ is the projection. Since $c_1({\cal P}_{{\cal C}})$ is the divisor
in ${\cal C}\tildemes_S{\cal J}$ that has degree zero on every fiber of $p_2$, we get $\tau_1({\cal C})=0$.
Now the vanishing of $\widetilde{X}_{1,0}({\cal C})$ follows from Lemma \ref{four-lem}.
\noindent
(ii) This is an immediate consequence of relations \operatorname{e}qref{X-rel-eq} and of part (i).
\operatorname{e}d
\noindent
{\it Proof of Theorem \ref{four-thm}}.
(i) These relations follow from Lemma \ref{sl2-lem} and from the observation that
$\widetilde{X}_{0,0}(\operatorname{e}ta)=g\cdotot\operatorname{id}$ (since it is the product with the pull-back of $\pi_*(\operatorname{e}ta)=g\cdotot [S]$).
\noindent
(ii) From Lemma \ref{four-lem} we get
\betaegin{equation}\lambdabel{F-f-e-eq}
F f F^{-1}=-e.
\operatorname{e}nd{equation}
Recall that $-f$ is the Pontryagin product with
the class $a_2$ on ${\cal J}$ defined by $[m]_*[{\cal C}]=\sum_i m^i a_i$. Replacing $m$ by $-m$ we see
that $[-1]_*a_2=a_2$. Hence, $f$ commutes with $[-1]^*$. Now the identity
$F^2=(-1)^g[-1]_*$ (see \cite{DM}, Cor.2.22) implies that $F^2$ commutes with $f$. Therefore, from
\operatorname{e}qref{F-f-e-eq} we get
$$FeF^{-1}=-f.$$
On the other hand, from
Lemma \ref{sl2-lem}(ii) we deduce by induction that
\betaegin{equation}\lambdabel{ad-eq}
\betaegin{array}{l}
\operatorname{ad}(f)^k \frac{\widetilde{X}_{0,n+k}(a)}{(n+k)!}=\frac{X_{k,n}(a)}{n!},\\
\operatorname{ad}(e)^k \frac{\widetilde{X}_{n+k,0}(a)}{(n+k)!}=\frac{X_{n,k}(a)}{n!}.
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
Now, combining all of this with Lemma \ref{four-lem} we get
$$F\frac{X_{n,k}(a)}{n!}F^{-1}=\operatorname{ad}(-f)^k(F\frac{\widetilde{X}_{n+k,0}(a)}{(n+k)!}F^{-1})=
(-1)^k\operatorname{ad}(f)^k\frac{\widetilde{X}_{0,n+k}(a)}{(n+k)!}=(-1)^k\frac{X_{k,n}(a)}{n!}$$
as required. The formulas for $[e,X_{n,k}(a)]$ and $[f,X_{n,k}(a)]$ follow immediately
from \operatorname{e}qref{ad-eq}.
\operatorname{e}d
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\section{Divided powers}
\lambdabel{div-sec}
In this section we construct and study divided powers of operators
$P_{n,0}({\cal C})$ and $P_{0,n}({\cal C})$. Then we use them to define an action of a certain
${\Bbb Z}$-form of an algebra of differential operators in two variables on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$.
We will also construct a ${\Bbb Z}$-version of the Lefschetz $\operatorname{sl}_2$-action on ${\Bbb C}H^*(C^{[N]})$
(see Theorem \ref{Lefschetz-thm}).
Let us start with divided powers of $P_{n,0}({\cal C})$. By definition, we have
$$P_{n,0}({\cal C})(x)=\delta_n * x,$$
where $\delta_n={\cal D}e_{n*}([{\cal C}])\subset{\Bbb C}H_1({\cal C}^{[n]})$ is the class of the diagonal.
Thus, we can set
\betaegin{equation}\lambdabel{div-eq1}
P_{n,0}({\cal C})^{[d]}(x)=\delta_n^{[d]}*x,
\operatorname{e}nd{equation}
where
$$\delta_n^{[d]}={\cal D}e_{n*}^{[d]}([{\cal C}^{[d]}])\in{\Bbb C}H_d({\cal C}^{[nd]}),$$
$${\cal D}e_n^{[d]}:{\cal C}^{[d]}\to {\cal C}^{[nd]}: D\mapsto nD.$$
Note that $d!\delta_n^{[d]}=\delta_n^{*d}$ --- the $d$th power of $\delta_n$ with the respect to the
Pontryagin product. Hence, $d!P_{n,0}({\cal C})^{[d]}=P_{n,0}({\cal C})^d$.
To describe the divided powers of $P_{0,n}({\cal C})$ let us introduce a new binary operation on
${\Bbb C}H^*({\cal C}^{[\betaullet]})$ as follows. For $a\in{\Bbb C}H^*({\cal C}^{[k]})$ and $x\in{\Bbb C}H^*({\cal C}^{[N]})$ set
$$i_a(x)=p_{2*}(p_1^*a\cdotot\alpha_{k,N-k}^*x)\in{\Bbb C}H^*({\cal C}^{[N-k]}),$$
where $p_1,p_2$ are projections of the product ${\cal C}^{[k]}\tildemes_S {\cal C}^{[N-k]}$ to its factors.
Then it is easy to see that
$$P_{0,n}({\cal C})(x)=i_{\delta_n}(x).$$
Also, it is straightforward to check that
$$i_{a*b}=i_a\circ i_b.$$
Thus, it is natural to set
\betaegin{equation}\lambdabel{div-eq2}
P_{0,n}({\cal C})^{[d]}(x)=i_{\delta_n^{[d]}}(x),
\operatorname{e}nd{equation}
so that we have $d!P_{0,n}({\cal C})^{[d]}=P_{0,n}({\cal C})^d$.
Below we use the notation from the Introduction.
Let $A$ be a supercommutative algebra $A$ with a unit and a distinguished
even element ${\betaf a}_0\in A$. We are going to define two extensions of the universal enveloping algebra
of ${\cal D}(A,{\betaf a}_0)$ by adding two families of divided powers.
\betaegin{lem}\lambdabel{div-sum-lem}
(i) Let ${\frak g}$ be a Lie algebra over ${\Bbb Z}$. Then for $x,y\in{\frak g}$, the following relations hold in
$U({\frak g})$:
$$x^{d}y=\sum_{i=0}^d {d\choose i}(\operatorname{ad} x)^{i}(y)x^{d-i},$$
$$yx^{d}=\sum_{i=0}^d {d\choose i}x^{d-i}(-\operatorname{ad} x)^{i}(y) $$
for all $d\ge 1$.
\noindent (ii) The following relations hold in $U({\cal D}(A,{\betaf a}_0))$:
$${\betaf P}_{m,k}(a){\betaf P}_{n,0}(1)^{d}=\sum_{j\le i}(-1)^{i-j}\frac{i!d!}{j!(d-j)!}{k\choose i}A_j(i,n){\betaf P}_{n,0}(1)^{d-j}
{\betaf P}_{m+nj-i,k-i}(a{\betaf a}_0^{i-j}),$$
$${\betaf P}_{0,n}(1)^{d}{\betaf P}_{m,k}(a)=\sum_{j\le i}(-1)^{i-j}\frac{i!d!}{j!(d-j)!}{m\choose i}A_j(i,n)
{\betaf P}_{m-i,k+nj-i}(a{\betaf a}_0^{i-j}){\betaf P}_{0,n}(1)^{d-j},$$
where
$$A_d(i,n)=\sum_{i_1+\ldots+i_d=i,i_s\ge1}{n\choose i_1}\ldots {n\choose i_d}.$$
We also use the convention $x^i=0$ for $i<0$ (so that $j\le d$ in both sums).
\operatorname{e}nd{lem}
\noindent {\it Proof} . (i) This is easily checked by induction in $d$.
\noindent (ii) Using the commutation relations in ${\cal D}(A,{\betaf a}_0)$ one can check by induction in $d$ that
$$\operatorname{ad}({\betaf P}_{0,n}(1))^d({\betaf P}_{m,k}(a))=\sum_{i\ge d}(-1)^{i-d}i!{m\choose i}A_d(i,n)
{\betaf P}_{m-i,k+nd-i}(a\cdotot {\betaf a}_0^{i-d}),$$
$$\operatorname{ad}(-{\betaf P}_{n,0}(1))^d({\betaf P}_{m,k}(a))=\sum_{i\ge d}(-1)^{i-d}i!{k\choose i}A_d(i,n)
{\betaf P}_{m+nd-i,k-i}(a\cdotot {\betaf a}_0^{i-d}).$$
Now the required relations follow from (i).
\operatorname{e}d
\betaegin{defi}
Let us denote by $\widetilde{U}_1(A,{\betaf a}_0)$ (resp., $\widetilde{U}_2(A,{\betaf a}_0)$)
the superalgebra over ${\Bbb Z}$ with generators
$({\betaf P}_{m,k}(a))$, $m\ge 0, k\ge 0$, depending additively on $a\in A$, and $({\betaf P}_{n,0}(1)^{[d]})$
(resp., $({\betaf P}_{0,n}(1)^{[d]})$), $n\ge 1$,
$d\ge 0$, subject to the following relations:
\noindent
(i) the supercommutator relations of ${\cal D}(A,{\betaf a}_0)$ for $({\betaf P}_{m,k}(a))$;
\noindent
(ii) $d!{\betaf P}_{n,0}(1)^{[d]}={\betaf P}_{n,0}(1)^d$ (resp., $d!{\betaf P}_{0,n}(1)^{[d]}={\betaf P}_{0,n}(1)^d$); \\
${\betaf P}_{n,0}(1)^{[d_1]}{\betaf P}_{n,0}(1)^{[d_2]}={d_1+d_2\choose d_1}{\betaf P}_{n,0}(1)^{[d_1+d_2]}$
(resp.,
${\betaf P}_{0,n}(1)^{[d_1]}{\betaf P}_{0,n}(1)^{[d_2]}={d_1+d_2\choose d_1}{\betaf P}_{0,n}(1)^{[d_1+d_2]}$);
\noindent
(iii)
${\betaf P}_{m,k}(a){\betaf P}_{n,0}(1)^{[d]}=\sum_{j\le i}(-1)^{i-j}\frac{i!}{j!}{k\choose i}A_j(i,n){\betaf P}_{n,0}(1)^{[d-j]}
{\betaf P}_{m+nj-i,k-i}(a{\betaf a}_0^{i-j})$,
$$(\text{resp., }
{\betaf P}_{0,n}(1)^{[d]}{\betaf P}_{m,k}(a)=\sum_{j\le i}(-1)^{i-j}\frac{i!}{j!}{m\choose i}A_j(i,n)
{\betaf P}_{m-i,k+nj-i}(a{\betaf a}_0^{i-j}){\betaf P}_{0,n}(1)^{[d-j]}).$$
\operatorname{e}nd{defi}
\betaegin{thm}\lambdabel{divided-powers-thm} The action of ${\cal D}({\Bbb C}H^*({\cal C}),K)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$
extends to the action of $\widetilde{U}_1({\Bbb C}H^*({\cal C}),K)$ (resp., $\widetilde{U}_2({\Bbb C}H^*({\cal C}),K)$), such that
the action of ${\betaf P}_{n,0}({\cal C})^{[d]}$ (resp., ${\betaf P}_{0,n}({\cal C})^{[d]}$)
is given by \operatorname{e}qref{div-eq1} (resp., \operatorname{e}qref{div-eq2}). Furthermore, these relations hold on the level of correspondences. Also, similar statements hold for cohomology.
\operatorname{e}nd{thm}
First, we need to calculate some intersection-products.
\betaegin{lem}\lambdabel{inter-mult-lem2}
Recall that for each $m\ge 0$ we denote by $Z_{m,N}\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ the image of
$t_{m,N}:{\cal C}\tildemes_S{\cal C}^{[N-m]}\to{\cal C}\tildemes_S{\cal C}^{[N]}$.
For $m\le n$ we can consider the fine intersection-product
$[Z_{m,N}]\cdotot [Z_{n,N}]\in{\Bbb C}H^m(Z_{n,N})$. We have the following formula:
$$[Z_{m,N}]\cdotot [Z_{n,N}]=\sum_{i\ge 0}(-1)^i i!{m\choose i}{n\choose i} p_1^*K^i\cdotot [Z_{m+n-i,N}],$$
where $p_1:Z_{n,N}\to {\cal C}$ is the natural projection.
\operatorname{e}nd{lem}
\noindent {\it Proof} . We have an isomorphism $t_{n,N}:{\cal C}\tildemes {\cal C}^{[N-n]}\widetilde{\to} Z_{n,N}$.
Under this isomorphism the intersection-product in question becomes $t_{n,N}^*[Z_{m,N}]$, and the
required formula is equivalent to
$$t_{n,N}^*[Z_{m,N}]=\sum_{i\ge 0}(-1)^i{m\choose i}{n\choose i} p_1^*K^i\cdotot [Z_{m-i,N-n}].$$
For $m=1$ this boils down to the identity
$$t_{n,N}^*[{\cal D}_N]={\cal D}_{N-n}-n p_1^*K$$
that follows easily from \operatorname{e}qref{D-res-eq}. To deduce the case of general $m$ we use Lemma \ref{Z-lem}.
\operatorname{e}d
\betaegin{lem}\lambdabel{Pi-d-lem}
Consider the cartesian diagram
\betaegin{diagram}
{\Bbb P}i_d(i_1,\ldots,i_n) &\rTo{} & {\cal C}^{[d]}\\
\dTo{} & &\dTo{{\cal D}e_n}\\
{\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]} &\rTo{s_{i_1,\ldots,i_n;d}}& ({\cal C}^{[d]})^n,
\operatorname{e}nd{diagram}
where ${\cal D}e_n$ is the diagonal embedding, while $s_{i_1,\ldots,i_n;d}$
is given by $(p,D_1,\ldots,D_n)\mapsto (i_1p+D_1,\ldots,i_np+D_n)$.
For each $j\ge\max(i_1,\ldots,i_n)$ we have a natural map
$$q^j:{\cal C}\tildemes_S {\cal C}^{[d-j]}\to{\Bbb P}i_d(i_1,\ldots,i_n):(p,D)\mapsto (p,(j-i_1)p+D,\ldots,(j-i_n)p+D,jp+D),$$
where we view ${\Bbb P}i_d(i_1,\ldots,i_n)$ as a subvariety of
${\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}\tildemes_S {\cal C}^{[d]}$.
Set $i=\sum_{s=1}^n i_s$. Then we have
$$s_{i_1,\ldots,i_n;d}^![{\cal C}^{[d]}]=
\sum_{j\ge 0} (-1)^j a(i_1,\ldots,i_n;j)\cdotot p_1^*K^j\cdotot q^{i-j}_*[C\tildemes C^{[d-i+j]}]$$
where the coefficients $a(i_1,\ldots,i_n;j)$ are defined recursively by
\betaegin{equation}\lambdabel{a-rec-eq}
\betaegin{array}{l}
a(i_1,\ldots,i_n;j)=\sum_{k=0}^j k!{i_1\choose k}{i_2+\ldots+i_n-j+k\choose k}a(i_2,\ldots,i_n,j-k),\\
a(i_1;j)=\deltalta_{j,0}
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
(note that $a(i_1,\ldots,i_n;j)=0$ unless $j\le i-\max(i_1,\ldots,i_n)$).
\operatorname{e}nd{lem}
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\betaegin{thm}\lambdabel{divided-powers-thm} The action of ${\cal D}({\Bbb C}H^*({\cal C}),K)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$
extends to the action of $\widetilde{U}_1({\Bbb C}H^*({\cal C}),K)$ (resp., $\widetilde{U}_2({\Bbb C}H^*({\cal C}),K)$), such that
the action of ${\betaf P}_{n,0}({\cal C})^{[d]}$ (resp., ${\betaf P}_{0,n}({\cal C})^{[d]}$)
is given by \operatorname{e}qref{div-eq1} (resp., \operatorname{e}qref{div-eq2}). Furthermore, these relations hold on the level of correspondences. Also, similar statements hold for cohomology.
\operatorname{e}nd{thm}
First, we need to calculate some intersection-products.
\betaegin{lem}\lambdabel{inter-mult-lem2}
Recall that for each $m\ge 0$ we denote by $Z_{m,N}\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ the image of
$t_{m,N}:{\cal C}\tildemes_S{\cal C}^{[N-m]}\to{\cal C}\tildemes_S{\cal C}^{[N]}$.
For $m\le n$ we can consider the fine intersection-product
$[Z_{m,N}]\cdotot [Z_{n,N}]\in{\Bbb C}H^m(Z_{n,N})$. We have the following formula:
$$[Z_{m,N}]\cdotot [Z_{n,N}]=\sum_{i\ge 0}(-1)^i i!{m\choose i}{n\choose i} p_1^*K^i\cdotot [Z_{m+n-i,N}],$$
where $p_1:Z_{n,N}\to {\cal C}$ is the natural projection.
\operatorname{e}nd{lem}
\noindent {\it Proof} . We have an isomorphism $t_{n,N}:{\cal C}\tildemes {\cal C}^{[N-n]}\widetilde{\to} Z_{n,N}$.
Under this isomorphism the intersection-product in question becomes $t_{n,N}^*[Z_{m,N}]$, and the
required formula is equivalent to
$$t_{n,N}^*[Z_{m,N}]=\sum_{i\ge 0}(-1)^i{m\choose i}{n\choose i} p_1^*K^i\cdotot [Z_{m-i,N-n}].$$
For $m=1$ this boils down to the identity
$$t_{n,N}^*[{\cal D}_N]={\cal D}_{N-n}-n p_1^*K$$
that follows easily from \operatorname{e}qref{D-res-eq}. To deduce the case of general $m$ we use Lemma \ref{Z-lem}.
\operatorname{e}d
\betaegin{lem}\lambdabel{Pi-d-lem}
Consider the cartesian diagram
\betaegin{diagram}
{\Bbb P}i_d(i_1,\ldots,i_n) &\rTo{} & {\cal C}^{[d]}\\
\dTo{} & &\dTo{{\cal D}e_n}\\
{\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]} &\rTo{s_{i_1,\ldots,i_n;d}}& ({\cal C}^{[d]})^n,
\operatorname{e}nd{diagram}
where ${\cal D}e_n$ is the diagonal embedding, while $s_{i_1,\ldots,i_n;d}$
is given by $(p,D_1,\ldots,D_n)\mapsto (i_1p+D_1,\ldots,i_np+D_n)$.
For each $j\ge\max(i_1,\ldots,i_n)$ we have a natural map
$$q^j:{\cal C}\tildemes_S {\cal C}^{[d-j]}\to{\Bbb P}i_d(i_1,\ldots,i_n):(p,D)\mapsto (p,(j-i_1)p+D,\ldots,(j-i_n)p+D,jp+D),$$
where we view ${\Bbb P}i_d(i_1,\ldots,i_n)$ as a subvariety of
${\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}\tildemes_S {\cal C}^{[d]}$.
Set $i=\sum_{s=1}^n i_s$. Then we have
$$s_{i_1,\ldots,i_n;d}^![{\cal C}^{[d]}]=
\sum_{j\ge 0} (-1)^j a(i_1,\ldots,i_n;j)\cdotot p_1^*K^j\cdotot q^{i-j}_*[C\tildemes C^{[d-i+j]}]$$
where the coefficients $a(i_1,\ldots,i_n;j)$ are defined recursively by
\betaegin{equation}\lambdabel{a-rec-eq}
\betaegin{array}{l}
a(i_1,\ldots,i_n;j)=\sum_{k=0}^j k!{i_1\choose k}{i_2+\ldots+i_n-j+k\choose k}a(i_2,\ldots,i_n,j-k),\\
a(i_1;j)=\deltalta_{j,0}
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
(note that $a(i_1,\ldots,i_n;j)=0$ unless $j\le i-\max(i_1,\ldots,i_n)$).
\operatorname{e}nd{lem}
\noindent {\it Proof} . Note that for $n=1$ we have ${\Bbb P}i_d(i_1)={\cal C}\tildemes_S {\cal C}^{[d-i_1]}$, and
the formula holds trivially. For $n>1$ we have the following commutative diagram with
cartesian squares:
\betaegin{diagram}
{\Bbb P}i_d(i_1,\ldots,i_n) &\rTo{} & {\Bbb P}i_d(i_2,\ldots,i_n) &\rTo{} & {\cal C}^{[d]}\\
\dTo{} & & \dTo{} & &\dTo{{\cal D}e_n}\\
{\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}
&\rTo{t_{i_1,d}\tildemes\operatorname{id}\tildemes\ldots\tildemes\operatorname{id}} &
{\cal C}\tildemes_S {\cal C}^{[d]}\tildemes_S {\cal C}^{[d-i_2]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]} &\rTo{}&
({\cal C}^{[d]})^n\\
\dTo{p_{12}} & &\dTo{p_{12}}\\
{\cal C}\tildemes_S {\cal C}^{[d-i_1]} &\rTo{t_{i_1,d}} & {\cal C}\tildemes_S {\cal C}^{[d]}
\operatorname{e}nd{diagram}
where the second arrow in the second row is induced by $s_{i_2,\ldots,i_n;d}$.
It follows that
$$s_{i_1,\ldots,i_n;d}^![C^{[d]}]=t_{i_1,d}^!s_{i_2,\ldots,i_n;d}^![C^{[d]}].$$
Now the required equality and the recursive formula for the coefficients $a(i_1,\ldots,i_n;j)$ follow easily by induction using Lemma \ref{inter-mult-lem2}.
\operatorname{e}d
\betaegin{lem}\lambdabel{inter-mult-lem3}
Consider the cartesian square
\betaegin{diagram}
Z^{d,[n]}_{k,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-k]}\\
\dTo{} & & \dTo{s_{k,N}}\\
{\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]} &\rTo{s^{[n]}_{d,N}} & {\cal C}^{[N]}
\operatorname{e}nd{diagram}
where $s^{[n]}_{d,N}(D_1,D_2)=nD_1+D_2$.
For every $i,j$ such that $i\le k$ and $i\le nj$ we have a closed embedding
$$q^{i,j}:{\cal C}\tildemes_S {\cal C}^{[d-j]}\tildemes_S {\cal C}^{[N-nd-k+i]}\to Z^{d,[n]}_{k,N}: (p,D_1,D_2)
\mapsto (jp+D_1,(k-i)p+D_2,p, (nj-i)p+nD_1+D_2),$$
where we view $Z^{d,[n]}_{k,N}$ as a subset of
${\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-k]}$.
Then we have the following formula for the intersection-product in the above diagram:
$$[{\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]}]\cdotot [{\cal C}\tildemes_S {\cal C}^{[N-k]}]=
\sum_{0\le j\le i\le k, i\le nj}(-1)^{i-j}\frac{i!}{j!}{k\choose i} A_j(i,n)
q^{i,j}_*(p_1^*K^{i-j}\cdotot [{\cal C}\tildemes_S {\cal C}^{[d-j]}\tildemes_S {\cal C}^{[N-nd-k+i]}]),$$
where we use the numbers $(A_j(i,n))$ introduced in Lemma \ref{div-sum-lem}(ii).
\operatorname{e}nd{lem}
\noindent {\it Proof} .
Consider the following commutative diagram with
cartesian squares:
\betaegin{diagram}
Z^{d,[n]}_{k,N} &\rTo{} & {\Bbb P}i_{k,(d)^n,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-k]}\\
\dTo{} & & \dTo{} & &\dTo{}\\
{\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]} &\rTo{{\cal D}e_n\tildemes\operatorname{id}} &
({\cal C}^{[d]})^n\tildemes_S {\cal C}^{[N-nd]} &\rTo{\alpha}& {\cal C}^{[N]}\\
\dTo{} & &\dTo{}\\
{\cal C}^{[d]} &\rTo{{\cal D}e_n}&({\cal C}^{[d]})^n
\operatorname{e}nd{diagram}
where ${\cal D}e_n$ is the diagonal embedding, the map $\alpha$
is given by the addition of divisors. We have to calculate
${\cal D}e_n^!\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]$.
Iterating Lemma \ref{mult-lem2} we obtain
$$\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=\sum_{i_1+\ldots+i_n=i\le k}\frac{k!}{i_1!\ldots i_n!(k-i)!}
[{\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}\tildemes_S {\cal C}^{[N-nd-k+i]}].$$
Next, by Lemma \ref{Pi-d-lem}
\betaegin{align*}
&{\cal D}e_n^![{\cal C}\tildemes_S{\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}]=
s_{i_1,\ldots,i_n;d}^![{\cal C}^{[d]}]=\\
&\sum_{l\ge 0}(-1)^l a(i_1,\ldots,i_n;l) p_1^*K^l\cdotot [{\cal C}\tildemes_S{\cal C}^{[d-i+l]}],
\operatorname{e}nd{align*}
where $i=i_1+\ldots+i_n$.
It follows that
$${\cal D}e_n^!\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=\sum_{0\le l\le i\le k}(-1)^l{k\choose i}
b(i,l;n) p_1^*K^l\cdotot [{\cal C}\tildemes_S{\cal C}^{[d-i+l]}\tildemes_S{\cal C}^{[N-nd-k+i]}],$$
where
$$b(i,l;n)=\sum_{i_1+\ldots+i_n=i}\frac{i!}{i_1!\ldots i_n!}a(i_1,\ldots,i_n;l).$$
It remains to show that $b(i,l;n)=\frac{i!}{(i-l)!}A_{i-l}(i,n)$. To this end we use the recursive formulas
$$b(i,l;n)=\sum_{0\le i_1\le i,0\le k\le l}\frac{i!(i-i_1-l+k)!}{(i-i_1)!(i_1-k)!(i-i_1-l)!k!}b(i-i_1,l-k;n-1),
\ \ \ b(i,l;1)=\delta_{l,0}$$
that follow immediately from \operatorname{e}qref{a-rec-eq}. Note that from the formula
${n\choose i}={n-1\choose i}+{n-1\choose i-1}$ one can derive a similar recursive formula for
$A_d(i,n)$:
$$A_d(i,n)=\sum_{r+s+t=d}\frac{d!}{r!s!t!}A_{d-r}(i-r-s,n-1).$$
Now the required equality follows easily by induction in $n$.
\operatorname{e}d
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0.4951.11
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\noindent {\it Proof} .
Consider the following commutative diagram with
cartesian squares:
\betaegin{diagram}
Z^{d,[n]}_{k,N} &\rTo{} & {\Bbb P}i_{k,(d)^n,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-k]}\\
\dTo{} & & \dTo{} & &\dTo{}\\
{\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]} &\rTo{{\cal D}e_n\tildemes\operatorname{id}} &
({\cal C}^{[d]})^n\tildemes_S {\cal C}^{[N-nd]} &\rTo{\alpha}& {\cal C}^{[N]}\\
\dTo{} & &\dTo{}\\
{\cal C}^{[d]} &\rTo{{\cal D}e_n}&({\cal C}^{[d]})^n
\operatorname{e}nd{diagram}
where ${\cal D}e_n$ is the diagonal embedding, the map $\alpha$
is given by the addition of divisors. We have to calculate
${\cal D}e_n^!\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]$.
Iterating Lemma \ref{mult-lem2} we obtain
$$\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=\sum_{i_1+\ldots+i_n=i\le k}\frac{k!}{i_1!\ldots i_n!(k-i)!}
[{\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}\tildemes_S {\cal C}^{[N-nd-k+i]}].$$
Next, by Lemma \ref{Pi-d-lem}
\betaegin{align*}
&{\cal D}e_n^![{\cal C}\tildemes_S{\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}]=
s_{i_1,\ldots,i_n;d}^![{\cal C}^{[d]}]=\\
&\sum_{l\ge 0}(-1)^l a(i_1,\ldots,i_n;l) p_1^*K^l\cdotot [{\cal C}\tildemes_S{\cal C}^{[d-i+l]}],
\operatorname{e}nd{align*}
where $i=i_1+\ldots+i_n$.
It follows that
$${\cal D}e_n^!\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=\sum_{0\le l\le i\le k}(-1)^l{k\choose i}
b(i,l;n) p_1^*K^l\cdotot [{\cal C}\tildemes_S{\cal C}^{[d-i+l]}\tildemes_S{\cal C}^{[N-nd-k+i]}],$$
where
$$b(i,l;n)=\sum_{i_1+\ldots+i_n=i}\frac{i!}{i_1!\ldots i_n!}a(i_1,\ldots,i_n;l).$$
It remains to show that $b(i,l;n)=\frac{i!}{(i-l)!}A_{i-l}(i,n)$. To this end we use the recursive formulas
$$b(i,l;n)=\sum_{0\le i_1\le i,0\le k\le l}\frac{i!(i-i_1-l+k)!}{(i-i_1)!(i_1-k)!(i-i_1-l)!k!}b(i-i_1,l-k;n-1),
\ \ \ b(i,l;1)=\delta_{l,0}$$
that follow immediately from \operatorname{e}qref{a-rec-eq}. Note that from the formula
${n\choose i}={n-1\choose i}+{n-1\choose i-1}$ one can derive a similar recursive formula for
$A_d(i,n)$:
$$A_d(i,n)=\sum_{r+s+t=d}\frac{d!}{r!s!t!}A_{d-r}(i-r-s,n-1).$$
Now the required equality follows easily by induction in $n$.
\operatorname{e}d
\noindent
{\it Proof of Theorem \ref{divided-powers-thm}.}
Relations of type (ii) are easy to check, so we will concentrate on relations of type (iii).
In the notation of Lemma \ref{inter-mult-lem3}
the composition $P_{m,k}(a)\circ P_{n,0}({\cal C})^{[d]}$ acting on ${\cal C}^{[N-nd]}$ is given by
$x\mapsto q_{2*}(w\cdotot p_{{\cal C}}^*a\cdotot q_1^*x)$, where
$w\in{\Bbb C}H^*(Z^{d,[n]}_{k,N})$ is the intersection-product computed in this lemma,
$q_1$ is the composition
$$Z^{d,[n]}_{k,N}\to {\cal C}^{[d]}\tildemes_{{\cal S}} {\cal C}^{[N-nd]}\stackrel{p_2}{\to}{\cal C}^{[N-nd]},$$
$q_2$ is the composition
$$
\betaegin{diagram}
Z^{d,[n]}_{k,N} &\rTo &{\cal C}\tildemes_S{\cal C}^{[N-k]} &\rTo{s_{m,N-k+m}} &{\cal C}^{[N-k+m]},
\operatorname{e}nd{diagram}
$$
and $p_{{\cal C}}:Z^{d,[n]}_{k,N}\to{\cal C}$ is the natural projection.
The formula for $w$ leads to the expression for the $P_{m,k}(a)\circ P_{n,0}({\cal C})^{[d]}$ as the
linear combination of the operators defined by the cycles $p_{{\cal C}}^*(a\cdotot K^{i-j})$ over the correspondences
$$
\betaegin{diagram}
& & {\cal C}\tildemes_S{\cal C}^{[d-j]}\tildemes_S{\cal C}^{[N-nd-k+i]} & \\
&\ldTo{s_{k-i,N-nd}p_{13}} & &\rdTo{q} &\\
{\cal C}^{[N-nd]} & & & &{\cal C}^{[N-k+m]}
\operatorname{e}nd{diagram}
$$
where $q(p,D_1,D_2)=(m+nj-i)p+nD_1+D_2$. It is easy to see that the same correspondence
arises when computing $P_{n,0}({\cal C})^{[d-j]}\circ P_{m+nj-i,k-i}(a\cdotot K^{i-j})$ and that the coefficients
match. The second relation of type (iii) is checked similarly: the operators involved in it are defined
by the transposes of the above correspondences.
\operatorname{e}d
\betaegin{ex} In the case of a trivial family ${\cal C}=C\tildemes S$ the relations established in
Theorem \ref{divided-powers-thm} take form
$$P_{m,k}(a)P_{n,0}(C\tildemes S)^{[d]}=
\sum_{i=0}^d{k\choose i} n^i P_{n,0}(C\tildemes S)^{[d-i]}P_{m+i(n-1),k-i}(a),$$
$$P_{0,n}(C\tildemes S)^{[d]}P_{m,k}(a)=\sum_{i=0}^d{m\choose i}n^i P_{m-i,k+i(n-1)}(a)
P_{0,n}(C\tildemes S)^{[d-i]},$$
where $a\in{\Bbb C}H^*(C\tildemes S)$.
\operatorname{e}nd{ex}
\betaegin{rems} 1. One should be able to establish also some commutation relations between
$P_{n_1,0}({\cal C})^{[d_1]}$ and $P_{0,n_2}({\cal C})^{[d_2]}$. The simplest example of such relations
is given in Proposition \ref{div-com-prop} below.
\noindent 2. For other operators $P_{k,m}({\cal C})$
one should be able construct some modified divided powers.
For example, we have $P_{1,1}({\cal C})(x)=Nx$
for $x\in{\Bbb C}H^*({\cal C}^{[N]})$, so one cannot construct $P_{1,1}({\cal C})^2/2$, however, one can construct
$(P_{1,1}({\cal C})^2-P_{1,1}({\cal C}))/2$.
\operatorname{e}nd{rems}
\betaegin{prop}\lambdabel{div-com-prop}
All the operators in the family $\{P_{1,0}({\cal C})^{[d]}\ |\ d\ge 1\}\cup
\{P_{0,1}({\cal C})^{[d]}\ |\ d\ge 1\}$ commute with each other (on the level of correspondences).
\operatorname{e}nd{prop}
\noindent {\it Proof} . The fact that the operators within each set commute with each other follows from the commutativity
of the Pontryagin product. Now let us check that $P_{1,0}({\cal C})^{[d_1]}$ commutes with
$P_{0,1}({\cal C})^{[d_2]}$.
The composition $P_{0,1}({\cal C})^{[d_2]}\circ P_{1,0}({\cal C})^{[d_1]}$ acting on ${\Bbb C}H^*({\cal C}^{[N]})$
is given by the following correspondence ${\Bbb P}i$ from ${\cal C}^{[N]}$ to ${\cal C}^{[N+d_1-d_2]}$ equipped
with a class of dimension $N+d_1$:
$${\Bbb P}i=\{(D_1,E_1,D_2,E_2)\in{\cal C}^{[d_1]}\tildemes_S{\cal C}^{[N]}\tildemes_S{\cal C}^{[d_2]}\tildemes_S
{\cal C}^{[N+d_1-d_2]}\ |\ D_1+E_1=D_2+E_2\},$$
where the map $q:{\Bbb P}i\to{\cal C}^{[N]}\tildemes_S{\cal C}^{[N+d_1-d_2]}$ sends $(D_1,E_1,D_2,E_2)$ to
$(E_1,E_2)$. This correspondence is equipped with the natural intersection-product class of dimension
$N+d_1$
\betaegin{equation}\lambdabel{div-com-inter-class}
[{\cal C}^{[d_1]}\tildemes_S{\cal C}^{[N]}]\cdotot [{\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N+d_1-d_2]}].
\operatorname{e}nd{equation}
On the other hand, the composition $P_{1,0}({\cal C})^{[d_1]}\circ P_{0,1}({\cal C})^{[d_2]}$ is given by the correspondence
$$
\betaegin{diagram}
& &{\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N-d_2]}\tildemes_S{\cal C}^{[d_1]}& &\\
&\ldTo{\alpha_{d_2,N-d_2}p_{12}}& &\rdTo{\alpha_{N-d_2,d_1}p_{23}}&\\
{\cal C}^{[N]}& & & &{\cal C}^{[N+d_1-d_2]}
\operatorname{e}nd{diagram}
$$
The natural map
$${\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N-d_2]}\tildemes_S{\cal C}^{[d_1]}\to{\Bbb P}i:(D_2,E',D_1)\mapsto
(D_1,D_2+E',D_2,D_1+E')$$
is an isomorphism onto an irreducible component ${\Bbb P}i_0\subset{\Bbb P}i$.
Other irreducible components ${\Bbb P}i_i\subset{\Bbb P}i$ are numbered by $i$, such that $1\le i\le\min(d_1,d_2)$.
Namely, ${\Bbb P}i_i$ is the image of the map
$${\cal C}^{[i]}\tildemes_S{\cal C}^{[d_2-i]}\tildemes_S{\cal C}^{[N-d_2+i]}\tildemes_S{\cal C}^{[d_1-i]}\to{\Bbb P}i:
(D_0,D'_2,E',D'_1)\mapsto (D_0+D'_1,D'_2+E', D_0+D'_2,D'_1+E').$$
Since the composition of this map with $q:{\Bbb P}i\to{\cal C}^{[N]}\tildemes_S{\cal C}^{[N+d_1-d_2]}$
factors through ${\cal C}^{[d_2-i]}\tildemes_S{\cal C}^{[N-d_2+i]}\tildemes_S{\cal C}^{[d_1-i]}$, we derive
that $q({\Bbb P}i_i)$ has dimension $\le N+d_1-i$. Therefore, the components ${\Bbb P}i_i$ with $i\ge 1$
give zero contribution to the composition $P_{0,1}({\cal C})^{[d_2]}\circ P_{1,0}({\cal C})^{[d_1]}$.
It remains to prove that $[{\Bbb P}i_0]$ appears in the intersection-product \operatorname{e}qref{div-com-inter-class}
with multiplicity $1$. To this end we can replace ${\cal C}^{[d_2]}$ by the cartesian product ${\cal C}^{d_2}$
and then use iteratively Lemma \ref{mult-lem2} (in the case $m=1$).
\operatorname{e}d
Let us denote by ${\cal D}_{t,u,{\Bbb Q}}={\Bbb Q}[t,u,\partial_t,\partial_u]$ the algebra of differential operators in two variables.
Let us also denote by
$${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]\subset{\cal D}_{t,u,{\Bbb Q}}$$
the subalgebra over ${\Bbb Z}$ generated by $t$, $\partial_u$ and by the divided powers of $u$ and $\partial_t$.
We will also consider the ${\Bbb Z}$-subalgebras ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$ and ${\Bbb Z}[u^{[\betaullet]},\partial_u]$ in
this algebra.
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