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In 1990 J-L. Krivine introduced the notion of storage operators. They are $\l$-terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in $AF2$ type system for storage operators using Gődel translation of classical to intuitionistic logic.\
In order to modelize the control operators, J-L. Krivine has extended the system $AF2$ to the classical logic. In his system the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system $AF2$ can be used to find the values of classical integers.\
In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. We present also a similar result in the M. Parigot’s $\l \m$-calculus.
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=Symbol at 16pt =Symbol at 10pt ł Ł PS. Ø v
\[section\] \[section\] \[section\] \[section\]
**Mixed Logic and Storage Operators\
**
**Karim NOUR\
LAMA - Equipe de Logique, Université de Chambéry\
73376 Le Bourget du Lac\
| 1 | member_40 |
e-mail [email protected]\
**
Introduction
============
In 1990, J.L. Krivine introduced the notion of storage operators (see \[4\]). They are closed $\l$-terms which allow, for a given data type (the type of integers, for example), to simulate in $\l$-calculus the “call by value” in a context of a “call by name” (the head reduction) and they can be used in order to modelize assignment instructions. J.L. Krivine has shown that the formula $\q x
\{ N$\*$[x] \f \neg\neg N[x] \}$ is a specification for storage operators for Church integers : where $N[x]$ is the type of integers in $AF2$ type system, and the operation $*$ is the simple Gődel translation from classical to intuitionistic logic which associates to every formula $F$ the formula $F$\* obtained by replacing in $F$ every atomic formula by its negation (see \[3\]).\
The latter result suggests many questions :
- Why do we need a Gődel translation ?
- Why do we need the type $N$\*$[x]$ which characterize a class larger than integers ?
In order to modelize the control operators, J-L. Krivine has extended the system $AF2$ to the classical logic (see \[6\]). His method is very simple : it consists of adding a new | 1 | member_40 |
constant, denoted by $C$, with the declaration $C : \q X \{ \neg \neg X \f X \}$ which axiomatizes classical logic over intuitionistic logic. For the constant $C$, he adds a new reduction rule : $(C t t_1 ... t_n) \f (t \quad \l x (x \quad t_1 ... t_n))$ which is a particular case of a rule given by Felleisen for control operator (see \[1\]). In this system the property of the unicity of integers representation is lost, but J-L. Krivine has shown that storage operators typable in the intuitionistic system $AF2$ can be used to find the values of classical integers [^1](see \[6\]).\
The latter result suggests also many questions :
- What is the relation between classical integers and the type $N$\*$[x]$ ?
- Why do we need intuitionistic logic to modelize the assignment instruction and classical logic to modelize the control operators ?
In this paper, we present a new classical type system based on a logical system called mixed logic. This system allows essentially to distinguish between classical proofs and intuitionistic proofs. We prove that, in this system, we can characterize, by types, the storage operators and the control operators. This results give some | 1 | member_40 |
answers to the previous questions.\
We present at the end (without proof) a similar result in the M. Parigot’s $\l
\m$-calculus.\
**Acknowledgement. We wish to thank J.L. Krivine, and C. Paulin for helpful discussions. We don’t forget the numerous corrections and suggestions from R. David and N. Bernard.**
Pure and typed $\l$-calculus
============================
- Let $t,u,u_1,...,u_n$ be $\l$-terms, the application of $t$ to $u$ is denoted by $(t)u$. In the same way we write $(t)u_1...u_n$ instead of $(...((t)u_1)...)u_n$.
- $Fv(t)$ is the set of free variables of a $\l$-term $t$.
- The $\b$-reduction (resp. $\b$-equivalence) relation is denoted by $u \f\sb{\b} v$ (resp. $u
\simeq\sb{\b} v$).
- The notation $\s(t)$ represents the result of the simultaneous substitution $\s$ to the free variables of $t$ after a suitable renaming of the bounded variables of $t$.
- We denote by $(u)^n v$ the $\l$-term $(u)...(u)v$ where $u$ occurs $n$ times, and $\sou{u}$ the sequence of $\l$-terms $u_1,...,u_n$. If $\sou{u} = u_1,...,u_n$ $n \geq 0$, we denote by $(t)\sou{u}$ the $\l$-term $(t)u_1...u_n$.
- Let us recall that a $\l$-term $t$ either has a head redex \[i.e. $t=\l x_1 ...\l
x_n (\l x u) v v_1 ... v_m$, the head redex being $(\l x | 1 | member_40 |
u) v$\], or is in head normal form \[i.e. $t=\l x_1 ...\l x_n (x) v_1 ... v_m$\]. The notation $u \p v$ means that $v$ is obtained from $u$ by some head reductions. If $u \p v$, we denote by $h(u,v)$ the length of the head reduction between $u$ and $v$.
(see\[3\])\
1) If $u \p v$, then, for any substitution $\s$, $\s(u) \p \s(v)$, and $h(\s(u),\s(v))$=h(u,v).\
2) If $u \p v$, then, for every sequence of $\l$-terms $\sou{w}$, there is a $w$, such that $(u)\sou{w} \p w$, $(v)\sou{w} \p w$, and $h((u)\sou{w},w)=h((v)\sou{w},w)+h(u,v)$.
**Remark. Lemma 2.1 shows that to make the head reduction of $\s(u)$ (resp. of $(u)\sou{w}$) it is equivalent - same result, and same number of steps - to make some steps in the head reduction of $u$, and after make the head reduction of $\s(v)$ (resp. of $(v)\sou{w}$). $\Box$**
- The types will be formulas of second order predicate logic over a given language. The logical connectives are $\perp$ (for absurd), $\f$, and $\q$. There are individual (or first order) variables denoted by $x,y,z,...,$ and predicate (or second order) variables denoted by $X,Y,Z,....$
- We do not suppose that the language has a special constant for equality. | 1 | member_40 |
Instead, we define the formula $u=v$ (where $u,v$ are terms) to be $\q Y(Y(u) \f Y(v))$ where $Y$ is a unary predicate variable. Such a formula will be called an equation. We denote by $a \approx
b$, if $a=b$ is a consequence of a set of equations.
- The formula $F_1 \f (F_2 \f(...\f (F_n \f G)...))$ is also denoted by $F_1,F_2,...,F_n \f G$. For every formula $A$, we denote by $\neg A$ the formula $A \f \perp$. If $\sou{v} = v_1,...,v_n$ is a sequence of variables, we denote by $\q \sou{v} A$ the formula $\q v_1...\q v_n A$.
- Let $t$ be a $\l$-term, $A$ a type, $\G = x_1 : A_1 ,..., x_n : A_n$ a context, and $E$ a set of equations. We define by means of the following rules the notion “$t$ is of type $A$ in $\G$ with respect to $E$” ; this notion is denoted by $\G\v_{AF2} t:A$ :
<!-- -->
- \(1) $\G\v_{AF2} x_i:A_i$ $1\leq i\leq n$.
- \(2) If $\G,x:A \v_{AF2} t:B$, then $\G\v_{AF2} \l xt:A \f B$.
- \(3) If $\G\v_{AF2} u:A \f B$, and $\G\v_{AF2} v:A$, then $\G\v_{AF2} (u)v:B$.
- \(4) If $\G\v_{AF2} t:A$, and $x$ is not free in | 1 | member_40 |
$\G$, then $\G\v_{AF2} t:\q xA$.
- \(5) If $\G\v_{AF2} t:\q xA$, then, for every term $u$, $\G\v_{AF2} t:A[u/x]$.
- \(6) If $\G\v_{AF2} t:A$, and $X$ is not free in $\G$, then $\G\v_{AF2} t:\q XA$.
- \(7) If $\G\v_{AF2} t:\q XA$, then, for every formulas $G$, $\G\v_{AF2} t:A[G/X]$.
- \(8) If $\G\v_{AF2} t:A[u/x]$, and $u \approx v$, then $\G\v_{AF2}
t:A[v/x]$.
This typed $\l$-calculus system is called $AF2$ (for Arithmétique Fonctionnelle du second ordre).
(see \[2\]) The $AF2$ type system has the following properties :\
1) Type is preserved during reduction.\
2) Typable $\l$-terms are strongly normalizable.
We present now a syntaxical property of system $AF2$ that we will use afterwards.
(see \[8\]) If in the typing we go from $\G\v_{AF2} t:A$ to $\G\v_{AF2} t:B$, then we may assume that we begin by the $\q$-elimination rules, then by the equationnal rule, and finally by the $\q$-introduction rules.
- We define on the set of types the two binary relations $\lhd$ and $\approx$ as the least reflexive and transitive binary relations such that :
- - $\q xA \lhd A[u/x]$, if $u$ is a term of language ;
- - $\q XA \lhd A[F/X]$, if $F$ is a formula of language ;
- | 1 | member_40 |
- $A \approx B$ if and only if $A=C[u/x]$, $B=C[v/x]$, and $u \approx v$.
Pure and typed $\l C$-calculus
==============================
The $C2$ type system
--------------------
We present in this section the J-L. Krivine’s classical type system.
- We add a constant $C$ to the pure $\l$-calculus and we denote by $\L C$ the set of new terms also called $\l C$-terms. We consider the following rules of reduction, called rules of head $C$-reduction.
- 1\) $(\l x u) t t_1 ... t_n \f (u[t / x]) t_1 ... t_n$ for every $u, t, t_1,...,t_n \in
\L C$.
- 2\) $(C) t t_1 ... t_n \f (t) \l x (x)t_1 ... t_n$ for every $ t, t_1,...,t_n
\in \L C$, $x$ being a $\l$-variable not appearing in $t_1,...,t_n$.
<!-- -->
- For any $\l C$-terms $t,t'$, we shall write $t \p_C t'$ if $t'$ is obtained from $t$ by applying these rules finitely many times. We say that $t'$ is obtained from $t$ by head $C$-reduction.
- A $\l C$-term $t$ is said $\b$-normal if and only if $t$ does not contain a $\b$-redex.
- A $\l C$-term $t$ is said $C$-solvable if and only if $t \p_C (f)t_1,...,t_n$ where $f$ | 1 | member_40 |
is a variable.
It is easy to prove that : if $t \p_C t'$, then, for any substitution $\s$, $\s (t) \p_C \s
(t')$.
- We add to the $AF2$ type system the new following rule :
\(0) $\G \v C : \q X \{ \neg \neg X \f X \}$
This rule axiomatizes the classical logic over the intuitionistic logic. We call $C2$ the new type system, and we write $\G \v_{C2} t : A$ if $t$ is of type $A$ in the context $\G$.
It is clear that $\G \v_{C2} t : A$ if and only if $\G , C : \q X \{ \neg \neg X \f X \} \v_{AF2} t :
A$.
(see \[6\])\
1) If $\G \v_{C2} t:A$, and $t \f_{\b} t'$, then $\G \v_{C2} t':A$.\
2) If $\G \v_{C2} t:\perp$, and $t \p_C t'$, then $\G \v_{C2} t':\perp$.\
3) If $A$ is an atomic type, and $\G \v_{C2} t:A$, then $t$ is $C$-solvable.
The $M2$ type system
--------------------
In this section, we present the system $M2$. This system allows essentialy to distinguish between classical proofs and intuitionistic proofs\
We assume that for every integer $n$, there is a countable set of special $n$-ary second | 1 | member_40 |
order variables denoted by $X_C,Y_C,Z_C$...., and called classical variables.\
Let $X$ be an $n$-ary predicate variable or predicate symbol. A type $A$ is said to be ending with $X$ if and only if $A$ is obtained by the following rules :
- - $X(t_1,...,t_n)$ ends with $X$;
- - If $B$ ends with $X$, then $A \f B$ ends with $X$ for every type $A$ ;
- - If $A$ ends with $X$, then $\q vA$ ends with $X$ for every variable $v$.\
A type $A$ is said to be a classical type if and only if $A$ ends with $\perp$ or a classical variable.\
We add to the $AF2$ type system the new following rules :
- (0$'$) $\G\v C : \q X_C \{ \neg \neg X_C \f X_C \}$
- (6$'$) If $\G\v t:A$, and $X_C$ has no free occurence in $\G$, then $\G\v t: \q
X_C A$.
- (7$'$) If $\G\v t: \q X_C A$, and $G$ is a classical type, then $\G\v t:A[G/ X_C]$.
We call $M2$ the new type system, and we write $\G \v_{M2} t:A$ if $t$ is of type $A$ in the context $\G$.\
We extend the definition of $\lhd$ by : $\q | 1 | member_40 |
X_C A \lhd A[G / X_C]$ if $G$ is a classical type.
If $A$ is a classical type and $A \lhd B$ (or $A \approx B$), then $B$ is a classical type.
**Proof Easy. $\Box$**
The logical properties of $M2$
------------------------------
We denote by $LAF2$, $LC2$, and $LM2$ the underlying logic systems of respectively $AF2$, $C2$, and $M2$ type systems.\
With each classical variable $X_C$, we associate a special variable $X^{\ast}$ of $AF2$ having the same arity as $X_C$. For each formula $A$ of $LM2$, we define the formula $A$\* of $LAF2$ in the following way :
- - If $A=D(t_1,...,t_n)$ where $D$ is a predicate symbol or a predicate variable, then $A$\*=$A$ ;
- - If $A=X_C(t_1,...,t_n)$, then $A$\*$=\neg X^{\ast}(t_1,...,t_n)$ ;
- - If $A=B \f C$, then $A$\*$=B$\*$ \f C$\* ;
- - If $A=\q xB$, then $A$\*=$\q xB$\*.
- - If $A=\q XB$, then $A$\*=$\q XB$\*.
- - If $A=\q X_C B$, then $A$\*=$\q X^{\ast} B$\*.
$A$\* is called the Gődel translation of $A$.
If $G$ is a classical type of $LM2$, then $\v_{LAF2} \neg \neg G$\*$ \equi G$\*.
**Proof It is easy to prove that $\v_{LAF2} G$\*$ \f \neg \neg G$\*.\
We prove $\v_{LAF2} \neg \neg | 1 | member_40 |
G$\*$ \f G$\* by induction on $G$.**
- - If $G = \perp$, then $G$\*=$\perp$, and $\v_{LAF2} ((\perp \f \perp) \f \perp) \f
\perp$.
- - If $G = X_C (t_1,...,t_n)$, then $G$\*=$\neg X^{\ast}(t_1,...,t_n)$, and $\v_{LAF2} \neg
\neg \neg X^{\ast}(t_1,...,t_n) \f \neg X^{\ast}(t_1,...,t_n)$.
- - If $G = A \f B$, then $B$ is a classical type and $G$\* = $A$\* $\f$ $B$\*. By the induction hypothesis, we have $\v_{LAF2} \neg \neg B$\*$ \f B$\*. Since $\v_{LAF2} \neg \neg
(A$\*$\f B$\*) $\f$ $(\neg \neg A$\*$\f \neg \neg B$\*), we check easily that $\v_{LAF2}
\neg \neg (A$\* $\f B$\*) $\f (A$\* $\f B$\*).
- - If $G = \q vG'$ where $v=x$ or $v=X$, then $G'$ is a classical type and $G$\*=$\q vG'$\*. By the induction hypothesis, we have $\v_{LAF2}
\neg \neg G'$\*$ \f G'$\*. Since $\v_{LAF2} \neg \neg \q vG'$\* $\f$ $\q v \neg \neg G'$\*, we check easily that $\v_{LAF2} \neg \neg \q vG'$\* $\f \q vG'$\*.
- - If $G = \q X_C G'$, then $G'$ is a classical type and $G$\*=$\q X^{\ast} G'$\*. By the induction hypothesis, we have $\v_{LAF2} \neg \neg G'$\*$ \f G'$\*. Since $\v_{LAF2} \neg \neg \q
X^{\ast} G'$\* $\f$ $\q X^{\ast} \neg \neg | 1 | member_40 |
G'$\*, we check easily that $\v_{LAF2} \neg
\neg \q X^{\ast} G'$\* $\f \q X^{\ast} G'$\*. $\Box$
Let $A,G$ be formulas of $LM2$, $t$ a term, $x$ a first order variable, and $X$ a second order variable. We have :\
1) $(A[t/x])$\*$= A$\*$[t/x]$.\
2) $(A[G/X])$\*$=A$\*$[G$\*$/X]$.
**Proof By induction on $A$. $\Box$**
Let $A$ be a formula of $LM2$, $G$ a classical type, and $X_C$ a classical variable.\
$\v_{LAF2} (A[G/X_C])$\*$ \equi A$\*$[\neg G$\*$/X_C]$.
**Proof By induction on $A$.**
- - If $A = D(t_1,...,t_n)$ where $D$ is a predicate variable or a predicate symbol, then $A$\*=$A$, and $\v_{LAF2} A \equi A$.
- - If $A = X_C (t_1,...,t_n)$, then $A$\*=$\neg X^{\ast}(t_1,...,t_n)$, and, by Lemma 3.2, $\v_{LAF2} \neg \neg G$\*$ \equi G$\*.
- - If $A = B \f C$, then $A$\* = $B$\* $\f$ $C$\*. By the induction hypothesis, we have $\v_{LAF2} (B[G/X_C])$\*$ \equi B$\*$[\neg G$\*$/X_C]$ and $\v_{LAF2}
(C[G/X_C])$\*$ \equi C$\*$[\neg G$\*$/X_C]$. Therefore $\v_{LAF2} \{ (B[G/X_C])$\*$
\f (B[G/X_C])$\*$\} \equi \{ B$\*$[\neg G$\*$/X_C] \f C$\*$[\neg G$\*$/X_C] \}$.
- - If $A = \q vA'$, where $v=x$ or $v=X$, then $A$\*=$\q vA'$\*. By the induction hypothesis, we have $\v_{LAF2} (A'[G/X_C])$\*$ \equi A'$\*$[\neg G$\*$/X_C]$. Therefore $\v_{LAF2}
(\q vA'[G/X_C])$\*$ \equi \q vA'$\*$[\neg G$\*$/X_C]$.
- - If | 1 | member_40 |
$A = \q Y_C A'$, then $A$\*=$\q Y^{\ast} A'$\*. By the induction hypothesis, we have $\v_{LAF2} (A'[G/X_C])$\* $\equi A'$\*$[\neg G$\*$/X_C]$. Therefore $\v_{LAF2}
(\q Y_C A'[G/X_C])$\*$ \equi$ $ (\q Y_C A')$\*$[\neg G$\*$/X_C]$. $\Box$
If $A_1,...,A_n \v_{LM2} A$, then $A_1$\*$,...,A_n$\* $\v_{LAF2} A$\*.
**Proof By induction on the proof of $A$ and using Lemmas 3.2, 3.3, and 3.4. $\Box$**
Let $A,A_1,...,A_n$ be formulas of $LAF2$.\
$A_1,...,A_n \v_{LM2} A$ if and only if $A_1,...,A_n \v_{LAF2} A$.
**Proof We use Theorem 3.2. $\Box$\
With each predicate variable $X$ of $C2$, we associate a classical variable $X_C$ having the same arity as $X$. For each formula $A$ of $LC2$, we define the formula $A^C$ of $M2$ in the following way :**
- - If $A=D(t_1,...,t_n)$ where $D$ is a constant symbol, then $A^C=A$ ;
- - If $A=X(t_1,...,t_n)$ where $X$ is a predicate symbol, then $A^C=X_C(t_1,...,t_n)$ ;
- - If $A=B \f C$, then $A^C=B^C \f C^C$ ;
- - If $A=\q xB$, then $A^C=\q xB^C$ ;
- - If $A=\q XB$, then $A^C=\q X_CB^C$.
$A^C$ is called the classical translation of $A$.
Let $A_1,...,A_n,A$ be formulas of $LC2$.\
$A_1,...,A_n \v_{LC2} A$ if and only if $A_1^C,...,A_n^C \v_{LM2} A^C$.
**Proof By induction on the proof of | 1 | member_40 |
$A$. $\Box$**
Properties of $M2$ type system
==============================
By corollary 3.1, we have that a formula is provable in system $LAF2$ if and only if it is provable in system $LC2$. This resultat is not longer valid if we decorate the demonstrations by terms. We will give some conditions on the formulas in order to obtain such a result.\
We define two sets of types of $AF2$ type system : $\O^+$ (set of $\q$-positive types), and $\O^-$ (set of $\q$-negative types) in the following way :
- - If $A$ is an atomic type, then $A \in \O^+$, and $A \in \O^-$ ;
- - If $T \in \O^+$, and $T' \in \O^-$, then, $T' \f T \in \O^+$, and $T \f T' \in \O^-$ ;
- - If $T \in \O^+$, then $\q x T \in \O^+$ ;
- - If $T \in \O^-$, then $\q x T \in \O^-$ ;
- - If $T \in \O^+$, then $\q X T \in \O^+$ ;
- - If $T \in \O^-$, and $X$ has no free occurence in $T$, then $\q X T \in \O^-$.
1\) If $A \in \O^+$ (resp. $A \in \O^-$) and $A \approx B$, then $B | 1 | member_40 |
\in \O^+$ (resp. $B \in \O^-$).\
2) If $A \in \O^-$ and $A \lhd B \f C$, then $B \in \O^+$ and $C \in \O^-$.
**Proof Easy. $\Box$**
Let $A_1,...,A_n$ be $\q$-negative types, $A$ a $\q$-positive type of $AF2$ which does not end with $\perp$, $B_1,...,B_m$ classical types, and $t$ a $\b$-normal $\l C$-term.\
If $\G = x_1:A_1,...,x_n:A_n,y_1:B_1,...,y_m:B_m \v_{M2} t:A$, then $t$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n \v_{AF2} t:A$.
**Proof We argue by induction on $t$.**
- - If $t$ is a variable, we have two cases :
- - If $t=x_i$ $1 \leq i \leq n$, this is clear.
- - If $t=y_j$ $1 \leq j \leq m$, then $A=\q \sou{v} B$ where $B_j \lhd B'_j$ and $B'_j
\approx B$. Therefore, by Lemma 3.1, $A$ is a classical type. A contradiction.
- - If $t=\l x u$, then $\G,x:E \v_{M2} u:F$, and $A=\q \sou{v}( E' \f F')$ where $E \approx
E'$, $F \approx F'$ and $\sou{v}$ does not appear in $\G$. First, by Lemma 4.1, $E \in \O^-$ and $F \in \O^+$, and then, by the induction hypothesis, $u$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n,x:E \v_{AF2} u:F$. Therefore $t$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n \v_{AF2} t:A$.
- | 1 | member_40 |
- If $t=(x)u_1 ... u_r$ $r \geq 1$, we have two cases :
- - If $t=x_i$ $1 \leq i \leq n$, then $A_i \lhd B_1 \f C_1$, $C'_i \lhd B_{i+1} \f
C_{i+1}$ $1 \leq i \leq r-1$, $C'_r \lhd D$, $A = \q vD'$, where $C'_i \approx C_i$ $1
\leq i \leq r$, $D' \approx D$, and $\G \v_{M2} u_i:B_i$ $1 \leq i \leq r$. Since $A_i$ is a $\q$-negative types, we prove (by induction and using Lemma 4.1) that for all $1 \leq i \leq r$ $B_i$ is a $\q$-positive types. By the induction hypothesis we have $u_i$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n \v_{AF2} u_i:B_i$. Therefore $t$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n \v_{AF2} t:A$.
- - If $t=y_j$ $1 \leq j \leq m$, then $B_j \lhd B_1 \f
C_1$, $C'_i \lhd B_{i+1} \f C_{i+1}$ $1 \leq i \leq r-1$, $C'_r \lhd D$, $A = \q vD'$, where $C'_i \approx C_i$ $1 \leq i \leq r$, $D' \approx D$, and $\G \v_{M2} u_i:B_i$ $1 \leq i
\leq r$. Therefore, by Lemma 3.1, $A$ is a classical type. A contradiction.
- - If $t=(C)uu_1 ... u_r$ $r \geq 0$, then there is a classical type $E$ such that | 1 | member_40 |
$\G \v_{M2}
u:\neg \neg E$, $E \lhd B_1 \f C_1$, $C'_i \lhd B_{i+1} \f C_{i+1}$ $1 \leq i \leq r-1$, $C'_r \lhd
D$, $A = \q vD'$, where $C'_i \approx C_i$ $1 \leq i \leq r$, $D' \approx D$, and $\G \v_{M2}
u_i:B_i$ $1 \leq i \leq r$. Therefore, by Lemma 3.1, $A$ is a classical type. A contradiction. $\Box$
Let $A$ be a $\q$-positive type of $AF2$ and $t$ a $\b$-normal $\l C$-term.\
If $\v_{M2} t:A$, then $t$ is a normal $\l$-term, and $\v_{AF2} t:A$.
**Proof We use Theorem 4.1. $\Box$\
As for relation betwen the systems $C2$ and $M2$, we have the following result.**
Let $A_1,...,A_n,A$ be types of $C2$, and $t$ a $\l C$-term.\
$A_1,...,A_n \v_{C2} t:A$ if and only if $A_1^C,...,A_n^C \v_{M2} t:A^C$.
**Proof By induction on the typing of $t$. $\Box$**
The integers
============
- Each data type can be defined by a second order formula. For example, the type of integers is the formula : $N[x]= \q X \{ X(0), \q y(X(y) \f X(sy)) \f X(x) \}$ where $X$ is a unary predicate variable, $0$ is a constant symbol for zero, and $s$ is a unary function symbol for successor. The formula $N[x]$ means | 1 | member_40 |
semantically that $x$ is an integer if and only if $x$ belongs to each set $X$ containing $0$ and closed under the successor function $s$.\
The $\l$-term $\so{0} = \l x \l fx$ is of type $N[0]$ and represents zero.\
The $\l$-term $\so{s} = \l n\l x\l
f(f)((n)x)f$ is of type $\q y(N[y] \f N[s(y)])$ and represents the successor function.
- A set of equations $E$ is said to be adequate with the type of integers if and only if :
- - $s(a) \not \approx 0$ ;
- - If $s(a) \approx s(b)$ , then so is $a \approx b$.
In the rest of the paper, we assume that all sets of equations are adequate with the type of integers.
- For each integer $n$, we define the Church integer $\so{n}$ by $\so{n} = \l x\l f(f)^n x$.
The integers in $AF2$
---------------------
The system $AF2$ has the property of the unicity of integers representation.
(see \[2\]) Let $n$ be an integer. If $\v_{AF2} t :N[s^n (0)]$, then $t \simeq\sb{\b}
\so{n}$.
The propositional trace $N=\q X \{ X,(X \f X) \f X \}$ of $N[x]$ also defines the integers.
(see \[2\]) If $\v_{AF2} t :N$, then, for a certain | 1 | member_40 |
$n$, $t \simeq\sb{\b} \so{n}$.
**Remark A very important property of data type is the following (we express it for the type of integers) : in order to get a program for a function $f : N \f N$ it is sufficient to prove $\v
\q x ( N[x] \f N[f(x)] )$. For example a proof of $\v \q x ( N[x] \f N[p(x)] )$ from the equations $p(0)=0$, $p(s(x))=x$ gives a $\l$-term for the predecessor in Church intergers (see \[2\]). $\Box$**
The integers in $C2$
--------------------
The situation in system $C2$ is more complex. In fact, in this system the property of unicity of integers representation is lost and we have only one operational characterization of these integers.\
Let $n$ be an integer. A classical integer of value $n$ is a closed $\l C$-term $\th_n$ such that $\v_{C2} \th_n :N[s^n(0)]$.
(see \[6\] and \[12\]) Let $n$ be an integer, and $\th_n$ a classical integer of value $n$.
- - if $n=0$, then, for every distinct variables $x,g,y$ : $(\th_n) x g y \p_C (x) y$ ;
- - if $n \not = 0$, then there is $m \geq 1$ and a mapping $I
: \{0,...,m \} \f N$, such that | 1 | member_40 |
for every distinct variables $x,g,x_0,x_1,...,x_m$ :
- $(\th_n) x g x_0 \p_C (g) t_1 x_{r_0}$ ;
- $(t_i) x_i \p_C (g) t_{i+1} x_{r_i}$ $1\leq i\leq m$ ;
- $(t_m) x_m \p_C (x) x_{r_m}$ ;
where $I(0)=n$, $I(r_m)=0$, and $I(i+1)=I(r_i)-1$ $0\leq i\leq m-1$.
We will generalize this result.\
Let $O$ be a particular unary predicate symbol. The typed system $C2_O$ is the typed system $C2$ where we replace the rules (2) and (7) by :
- $(2_O)$ If $\G,x:A \v_{C2_O} t:B$, $A$ and $B$ are not ending with $O$, then $\G
\v_{C2_O} \l xt:A \f B$.
- $(7_O)$ If $\G\v_{C2_O} t:\q X A$, and $G$ is not ending with $O$, then $\G \v_{C2_O}
t:A[G/X]$.
We define on the types of $C2_O$ a binary relation $\lhd_O$ as the least reflexive and transitive binary relation such that :
- $\q xA \lhd_O A[u/x]$ if $u$ is a term of language ;
- $\q XA \lhd_O A[G/X]$ if $G$ is a type which is not ending with $O$.
a\) If $\G \v_{C2_O} t:\perp$, and $t \p_C t'$, then $\G \v_{C2_O} t':\perp$.\
b) If $\G \v_{C2_O} t:A$, and $A$ is an atomic type, then $t$ is $C$-solvable.
**Proof a) It is enough to do the | 1 | member_40 |
proof for one step of reduction. We have two cases :**
- - If $t=(\l xu)vv_1...v_m$, then $t'=(u[v/x])v_1...v_m$, $\G,x:F \v_{C2_O} u:G$, $F$ and $G$ are not ending with $O$, $G'\lhd_O F_1 \f G_1$, $G'_j \lhd_O F_{j+1} \f G_{j+1}$ $1 \leq j
\leq m-1$, $G_m \approx \perp$, $G_j \approx G'_j$ $1 \leq j \leq m-1$, $\G \v_{C2_O} v:F$, and $\G \v_{C2_O} v_j:F_j$ $1 \leq j \leq m$. It is easy to check that $\G \v_{C2_O} u[v/x]:G$, then $\G \v_{C2_O} t':\perp$.
- - If $t=(C)vv_1...v_m$, then $t'=(v)\l x(x)v_1...v_m$, and there is a type $A$ which is not ending with $O$ such that : $A'\lhd_O F_1 \f G_1$, $G'_j \lhd_O F_{j+1} \f G_{j+1}$ $1 \leq j
\leq m-1$, $G_m \approx \perp$, $A \approx A'$, $G_j \approx G'_j$ $1 \leq j \leq m$, $\G \v_{C2_O}
v:\neg \neg A$, and $\G \v_{C2_O} v_j:F_j$ $1 \leq j \leq m$. It is easy to check that $\G,x:A \v_{C2_O} (x)v_1...v_m:\perp$, but $A$ is not ending with $O$, then $\G \v_{C2_O}
\l x(x)v_1...v_m:\neg A$, and $\G \v_{C2_O} t':\perp$.
b\) Indeed, a typing of $C2_O$ may be seen as a typing of $C2$. $\Box$
a\) If $\G \v_{C2_O} t:O(a)$, and $t \p_C t'$, then $t=t'$.\
b) If $\G=y_1:A_1,...,y_n:A_n,x_1:O(a_1),...,x_m:O(a_m) \v_{C2_O} t:O(a)$, | 1 | member_40 |
and all $A_i$ $1 \leq i\leq
n$ are not ending with $O$, then $t$ is one of $x_i$, and $a_i \approx a$ $1 \leq i \leq n$.
**Proof a) It is enough to do the proof for one step of reduction. We have two cases :**
- - If $t=(\l xu)vv_1...v_m$, then $t'=(u[v/x])v_1...v_m$, $\G,x:F \v_{C2_O} u:G$, $F$ and $G$ are not ending with $O$, $G'\lhd_O F_1 \f G_1$, $G'_j \lhd_O F_{j+1} \f G_{j+1}$ $1 \leq j
\leq m-1$, $G_m \approx O(a)$, $G_j \approx G'_j$ $1 \leq j \leq m-1$, $\G \v_{C2_O} v:F$, and $\G
\v_{C2_O} v_j:F_j$ $1 \leq j \leq m$. Therefore $G_j$ $1 \leq j \leq m$ is not ending with $O$, which is impossible since $G_m \approx O(a)$.
- - If $t=(C)vv_1...v_m$, then $t'=(v)\l x(x)v_1...v_m$, and there is a type $A$ which is not ending with $O$ such that : $A'\lhd_O F_1 \f G_1$, $G'_j \lhd_O F_{j+1} \f G_{j+1}$ $1 \leq j
\leq m-1$, $G_m \approx O(a)$, $A \approx A'$, $G_j \approx G'_j$ $1 \leq j \leq m$, $\G \v_{C2_O}
v:\neg \neg A$, and $\G \v_{C2_O} v_j:F_j$ $1 \leq j \leq m$. $A$ is not ending with $O$, therefore $G_j$ $1 \leq j \leq m$ is not ending | 1 | member_40 |
with $O$, which is impossible since $G_m \approx O(a)$.
b\) By Lemma 5.1, we have $t \p_C (f)t_1...t_r$, and, by a), $t=(f)t_1...t_r$. Therefore $\G
\v_{C2_O} (f)t_1...t_r:O(a)$.
- - If $f=x_i$ $1 \leq i \leq m$, then $r=0$, $t=x_i$, and $O(a_i) \approx O(a)$, then $a_i
\approx a$.
- - If $f=y_j$ $1 \leq j \leq k$, then $A_j \lhd_O F_1 \f G_1$, $G'_k \lhd_O F_{k+1} \f
G_{k+1}$ $1 \leq k \leq r-1$, $G_r \approx O(a)$, $G_k \approx G'_k$ $1 \leq k \leq r$, and $\G\v_{C2_O} t_k:F_k$ $1 \leq k \leq r$. Since $A_j$ is not ending with $O$, then $G_k$ $1 \leq
k \leq r$ is not ending with $O$, which is impossible since $Cr \approx O(a)$. $\Box$
Let $V$ be the set of variables of $\l C$-calculus.\
Let $P$ be an infinite set of constants called stack constants [^2].\
We define a set of $\l C$-terms $\L CP$ by :
- - If $x \in V$, then $x \in \L CP$ ;
- - If $t \in \L CP$, and $x \in V$, then $\l xt \in \L CP$ ;
- - If $t \in \L CP$, and $u \in \L CP \bigcup P$, then $(t)u \in \L CP$.
In | 1 | member_40 |
other words, $t \in \L CP$ if and only if the stack constants are in argument positions in $t$.\
Let $\s$ be a function defined on $V \bigcup P$ such that :
- - If $x \in V$, then $\s (x) \in \L CP$ ;
- - If $p \in P$, then $\s (p)=\sou{t}=t_1,...,t_n$, $n \geq 0$, $t_i \in \L CP \bigcup P$ $1 \leq i \leq n$.
We define $\s(t)$ for all $t \in \L CP$ by :
- - $\s ((u)v)=(\s (u))\s (v)$ if $v \not \in P$ ;
- - $\s (\l xu)=\l x \s (u)$ ;
- - $\s ((t)p)=(t)\sou{t}$ if $\s (p)=\sou{t}$.
$\s$ is said to be a $P$-substitution.\
We consider, on the set $\L CP$, the following rules of reduction :
- 1\) $(\l xu)tt_1...t_n \f (u[t/x])t_1...t_n$ for all $u,t \in \L CP$ and $t_1,...,t_n \in \L
CP \bigcup P$ ;
- 2\) $(C)tt_1...t_n \f (t)\l x(x)t_1...t_n$ for all $t \in \L CP$ and $t_1,...,t_n \in \L CP
\bigcup P$, and $x$ being $\l$-variable not appearing in $t_1,...,t_n$.
For any $t,t' \in \L CP$, we shall write $t \rhd_C t'$, if $t'$ is obtained from $t$ by applying these rules finitely many times.
If | 1 | member_40 |
---
abstract: 'In this paper, we classify Gorenstein stable log surfaces with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1$.'
address: 'Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084'
author:
- Jingshan Chen
title: ' Gorenstein stable log surfaces with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1$'
---
Introduction
============
KSBA stable (log) surfaces are the two-dimensional analogues of stable (pointed) curves. They are the fundamental objects in compactifying the moduli spaces of smooth surfaces of general type.
In general, stable (log) surfaces are difficult to classify. We may first focus on Gorenstein ones, i.e. $K_X$ (resp. $K_X+\Lambda$) being Cartier. In [@LR13], Liu and Rollenske give several inequalities for the invariants of Gorenstein stable log surfaces. One important inequality among them is the stable log Noether inequality $(K_X+\Lambda)^2\ge p_g(X,\Lambda)-2$ (see [@LR13 Thm 4.1]). This can be rephrased as $\Delta(X,K_X+\Lambda)\ge 0 $, where $\Delta$ is Fujita’s $\Delta$-genus.
Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=0$ have been classified in [@LR13]. Normal Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=1$ have been classified in [@Chen18]. Here we continue to classify non-normal Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=1$. The main result is as follows.
\[Main Theorem \] Let $(X,\Lambda)$ be a Gorenstein stable log surface with $\Delta(X,K_X+\Lambda)=1$ which is not normal.
If $X$ is irreducible, let $\pi\colon \bar{X}\to X$ be | 1 | member_41 |
the normalization map. Then
- either $\Delta(\bar{X},\pi^*(K_X+\Lambda))=1$ and $(X,\Lambda)$ is as in Thm \[delta1,1\],
- or $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$ and $(\bar{X},\bar{\Lambda}+\bar{D})$ is as in Thm \[delta1,0\].
If $X$ is reducible, write $X=\bigcup X_i$, where $X_i$ is an irreducible component. Then $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$ or $0$ for each component $X_i$, $X$ has a unique minimal connected component $U$ such that $\Delta(U,(K_X+\Lambda)|_{U})=1$, $X\setminus U$ is composed with several trees $T_j$ with $\Delta(T_j,(K_X+\Lambda)|_{T_j})=0$ and $X$ is glued by $U$ and $T_j$ along lines.
Moreover, $U$ is one of the followings:
- $X=U$ is a string of log surfaces $X_i$ with $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$ and $|(K_{X}+\Lambda)|_{X_i}|$ composed with a pencil of elliptic curves. The connecting curves are all fibers. Moreover, in this case $|K_X+\Lambda|$ is composed with a pencil of elliptic curves.
- $U$ is a string of surfaces glued along lines. The end surfaces $X_i$ of the string $U$ are non-normal with $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$.
- $U$ is composed with a single irreducible log surface $X_k$ with $\Delta(X_k,(K_X+\Lambda)|_{X_k})=1$ and $(K_{X}+\Lambda)|_{X_k}$ very ample. Moreover, in this case $K_X+\Lambda$ is very ample.
- $U=X_j\cup X_k $ where $X_j$, $X_k$ are two Gorenstein log surfaces with $\Delta(X_j,(K_X+\Lambda)|_{X_j})=\Delta(X_k,(K_X+\Lambda)|_{X_k})=0$. All the connecting curves of $X$ are lines except $X_j\cap X_k$.
- $U$ is a cycle of | 1 | member_41 |
log surface $X_i$ with $\Delta(X_i,(K_X+\Lambda)|_{X_i})=0$. All the connecting curves of $X$ are lines.
Now we give a brief account of each section. In \[prelim\], we recall some definitions and facts about Gorenstein stable log surfaces. In \[zero-Delta-genus\], we recall the definition of Fujita’s $\Delta$-genus and include some results about normal Gorenstein stable log surfaces $(X,\Lambda)$ with $\Delta(X,K_X+\Lambda)=0$ or $1$. In \[non-normal\], we include some results about non-normal Gorenstein stable log surfaces. In \[nnorm\], we deal with the case that $X$ is non-normal and irreducible. In \[reducible-stable\], we deal with the case that $X$ is reducible. Finally we describe Gorenstein stable surfaces with $K_X^2= p_g-1$.
Acknowledgements: {#acknowledgements .unnumbered}
-----------------
I am grateful to Prof. Jinxing Cai and Prof. Wenfei Liu for their instructions. I would also thank the anonymous referee for helpful advices and suggestions.
Notations and conventions
-------------------------
We work exclusively with schemes of finite type over the complex numbers $\mathbb{C}$.
- A surface is a connected reduced projective schemes of pure dimension two.
- By abuse of notation, we sometimes do not distinguish a Cartier divisor $D$ and its associated invertible sheaf $\mathcal{O}_X(D)$.
- $\Sigma_d$ denotes a Hirzebruch surface, which admits a $\mathbb{P}^1$ fibration over $\mathbb{P}^1$. We denote $\Gamma$ | 1 | member_41 |
as a fiber. It has a unique 1-section $\Delta_0$ whose self-intersection is $-d$.
- We use ’$\equiv$’ to denote linear equivalent relation of divisors.
- If $D$ is a Cartier divisor on $X$, then we denote $\Phi_{|D|}\colon X\dashrightarrow \mathbb{P}:=|D|^*$ as the rational map defined by the linear system $|D|$.
- A line $l$ on a variety $X$ with respect to $\mathcal{O}_X(1)$ is a rational curve such that $l\cdot \mathcal{O}_X(1)=1$.
Preliminaries {#prelim}
=============
Let $X$ be a demi-normal surface, i.e. $X$ satisfies $S_2$ and is at worst ordinary double at any generic point of codimension 1. Denote $\pi\colon \bar X \to X$ as the normalisation map of $X$. The conductor ideal $ \mathrm{\mathcal{H}om}_{\mathcal{O}_X}(\pi_*\mathcal{O}_{\bar{X}}, \mathcal{O}_X)$ is an ideal sheaf both on $X$ and $\bar{X}$ and hence defines subschemes $D\subset X \text{ and } \bar D\subset \bar X$, both reduced and of pure codimension 1; we often refer to $D$ as the non-normal locus of $X$.
Let $\Lambda$ be a reduced curve on $X$ whose support does not contain any irreducible component of $D$. Then the strict transform $\bar \Lambda$ in the normalization is well defined. We have $\pi^*(K_X+\Lambda)=K_{\bar{X}}+\bar D+\bar \Lambda$ and $(K_X+\Lambda)^2 = (K_{\bar X}+\bar D+\bar \Lambda)^2$.
\[defin: slc\] We call a | 1 | member_41 |
pair $(X, \Lambda)$ as above a *log surface*; $\Lambda$ is called the (reduced) boundary.
A log surface $(X,\Lambda)$ is said to have *semi-log-canonical (slc)* singularities if it satisfies the following conditions:
1. $K_X + \Lambda$ is $\mathbb{Q}$-Cartier, that is, $m(K_X+\Lambda)$ is Cartier for some $m\in\mathbb{Z}^{>0}$; the minimal such $m$ is called the (global) index of $(X,\Lambda)$.
2. The pair $(\bar X, \bar \Lambda+\bar D)$ has log-canonical singularities.
The pair $(X,\Lambda)$ is called stable log surface if in addition $K_X+\Lambda$ is ample. A stable surface is a stable log surface with empty boundary.
By abuse of notation we say $(X, \Lambda)$ is a Gorenstein stable log surface if the index is equal to one, i.e. $K_X+\Lambda$ is an ample Cartier divisor.
Gorenstein slc singularities and semi-resolutions
-------------------------------------------------
Normalizing a demi-normal surface looses all information on the gluing in codimension one. Often it is better to work on a simpler but still non-normal surface.
A surface $X$ is called semi-smooth if every singularity of $X$ is either double normal crossing or a pinch point [^1]. The normalization of a semi-smooth surface is smooth.
A morphism of demi-normal surfaces $f\colon Y\rightarrow X$ is called a semi-resolution if the following conditions are satisfied:
1. | 1 | member_41 |
$Y$ is semi-smooth;
2. $f$ is an isomorphism over the semi-smooth open subscheme of $X$;
3. $f$ maps the singular locus of $Y$ birationally onto the non-normal locus of $X$.
A semi-resolution $f\colon Y\rightarrow X$ is called minimal if no $(-1)$-curve is contracted by $f$, that is, there is no exceptional curve $E$ such that $E^2 =K_Y\cdot E = -1$.
Semi-resolutions always exist and one can also incorporate a boundary [@KollarSMMP 10.5].
\[rem: classification of sings\] Semi-log-canonical surface singularities have been classified in terms of their resolution graphs, at least for reduced boundary [@KSB88].
Let $x\in (X, \Lambda)$ be a Gorenstein slc singularity with minimal log semi-resolution $f\colon Y\to X$. Then it is one of the followings (see [@Kollar-Mori Ch. 4], [@KollarSMMP Sect. 3.3], [@kollar12] and [@LR13]):
Gorenstein lc singularities, $\Lambda=0$
: In this case $x\in X$ is a canonical singularity, or a simple elliptic respectively cusp singularity. For the latter the resolution graph is a smooth elliptic curve, a nodal rational curve, or a cycle of smooth rational curves (see also [@Lau77] and [@Reid97 Ch. 4]).
Gorenstein lc singularities, $\Lambda\neq 0$
: Since the boundary is reduced, $\Lambda$ has at most nodes. If $\Lambda$ is smooth so is $X$ | 1 | member_41 |
because of the Gorenstein assumption.
If $\Lambda$ has a node at $x$ then $x$ is a smooth point of $X$ or $(X, \Lambda)$ is a general hyperplane section of a cyclic quotient singularity. In the minimal log resolution the dual graph of the exceptional curves is $$\bullet \ {-}\ c_1 \ -\ \cdots \ - \ c_n \ {-}
\ \bullet \qquad (c_i\geq1),$$ where $c_i$ represents a smooth rational curve of self-intersection $-c_i$ and each $\bullet$ represents a (local) component of the strict transform of $\Lambda$. If $c_i=1$ for some $i$ then $n=1$ and $\Lambda$ is a normal crossing divisor in a smooth surface.
non-normal Gorenstein slc singularities, $\Lambda=0$
: We describe the dual graph of the $f$-exceptional divisors over $x$: analytically locally $X$ consists of $k$ irreducible components, on each component we have a resolution graph as in the previous item, and these are glued together where the components intersect. In total we have a cycle of smooth rational curve.
non-normal Gorenstein slc singularities, $\Lambda\neq 0$
: The difference to the previous case is that the local components are now glued in a chain and the ends of the chain intersect the strict transform of the boundary. In this | 1 | member_41 |
case $X$ itself might not even be $\mathbb{Q}$-Gorenstein.
Normal Gorenstein stable log surfaces with small $\Delta$-genus {#zero-Delta-genus}
===============================================================
Let $X$ be a variety and $\mathcal{L}$ be an ample line bundle on it. Fujita introduced several invariants for such polarized varieties. One important of them is the $\Delta$-genus $\Delta(X,\mathcal{L}):=\mathcal{L}^{\dim X}-h^0(X,\mathcal{L})+\dim X$.
\[gepg-2\] Let $(X,\Lambda)$ be a normal irreducible Gorenstein stable log surface. Then $\Delta(X,K_X+\Lambda)\ge 0$.
Moreover if ’=’ holds, then $(X,\Lambda)$ is one of the followings:
- $X$ is $\mathbb{P}^{2}$, $\mathcal{O}_X(K_X+\Lambda) =\mathcal{O}_{\mathbb{P}^{2}}(1)$ and $\Lambda\in|\mathcal{O}_{\mathbb{P}^{2}}(4)|$;
- $X$ is $\mathbb{P}^{2}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^{2}}(2)$ and $\Lambda\in|\mathcal{O}_{\mathbb{P}^{2}}(5)|$;
- $X$ is $\Sigma_d$, $K_X+\Lambda\equiv \Delta_0+\frac{N+d-1}{2}\Gamma$ and $\Lambda\in|3\Delta_0+\frac{N+3d+3}{2}\Gamma|$; ($N-d-3\ge0$ is an even number);
- $X$ is a singular quadric $C_2$ in $\mathbb{P}^{3}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{C_2}(1)=\mathcal{O}_{\mathbb{P}^{2}}(1)|_{C_2}$ and $\Lambda\in|\mathcal{O}_{C_2}(3)|$;
- $X$ is a cone $C_{N-1}\hookrightarrow \mathbb{P}^{N}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^{N}}(1)|_{C_{N-1}}$ and the proper transformation $\bar{\Lambda}$ in the minimal resolution $\Sigma_{N-1}$ is linearly equivalent to $2\Delta_0+2N\Gamma$. ($N>3$)
\[delta-genus-one\] Let $(X,\Lambda)$ be a normal Gorenstein stable log surface with $(K_X+\Lambda)^2= p_g(X,\Lambda)-1$.
Then $(X,\Lambda)$ is one of the followings:
- $X$ is a double cover of $\mathbb{P}^2$. $\Phi_{|K_X+\Lambda|}\colon X\to \mathbb{P}^2$ is the double covering map. The branch curve $B\in |\mathcal{O}_{\mathbb{P}^2}(2k)|$ is a reduced curve which admits curve singularities of lc double-covering type, and $\Lambda\in |{\Phi_{|K_X+\Lambda|}}^*\mathcal{O}_{\mathbb{P}^2}(4-k)|$. ($p_g(X,\Lambda)=3$, and $k=2,3,4$)
- $X$ | 1 | member_41 |
is a quadric in $\mathbb{P}^3$, and $\Lambda\in |\mathcal{O}_{\mathbb{P}^3}(4)|_X|$. ($p_g(X,\Lambda)=6$)
- $X$ is $\mathbb{P}^2$ blown up at $k$ points (possible infinitely near), and $\Lambda\in |-2K_X|$. ($p_g(X,\Lambda)=10-k$, and $k=0,1,...,7$)
- $X$ is a cone over an elliptic curve of degree $N$ in $\mathbb{P}^{N-1}$, and $\Lambda\in |-2K_X|$. ($p_g(X,\Lambda)=N$)
- $X$ is obtained from $\tilde{X}$ by contracting a $(-n)$ curve $G$, where $n=p_g(X,\Lambda)-1$ and $\tilde{X}$ is an elliptic surface (possible singular) with $G$ as the rational zero section. Every elliptic fiber of $\tilde{X}$ is irreducible. $\Lambda$ is the image of a sum of two different elliptic fibers which admit at worst $A_n$ type singularities.
- $X$ is a (possibly singular) Del Pezzo surface of degree 1, namely $X$ has at most canonical singularities and elliptic singularities, $-K_X$ is ample and $K_X^2=1$. The curve $\Lambda$ belongs to the system $|-2K_X|$, and $p_a(\Lambda)=2$. $p_g(X,\Lambda)=2$.
- $\Lambda=0$, $|K_X+\Lambda|$ is composed with a pencil of genus $2$ curves. $X$ is canonically embedded as a hypersurface of degree 10 in the smooth locus of $\mathbb{P}(1,1,2,5)$. $p_g(X)=2$.
In Theorem 1.1(1) and Theorem 4.3(i), $k=1,2,3,4$ should be corrected by $k=2,3,4$. The case $k=1$ is excluded as $\Phi_{|K_X+\Lambda|}$ would be an embedding.
Non-normal Gorenstein stable log surfaces {#non-normal}
=========================================
Let $(X,\Lambda)$ be | 1 | member_41 |
a Gorenstein stable log surface which is non-normal. $D$ is the non-normal locus of $X$.
Let $C\subset D$ be a subcurve of the non-normal locus of a Gorenstein stable log surface $(X,\Lambda)$ and $\nu\colon:\tilde{X}\to X$ be the partial normalization of $X$ along $C$. Denote $\tilde{C}$ and $\tilde{\Lambda}$ as the proper transformation of $C$ and $\Lambda$.
Then $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ is a Gorenstein stable log surface.
We first notice that $\nu^*(K_X+\Lambda)=K_{\tilde{X}}+\tilde{\Lambda}+\tilde{C}$ is ample and Cartier. Second it is easy to verify that $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ still has Gorenstein slc singularities only by the classification of Gorenstein slc singularities. Therefore $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ is a Gorenstein stable log surface as well.
The map $\nu|_{\tilde C}\colon \tilde{C} \to C$ is generically a double cover and thus induces a rational involution $\tau$ on $\tilde{C}$.
\[prop: descend section\] If $(X, \Lambda)$ is a non-normal Gorenstein stable log surface and $\nu$, $C$, $\tilde{C}$, $\tau$ are defined as above, then $\nu^*H^0(X, K_X+\Lambda)\subset H^0(\tilde{X}, \nu^*(K_X+\Lambda)))$ is the subspace of those sections $s$ whose restriction to $\tilde{C}$ is $\tau$-anti-invariant.
\[separateNonnormalCurve\] It is easy to see that $\nu^*H^0(X, K_X+\Lambda)$ can not separate $\tilde{C}$. Actually, $\Phi_{\nu^*H^0(X, K_X+\Lambda)}|_{\tilde{C}}\circ \tau=\Phi_{\nu^*H^0(X, K_X+\Lambda)}|_{\tilde{C}}$ on $\tilde{C}$.
\[fibersection\] Let $(X,\Lambda)$ be a reducible Gorenstein stable log surface such that $X=X_1\cup X_2$ and $C:=X_1\cap | 1 | member_41 |
X_2$ is the connecting curve. Write $(K_X+\Lambda)|_{X_i}:=(\pi^*(K_X+\Lambda))|_{X_i}=K_{X_i}+C+\Lambda|_{X_i}$.
Let $\mathcal{R}_{X_i\to C}\colon H^0(X_i,(K_X+\Lambda)|_{X_i})\to H^0(C,(K_X+\Lambda)|_C)$ be the restriction map. Then we have the following fiber product diagram of vector spaces $$\xymatrix{
H^0(X, K_X+\Lambda) \ar[r]\ar[d] & H^0(X_1,(K_X+\Lambda)|_{X_1})\ar[d]^{\mathcal{R}_{X_1\to C}}\\
H^0(X_2,(K_X+\Lambda)|_{X_2})\ar[r]^{-\mathcal{R}_{X_2\to C}}& H^0(C, (K_X+\Lambda)|_C)
}.$$
Moreover, denote $r_{X_i\to C}((K_X+\Lambda)|_{X_i}):=\dim\mathcal{R}_{X_i\to C}(H^0(X_i,(K_X+\Lambda)|_{X_i}))$. We have $$\begin{aligned}
h^0(X, K_X+\Lambda)\le& h^0(X_1,(K_X+\Lambda)|_{X_1})+h^0(X_2,(K_X+\Lambda)|_{X_2})\\
&-\max\{r_{X_1\to C}((K_X+\Lambda)|_{X_1}),r_{X_2\to C}((K_X+\Lambda)|_{X_2})\}.\end{aligned}$$
irreducible Non-normal Gorenstein stable surfaces with $\Delta(X,K_X+\Lambda)=1$ {#nnorm}
================================================================================
Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface and $\bar{X}$, $\pi$, $D$, $\bar{D}$, $\tau$ defined as before. Since $H^0(X,K_X+\Lambda)\cong \pi^*H^0(X,K_X+\Lambda) \subset H^0(\bar X,\pi^*(K_X+\Lambda))$, we have $\Delta(X,K_X+\Lambda)\ge \Delta(\bar{X},\pi^*(K_X+\Lambda))$, and ’=’ holds if and only if $\pi^*H^0(X,K_X+\Lambda) = H^0(\bar X,\pi^*(K_X+\Lambda))$.
\[nonnorm&irred\] Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface.
Then $\Delta(X,K_X+\Lambda)\ge 1$.
We only need to show that the case $\Delta(X,K_X+\Lambda)=\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$ does not occur. First, $h^0(X,K_X+\Lambda)=h^0(\bar{X},\pi^*(K_X+\Lambda))$ implies every section in $H^0(\bar{X},\pi^*(K_X+\Lambda))$ is $\tau$-anti-invariant restricting to $\bar{D}$. Hence $H^0(\bar{X},\pi^*(K_X+\Lambda))$ can not separate points of $\bar{D}$ by Remark \[separateNonnormalCurve\]. However, $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$ implies $\pi^*(K_X+\Lambda)$ is very ample by Cor \[gepg-2\], therefore it will separate points of $\bar{D}$, a contradiction.
\[delta1,1\]
Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface as before. Assume $\Delta(X,K_X+\Lambda)=\Delta(\bar{X},\pi^*(K_X+\Lambda))=1$. Then
- either $X$ is a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branched | 1 | member_41 |
curve is $2C+B\in |\mathcal{O}_{\mathbb{P}^2}(2m+2k)|$, where $C,B$ are reduced curves of degree $m,2k$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$. $k=2,3$. $0<m\le 4-k$; ($p_g(X,\Lambda)=3$)
- or $\Lambda=0$, $K_X^2=(K_{\bar{X}}+\bar D)^2=1$. $(\bar{X},\bar{D})$ is a normal Gorenstein stable log surface as in Thm \[delta-genus-one\] (6). $\bar D$ is a 2 section with $p_a(\bar D)=2$. $\tau$ is induced from the double covering $\bar D \to \mathbb{P}^1$. ($p_g(X)=2$)
We see that $\pi^*H^0(X,K_X+\Lambda)=H^0(\bar{X},\pi^*(K_X+\Lambda))$ and $(\bar{X},\bar D+\bar{\Lambda})$ is a stable log surface as in Thm \[delta-genus-one\].
First $\Phi_{|\pi^*(K_X+\Lambda)|}$ is not an embedding by Remark \[separateNonnormalCurve\]. Second $\Phi_{|\pi^*(K_X+\Lambda)|}$ is not composed with a pencil of genus 2 curves since $\bar D+\bar{\Lambda}\not= 0$.
If $\Phi_{|\pi^*(K_X+\Lambda)|}$ is composed with a pencil of elliptic curves, $\bar{D}+\bar{\Lambda}$ is either a 2-section or a sum of two elliptic fibers of the fibration map. For the former case, we see that $\bar{\Lambda}=0$ and $\bar{D}$ is a genus 2 curve by Remark \[separateNonnormalCurve\]. Moreover, $(K_{\bar X}+\bar D)^2=1$. $\tau$ is induced from the double map $\bar{D} \to \mathbb{P}^1$. For the latter case, we see that either $\bar{D}=F_1+F_2$ and $\bar{\Lambda}=0$, or $\bar{D}=F_1$ and $\bar{\Lambda}\not=0$, where $F_i$ is a fiber of the fibration map. Both cases can be excluded since the $F_1\cap F_2$ or $F_1\cap\bar{\Lambda}$ on $\bar{X}$ can not be glued into a Gorenstein | 1 | member_41 |
slc singularity on $X$.
If $\bar{X}$ is a double cover of $\mathbb{P}^2$, then $\bar{D}+\bar{\Lambda}\in |\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^*\mathcal{O}_{\mathbb{P}^2}(4-k)|$, $k=2,3$. Remark \[separateNonnormalCurve\] indicates $\bar{D}=\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^{-1}(C)$, where $C$ is a curve on $\mathbb{P}^2$. Denote $m$ as the degree of $C$. Then $\bar{\Lambda}\in |\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$. We have the following commutative diagram: $$\xymatrix{
\bar{D}\ar@{^{(}->}[r]\ar[d] & \bar{X}\ar[d]^{\pi}\ar[rr]^{\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}} && \mathbb{P}^2\\
D\ar@{^{(}->}[r] & X \ar[rru]_{\Phi_{|K_{X}+\Lambda|}}
}.$$
We see that $X$ is also a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve is $2C+B\in |\mathcal{O}_{\mathbb{P}^2}(2m+2k)|$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$.
Next we consider those irreducible non-normal Gorenstein stable log surfaces $(X,\Lambda)$ with $\Delta(X,K_X+\Lambda)=1$ and $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$.
We first notice that $\pi^*H^0(X,K_X+\Lambda)\subset H^0(\bar{X},\pi^*(K_X+\Lambda))$ is of codimension one. Hence it has a base point $c\in \mathbb{P}(H^0(\bar{X},\pi^*(K_X+\Lambda)))$. We have the following commutative diagram: $$\xymatrix{
\bar{X} \ar@{^{(}->}[rrr]^<(0.3){\Phi_{|\pi^*(K_X+\Lambda)|}}\ar[d]^{\pi} & & & \mathbb{P}(H^0(\bar{X},\pi^*(K_X+\Lambda)))\ar@{-->}[d]^{pr_c}\\
X\ar@{-->}[rrr]^<(0.3){\Phi_{|K_X+\Lambda|}}& & & \mathbb{P}(\pi^*H^0(X,K_X+\Lambda))
}.$$
We regard $\Phi_{|\pi^*(K_X+\Lambda)|}$ as an inclusion. To describe $pr_c|_{\bar{X}}$, we use the following theorem (see [@Ber06 Thm 2.1]):
\[linearsectionthm\] Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^r$ and $L$ be a linear subspace of $\mathbb{P}^r$ of dimension $s\le r$ such that $L\cap X$ is a 0-dimensional scheme $\zeta$. Then $\mathrm{length}\, \zeta \le \Delta(X,\mathcal{O}(1))+s+1$.
\[lineIntersecting\] Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^r$ with $\Delta(X,\mathcal{O}(1))=0$ and $L$ is a line intersecting $X$ | 1 | member_41 |
along a 0-dimensional scheme $\zeta$. Then $\mathrm{length}\, \zeta\le 2$.
Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^N$ with $\Delta(X,\mathcal{O}(1))=0$ and $c\in \mathbb{P}^N$ be a point outside $X$. Denote $pr_c\colon \mathbb{P}^N\dashrightarrow \mathbb{P}^{N-1}$ as the projection from $c$. Assume $pr_c|_X$ is birational and is not an isomorphism. Denote by $W$ the image of $pr_c|_X$. Then $W$ is non-normal and $pr_c|_X\colon X\to W$ is a normalization map. The non-normal locus $D$ of $W$ is a line in $\mathbb{P}^{N-1}$. The pre-image $\bar{D}\colon = pr_c|_X^{-1}(D)$ is described as follows:
- if $X$ is $\mathbb{P}^2$ embedded in $\mathbb{P}^5$, then $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$.
- if $X$ is a cone $C_{N-1}\hookrightarrow \mathbb{P}^N$, then $\bar{D}$ is a sum of two rulings.
- if $X$ is $\Sigma_d\hookrightarrow \mathbb{P}^N$, either $\bar{D}\in |\Delta_0+\Gamma|$, $N=d+3$, or $\bar{D}=\Delta_0$, $N=d+5$.
$pr_c|_X\colon X\to W$ is a normalization map by Cor \[lineIntersecting\].
We show that $D$ is a line in $\mathbb{P}^{N-1}$ by contradiction hypothesis. Assume $D$ is not a line, then there are two points $p,q\in D$ such that the line $L_{p,q}$ passing $p,q$ is not contained in $D$. Let $H_{p,q}\subset \mathbb{P}^N$ be the pre-image of $L_{p,q}$ with respect to $pr_c$. Then $H_{p,q}\cap X$ has length great than $4$ which contradicts Thm \[linearsectionthm\]. Therefore $D$ is a | 1 | member_41 |
line in $\mathbb{P}^{N-1}$.
Finally, we describe $\bar{D}$. For the case where $X$ is a cone $C_{N-1}$, $\bar{D}$ must be a sum of two rulings. This can be shown by considering the plane $H_{v, L}\subset \mathbb{P}^{N-1}$ containing $v,L$, where $v$ is the vertex of $C_{N-1}$ and $L$ is a secant line of $X$ passing through $c$.
For the case where $X$ is $\mathbb{P}^2$ embedded in $\mathbb{P}^5$, we have $\bar{D}\cdot \mathcal{O}_{X}(1)=2$ since $D$ is a line in $\mathbb{P}^{N-1}$. Hence $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$.
For the case where $X$ is $\Sigma_d\hookrightarrow \mathbb{P}^N$, similarly as the former case we have $\bar{D}\cdot \mathcal{O}_{X}(1)=\bar{D}\cdot(\Delta_0+\frac{N+d-1}{2}\Gamma)=2$. Hence we have either $\bar{D}\in |\Delta_0+\Gamma|$, $N=d+3$, or $\bar{D}=\Delta_0$, $N=d+5$.
We first consider the cases where $c\in \bar{X}$.
Case I) $\bar{X}$ is $\mathbb{P}^2$ and $\pi^*(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^2}(1)$. Then $\bar{\Lambda}+\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$ and $pr_c|_{\bar{X}}$ is a fibration over $\mathbb{P}^1$. By the classification of Gorenstein slc singularities, we see that $\bar{\Lambda}\cdot\bar{D}$ is even. Hence we have either $\bar{\Lambda},\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(2)|$ or $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$ and $\bar{\Lambda}=0$. Examples will be given later. We note further that $c\not\in\bar{D}$, otherwise it can not be glued into a Gorenstein slc singularity. Case II) $\bar{X}$ is $C_{N-1}$ and $c$ is the vertex. We see that $\bar{\Lambda}$ is two rulings and $\bar{D}\in \mathcal{O}_{C_{N-1}}(2)$.
Case III) $\bar{X}$ is | 1 | member_41 |
$C_{N-1}$ and $c$ is not the vertex. We show that this does not occur. By Cor \[lineIntersecting\] $pr_c|_{\bar{X}}$ is an isomorphism outside the ruling $l$ passing $c$. Hence $\bar{D}$ must be $l$. However this is impossible as the vertex of $C_{N-1}$ would not be glued into a Gorenstein slc singularity.
Case IV) $\bar{X}$ is $\Sigma_{d}$. We show that this does not occur as well. Similarly as in Case III), we see that $\bar{D}$ should be a ruling $\Gamma$ passing through $c$. Hence $\bar{D}\cdot \bar{\Lambda}=3$, which is impossible.
Case V) $\bar{X}$ is $\mathbb{P}^2$ embedding in $\mathbb{P}^5$. We show that this does not occur. Since by Cor \[lineIntersecting\], we see that $pr_c|_{\bar{X}}$ is an isomorphism outside $c$, which is impossible.
Next we consider the cases where $c\not\in \bar{X}$. We see $pr_c|_{\bar{X}}$ is a morphism of degree at most two by Cor \[lineIntersecting\] and it contracts no curve on $\bar{X}$.
Case VI) $pr_c|_{\bar{X}}$ is a double cover. In this case $\bar{X}$ is a quadric in $\mathbb{P}^3$. Denote $pr_c|_{\bar{X}}$ as $\delta$. The branch curve $B\in |\mathcal{O}_{\mathbb{P}^2}(2)|$. $K_{\bar{X}}+\bar{D}+\bar{\Lambda}=\mathcal{O}_{\bar{X}}(1)=\delta^*\mathcal{O}_{\mathbb{P}^2}(1)$. Hence $\bar{D}+\bar{\Lambda}\in |\delta^*\mathcal{O}_{\mathbb{P}^2}(3)|=|\mathcal{O}_{\bar{X}}(3)|$. $\bar{D}$ is the pre-image of a curve $C$ on $\mathbb{P}^2$ under $pr_c|_{\bar{X}}$. Denote $m$ as the degree of $C$. $m\le 3$. We have the | 1 | member_41 |
following commutative diagram: $$\xymatrix{
\bar{D}\ar@{^{(}->}[r] &\bar{X}\ar[d]^{\pi}\ar[drr]^{pr_c|_{\bar{X}}}\ar@{^{(}->}[rr]^{\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}} && \mathbb{P}^3 \ar@{-->}[d]^{pr_c}\\
& X \ar[rr]_{\Phi_{|K_{X}+\Lambda|}} && \mathbb{P}^2
}.$$ We see that $X$ is also a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve ia $2C+B$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(3-m)|$.
If $pr_c|_{\bar{X}}$ is birational, it must be a normalisation map. Its image must be $X$ and $\pi=pr_c|_{\bar{X}}$.
Case VII) $\bar{X}$ is $\mathbb{P}^2$ embedding in $\mathbb{P}^5$. The non-normal locus $D$ is a line in $\mathbb{P}^5$ and $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$. $\bar{\Lambda}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$.
Case VIII) $\bar{X}$ is $C_{N-1}$, $N>3$. $\bar{D}$ is two rulings. $\bar{\Lambda}$ is linearly equivalent to $2H_{\infty}$ where $H_{\infty}$ is a hyperplane.
Case IX) $\bar{X}$ is $\Sigma_{d}$ embedded in $\mathbb{P}^N$ , $N>3$. Either $N=d+3$, $\bar{D}\in |\Delta_0+\Gamma|$ and $\bar{\Lambda}\in |2\Delta_0+(2d+2)\Gamma|$; or $N=d+5$, $\bar{D}=\Delta_0$ and $\bar{\Lambda}\in |2\Delta_0+(2d+4)\Gamma|$.
We summarize the above results in the following theorem:
\[delta1,0\] Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface as before. Assume $\Delta(X,K_X+\Lambda)=1$ and $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$. Then there are the following possibilities:
- $\bar{X}$ is $\mathbb{P}^2$. $\bar{\Lambda}\in |\mathcal{O}_{\mathbb{P}^2}(2)|$ and $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(2)|$. Moreover, $c\not\in \bar{D}$. ($p_g(X,\Lambda)=2$)
- $\bar{X}$ is $\mathbb{P}^2$. $\bar{\Lambda}=0$ and $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. Moreover, $c\not\in \bar{D}$. ($p_g(X,\Lambda)=2$)
- $\bar{X}$ is $C_{N-1}$. $\bar{\Lambda}$ is two rulings and $\bar{D}\in \mathcal{O}_{C_{N-1}}(2)$. $c$ is the vertex of $C_{N-1}$. ($p_g(X,\Lambda)=N$)
- $\bar{X}$ is a quadric in | 1 | member_41 |
$\mathbb{P}^3$. $\bar{D}\in |\mathcal{O}_{\bar{X}}(m)|$ and $\bar{\Lambda} \in |\mathcal{O}_{\bar{X}}(3-m)|$. $X$ is a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve is $2C+B$, where $C$,$B$ are reduced curves of degree $m$, $2$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(3-m)|$. $0<m \le 3$. ($p_g(X,\Lambda)=3$)
- $X$ is the projection image of a Veronese embedding of $\mathbb{P}^2$. $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$ and $\bar{\Lambda}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. ($p_g(X,\Lambda)=5$)
- $X$ is the projection image of $C_{N-1}\subset\mathbb{P}^N$, $N>3$. $\bar{D}$ is a sum of two lines passing the vertex, and $\bar{\Lambda}\in \mathcal{O}_{C_{N-1}}(2)$. ($p_g(X,\Lambda)=N$)
- $X$ is the projection image of $\Sigma_d$ embedded in $\mathbb{P}^N$, $N>3$. $D\subset X$ is a line. We have either $N=d+3$, $\bar{D}\in |\Delta_0+\Gamma|$ and $\bar{\Lambda}\in |2\Delta_0+(2d+2)\Gamma|$, or $N=d+5$, $\bar{D}=\Delta_0$ and $\bar{\Lambda}\in |2\Delta_0+(2d+4)\Gamma|$. ($p_g(X,\Lambda)=N$)
Let $\bar{D}$ be a smooth quadric in $\mathbb{P}^2$ defined by $x^2+y^2+z^2=0$. $\tau$ acts on $\bar{D}$ by $x\mapsto -x, y \mapsto -y, z\mapsto z$. Let $\bar{\Lambda}=L_{x=0}+L_{y=0}$, where $L_{x=0}$, $L_{y=0}$ are two lines. Gluing $\mathbb{P}^2$ along $\bar{D}$ by $\tau$ we get a non-normal Gorenstein stable log surface $(X,\Lambda)$ with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1=1$.
Let $\bar{D}$ be a smooth quartic in $\mathbb{P}^2$ defined by $x^4+y^4+z^4=0$. $\tau$ acts on $\bar{D}$ by $x\mapsto -x, y \mapsto -y, z\mapsto z$. Gluing $\mathbb{P}^2$ along $\bar{D}$ by $\tau$ we get a non-normal Gorenstein stable surface $X$ with $K_X^2=p_g-1=1$.
\[irrnonnormalstabledelta1\] Let | 1 | member_41 |
$X$ be an irreducible non-normal Gorenstein stable surface with $\Delta(X,K_X)=1$. Then $X$ is one of the followings:
- $X$ is a double cover of $\mathbb{P}^2$. The branch curve is $2C+B$, where $C$,$B$ are reduced curves of degree $4-k$, $2k$. $k=2,3$. ($p_g(X)=3$)
- $X$ is obtained from a log surface $(\bar{X},\bar{D})$ by gluing the 2-section $\bar{D}$. $(\bar{X},\bar{D})$ is a normal Gorenstein stable log surface as in Thm \[delta-genus-one\] (6). ($p_g(X)=2$)
- $\bar{X}$ is $\mathbb{P}^2$. $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. ($p_g(X)=2$)
- $\bar{X}$ is a quadric in $\mathbb{P}^3$. $\bar{D}\in |\mathcal{O}_{\bar{X}}(3)|$. ($p_g(X)=3$)
reducible Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=1$ {#reducible-stable}
=======================================================================
In this section we consider reducible Gorenstein stable log surfaces. They are glued by irreducible ones along some connecting curves.
\[restsecions\] Let $X$ be a connected $S_2$ scheme of pure dimension, $\mathcal{L}$ be an invertible sheaf such that $\dim \mathrm{Bs} |\mathcal{L}|<\dim X-1$ and $C$ be a subscheme of codimension 1. Then $$\begin{aligned}
r_{X\to C}(\mathcal{L})=\dim <\Phi_{|\mathcal{L}|}(C)>+1,\end{aligned}$$ where $<\Phi_{|\mathcal{L}|}(C)>$ is the projective subspace of $|\mathcal{L}|^*$ spanned by $\Phi_{|\mathcal{L}|}(C)$.
Denote $\mathbb{P}:=|\mathcal{L}|^*$. We have the following commutative diagram: $$\xymatrix{
H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(1))\ar[rr]^<(0.2){\mathcal{R}_{\mathbb{P}\to \Phi_{|\mathcal{L}|}(C)}}\ar[d]^{\cong}_{\Phi_{|\mathcal{L}|}^*} & & H^0(\Phi_{|\mathcal{L}|}(C),\mathcal{O}_{\mathbb{P}}(1)|_{\Phi_{|\mathcal{L}|}(C)})\ar@{^{(}->}[d]_{\Phi_{|\mathcal{L}|}^*} \\
H^0(X,\mathcal{L})\ar[rr]^{\mathcal{R}_{X\to C}}& & H^0(C,\mathcal{L}|_C)
}.$$ Therefore $r_{X\to C}(\mathcal{L})=r_{\mathbb{P}\to \Phi_{|\mathcal{L}|}(C)}(\mathcal{O}_{\mathbb{P}}(1))=\dim <\Phi_{|\mathcal{L}|}(C)>+1$.
Still we call a curve $C$ on a demi-normal scheme $X$ a *[line]{} if | 1 | member_41 |
the proper transformation $\bar{C}$ is a line on the normalization $\bar{X}$ of $X$.*
\[restdim\] Let $(X,\Lambda)$ be an irreducible Gorenstein stable log surface with a reduced curve $C$ on it. Then
- if $X$ is normal and $\Delta(X,K_X+\Lambda)=0$, then $r_{X\to C}(K_X+\Lambda)\ge2$. Moreover ’=’ holds if and only if $C$ is a line on $X$.
- if $X$ is normal and $\Delta(X,K_X+\Lambda)=1$, then $r_{X\to C}(K_X+\Lambda)\ge1$.
Moreover, ’=’ holds if and only if $|K_X+\Lambda|$ is composed with a pencil and $C$ is a fiber on $X$.
If $|K_X+\Lambda|$ is not composed with a pencil, then $r_{X\to C}(K_X+\Lambda)\ge2$ and ’=’ holds if and only if $\Phi_{|K_X+\Lambda|}(C)$ is a line in $|K_X+\Lambda|^*$.
- if $X$ is non-normal and $\Delta(X,K_X+\Lambda)=1$, then $r_{X\to C}(K_X+\Lambda)\ge1$. Moreover, ’=’ holds if and only if $|K_X+\Lambda|$ has a base point, and $C$ is a line passing through the base point.
\(i) and (ii) follows from Thm \[gepg-2\], Thm \[delta-genus-one\] and Lemma \[restsecions\]. To prove (iii), we note that $H^0(X,K_X+\Lambda)\cong \pi^* H^0(X,K_X+\Lambda)$, which corresponds to the space of hyperplane sections passing through the base point $c$ of $\pi^* H^0(X,K_X+\Lambda)$ in $ \mathbb{P}:=|\pi^*(K_X+\Lambda)|^*$. Denote $pr_c\colon \mathbb{P}\dashrightarrow \mathbb{P}':=|K_X+\Lambda|^*$ as the projection from the point $c$.
We have the following commutative diagram: $$\xymatrix{
\bar{C}\ar@{^{(}->}[r]\ar[d] | 1 | member_41 |
& \bar{X}\ar@{^{(}->}[rr]^{\Phi_{|\pi^*(K_X+\Lambda)|}}\ar@{-->}[rrd]^{\Phi'}\ar[d]_{\pi}& & \mathbb{P}\ar@{-->}[d]^{pr_c}\\
C\ar@{^{(}->}[r] &X\ar@{-->}[rr]_{\Phi_{|K_X+\Lambda|}} &&\mathbb{P}',
}$$ where $\Phi'$ is a map defined by the partial linear system $\pi^*H^0(X,K_X+\Lambda)$. We regard $\Phi_{|\pi^*(K_X+\Lambda)|}$ as an inclusion. By Lemma \[restsecions\], $r_{X\to C}(K_X+\Lambda)\ge1$. If ’=’ holds, $\Phi_{|K_X+\Lambda|}(C)$ is a point. Hence $pr_c(\bar{C})$ is a point, which implies $\bar{C}$ is a line passing $c$. Hence $C$ is a line passing through $\pi(c)$, which is the base point of $|K_X+\Lambda|$.
\[2comps\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface. Assume $X=X_1\cup X_2$, where $X_1$ is connected and $X_2$ is irreducible. Denote $C:=X_1\cap X_2$ as the connecting curve of $X_1$ and $X_2$.
Then:
- $\Delta(X,K_X+\Lambda)\ge \Delta(X_1,(K_X+\Lambda)|_{X_1})$.
- if ’=’ holds and $\Delta(X_2,(K_X+\Lambda)|_{X_2})\le 1$, then
- either $X_2$ is non-normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=1$. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=1$. $C$ is a line on $X_2$ passing through the base point of $|(K_X+\Lambda)|_{X_2}|$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 1$.
- $X_2$ is normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$, $|(K_X+\Lambda)|_{X_2}|$ is very ample. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=2$. $C$ is a line on $X_2$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 2$;
- or $X_2$ is normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=1$, $|(K_X+\Lambda)|_{X_2}|$ is composed with a pencil of elliptic curves. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=1$. $C$ is a fiber on $X_2$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 1$.
$(K_X+\Lambda)^2=(K_X+\Lambda)|_{X_1}^2+(K_X+\Lambda)|_{X_2}^2$ together with Cor \[fibersection\] gives $$\label{deltaIneq}
\begin{split}
\Delta(X,K_X+\Lambda) & \ge \Delta(X_1,(K_X+\Lambda)|_{X_1})+\Delta(X_2,(K_X+\Lambda)|_{X_2})\\
&+\max\{r_{X_1\to C}((K_X+\Lambda)|_{X_1}),r_{X_2\to | 1 | member_41 |
C}((K_X+\Lambda)|_{X_2})\}-2.
\end{split}$$ By Lemma \[restdim\] we see that $\Delta(X_2,(K_X+\Lambda)|_{X_2})+r_{X_2\to C}((K_X+\Lambda)|_{X_2})\ge 2$. Thus $\Delta(X,K_X+\Lambda) \ge \Delta(X_1,(K_X+\Lambda)|_{X_1})$.
If ’=’ holds, then $$\begin{aligned}
\max\{r_{X_1\to C}((K_X+\Lambda)|_{X_1}),r_{X_2\to C}((K_X+\Lambda)|_{X_2})\}= 2- \Delta(X_2,(K_X+\Lambda)|_{X_2}).
\end{aligned}$$ Therefore, by Lemma \[restdim\], $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=2$, if $\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=1$, if $\Delta(X_2,(K_X+\Lambda)|_{X_2})=1$. Then other statements of (ii) follow from Lemma \[restdim\].
We then have some corollaries.
\[genus-like\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface. Assume $Y\subset X$ is a connected subsurface. Then we have $\Delta(X,K_X+\Lambda)\ge \Delta(Y,(K_X+\Lambda)|_{Y})$.
If $X_i\subset X$ is an irreducible surface connected to $Y$, $\Delta(Y,(K_X+\Lambda)|_{Y})\le \Delta(Y\cup X_i,(K_X+\Lambda)|_{Y\cup X_i})$ by Lemma \[2comps\]. Then by induction hypothesis, we have $\Delta(Y,(K_X+\Lambda)|_{Y})\le \Delta(X,K_X+\Lambda)$.
Let $(X,\Lambda)$ be a reducible Gorenstein stable log surface. Write $X=\bigcup X_i$. We say that $\Phi:=\Phi_{K_X+\Lambda}$ separates $X_i$ and $X_j$, if $\Phi(X_i\setminus X_i\cap X_j)\cap \Phi(X_j\setminus X_i\cap X_j)=\emptyset$.
\[globalsection\] Let $(X,\Lambda)$ be a connected reducible Gorenstein stable log surface with $X=X_1\cup X_2$ such that $X_1$ is connected and $X_2$ is irreducible. Then
- We have the following commutative diagram: $$\xymatrix{
0 \ar[r] & \ker \mathcal{R}_{X\to X_1} \ar[r]\ar[d]^{\cong} & H^0(X,K_X+\Lambda) \ar[r]^{\mathcal{R}_{X\to X_1}}\ar[d]^{\mathcal{R}_{X\to X_2}} &H^0(X_1,(K_X+\Lambda)|_{X_1})\ar[d]^{\mathcal{R}_{X_1\to X_1\cap X_2}}\\
0 \ar[r] & \ker\mathcal{R}_{X_2\to X_1\cap X_2} \ar[r] & H^0(X_2,(K_X+\Lambda)|_{X_2}) \ar[r]^{\mathcal{R}_{X_2\to X_1\cap X_2}} &H^0(X_1\cap X_2,(K_X+\Lambda)|_{X_1\cap X_2})
.
}$$
- If $\mathrm{im}\, \mathcal{R}_{X_1\to X_1\cap X_2}\subset \mathrm{im} \, \mathcal{R}_{X_2\to X_1\cap | 1 | member_41 |
X_2}$, then $\mathcal{R}_{X\to X_1}$ is surjective.
- If $(K_X+\Lambda)|_{X_2}$ is very ample and $X_1\cap X_2$ is a line on $X_2$, then $\mathcal{R}_{X\to X_1}$ is surjective and $\Phi_{|K_X+\Lambda)|}$ separates $X_1$ and $X_2$.
- If each $(K_X+\Lambda)|_{X_i}$ is very ample and $X_1\cap X_2$ is a line on each $X_i$, then $K_X+\Lambda$ is very ample.
\(i) and (ii) is obvious by chasing the diagram.
For (iii), $X_1\cap X_2$ is a line implies $\mathcal{R}_{X_2\to X_1\cap X_2}$ is surjective. Then any section in $H^0(X_1,(K_X+\Lambda)|_{X_1})$ can be extended into a section in $H^0(X,K_X+\Lambda)$. Therefore $\mathcal{R}_{X\to X_1}$ is surjective. Next $\ker \mathcal{R}_{X_2\to X_1\cap X_2}$ is nontrivial and its base part is the line $X_1\cap X_2$. Then $\ker \mathcal{R}_{X\to X_1}$ is nontrivial and its base part is $X_1$. Therefore $\Phi_{|K_X+\Lambda)|}$ separates $X_1$ and $X_2$.
For (iv) we see that $\mathcal{R}_{X\to X_i}$ is surjective and the kernel is nontrivial by (ii). Then we have plenty of sections to separate points and tangents, which implies $K_X+\Lambda$ is very ample.
\[equaldelta\] Let $(X,\Lambda)$ be a log surface as in Lemma \[2comps\]. We assume further $X_1$ is irreducible and $\Delta(X,K_X+\Lambda)= \Delta(X_i,(K_X+\Lambda)|_{X_i})\le 1$ for $i=1,2$. Then
- if $\Delta(X,K_X+\Lambda)=0$, then $X_1\cap X_2$ is a line on each $X_i$. Moreover, $K_X+\Lambda$ is very ample.
| 1 | member_41 |
- if $\Delta(X,K_X+\Lambda)=1$, then
- either $X_1$ and $X_2$ are both normal. Each $|(K_X+\Lambda)|_{X_i}|$ is composed with a pencil. $X_1\cap X_2$ is a fiber on each $X_i$. $|K_X+\Lambda|$ is composed with a pencil as well.
- or $X_1$ and $X_2$ are both non-normal. The base points of $|(K_X+\Lambda)|_{X_i}|$ coincide into the unique base point of $|K_X+\Lambda|$. $X_1\cap X_2$ is a line passing through the base point of $|(K_X+\Lambda)|_{X_i}|$ on each $X_i$.
\(i) follows from Lemma \[2comps\] and Lemma \[globalsection\].
For (ii), applying Lemma \[2comps\] we see that either $X_i$ is non-normal or $(K_X+\Lambda)|_{X_i}$ is composed with a pencil. It is easy to see that either $X_1$, $X_2$ are both non-normal or $(K_X+\Lambda)|_{X_1}$, $(K_X+\Lambda)|_{X_2}$ are both composed with a pencil of elliptic curve, since the geometric genus of $X_1\cap X_2$ on $X_1$ or $X_2$ should coincide.
If $(K_X+\Lambda)|_{X_1}$, $(K_X+\Lambda)|_{X_2}$ are both composed with a pencil of elliptic curve, then by Lemma \[2comps\] each $(K_X+\Lambda)|_{X_i}$ is composed with a pencil and $X_1\cap X_2$ is a fiber on each $X_i$. Therefore $|K_X+\Lambda|$ is composed with a pencil as well. If $X_1$ and $X_2$ are both non-normal, then by Lemma \[restdim\], $X_1\cap X_2$ is a line passing through the base point of $|(K_X+\Lambda)|_{X_i}|$ on | 1 | member_41 |
each $X_i$. These two base points must coincide, otherwise no nontrivial section in $H^0(X_i,(K_X+\Lambda)|_{X_i})$ can be glued into a section of $H^0(X,K_X+\Lambda)$.
\[decrease1\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface which has two irreducible components $X_1$, $X_2$. Assume further $\Delta(X_1,(K_X+\Lambda)|_{X_1})=\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$ and $\Delta(X,K_X+\Lambda)=1$. Then $$r_{X_1\to X_1\cap X_2}((K_X+\Lambda)|_{X_1})=r_{X_2\to X_1\cap X_2}((K_X+\Lambda)|_{X_2})=3.$$
By (\[deltaIneq\]), we have $\max\{r_{X_1\to X_1\cap X_2}((K_X+\Lambda)|_{X_1}),r_{X_2\to X_1\cap X_2}((K_X+\Lambda)|_{X_2}\}\le 3$. Moreover $r_{X_i\to X_1\cap X_2}((K_X+\Lambda)|_{X_i}\ge 2$ by Lemma \[restdim\]. Thus there must be one $r_{X_i\to X_1\cap X_2}((K_X+\Lambda)|_{X_i}=3$.
Now we suppose $r_{X_1}(X_1\cap X_2)=2$ and $r_{X_2}(X_1\cap X_2)=3$ for a contradiction. Then $X_1\cap X_2$ is a line on $X_1$ and not a line on $X_2$. Thus nonzero elements of $\mathrm{im}\mathrm{Res}_{X_i|X_1\cap X_2}$ have different degrees. Hence only those sections in $H^0(X_i,(K_X+\Lambda)|_{X_i})$ vanishing on $X_1\cap X_2$ can be glued together. Therefore $$\begin{aligned}
p_g(X,\Lambda)&=\dim\ker \mathcal{R}_{X_1\to X_1\cap X_2}+\dim \ker \mathcal{R}_{X_2\to X_1\cap X_2}\\
&\le p_g(X_1,(K_X+\Lambda)|_{X_1})-2+p_g(X_2,(K_X+\Lambda)|_{X_2})-2\\
&\le (K_X+\Lambda)^2,
\end{aligned}$$ a contradiction. This completes the proof.
\[diffdelta\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface. Assume $X=X_1\cup X_2$, where $X_i$ is irreducible. Assume further $\Delta(X,K_X+\Lambda)= \Delta(X_1,(K_X+\Lambda)|_{X_1})=1$ and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$.
Then
- either each $(K_X+\Lambda)|_{X_i}$ is very ample and on each $X_i$ the curve $X_1\cap X_2$ is a line on each $X_i$. $K_X+\Lambda$ is very ample;
- or $X_1$ is non-normal. $X_1\cap | 1 | member_41 |
---
abstract: 'A scattering problem (or more precisely, a transmission-reflection problem) of linearized excitations in the presence of a dark soliton is considered in a one-dimensional nonlinear Schrödinger system with a general nonlinearity: $ \mathrm{i}\partial_t \phi = -\partial_x^2 \phi + F(|\phi|^2)\phi $. If the system is interpreted as a Bose-Einstein condensate, the linearized excitation is a Bogoliubov phonon, and the linearized equation is the Bogoliubov equation. We exactly prove that the perfect transmission of the zero-energy phonon is suppressed at a critical state determined by Barashenkov’s stability criterion \[Phys. Rev. Lett. 77, (1996) 1193.\], and near the critical state, the energy-dependence of the reflection coefficient shows a saddle-node type scaling law. The analytical results are well supported by numerical calculation for cubic-quintic nonlinearity. Our result gives an exact example of scaling laws of saddle-node bifurcation in time-reversible Hamiltonian systems. As a by-product of the proof, we also give all exact zero-energy solutions of the Bogoliubov equation and their finite energy extension.'
address: 'Department of Basic Science, The University of Tokyo, Tokyo 153-8902, Japan'
author:
- 'Daisuke A. Takahashi'
bibliography:
- 'genNLS.bib'
title: 'Soliton-phonon scattering problem in 1D nonlinear Schrödinger systems with general nonlinearity'
---
Nonlinear Schrödinger equation ,Bose-Einstein condensate ,Bogoliubov | 1 | member_42 |
equation ,saddle-node bifurcation ,universal scaling laws ,cubic-quintic nonlinear Schrödinger equation
Introduction
============
In this paper, we solve a scattering problem of linearized excitations in the presence of a dark soliton in one-dimensional(1D) nonlinear Schrödinger (NLS) equation with a general nonlinearity: $$\begin{aligned}
\mathrm{i}\partial_t\phi = -\partial_x^2\phi+F(|\phi|^2)\phi, \label{eq:intro001}
\end{aligned}$$ and discuss the physical and mathematical significance of our results. For a schematic picture, see Fig. \[introfigure\]. The precise mathematical definition of the problem will be given in Sec. \[sec:fundamental\]. If we regard the system as a Bose-Einstein condensate(BEC), the linearized excitation is a Bogoliubov phonon, so the problem can be also called a soliton-phonon scattering problem, as this paper entitled.\
The NLS equation (\[eq:intro001\]) has a great number of applications in nonlinear optics, superconductors, magnetism, BECs, and so on. Particularly, much attention has been focused on the experimental realizations of BECs in laser-cooled ultracold atoms for more than a decade, because of high-controllability of the system parameters. By using elongated laser beams, low-dimensional systems are realized, and a dark soliton can be created via the phase imprinting method[@BurgerPhaseImprinting]. The Bogoliubov theory is also well confirmed[@Andrews; @Stamper; @Steinhauer].\
It is known that 1D NLS with a *cubic* nonlinearity is completely integrable[@ZakharovShabat; @ZakharovShabat2]. Because of | 1 | member_42 |
integrability, the linearized equation is also solved exactly[@ChenChenHuang; @Kovrizhin], and the phonon excitations are shown to be completely reflectionless against a soliton for *any* excitation energy. Thus, the problem is trivial in this case. However, when the nonlinear term is generalized, the phonon has a finite reflection coefficient in general. It is worthy to note that the soliton decay dynamics in the laser-trapped quasi-1D BEC has been well explained by the *quintic* term, which appears as a second-order perturbation of the trapping effect[@Muryshev; @SinhaChernyKovrizhinBrand; @KhaykovichMalomed], and yields the frictional force between thermal excitation clouds and solitons[@Muryshev; @SinhaChernyKovrizhinBrand]. Thus, knowing the scattering properties between solitons and linearized excitations is essential to understand and control the transport of solitons, that is, the transport of stable wave packets. We also mention that the theory of nonpolynomial NLS equation is formulated to describe the confinement effect[@SalasnichParolaReatto; @MateoDelgado; @MateoDelgadoAnnPhys; @Salasnich]. The quintic NLS also appears in an effective mean-field description of the Tonks-Girardeau gas[@Kolomeisky].\
The NLS equation with an integrability breaking factor is also interesting from the viewpoint of an infinite-dimensional dynamical system and the bifurcation theory. When the potential barrier is added in the cubic NLS equation, there exist stable and unstable stationary supercurrent-flowing | 1 | member_42 |
solutions[@BaratoffBlackburnSchwartz; @Hakim], if the condensate velocity does not exceed a certain critical value. Near the critical point, which separates the stable branch and the unstable branch, it is known that many physical quantities obey saddle-node type scaling, such as an emission period of dark solitons[@Hakim; @PhamBrachet], an eigenvalue of a growing mode for the unstable solution[@PhamBrachet], and a transmission coefficient of linearized excitations[@Kovrizhin; @Kagan; @DanshitaYokoshiKurihara]. It is quite nontrivial that the time-reversible Hamiltonian system exhibits the scaling behaviors of saddle-node type, since this bifurcation is normally understood to emerge in time-irreversible phenomena. However, it is not easy to prove these properties analytically or exactly, because of the infinite dimensional character of the system.\
On the other hand, as another way to break the integrability, one can consider the generalization of the nonlinearity, that is what we will consider in the present paper. When the nonlinear term includes a competing interaction, the dark soliton is no longer always stable. One typical example of an unstable dark soliton is a “bubble” in a cubic-quintic NLS (CQNLS) system[@BarashenkovMakhankov; @BarashenkovPanova]. (See also [@JYang].) The most general criterion for the stability of the dark soliton has been shown by Barashenkov[@Barashenkov], and the critical velocity of | 1 | member_42 |
the soliton is determined by $ \partial P/\partial v=0 $, where $ v $ is a velocity of the soliton and $ P $ is a renormalized momentum. The existence of the critical velocity lower than the sound velocity and the separation of stable and unstable regions are similar to the phenomena of superflows against a potential barrier. Therefore, we can expect some scaling behavior near the critical state. Furthermore, in the present case, the preserved translational symmetry of the fundamental equation makes it possible to access the problem analytically.\
In this paper, we solve the scattering problem of linearized phonon excitations, and exactly show the following: (i) At the critical state determined by Barashenkov’s criterion[@Barashenkov], the transparency of the zero-energy phonon is suppressed, and only partial transmission occurs. (ii) Near the critical state, the energy-dependence of the reflection coefficient of low-energy phonons shows saddle-node scaling behavior, regarding the renormalized momentum as a parameter of a normal form of saddle-node bifurcation. The obtained analytical results are well confirmed by comparison with the numerical results of CQNLS equation. Our result gives an exact example of scaling laws of saddle-node bifurcation in time-reversible Hamiltonian systems. The proof is based on the exact | 1 | member_42 |
low-energy expansion of the solution of the linearized equation. Since the exact zero-energy solutions given in this paper are quite general, we believe that our method will also be useful to derive other low-energy physical properties.\
The organization of the paper is as follows. In Sec. \[sec:fundamental\], we introduce fundamental equations and see the fundamental properties. The definition of the transmission-reflection problem is also given. In Sec. \[sec:mainresult\], we give a main result and verify it by numerical study of CQNLS equation. Sections \[sec:proof1\] and \[sec:proof2\] are devoted to the proof of main results. Discussions, future perspectives, and conclusions are given in Sec. \[sec:summary\]. Some mathematically technical formulae are treated in Appendices.
![\[introfigure\]A schematic picture of the problem that we consider in this paper. $ p $ represents a half of the velocity of the dark soliton. The problem is always considered in the comoving frame of the soliton. $ \mathrm{e}^{\mathrm{i}k_1x} $ is an incident wave of a linearized excitation, $ t\mathrm{e}^{\mathrm{i}k_1x} $ is a transmitted wave, and $ r\mathrm{e}^{\mathrm{i}k_2x} $ is a reflected wave. For more detailed definitions of each quantity, see Sec. \[sec:fundamental\].](introfiguretex2img.eps)
Fundamental Equations and Definition of the Problem {#sec:fundamental}
===================================================
Fundamental equations
---------------------
We begin with | 1 | member_42 |
the NLS equation with a general nonlinearity $$\begin{aligned}
\mathrm{i}\partial_t\phi=-\partial_x^2\phi+F(|\phi|^2)\phi. \label{eq:nls}
\end{aligned}$$ Here, $ F(\rho) $ is a real-valued function such that $ F(0)=0 $. The energy functional (Hamiltonian) which yields this equation is $$\begin{aligned}
H = \int\!\mathrm{d}x \left(|\partial_x\phi|^2+U(|\phi|^2)\right),
\end{aligned}$$ where $$\begin{aligned}
U(\rho) = \int_0^\rho\mathrm{d}\rho' F(\rho').
\end{aligned}$$ Letting $ \phi=\phi+\delta\phi $ in Eq. (\[eq:nls\]), and discarding higher order terms of $ \delta\phi $, one obtains the following linearized equation: $$\begin{aligned}
\mathrm{i}\partial_t\delta\phi=\left[-\partial_x^2+ F(|\phi|^2)+|\phi|^2F'(|\phi|^2) \right]\delta\phi+\phi^2F'(|\phi|^2)\delta\phi^*.
\end{aligned}$$ Writing $ \delta\phi=u $ and $ -\delta\phi^*=v $, one obtains $$\begin{aligned}
\mathrm{i}\partial_t\begin{pmatrix}u\\v\end{pmatrix}=\mathcal{L}\begin{pmatrix}u\\v\end{pmatrix}, \label{eq:tdbogo}
\end{aligned}$$ where $ \mathcal{L} $ is a $ 2\times2 $ matrix operator whose components are $$\begin{aligned}
\mathcal{L}_{11}^{}&=-\mathcal{L}_{22}^{}=-\partial_x^2+F(|\phi|^2)+|\phi|^2F'(|\phi|^2), \\
\mathcal{L}_{12}^{}&=-\mathcal{L}_{21}^*=-\phi^2F'(|\phi|^2).
\end{aligned}$$ We use the notation $ (u,v) $ since it is commonly used by condensed matter physicists. If we interpret the system as BEC, this equation is the Bogoliubov equation which describes the Bogoliubov phonon (or Bogoliubov quasiparticle) [@Bogoliubov]. (As a review or a textbook, see, e.g., [@FetterWalecka; @DalfovoGiorginiPitaevskiiStringari; @PethickSmith].) For this reason, henceforth, we call $ \phi $ the condensate wavefunction or the order parameter, and $ (u,v) $ the Bogoliubov (quasiparticle) wavefunction, though the NLS equation itself has more applications in various fields.\
Henceforth we mainly consider the stationary (i.e., time-independent) problem. The | 1 | member_42 |
stationary NLS equation with chemical potential $ \mu $ is obtained by setting $ \phi(x,t)=\phi(x)\mathrm{e}^{-\mathrm{i}\mu t} $: $$\begin{aligned}
(-\mu-\partial_x^2+F(|\phi|^2))\phi=0. \label{eq:nls2}
\end{aligned}$$ As will be seen, the value of $ \mu $ is fixed by the asymptotic form of $ \phi $ . The stationary Bogoliubov equation with eigenenergy $ \epsilon $ is obtained by setting $ u(x,t)=u(x)\mathrm{e}^{-\mathrm{i}(\epsilon+\mu)t} $ and $ v(x,t)=v(x)\mathrm{e}^{-\mathrm{i}(\epsilon-\mu)t} $: $$\begin{aligned}
\epsilon\begin{pmatrix}u\\v\end{pmatrix}=\mathcal{L}_\mu\begin{pmatrix}u\\v\end{pmatrix} \label{eq:bogos}
\end{aligned}$$ with $$\begin{aligned}
\mathcal{L}_\mu:= \mathcal{L}+\begin{pmatrix}-\mu &0 \\ 0& \mu \end{pmatrix}. \label{eq:bogos2}
\end{aligned}$$
Bogoliubov phonons in a uniform condensate {#subsec:uniform}
------------------------------------------
Let us derive the dispersion relation (the energy-momentum relation) of Bogoliubov phonons when the condensate is flowing uniformly: $ \phi(x) = \sqrt{\rho_\infty}\mathrm{e}^{\mathrm{i}(px+\varphi)} $. In order for this $ \phi(x) $ to be the solution of Eq. (\[eq:nls2\]), the chemical potential must be $$\begin{aligned}
\mu=p^2+F(\rho_\infty). \label{eq:cp}
\end{aligned}$$ The four solutions of Bogoliubov equation (\[eq:bogos\]) are given by $$\begin{aligned}
w_i(x,\varphi):= \begin{pmatrix} \bar{u}_i\, \mathrm{e}^{\mathrm{i}(px+\varphi)} \\ \bar{v}_i\, \mathrm{e}^{-\mathrm{i}(px+\varphi)} \end{pmatrix} \mathrm{e}^{\mathrm{i}k_i x}, \label{eq:bogouniform}
\end{aligned}$$ where $ i=1,\,2,\,3, \text{ and }4 $, and the wavenumber $ k_i $s are the roots of the following quartic equation: $$\begin{aligned}
(\epsilon-2kp)^2=k^2(k^2+2\rho_\infty F'(\rho_\infty)). \label{eq:uniformdisp}
\end{aligned}$$ Equation (\[eq:uniformdisp\]) gives the dispersion relation, and from this dispersion one can see that a half of the Landau’s critical | 1 | member_42 |
velocity (or a half of the sound wave velocity) is given by $$\begin{aligned}
p_{\text{L}}=\sqrt{\frac{\rho_\infty F'(\rho_\infty)}{2}}. \label{eq:defofpl}
\end{aligned}$$ (Note that the sound wave velocity is not $ p_{\text{L}} $ but $ 2p_{\text{L}} $; see the next subsection.) Since $ p_{\text{L}} $ must be real, in order for the uniform condensate to be stable, $ F'(\rho_\infty)>0 $ must hold. The coefficients $ \bar{u}_i $ and $ \bar{v}_i $ can be, e.g., chosen as follows: $$\begin{aligned}
\bar{u}_{i}&=\sqrt{1+\frac{\epsilon-2p k_i^{}+k_i^2}{2p_{\text{L}}^2}}, \label{eq:ucoeff}\\
\bar{v}_{i}&=\sqrt{1-\frac{\epsilon-2p k_i^{}-k_i^2}{2p_{\text{L}}^2}}. \label{eq:vcoeff}
\end{aligned}$$ When $ \epsilon>0 $ and $ -p_{\text{L}}<p<p_{\text{L}} $, the quartic equation (\[eq:uniformdisp\]) has one real positive root, one real negative root, and two complex roots conjugate to each other. We call a real positive (negative) root $ k_1 \ (k_2) $, and a complex root with positive (negative) imaginary part $ k_3 \ (k_4) $. The low-energy expansions of them are given by $$\begin{aligned}
k_1 &= \frac{\epsilon}{2(p+p_{\text{L}})}+O(\epsilon^3), \label{eq:kexpand1} \\
k_2 &= \frac{\epsilon}{2(p-p_{\text{L}})}+O(\epsilon^3), \label{eq:kexpand2} \\
k_3 &= 2\mathrm{i}\sqrt{p_{\text{L}}^2-p^2}+\frac{p\epsilon}{2(p_{\text{L}}^2-p^2)}+O(\epsilon^2), \label{eq:kexpand3} \\
k_4 &= -2\mathrm{i}\sqrt{p_{\text{L}}^2-p^2}+\frac{p\epsilon}{2(p_{\text{L}}^2-p^2)}+O(\epsilon^2). \label{eq:kexpand4}
\end{aligned}$$ $ w_1 $ and $ w_2 $ are plane wave solutions propagating in the positive and negative directions, respectively. $ w_3 $ and $ w_4 $ are exponentially divergent unphysical solutions.
Dark soliton solution in comoving | 1 | member_42 |
frame
---------------------------------------
Let us consider the dark soliton solution of stationary NLS Eq. (\[eq:nls2\]) in the comoving frame of the soliton. In this coordinate, the soliton is static but the surrounding condensate is flowing. Let us seek the solution with the asymptotic form $$\begin{aligned}
\phi(x\rightarrow \pm\infty)=\sqrt{\rho_\infty}\mathrm{e}^{\mathrm{i}(px\pm\frac{\delta}{2})}. \label{eq:nlsdsasym}
\end{aligned}$$ It should be noted that the velocity of the soliton is not $ -p $ but $ -2p $, because the Galilean covariance of NLS equation leads to the following property: $$\begin{aligned}
\begin{split}
&\text{$ \phi(x,t) $ is a solution.} \\
\leftrightarrow \quad & \text{$\tilde{\phi}(x,t,\alpha)=\phi(x+2\alpha t,t)\mathrm{e}^{-\mathrm{i}\alpha x}\mathrm{e}^{-\mathrm{i}\alpha^2t}$ is a solution.}
\end{split}\label{eq:galilei}
\end{aligned}$$ So, if one has the solution in the form $ \phi(x,t)=\mathrm{e}^{-\mathrm{i}\mu t}\mathrm{e}^{\mathrm{i}px}f(x) $, the corresponding soliton-moving solution is given by $ \tilde{\phi}(x,t,p)=\mathrm{e}^{-\mathrm{i}(\mu-p^2)t}f(x+2pt) $. However, for brevity, we sometimes call $ p $ “velocity”, ignoring the difference of twice factor.\
From the conservation laws of mass and momentum, one can immediately find two integration constants: $$\begin{aligned}
j&=\frac{\phi^*\phi_x-\phi\phi_x^*}{2\mathrm{i}}, \\
j_m &= |\phi_x|^2+\mu|\phi|^2-U(|\phi|^2). \label{eq:genjm}
\end{aligned}$$ Here $ j $ is a mass current density and $ j_m $ is a momentum current density. Let us write the density and the phase of the condensate as $ \phi=\sqrt{\rho}\mathrm{e}^{\mathrm{i}S} $. Taking account of the asymptotic form | 1 | member_42 |
(\[eq:nlsdsasym\]), the chemical potential $ \mu $ becomes the same as (\[eq:cp\]), and the above constants are determined as $$\begin{aligned}
j&=\rho_\infty p, \label{eq:dsj} \\
j_m&=2\rho_\infty p^2+\rho_\infty F(\rho_\infty)-U(\rho_\infty). \label{eq:dsjm}
\end{aligned}$$ The conservation laws are then rewritten as $$\begin{aligned}
S_x &= \frac{j}{\rho} =\frac{\rho_\infty p}{\rho} \ \leftrightarrow \ S = p\int_0^x\frac{\rho_\infty\mathrm{d}x}{\rho}, \label{eq:phasecond} \\
\frac{(\rho_x)^2}{4} &= -p^2(\rho_\infty-\rho)^2+\rho\left[ U(\rho)-U(\rho_\infty)-(\rho-\rho_\infty)U'(\rho_\infty) \right]. \label{eq:momconsrv}
\end{aligned}$$ Thus, one can at least obtain the formal solution $$\begin{aligned}
\pm 2(x-x_0) = \int\!\!\frac{\mathrm{d}\rho}{\sqrt{\text{R.H.S. of Eq. (\ref{eq:momconsrv})}}},
\end{aligned}$$ even though it is not easy in general to carry out this integration and obtain the solution in closed form “$ \rho(x)=\dots $”. Henceforth we do not need this formal solution, but we assume the existence of a dark soliton solution which has no singularity and satisfies the asymptotic condition (\[eq:nlsdsasym\]).\
From Eq. (\[eq:phasecond\]), The phase shift $ \delta $ in Eq. (\[eq:nlsdsasym\]) can be written down explicitly: $$\begin{aligned}
\delta = p\int_{-\infty}^\infty\!\!\mathrm{d}x\left( \frac{\rho_\infty}{\rho}-1 \right).
\end{aligned}$$ We also introduce the symbol for the particle number of the dark soliton for later convenience: $$\begin{aligned}
N := \int_{-\infty}^\infty\!\!\mathrm{d}x\left( \rho-\rho_\infty \right)<0.
\end{aligned}$$
Barashenkov’s criterion
-----------------------
The stability of the soliton is described by the following renormalized momentum[@Barashenkov]: $$\begin{aligned}
P = \int_{-\infty}^\infty\mathrm{d}x\left( \frac{\tilde{\phi}^*\tilde{\phi}_x-\tilde{\phi}\tilde{\phi}^*_x}{2\mathrm{i}} \right)\left( 1-\frac{\rho_\infty}{|\tilde{\phi}|^2} \right)
\end{aligned}$$ Here $ \tilde{\phi}(x,t) | 1 | member_42 |
$ is the dark soliton solution in the frame where the surrounding condensate is at rest and the soliton is moving. Writing the soliton velocity $ v $, the stability criterion for the dark solitons is expressed by $ \partial P/\partial v<0 $.\
We can rewrite the above integral by the density profile $ \rho(x) $: $$\begin{aligned}
\begin{split}
P &= -p\int_{-\infty}^\infty\mathrm{d}x\left( \rho_\infty-\rho \right)\left( \frac{\rho_\infty}{\rho}-1 \right) \\
&= -p N-\rho_\infty \delta.
\end{split}
\end{aligned}$$ Here remember that the soliton velocity is given by $ v=-2p $, as stated in the preceding subsection. The stability condition is rewritten as $ \partial (-P)/\partial p <0 $.
Definition of the scattering problem {#sec:defofsp}
------------------------------------
In this subsection, we define the transmission and reflection problem of Bogoliubov phonons shown in Fig. \[introfigure\]. Since the linearized equation does not satisfy simple particle number conservation, we must define transmission and reflection coefficients via the conservation of excitation energy. The conservation of excitation energy corresponds to the constancy of the following Wronskian:[@Kagan; @DanshitaYokoshiKurihara]: $$\begin{aligned}
W = u^*\partial_xu-u\partial_xu^*+v^*\partial_xv-v\partial_xv^*.
\end{aligned}$$ Let us assume that the asymptotic form of the condensate wavefunction $ \phi(x) $ is given by Eq. (\[eq:nlsdsasym\]). In this situation, sufficiently far from the origin, the Bogoliubov equations have | 1 | member_42 |
the plane wave (and exponentially decaying/diverging) solutions given by Eq. (\[eq:bogouniform\]) with $ \varphi=\pm\frac{\delta}{2} $.\
The solution of the scattering problem is defined by the one that has the following asymptotic form[@Kovrizhin; @Kagan; @DanshitaYokoshiKurihara]: $$\begin{aligned}
\begin{pmatrix}u \\ v \end{pmatrix} \rightarrow \begin{cases} w_1\left(x,-\frac{\delta}{2}\right)+r\, w_2\left(x,-\frac{\delta}{2}\right) & (x\rightarrow-\infty) \\ t\, w_1\left(x,+\frac{\delta}{2}\right) & (x\rightarrow+\infty). \end{cases} \label{eq:uvasymptotic}
\end{aligned}$$ Here the exponentially decaying waves, which are $ w_4(x,-\frac{\delta}{2}) $ in $ x\rightarrow -\infty $ and $ w_3(x,+\frac{\delta}{2}) $ in $ x\rightarrow+\infty $, can be also included. However, they are irrelevant in the definition of transmission and reflection coefficients. The calculation of $ W $ shows that $$\begin{aligned}
\frac{W(+\infty)}{2\mathrm{i}}=&|t|^2\left[ (k_{1}+p)|\bar{u}_{1}|^2+(k_{1}-p)|\bar{v}_{1}|^2 \right]\!, \\
\begin{split}
\frac{W(-\infty)}{2\mathrm{i}}=&(k_{1}+p)|\bar{u}_{1}|^2+(k_{1}-p)|\bar{v}_{1}|^2 \\
&\ +|r|^2\left[ (k_{2}+p)|\bar{u}_{2}|^2+(k_{2}-p)|\bar{v}_{2}|^2 \right].
\end{split}
\end{aligned}$$ Since $ W(+\infty)=W(-\infty) $, the transmission coefficient $ T $ and the reflection coefficient $ R $ are naturally defined as $$\begin{aligned}
T&=|t|^2, \label{eq:tc} \\
R&=\frac{(-k_{2}-p)|\bar{u}_{2}|^2+(-k_{2}+p)|\bar{v}_{2}|^2}{(k_{1}+p)|\bar{u}_{1}|^2+(k_{1}-p)|\bar{v}_{1}|^2}|r|^2. \label{eq:defofR}
\end{aligned}$$ By this definition, $ T+R=1 $ always holds. If one chooses the normalization $ \bar{u}_i $ and $ \bar{v}_i $ as Eq. (\[eq:ucoeff\]) and (\[eq:vcoeff\]), one can show $$\begin{aligned}
\frac{(-k_{2}-p)|\bar{u}_{2}|^2+(-k_{2}+p)|\bar{v}_{2}|^2}{(k_{1}+p)|\bar{u}_{1}|^2+(k_{1}-p)|\bar{v}_{1}|^2} = 1-\frac{pp_{\text{L}}\epsilon^2}{2(p^2-p_{\text{L}}^2)^3}+O(\epsilon^4).
\end{aligned}$$ So, if one is only interested in the leading order, one can approximate $ R \simeq |r|^2 $.
Summary of Main Result and Numerical Verification | 1 | member_42 |
{#sec:mainresult}
=================================================
In this section we present the main results of this paper and verify them by numerical study of CQNLS equation. The proof will be given in Secs. \[sec:proof1\] and \[sec:proof2\].
Main result {#subsec:mainresult}
-----------
In the scattering problem of linearized excitations defined in Subsec. \[sec:defofsp\], the amplitude of the reflected component $ r $ in Eq. (\[eq:uvasymptotic\]) is given by the following Padé approximant-like form: $$\begin{aligned}
r = \frac{- \mathrm{i}\epsilon(d+d_1P_p)+O(\epsilon^2)}{a P_p - \mathrm{i}\epsilon(b+b_1P_p)+O(\epsilon^2)}. \label{eq:mainr}
\end{aligned}$$ Here $ X_p:= \partial X/\partial p $ and $$\begin{aligned}
a &= 4p_{\text{L}}\rho_\infty, \\
b &= (N+pN_p)^2+(p_{\text{L}}N_p)^2, \\
d &= (N+pN_p)^2-(p_{\text{L}}N_p)^2, \\
b_1 &= N-\frac{p_{\text{L}}^2+p^2}{p_{\text{L}}^2-p^2}\widetilde{N}, \\
d_1 &= N+\widetilde{N}
\end{aligned}$$ with $$\begin{aligned}
\widetilde{N}:=p \frac{\partial N}{\partial p}-\rho_\infty\frac{\partial N}{\partial \rho_\infty}.
\end{aligned}$$ From (\[eq:mainr\]), the energy dependence of the reflection coefficient $ R $ (\[eq:defofR\]) becomes $$\begin{aligned}
R = \begin{cases} \displaystyle \left( \frac{d+d_1P_p}{aP_p} \right)^2\epsilon^2+O(\epsilon^4) & (P_p\ne0) \\ \displaystyle \left( \frac{d}{b} \right)^2+O(\epsilon^2) & (P_p=0). \end{cases} \label{eq:mainrR}
\end{aligned}$$ When $ P_p \ne0$, the zero-energy phonon transmits perfectly: $ \lim_{\epsilon\rightarrow0}R=0 $. On the other hand, when the soliton velocity reaches a critical value, i.e. $ P_p=0 $, this perfect transmission disappears. We note that the rational form expression (\[eq:mainr\]) makes it possible to unify the description of low-energy behaviors in | 1 | member_42 |
both critical and non-critical cases. If we use a simple Taylor series, the singular behavior at the critical velocity state cannot be expressed.\
The above (\[eq:mainrR\]) is one good result valid for any soliton velocity, even if it is far from the critical state. However, when the velocity comes close to the critical value, i.e., $ P_p $ comes close to zero, we can derive a more powerful scaling law as below. Let us assume that coefficients of $ \epsilon^n \ (n\ge2) $ in (\[eq:mainr\]) are all finite in the limit $ P_p\rightarrow0 $. and take the limit $ \epsilon\rightarrow 0 $ and $ P_p\rightarrow0 $ with a constraint $ \epsilon/P_p=\text{fix} $. We then obtain $$\begin{aligned}
r \rightarrow \frac{-\mathrm{i}d(\epsilon/P_p)}{a-\mathrm{i}b(\epsilon/P_p)},
\end{aligned}$$ and in the same limit, the *universal* form of reflection coefficient $$\begin{aligned}
\lim_{\substack{\epsilon\rightarrow0,\, P_p\rightarrow0,\\ \epsilon/P_p:\text{fix}}} R = \frac{d^2(\epsilon/P_p)^2}{a^2+b^2 (\epsilon/P_p)^2} \label{eq:scaledR}
\end{aligned}$$ follows. Here, the values of $ a, b, $ and $ d $ in the critical state must be substituted when we use Eq. (\[eq:scaledR\]). We remark that Eq. (\[eq:scaledR\]) contains only $ p $-derivatives, whereas the expression before taking the scaling limit contains $ \rho_\infty $-derivatives in addition to $ p $-derivatives.\
Let $ p_c $ be a | 1 | member_42 |
critical velocity of the dark soliton, i.e., $ P'(p_c)=0 $. The expansion of $ P $ near $ p=p_c $ gives $$\begin{aligned}
P(p) &\simeq P(p_c)+\frac{1}{2}P''(p_c)(p-p_c)^2+\dotsb, \\
\rightarrow |P_p| &= |P'(p)| \simeq |2P''(p_c)(P(p)-P(p_c))|^{1/2}.
\end{aligned}$$ Therefore we obtain $$\begin{aligned}
\frac{\epsilon}{|P_p|} \simeq \frac{\epsilon}{|2P''(p_c)(P(p)-P(p_c))|^{1/2}}. \label{eq:energyscale}
\end{aligned}$$ This is an expected scaling behavior from the normal form of saddle-node bifurcation[@GuckenheimerHolmes; @PhamBrachet], if we regard the renormalized momentum $ P $ as a parameter of normal form.
Comparison with Numerical Results in CQNLS System {#subsec:cqnls}
-------------------------------------------------
In this subsection, we numerically verify the analytical results of the preceding subsection in the CQNLS system. We first derive the expressions for the dark soliton solution and the renormalized momentum in Subsec. \[subsubsec:darksolitoncqnls\], and solve the scattering problems of linearized excitations for (i) the purely cubic case in Subsec. \[sec:purecubic\], (ii) the purely quintic case in Subsec. \[sec:purequintic\], and (iii) the case where a non-trivial critical velocity exists in Subsec. \[sec:criticalcase\].
### Dark soliton solution and renormalized momentum {#subsubsec:darksolitoncqnls}
In the CQNLS system, the nonlinear term is defined by $$\begin{aligned}
U(\rho) &= a_1\rho^2+a_2\rho^3, \\
F(\rho)&=U'(\rho) = 2a_1\rho+3a_2\rho^2,
\end{aligned}$$ and the NLS equation (\[eq:nls\]) has the cubic-quintic nonlinearity: $$\begin{aligned}
\mathrm{i}\partial_t\phi=-\partial_x^2\phi+2a_1|\phi|^2\phi+3a_2|\phi|^4\phi.
\end{aligned}$$ The stationary linearized equation, i.e., the stationary Bogoliubov equation | 1 | member_42 |
(\[eq:bogos\]) is given by $$\begin{aligned}
\!\!\!\!\!\!&\begin{pmatrix}-\partial_x^2-\mu+4a_1|\phi|^2+9a_2|\phi|^4 & -2a_1\phi^2-6a_2|\phi|^2\phi^2 \\ 2a_1\phi^{*2}+6a_2|\phi|^2\phi^{*2} & \!\! \partial_x^2+\mu-4a_1|\phi|^2-9a_2|\phi|^4 \end{pmatrix}\begin{pmatrix}u \\ v \end{pmatrix} \nonumber \\
&=\epsilon\begin{pmatrix}u\\v\end{pmatrix}. \label{eq:bogocqnls}
\end{aligned}$$ It is known that a bubble and unstable dark solitons appear when $ a_1<0 $ and $ a_2>0 $[@BarashenkovMakhankov; @BarashenkovPanova]. This case is considered in Subsec. \[sec:criticalcase\]. As shown in [@BarashenkovPanova], when the soliton velocity is smaller than the critical value, a small perturbation induces “nucleation dynamics”, and the soliton cannot preserve its shape any more. So, this instability is not convective but absolute.\
The Landau velocity (\[eq:defofpl\]) is given by $$\begin{aligned}
p_{\text{L}} = \sqrt{\rho_\infty(a_1+3a_2\rho_\infty)}.
\end{aligned}$$ The necessary condition $ a_1+3a_2\rho_\infty>0 $ follows for a uniform condensate to be stable. The dark soliton solution is given by $$\begin{aligned}
\phi(x,p,\rho_\infty) =\mathrm{e}^{\mathrm{i}px}\frac{\kappa\rho_0+\mathrm{i}p(\rho_\infty-\rho_0)\tanh\kappa x}{\sqrt{\rho_0(\kappa^2-a_2(\rho_\infty-\rho_0)^2\tanh^2\kappa x)}} \label{eq:dscqnls}
\end{aligned}$$ with $$\begin{aligned}
\kappa &=\sqrt{p_{\text{L}}^2-p^2}, \\
\rho_0 &=\rho(x=0)= \frac{-(2a_2\rho_\infty+a_1)+\sqrt{(2a_2\rho_\infty+a_1)^2+4a_2p^2}}{2a_2}. \label{eq:rho0cqnls}
\end{aligned}$$ See \[app:cqnls\] for a detailed derivation. Since $ \kappa $ and $ \rho_0 $ are the functions of $ (p,\rho_\infty) $, the dark soliton solution has two parameters $ (p,\rho_\infty) $. From (\[eq:rho0cqnls\]), $$\begin{aligned}
\lim_{p\rightarrow0}\rho_0=\begin{cases} 0 & (2a_2\rho_\infty+a_1>0) \\ \frac{1}{2a_2}|2a_2\rho_\infty+a_1| & (2a_2\rho_\infty+a_1<0), \end{cases}
\end{aligned}$$ so the bubble (= a non-topological dark soliton) appears when $ \rho_\infty<-a_1/(2a_2) $. A particle number of soliton $ N | 1 | member_42 |
$ and a phase difference $ \delta $ are calculated as $$\begin{aligned}
N &= -\frac{2}{\sqrt{a_2}}\tanh^{-1}\left[ \frac{\sqrt{a_2}(\rho_\infty-\rho_0)}{\kappa} \right],\label{eq:cqnlsN} \\
\delta &= 2 \tan^{-1}\left[ \frac{p(\rho_\infty-\rho_0)}{\rho_0\kappa} \right]. \label{eq:cqnlsdelta}
\end{aligned}$$ From them we can calculate the renormalized momentum $ -P=pN+\rho_\infty \delta $. An example is shown in Fig. \[fig:renP\]. The case where the unstable region exists is analyzed in Subsec. \[sec:criticalcase\] in detail.
 Plot of renormalized momentum $ -P/\rho_\infty=\delta+pN/\rho_\infty $ in CQNLS. Here we set $ (a_1,a_2)=(-1,1) $. The values of $ \rho_\infty $ of each curve are set, from top to bottom, $ \rho_\infty=0.55,\, 0.52,\, 0.502,\, 0.5,\, 0.498,\, 0.48,\, \text{ and } 0.45 $, respectively. The dark solitons are stable in the regions of the solid lines, while unstable in the regions of the dashed lines. The critical points are marked by black dots. The unstable soliton appears when $ \rho_\infty<-a_1/(2a_2)=0.5 $. ](renP102.eps)
### Purely cubic case {#sec:purecubic}
As a first example, let us consider the case $ a_2=0 $, i.e., the nonlinearity is purely cubic. As mentioned in the Introduction, the NLS equation is integrable in this case and the Bogoliubov phonons are reflectionless for any energy. Let us see that our analytical result Eq. (\[eq:mainrR\]) is consistent with these | 1 | member_42 |
known facts.\
Without loss of generality, we can set $ a_1=1 $. The dark soliton solution (\[eq:dscqnls\]) is then reduced to $$\begin{aligned}
\phi = \mathrm{e}^{\mathrm{i}px}\left( p+\mathrm{i}\kappa\tanh \kappa x \right), \quad \kappa = \sqrt{\rho_\infty-p^2}.
\end{aligned}$$ The exact solution of the linearized equation, i.e., the Bogoliubov equation is given by the squared Jost solution[@ChenChenHuang; @Kovrizhin]: $$\begin{aligned}
u &= \mathrm{e}^{\mathrm{i}(k_j+p)x}\left( \mathrm{i}\kappa\tanh \kappa x+\frac{k_j}{2}+\frac{\epsilon}{2k_j} \right)^2, \\
v &= \mathrm{e}^{\mathrm{i}(k_j-p)x}\left( \mathrm{i}\kappa\tanh \kappa x+\frac{k_j}{2}-\frac{\epsilon}{2k_j} \right)^2,
\end{aligned}$$ where $ k_j $s are given by the roots of the dispersion relation Eq. (\[eq:uniformdisp\]) with $ F'(\rho_\infty)=2 $.\
From the above explicit expression, it is obvious that the phonons are reflectionless. Therefore, the coefficient of $ \epsilon^2 $ in Eq. (\[eq:mainrR\]) must vanish. Let us check it. For the cubic case, it follows that $$\begin{aligned}
N=-2\kappa,\quad \delta = 2 \tan^{-1}\frac{\kappa}{p},
\end{aligned}$$ by taking the limit $ a_2\rightarrow0 $ of Eqs. (\[eq:cqnlsN\]) and (\[eq:cqnlsdelta\]). With the use of them, we can show $$\begin{gathered}
P_p = -N-pN_p-\rho_\infty \delta_{p}=4\kappa, \\
d=4(\kappa^2-3p^2),\quad d_1=\frac{-\kappa^2+3p^2}{\kappa}.
\end{gathered}$$ Thus we obtain $ d+d_1P_p=0 $, as expected. We also note that the soliton is always stable since $ -P_p<0 $ for all velocities.\
It is also possible to discuss the reflection properties when the quintic term is small | 1 | member_42 |
by expanding Eqs (\[eq:cqnlsN\]) and (\[eq:cqnlsdelta\]) with respect to $ a_2 $, but the expression is not so simple. In this case, one can derive an approximate formula valid not only for small energy but for arbitrary energy by the method given in Refs. [@Muryshev; @SinhaChernyKovrizhinBrand].
### Purely quintic case {#sec:purequintic}
Next, we treat the purely (self-defocusing) quintic case. As already mentioned, the quintic NLS equation is known to describe the dynamics of the Tonks-Girardeau gas[@Kolomeisky].\
Without loss of generality, we can set $ a_1=0 $ and $ a_2=1 $. Though we can also set $ \rho_\infty=1 $, we keep it for a moment because we need the differentiation of $ \rho_\infty $ to calculate the reflection coefficient (\[eq:mainrR\]). Eqs. (\[eq:cqnlsN\]) and (\[eq:cqnlsdelta\]) are reduced to $$\begin{aligned}
N &= -\tanh^{-1}\left[ \frac{\sqrt{3(1-y^2)}}{2} \right], \\
\delta &= 2 \tan^{-1}\left[ \frac{1-3y^2+\sqrt{1+3y^2}}{3y\sqrt{1-y^2}} \right] \\
\text{with}\quad y &:= \frac{p}{p_{\text{L}}} = \frac{1}{\sqrt{3}}\frac{p}{\rho_\infty}.
\end{aligned}$$ From them one can plot the renormalized momentum $ -P/\rho_\infty=\sqrt{3}yN+\delta $ and can show that $ -P_p<0 $ always holds, i.e., the soliton is always stable. One can also obtain the coefficient of $ \epsilon^2 $ in Eq. (\[eq:mainrR\]) as follows: $$\begin{aligned}
-\frac{d+d_1P_p}{aP_p} &= \frac{1}{4\sqrt{3}\rho_\infty^2}\frac{\gamma\tanh^{-1}\gamma}{\gamma+2(1-\gamma^2)\tanh^{-1}\gamma}, \label{eq:quinticcoeff} \\
\gamma&:=\frac{\sqrt{3(1-y^2)}}{2}. \end{aligned}$$
 Energy-dependence of | 1 | member_42 |
reflection coefficient $ R $ of linearized excitations for various soliton velocities in the purely quintic system. Here we set $ (a_1,a_2)=(0,1) $ and $ \rho_\infty=1 $. The Landau velocity is given by $ p_{\text{L}}=\sqrt{3}\rho_\infty=\sqrt{3} $. Parabolic curves represent the theoretical approximate expression (\[eq:mainrR\]) with (\[eq:quinticcoeff\]).](quinticRC002.eps)
The energy-dependence of the reflection coefficient $ R $ of linearized excitations is obtained by solving the Bogoliubov equation (\[eq:bogocqnls\]) numerically, and the results are shown in Fig. \[fig:quinticRC\]. We can verify that the expression (\[eq:mainrR\]) with (\[eq:quinticcoeff\]) is valid for low-energy region. From this figure we can also see that the soliton with zero velocity is the strongest scatterer. It is intuitively clear since the shape of the soliton becomes shallower and wider if the velocity of the soliton increases. However, this intuitive understanding is not always correct, as the integrable cubic case in Subsec. \[sec:purecubic\] and the instability-induced anomaly in Subsec. \[sec:criticalcase\] illustrate.
### The case with $ a_1<0 $ and $ a_2>0 $ {#sec:criticalcase}
Finally, we consider the case with $ a_1<0 $ and $ a_2>0 $, which is most interesting from the viewpoint of critical phenomena, since the soliton can become unstable and the reflection coefficient can show the singular | 1 | member_42 |
and scaling behavior.\
If both $ a_1 $ and $ a_2 $ are nonzero, we can set $ |a_1|=|a_2|=1 $ without loss of generality by the following scale transformation: $$\begin{gathered}
\bar{x}=\frac{x}{\xi},\ \bar{t}=\frac{t}{\xi^2},\ \bar{\phi}(\bar{x},\bar{t})=\frac{1}{\eta}\phi(x,t), \\
\xi=\frac{\sqrt{|a_2|}}{|a_1|},\ \eta=\sqrt{\frac{|a_1|}{|a_2|}}.
\end{gathered}$$ So we performed numerical calculations by setting $ (a_1,a_2)=(-1,1)$. Note that $ \rho_\infty $ cannot be normalized to be unity if we choose $ \xi $ and $ \eta $ as the above. Another choice of $ \eta $ is possible to normalize $ \rho_\infty=1 $, but in this case either $ a_1 $ or $ a_2 $ cannot be normalized.\
Using the dark soliton solution (\[eq:dscqnls\]), we numerically solved the stationary Bogoliubov equation (\[eq:bogocqnls\]), and constructed the solution with the asymptotic form (\[eq:uvasymptotic\]). Figure \[fig:RC\] shows the reflection coefficient with $ \rho_\infty=0.45 $ for various soliton velocities. We can observe that the zero-energy phonon transmits perfectly, unless the soliton velocity is equal to the critical one. When the soliton velocity comes close to the critical value, the slope of the reflection coefficient becomes very steep, and at the critical state, the perfect transmission eventually vanishes. The approximate expression (\[eq:mainrR\]) is good for sufficiently low excitation energy. Figure \[fig:scaledR\] shows the scaling behavior | 1 | member_42 |
of reflection coefficient $ R $. Here, based on Eq. (\[eq:energyscale\]), the horizontal axis is chosen to be the scaled energy $ \tilde{\epsilon}=\epsilon/\sqrt{2P''(p_c)(P(p)-P(p_c))} $. We can see that if the soliton velocity is close to the critical one, numerically calculated points are well fitted to the universal curve (\[eq:scaledR\]). Thus, the theoretical results are well confirmed in this example.
 Energy-dependence of reflection coefficient $ R $ of linearized excitations for various soliton velocities. Here we set $ (a_1,a_2)=(-1,1) $ and $ \rho_\infty=0.45 $. The critical velocity of the dark soliton is given by $ p_c = (0.206597\dots)\times p_{\text{L}}$. (See the lowest curve of Fig. \[fig:renP\].) A reflection coefficient of zero-energy phonon at the critical state is given by $ (d/b)^2\simeq 0.5718 $. Parabolic curves represent the theoretical approximate expression (\[eq:mainrR\]).](RC103.eps)
 Scaling behavior of reflection coefficient $ R $. Here we set $ (a_1,a_2)=(-1,1) $ and $ \rho_\infty=0.45 $. $ \tilde{\epsilon}:=\epsilon/\sqrt{2P''(p_c)(P(p)-P(p_c))} $ is a scaled excitation energy. “Theory” represents the universal form of reflection coefficient (\[eq:scaledR\]).](scaledR002.eps)
Proof – Step 1: Exact Zero-Energy Solutions {#sec:proof1}
===========================================
In this and the next section, we prove the main result. This section is particularly devoted to the construction of exact zero-energy | 1 | member_42 |
solutions. As an important tool, parameter derivatives are introduced.
Parameter derivative {#subsec:paradera}
--------------------
As seen in the asymptotic form (\[eq:nlsdsasym\]) or in the example of the CQNLS system in Subsec. \[subsec:cqnls\], the dark soliton solution has two parameters, i.e., $ (p,\rho_\infty) $. So we can consider two kinds of parameter derivatives:[^1] $ \partial_p\phi $ and $ \partial_{\rho_\infty}\phi $. We can use arbitrary coordinates to “label” the two-dimensional parameter space $ (\alpha,\beta)=(\alpha(p,\rho_\infty), \beta(p,\rho_\infty)) $, unless the Jacobian of coordinate transformation is singular. Obviously, the final result must not depend on the choice of coordinates. In order to make the story general, we always use these general parameter derivatives, and henceforth, we write the parameter derivative simply by the subscript, i.e., $ \phi_\alpha := \partial_\alpha\phi $ and $ \phi_\beta:=\partial_\beta\phi $. We also introduce the following symbol: $$\begin{aligned}
[A,B]_{\alpha\beta}:=A_\alpha B_\beta-A_\beta B_\alpha.
\end{aligned}$$ Note that the ratio $$\begin{aligned}
\frac{[A,B]_{\alpha\beta}}{[C,D]_{\alpha\beta}} \label{eq:invratio}
\end{aligned}$$ has a coordinate-free meaning, in other words, it is invariant under coordinate transformations of parameter space. We often construct coordinate-free solutions in such a ratio form.\
An immediate application of parameter derivatives is that one can obtain a particular solution of zero-energy Bogoliubov equation (\[eq:bogos\]). By differentiation of the stationary NLS eq. (\[eq:nls2\]), | 1 | member_42 |
one obtains $$\begin{aligned}
\mathcal{L}_\mu \begin{pmatrix} \phi_\alpha \\ -\phi^*_\alpha \end{pmatrix} = \mu_\alpha \begin{pmatrix} \phi \\ \phi^* \end{pmatrix}. \label{eq:1storder}
\end{aligned}$$ The same expression follows by replacement $ \alpha\rightarrow\beta $. Thus, taking the difference of double parameter derivatives, one can obtain the following zero-energy solution: $$\begin{aligned}
\mathcal{L}_\mu \begin{pmatrix} [\mu,\phi]_{\alpha\beta} \\ -[\mu,\phi^*]_{\alpha\beta} \end{pmatrix} = 0.
\end{aligned}$$ It must be emphasized that this solution exists even when a localized potential barrier is added, in other words, when the fundamental equation loses a translational symmetry. What we only need is two kinds of parameter derivatives. So, this solution is not a symmetry-originated zero-mode. (For a symmetry consideration, see \[app:symmetry\].)\
Some technical (but crucially important) identities are derived in \[app:idnty\]. Equation (\[eq:1storder\]) will be used again in the process of energy expansions.
Density fluctuation and phase fluctuation {#subsec:fg}
-----------------------------------------
Here we introduce notations for the linearized density fluctuations and phase fluctuations, and rewrite the Bogoliubov equation with respect to these variables. They are convenient for both calculations and physical interpretations. Through the symbols $ (u,v)=(\delta\phi,-\delta\phi^*) $, the density and phase fluctuations are expressed as $$\begin{aligned}
\delta \rho &= \delta(\phi\phi^*)=\delta\phi\phi^*+\phi\delta\phi^*=u\phi^*-v\phi \\
\delta S &= \delta\left( \frac{1}{2\mathrm{i}}\log\frac{\phi}{\phi^*} \right)=\frac{1}{2\mathrm{i}}\left( \frac{\delta\phi}{\phi}-\frac{\delta\phi^*}{\phi^*} \right)=\frac{1}{2\mathrm{i}}\left( \frac{u}{\phi}+\frac{v}{\phi^*} \right)
\end{aligned}$$ Therefore, if one defines $ f | 1 | member_42 |
---
abstract: 'New sensitive CO(2-1) observations of the 30 Doradus region in the Large Magellanic Cloud are presented. We identify a chain of three newly discovered molecular clouds we name KN1, KN2 and KN3 lying within 2–14pc in projection from the young massive cluster R136 in 30 Doradus. Excited H$_2$2.12$\mu$m emission is spatially coincident with the molecular clouds, but ionized Br$\gamma$ emission is not. We interpret these observations as the tails of pillar-like structures whose ionized heads are pointing towards R136. Based on infrared photometry, we identify a new generation of stars forming within this structure.'
author:
- 'Venu M. Kalari'
- 'M[ó]{}nica Rubio'
- 'Bruce G. Elmegreen'
- 'Viviana V. Guzm[á]{}n'
- 'Cinthya N. Herrera'
- Hans Zinnecker
title: |
Pillars of creation amongst destruction:\
Star formation in molecular clouds near R136 in 30 Doradus
---
Introduction {#sec:intro}
============
30 Doradus is a giant H[II]{} region in the Large Magellanic Cloud (LMC). The LMC is a local group dwarf galaxy that lies at a distance of 50kpc (Pietrzy[ń]{}ski et al. 2013), and has a mean stellar metallicity ($Z$) half of the Sun (Rolleston et al. 2002). 30 Doradus hosts the young massive cluster (YMC) R136. R136 is a $\sim$1.5-3Myr | 1 | member_43 |
YMC that encloses a total cluster mass in excess of 10$^5$$M_{\odot}$ within 10pc (Selman & Melnick 2013). The cluster contains roughly 200 massive stars ($>$8$M_{\odot}$) within a central region less than 6pc, whose radiation and mechanical feedback profoundly impact the surrounding medium (Schneider et al. 2017, subm.). R136 is the most massive YMC in our local neighbourhood that can be adequately resolved spatially (at 50kpc, the nominal distance to R136, 1$\arcsec$$\approx$0.25pc) enabling us observe individual objects at the star and molecular clump scale. This makes R136 an ideal laboratory to examine how feedback from massive stars affects further star formation (e.g. Dale et al. 2012).
The mechanical and radiation output from R136 has created a central cavity by sweeping the surrounding molecular clouds (labelled as clouds 6 and 10 in Fig.1) that extend up to 100pc along the northeast-southwest axis (Pellegrini et al. 2010). We adopt the cloud nomenclature of Johansson et al. (1998). Brightly illuminated arcs delineate the interfaces between the cold gas and the ionizing radiation, where subsequent generations of stars are thought to have been triggered (Walborn et al. 2002). Studies at optical (De Marchi et al. 2011; Kalari et al. 2014), near-infrared (nIR; Rubio et al. | 1 | member_43 |
1998; Brandner et al. 2001), mid-infrared (mIR; Whitney et al. 2008; Gruendl & Chu 2009; Walborn et al. 2013), far-infrared (fIR; Seale et al. 2014) and sub-millimeter (Johansson et al. 1998; Indebetouw et al. 2013) wavelengths have identified evidence for active star formation throughout the 30 Doradus nebula, consistent with the idea of multiple star formation episodes.
We focus on the stapler nebula that lies 2-14pc away from R136 (see Fig.1) in projection. The stapler nebula is the H[II]{} region including and surrounding the stapler shaped dark cloud that is seen in silhouette in the optical near R136. The nebula spans an area of 1.1$'\times$0.35$'$ centred on $\alpha$=05$^h$38$^m$40$^s$, $\delta\,=\,-$69$^{\rm \circ}$05$'$36$''$ and is elongated with a position angle of 35$^{\rm \circ}$. A candidate young stellar object (YSO) has been reported at the edge of the elongated dark cloud by Walborn et al. (2013; marked as S5 in that paper). The YSO is close to, but not coincident with a region of high density ($n>$10$^{6}$cm$^{-3}$) reported by Rubio et al. (2009) using CS line observations. Known infrared excess objects, some of which are thought to be disc/envelope bearing young stellar objects (YSOs) are dotted towards the edge of the dark cloud according | 1 | member_43 |
to Rubio et al. (1998; their Figure 3). The literature evidence for dense molecular gas and YSOs in the stapler nebula lying near R136 indicates that star formation may be ongoing, which deserves further study. In this paper we discuss the properties of molecular clouds in the stapler nebula, and examine whether new stars are being formed in these clouds.
This paper is organised as follows. In Section 2 we describe the data used in this study. The results from the analysis of CO(2-1) line observations are presented in Section 3. We discuss the results obtained from nIR emission line images of the stapler nebula in Section 4. Based on archival infrared photometry, we identify YSOs within the stapler nebula in Section 5. The picture obtained from our results is described in Section 6. In Section 7, a brief summary of our paper is presented along with future work arising from our results.
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\[fig:COa\]
Data
====
[CO(2-1)]{} observations
------------------------
We conducted a deep CO(2-1) survey centred on the 30 Doradus Nebula using the Swedish-ESO submillimeter telescope (SEST) across March 1997- January 2001. SEST was a 15m radio telescope located at La Silla, Chile. The angular resolution | 1 | member_43 |
at the CO(2-1) frequency of 230GHz is 23$\arcsec$, which corresponds to a projected size of 5.6pc at the distance to the LMC.
Observations were conducted in position switching mode using a reference point free of CO(2-1) (at $\alpha$=05$^{h}$37$^{m}$54$^s$, $\delta$=$-69^{\rm \circ}$04$'24''$) for sky subtraction. The backend narrow-band high-resolution spectrometer (HRS) was used with a bandwidth of 80MHz, and a frequency resolution of 41.7kHz, which translates to a velocity resolution of 0.054kms$^{-1}$ at the frequency of CO(2-1). The data were reduced using the GILDAS software [^1], with linear or third-order polynomial for baseline fitting. The resultant spectra were smoothed to a velocity resolution of 0.25kms$^{-1}$. The rms noise achieved in a single channel is 0.07K after 240s of integration. We mapped the 30 Doradus region with 10$\arcsec$ spacings and detected CO(2-1) emission across 30 Doradus, including the stapler nebula where CO(1-0) emission had not been previously detected (Pineda et al. 2009; Johansson et al. 1998). In Fig.1, the CO(2-1) contours are overlaid on a [*Hubble space telescope*]{} (HST) three colour optical $BVI$H$\alpha$ image of 30 Doradus, where the position of CO(2-1) emission with respect to the R136 cluster, and its ionized surroundings can be visualized. Our observations represent a five-fold increase in | 1 | member_43 |
sensitivity at twice the spatial resolution of previous CO(1-0) observations across the 30 Doradus nebula (see Pineda et al. 2009). Higher angular resolution observations of the 30 Doradus nebula are presented in Indebetouw et al. (2013), Anderson et al. (2014) and Nayak et al. (2016), but those data do not cover the region studied here.
Near-infrared emission line imaging
-----------------------------------
We obtained nIR imaging of the 30 Doradus region in H$_2$ 2.12$\mu$m narrowband filter, and the $K$s broadband filter using the ISAAC (Infrared spectrometer and Array Camera) imager mounted on 8m Melipal (UT3) telescope of the Very Large Telescope (VLT) situated in Paranal, Chile (program ID 078.C-0487A). Toward the R136 region, the mosaic covered a field of 5$\arcmin$$\times$5$\arcmin$ area. The average seeing measured from the images is $\sim$0.8$''$–1.1$''$. Observations were taken in the ABBA sequence with the sky image 30$'$ from the stapler nebula at $\alpha$=05$^h$39$^m$01$^s$, $\delta\,=\,-$69$^{\rm \circ}$42$'$36$''$ in a region free of nebulosity within the ISAAC field of view, ensuring adequate sky subtraction. The total integration time on source was 1 hour for the narrowband filter, and 100s for the $K$s broadband filter. Data were reduced using the ISAAC pipelines, with flux calibration carried out using Persson (1998) standards. | 1 | member_43 |
Astrometric calibration was refined using 2MASS. Bright stars are saturated in the emission line images and have a non-linear CCD response, meaning they cannot be completely subtracted. This leads to circular bright residuals (and in some cases vertical bleeding) in the emission line images. We excluded these regions from further analysis by masking them.
We used the Br$\gamma$ 2.165$\mu$m narrowband flux calibrated image from Yeh et al. (2015). The image was obtained using the NOAO Extremely Wide Field Infrared Imager (NEWFIRM) mounted on the 4m Victor Blanco telescope located at the Cerro Tololo Inter-American Observatory, Chile. The final Gaussian convolved image resolution for the Br$\gamma$ narrowband image is 1$\arcsec$, and is comparable to the VLT H$_2$2.12$\mu$m narrowband image.
Archival photometry
-------------------
mIR photometry of point sources in the stapler nebula were estimated by Gruendl & Chu (2009) from images at 3.6, 4.5, 5.8 and 8.0$\mu$m taken using the [*Spitzer*]{} space telescope IRAC (Infrared Array Camera) as part of the [*Spitzer*]{} legacy program SAGE (Spitzer Survey of the Large Magellanic Cloud: Surveying the Agents of a Galaxy’s Evolution; Meixner et al. 2006). The full width half maximum (FWHM) of the images is 1.6$\arcsec$, 1.7$\arcsec$, 1.7$\arcsec$ and 2$\arcsec$ respectively. Alternative photometry of | 1 | member_43 |
point sources from the same images are also presented by Whitney et al. (2008), but Whitney et al. (2008) miss a significant fraction of point sources in the 30 Doradus region, as their study is motivated towards detecting reliable sources throughout the LMC via pipeline analysis. Gruendl & Chu (2009) detect objects in the dense and nebulous surroundings using detailed aperture photometry (see Section 6.3 of Gruendl & Chu (2009) for a comparison). Photometry in the fIR at 100 and 160$\mu$m, and 250 and 350$\mu$m of point-like and extended sources in the stapler nebula is given in Seale et al (2014), using images taken by the [*Herschel*]{} space telescope PACS (Photoconductor Array Camera and Spectrometer), and SPIRE (Spectral and Photometric Imaging Receiver) instruments respectively as part of the [*Hershel*]{} large program HERITAGE (HERschel Inventory of The Agents of Galaxy Evolution; Meixner et al. 2013). The FWHM for these images are 8$\arcsec$, 12$\arcsec$, 18$\arcsec$ and 25$\arcsec$ respectively. We utilize the photometry from Gruendl & Chu (2009) and Seale et al. (2014) in this study.
Molecular clouds
================
Figure \[fig:CO\] shows the CO(2-1) integrated line emission over the velocity interval 235–270 kms$^{-1}$ as contours, superimposed on the HST composite $BVI$H$\alpha$ image. Strong | 1 | member_43 |
CO(2-1) emission from Cloud 10 (northeast region of the map) and Cloud 6 (southwest region of the map) previously reported by Johansson et al. (1998) is seen along the northeast-southwest axis.
We observe previously undetected CO(2-1) emission originating from the region located between Clouds 6 and 10, close to R136 (see Fig.\[fig:COa\]). The emission extends along the southeast-northwest direction in projection. The distance of the emission from R136 in projection is between 2pc (emission is located to the north of R136) to 14pc away (in the northwest direction from R136). This CO emission is approximately five times weaker than the CO emission observed in Clouds 6 and 10. This emission is coincident spatially to the stapler nebula in the optical image of Fig.\[fig:COa\]a. The emission is resolved in CO(2-1) velocity as a chain of small and weak clouds (Fig.\[fig:COa\]a,b). We define the stapler region by the extent of the CO(2-1) emission, which goes beyond the visible stapler shaped dark cloud in the optical. This boundary is marked in Fig.1 with a dashed rectangle. We named the CO clouds Knots (KN), as they form a chain separated in velocity, as demonstrated by the position velocity slice across the stapler nebula, and | 1 | member_43 |
the CO(2-1) spectra of each individual cloud shown in Fig.\[fig:spectra\]. By analysing the radial velocities and spatial distribution we found that the KN clouds are composed of three clouds we name KN1, KN2, and KN3 in order of decreasing Right Ascension (labelled in Fig.\[fig:COa\]a).
![[*Top*]{}: The stapler nebula blown up from the HST mosaic in Fig.1, with the stapler region is outlined with the dashed white box. The outermost CO(2-1) contours integrated over the 235–240kms$^{-1}$, and 245–250kms$^{-1}$ are shown in magenta and red respectively, with the second outermost contour of the 240–245kms$^{-1}$ also shown. The solid white line marks the position of the slice shown in the middle panel. [*Middle*]{}: Position velocity slice of the CO(2-1) cube in linear scale along the direction of the slit given by the solid white line in the top panel. The slit cuts along the centre of the stapler nebula. [*Bottom*]{}: CO(2-1) spectra of each cloud extracted from the region bounded by the contours shown in top panel. The dashed lines for each cloud is it’s $V_{\rm{lsr}}$ given in Table 1. []{data-label="fig:spectra"}](30dorfigcocloseupcss.jpg "fig:"){width="49.50000%"} ![[*Top*]{}: The stapler nebula blown up from the HST mosaic in Fig.1, with the stapler region is outlined with the dashed | 1 | member_43 |
white box. The outermost CO(2-1) contours integrated over the 235–240kms$^{-1}$, and 245–250kms$^{-1}$ are shown in magenta and red respectively, with the second outermost contour of the 240–245kms$^{-1}$ also shown. The solid white line marks the position of the slice shown in the middle panel. [*Middle*]{}: Position velocity slice of the CO(2-1) cube in linear scale along the direction of the slit given by the solid white line in the top panel. The slit cuts along the centre of the stapler nebula. [*Bottom*]{}: CO(2-1) spectra of each cloud extracted from the region bounded by the contours shown in top panel. The dashed lines for each cloud is it’s $V_{\rm{lsr}}$ given in Table 1. []{data-label="fig:spectra"}](pv2.jpg "fig:"){width="49.50000%"} ![[*Top*]{}: The stapler nebula blown up from the HST mosaic in Fig.1, with the stapler region is outlined with the dashed white box. The outermost CO(2-1) contours integrated over the 235–240kms$^{-1}$, and 245–250kms$^{-1}$ are shown in magenta and red respectively, with the second outermost contour of the 240–245kms$^{-1}$ also shown. The solid white line marks the position of the slice shown in the middle panel. [*Middle*]{}: Position velocity slice of the CO(2-1) cube in linear scale along the direction of the slit given by the solid | 1 | member_43 |
white line in the top panel. The slit cuts along the centre of the stapler nebula. [*Bottom*]{}: CO(2-1) spectra of each cloud extracted from the region bounded by the contours shown in top panel. The dashed lines for each cloud is it’s $V_{\rm{lsr}}$ given in Table 1. []{data-label="fig:spectra"}](spectra.pdf "fig:"){width="49.50000%"}
Physical properties
-------------------
After identifying each cloud, we computed the central velocity ($V_{\rm lsr}$) in the local standard of rest frame, and velocity width ($\sigma_{v}$) by fitting a Gaussian profile to the total cloud spectrum. The major and minor axis sizes of the profiles, in conjunction with the rms size of the beam were used to compute the deconvolved radius ($r$). Given the uncertainties on the Gaussian fit of the CO spectra found for each cloud are around 30%, we estimate the uncertainties on $r$ to be 15%.
### CO luminosity and mass
The CO cloud luminosity is computed as: $$L_{\rm CO}\, [K{\rm kms}^{-1}{\rm pc}^{-2}] = \rm{D}^2 \int_{\Omega} \int_v T_{\rm mb}(\nu) \, d\nu \,d\Omega$$ where D is the distance to the source in pc (adopted as 50kpc), $T_{\rm mb}$ the main beam temperature which is the antenna temperature corrected for the efficiency of the antenna ($T_{\rm mb} = T_{\textrm{A}}/\eta$), and $\Omega$ | 1 | member_43 |
is the solid angle of the subtended by the source.
The H$_2$ mass of the clouds can be calculated from the observed CO(1-0) luminosity assuming a linear conversion between the velocity integrated CO emission ($I_{\textrm{CO}}$) and the H$_2$ column density ($N_{{\rm H}_2}$); $$N_{\rm H{_2}} = X_{\textrm{CO}}\, [cm^{-2}(K{\rm kms^{-1}})^{-1}]\,\, I_{\textrm{CO}}\,[K{\rm kms^{-1}}],$$ where $X_{\textrm{CO}}$ is the CO-to-H$_2$ conversion factor (Bolatto et al. 2013; Roman-Duval et al. 2014). The total mass of H$_2$ ($M_{{\rm H}_2}$) is, $$M_{\textrm{H}_2} \,[M_\odot] = \alpha_{\textrm{CO}} {\rm D}^2\, [{\rm Mpc}] S_{\textrm{CO}},$$ where $$\alpha_{\textrm{CO}}\, [M_\odot {\rm Mpc}^{-2} ({\it Jy}{\rm kms}^{-1})^{-1}] = X_{\textrm{CO}} \frac{m_{\textrm{H}_2} c^2}{2 k \nu^2},$$ and the flux density $S_{\textrm{CO}}$ is, $$S_{\textrm{CO}}\,\, [{\it Jy}{\rm kms}^{-1}] = \int S_{\nu} \, dv.$$ The molecular gas mass is multiplied by 1.36 to include the Helium contribution. This method is calibrated for the $J$=1$\rightarrow$0 transition. We use a ratio between the CO$J$=2$\rightarrow$1 and $J$=1$\rightarrow$0 lines of 0.87 for 30 Doradus Cloud 10 (the North Eastern cloud; see Fig.1) found by Johannson et al. (1998) to estimate the CO(1-0) luminosity.
The conversion factor depends on both metallicity and the ambient radiation field intensity (Maloney 1988). As a consequence of strong radiation fields and poor self-shielding in low metallicity environments, the CO molecule is photo-dissociated | 1 | member_43 |
as it does not self-shield like the H$_2$ molecule. Therefore, there is less CO compared to the H$_2$ abundance in the Magellanic Clouds than in the Galaxy. This translates into higher values of $X_{\textrm{CO}}$ in the LMC compared to the Galaxy. In general the conversion factor increases with higher radiation fields and decreases with higher metallicities. We adopt the median conversion factor in the LMC compiled from the literature by Bolatto et al. (2013) of $$X_{\textrm{CO}} = 8.8 \pm 0.3 \times 10^{20} \,[\textrm{cm}^{-2} \textrm{(K kms}^{-1})^{-1}].$$ The adopted $X_{\textrm{CO}}$ factor is 3.8 times larger than the canonical Galactic $X_{\textrm{CO}}$ of Bolatto et al. (2013). Our adopted value is similar to that found by Herrera et al. (2013) when comparing molecular and dust mass estimates in the LMC N11 region; but higher than the value of $6 \times 10^{20} \,\textrm{cm}^{-2} \textrm{(K kms}^{-1})^{-1}$ reported by Roman-Duval et al. (2014). The resulting cloud masses if we adopted the Roman-Duval et al. (2014) $X_{\textrm{CO}}$ would be reduced by $\sim20$%.
### Virial mass
The virial mass ($M_{\rm{vir}}$) was computed assuming that each cloud is spherical, is in virial equilibrium and has a density ($\rho$) profile of the form $\rho \propto
r^{-1}$. The virial mass is given | 1 | member_43 |
by $$M_{\textrm{vir}} \,[M_\odot] = 190 \sigma_{v}^2\,[{\rm kms}^{-1}]\, r\, [{\rm pc}]
\label{eq:virial_mass2}$$ according to MacLaren et al. (1988). The results of our analysis for each cloud are given in Table 1.
From Table \[tab:clouds\_properties\] we see that the virial masses are a factor 3-6 larger than the masses derived from the integrated CO emission for the resolved molecular clouds in 30 Doradus. Therefore, the conversion factor between the H$_2$ column density and the CO intensity is, on average, 4.5 times the Galactic value. This translates into a conversion factor $X_{\textrm{CO}} = 1.0 \pm 0.4 \times 10^{21}$ cm$^{-2}$ (K km s$^{-1}$)$^{-1}$. [@israel97] estimated the H$_2$ column densities towards CO clouds in the LMC and SMC from far-infrared surface brightness, and derived, in units of $10^{21}$ cm$^{-2}$ (K km s$^{-1}$)$^{-1}$, $X_{\textrm{CO}} = 12 \pm 2$ and $X_{\textrm{CO}} = 1.3 \pm 0.2$ for the SMC and LMC, respectively. [@israel03] found, in units of $10^{20}$ cm$^{-2}$ (K km s$^{-1}$)$^{-1}$, $X_{\textrm{CO}} = 4.8 \pm 1.0$ and $X_{\textrm{CO}} =
4.3 \pm 0.6$ for the SMC and LMC, respectively. [@garay02] derived $X_{\textrm{CO}} = 6.4 \times 10^{20}$ cm$^{-2}$ (K km s$^{-1}$)$^{-1}$ for the Complex-37 in the LMC. [@johansson98] also derived conversion factors close to the canonical value for the | 1 | member_43 |
Galaxy to a factor of a few higher, although he found these values using CO($1-0$) observations whose sensitivity is lower than our CO($2-1$) data. There is a small difference between the conversion factor for 30 Doradus found in this work and the previously mentioned values found by other studies in the Magellanic clouds. However, one would expect to find a different conversion factor in the clouds of 30 Doradus than in the rest of the clouds of the LMC due to the extreme conditions in the environment, specially the strong radiation fields that photo-dissociate the CO molecule leaving large H$_2$ envelopes untraced by CO.
Analysis
--------
### Larson’s Laws
Molecular clouds in virial equilibrium follow the empirical power law relation $\sigma_{v} \propto r^{\alpha}$ (Larson 1981). $\alpha$ is generally agreed to be between 0.4–0.5 based on numerous molecular cloud surveys of the Milky Way, and external normal and dwarf Galaxies (Bolatto et al. 2008; Heyer et al. 2009). The observed value of $\alpha$ is oft explained by turbulence (McKee & Ostriker 2007; Lombardi et al. 2010). The velocity dispersion is considered to be a measure of the internal dynamics within the clouds, because the observed line profiles, averaged over a cloud, | 1 | member_43 |
have Gaussian shapes and the line widths are broader than the thermal line widths. It is a reasonable assumption to make that the line profiles are produced by turbulent motions of the gas inside the clouds (Solomon et al. 1987).
Figure\[fig:slw\]a displays the position of the KN clouds in the $\sigma_{v}$–$r$ diagram. Also shown are results from Heyer et al. (2009) summarising the canonical relation found for Galactic clouds of $\sigma_{v} = 0.72\,r^{0.5}$; and clouds in the 30 Doradus region (from Pineda et al. 2009; having a resolution of 43$\arcsec$; and from Nayak et al. (2016) having a resolution of 2$\arcsec$), and in the LMC (excluding 30 Doradus) from Wong et al. (2011) whose study had a spatial resolution of 45$\arcsec$. Note that the Pineda et al. (2009) and Nayak et al. (2016) clouds do not cover the central region of 30 Doradus near R136, and there are no spatial overlaps between the KN clouds and the clouds they identify. The molecular clouds associated with the stapler nebula lie above the canonical $\sigma_{v}$–$r$ relation for Galactic clouds. Interestingly, the other detected clouds in 30Doradus from Pineda et al. (2009) and Nayak et al. (2016) also lie above the canonical relation. | 1 | member_43 |
Although the departure from the relation is not at the same scale as the KN clouds, this might still indicate that the observed $\sigma_{v}$–$r$ relation might be a function of distance from R136, and a more global property of 30Doradus.
The position of the KN molecular clouds in the $\sigma_{v}$–$r$ diagram implies either of two scenarios; the clouds are collapsing or expanding, or the observed line widths are the manifestation of external pressures that keep the clouds in equilibrium. Since collapse velocities are generally only $\sim40$% larger than equilibrium velocity dispersions for a self-gravitating cloud, the observed large linewidths for the sizes of the KN clouds are probably not the result of collapse (Ballesteros-Paredes et al. 2011). Neither are they likely to result from expansion because there is no obvious shell or hole structure that usually accompanies expansion.
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Name R.A. Dec. $V_{\rm lsr}$ $\sigma_{\rm v}$ $L_{\rm CO}$ $r$ $M_{\rm H_2}$ $M_{\rm vir}$
(J2000) (J2000) (kms$^{-1}$) (kms$^{-1}$) (Kkms$^{-1}$pc$^2$) (pc) (10$^3\,M_{\odot}$) (10$^3\,M_{\odot}$)
30Dor-KN1 5$^h$38$^m$45$^s$ $-$69$^{\rm \circ}$06$'$00$''$ 237.2$\pm$0.1 4.0$\pm$0.7 154.2$\pm$21.3 3.93$\pm$0.58 2.7$\pm$0.2 10.9$\pm$3.3
30Dor-KN2 5$^h$38$^m$40$^s$ $-$69$^{\rm \circ}$05$'$30$''$ 244.4$\pm$0.2 4.4$\pm$0.1 243.6$\pm$3.7 3.35$\pm$0.5 4.2$\pm$0.3 11.25$\pm$1.8
30Dor-KN3 5$^h$38$^m$36$^s$ $-$69$^{\rm \circ}$05$'$30$''$ 250.0$\pm$0.2 5.0$\pm$0.3 320.8$\pm$44.3 2.96$\pm$0.44 5.3$\pm$0.4 12.83$\pm$2.5
----------- | 1 | member_43 |
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\
We examine whether the observed large $\sigma_{v}$ are the manifestation of external pressures necessary to keep the clouds in equilibrium. In Fig.\[fig:slw\]b, we plot the mass surface density ($\Sigma_{\rm H_2}$) against the ${\sigma_v}^2/r$ value of the KN clouds, along with those in the Milky Way from Heyer et al. (2009); in 30 Dor from Pineda et al. (2009) and Nayak et al. (2016); and in the LMC from Wong et al. (2011). Isolated virial clouds confined by self-gravity follow a linear relation in the ${\sigma_v}^2/r$ vs. $\Sigma_{\rm H_2}$ plot (for e.g. see Heyer et al. 2009). The values of clouds from the literature fall along this expectation. However, the KN clouds alone depart from the expected relation, and are likely confined by external pressure and not in virial equilibrium. Following the simplifying assumptions of Field et al. (2011), we plot isobars of external pressure (in terms of $P/k_{\rm B}$) according to their prescription. The lines reflect the external pressure necessary to confine clouds for a given ${\sigma_v}^2/r$ assuming clouds with a centrally concentrated internal density structure approximated by hydrostatic equilibrium. From Fig.\[fig:slw\]b, we see that external pressures of $\sim\,10^6$cm$^{-3}$K are necessary | 1 | member_43 |
to keep the KN clouds confined. These values are in general agreement with Chevance et al. (2016), who report that the stapler nebula is located in a region with gas pressure $\sim$ 0.85-1.2$\times 10^6$cm$^{-3}$K, with the peak found in KN2 (see Fig. 15 in Chevance et al. 2016).
### Variation in properties as a function of distance from R136
The $V_{\rm lsr}$, and $M_{{\rm H}_2}$ of each cloud are plotted as a function of projected distance from R136 in Fig.\[fig:distanced\]. The velocity of the clouds increases as a function of projected distance from R136. The velocity of the KN3 cloud loosely matches the radial velocities of the stars within R136[^2]. The KN1 cloud is closest in projection to R136 and is blue shifted with respect to the mean velocity of stars in the cluster. This suggests the KN1 cloud, (and likely KN2 and KN3 clouds as suggested by the lack of background stars) lie slightly in front of the cluster. The clouds appear to be moving away from the cluster as function of projected distance from it (Fig.\[fig:distanced\]a).
The $M_{{\rm H}_2}$ of each cloud also increases as the projected distance from R136 increases (Fig.\[fig:distanced\]b). The KN1 cloud mass is approximately | 1 | member_43 |
2 times lower than the KN3 cloud. Under the assumption that the molecular clouds detected in CO(2-1) were initially all of similar densities, and photoionization from R136 alone is evaporating the molecular cloud, then KN1 must be closer to R136 (considering it as the only source of external photoionization) because the ionizing flux decreases to the inverse square with distance.
Line of sight distances
-----------------------
Our main results concerning the detection of cold molecular gas near R136 suffers from possible projection effects. Although the cold molecular gas detected in the CO(2-1) observations lies within 2-14pc in projection of the R136 cluster, the actual distance may likely be further in the line of sight direction allowing for the clouds to possibly survive photoionisation. Chevance et al. (2016) analyse the physical distance of the CO gas in 30 Doradus to the stars, by comparing the incident radiation field on the gas modelled against fIR observations in fine structure lines of the emitted radiation field measured from the known massive star population. By comparing the luminosity of the photodissociation region (which forms the interface between the photoionizing radiation from the stars and the gas) against their predictions, they are able to constrain the | 1 | member_43 |
line of sight distance of the photodissociation regions from R136 with uncertainties of 4pc. Based on their results (Figure 20 in Chevance et al. 2016), the stapler nebula lies less than 20pc away in the line of sight direction from R136, which itself lies at the centre of a sphere of about 6pc in radius. This distance agrees well with the line of sight distance measured from line ratios of ionized lines in optical spectra by Pellegrini et al. (2010). From their Fig.12, we find that the distance of the KN clouds is less than 20pc away in our line of sight from R136.
We also consider that if the CO gas is close to R136, and coincident with the dust, there is likely to be a gradient (reflecting the gradient in the projected $V_{\rm lsr}$) in the dust temperature, and the total fIR luminosity arising from the photodissociation region. Such a gradient is visible in both the dust temperature maps (Guzman 2010), and also in the total fIR luminosity which peaks at KN2, and decreases towards KN3 (see Fig.1 of Chevance of et al. 2016). Therefore, although our observations are unable to resolve the line of sight distances to | 1 | member_43 |
the KN clouds from R136, based on corroboration from multiple independent sources in the literature, we find that the line of sight distance to the KN clouds from R136 is $\lesssim$20pc. The molecular clouds in the stapler nebula lie between 2-14pc in projected distance, and $\lesssim$20pc in the line of sight distance from R136.
Near-infrared emission line imaging
===================================
The H$_2$ 2.12$\mu$m emission line image is shown in Fig.\[fig:COa\]c, with the CO(2-1) contours overlaid. Strong H$_2$ emission is spatially coincident with the CO(2-1) emission of the molecular clouds, and shares similar morphology. The H$_2$ emission is clumpy, with numerous knots and a reticulated pattern. In contrast, detected ionized gas (Br$\gamma$) at the position of the CO(2-1) molecular clouds is weak and diffuse (see Fig.\[fig:COa\]d). This diffuse Br$\gamma$ emission is associated with filaments and arc-like structures of ionized gas vivid in H$\alpha$ (brown in Fig.\[fig:COa\]a). This convinces us that the strong H$_2$ nIR emission is the warmer component of the cold molecular gas traced by the CO detections, whereas the diffuse Br$\gamma$ emission lies slightly beyond the ionized surface of the molecular cloud although no clear demarcation is noted. The Br$\gamma$ morphology is not spatially coincident with the H$_2$ and CO(2-1) | 1 | member_43 |
emission.
The H$_2$/Br$\gamma$ ratio can be used to disentangle shock/collisionally excited H$_2$ from fluorescence excitation. This is because the shocks and collisional excitation affect primarily the H$_2$ gas, leaving the ratio of H$_2$/Br$\gamma$ above unity, whereas fluorescence acts on both the molecular and ionised gas leading to a ratio below unity. Using the absolute flux ratio, we find that the H$_2$/Br$\gamma$ ratio never exceeds 0.5 at an angular resolution of 1$\arcsec$ in the stapler nebula, agreeing with the findings of Yeh et al. (2015). This indicates that the excited nIR H$_2$ is primarily excited by the ultraviolet (UV) radiation from R136 acting on the surfaces of the molecular gas, with the filamentary Br$\gamma$ arising from the same source. The KN clouds therefore must lie in front of us given the morphology of the clumped H$_2$ emission, and lack of background stars. We note that it is possible that shock excited emission is prevalent on smaller scales (a few tenths of a parsec, or $\lesssim$0.5$\arcsec$ at the distance to 30 Doradus) caused by outflows from massive protostars residing within the KN molecular clouds, but our current angular resolution limitations in the nIR narrowband images ($\sim$1$\arcsec$) prevent us from examining the same | 1 | member_43 |
in detail. Future high angular resolution integral field unit (IFU) nIR spectroscopy would help towards constraining this further, as shocked gas will likely be offset in velocity.
A picture of ionization fronts emerges, with the photodissociation region extincted from our line of sight by the cold molecular gas. These structures could resemble the dense “pillars” or structures observed in the galaxy in regions such as M16 and NGC3603 (Sankrit & Hester 2000), but are smaller in scale at $\sim$0.1–0.3pc, and in the 30 Doradus Nebula by Pellegrini et al. (2010). From the observed emission line imaging and CO(2-1) observations it appears that the clouds are being ionised on the backside. We are viewing the KN molecular clouds face on, and they are likely the tail of pillar-like structures (we refer the reader to Pound 1998 for a description of pillar morphology; or to Fig.7) with the ionized head pointing towards R136. The observed velocity line widths of the CO(2-1) line are then likely caused due to the velocity gradient between the head and tail of pillars (e.g. Pound 1998). Following the Bertoldi (1989) analytical theory of photoevaporating clouds, during photoevaporation clouds form a cometary structure similar to pillars. Neutral gas | 1 | member_43 |