Partly-local domain-dependent almost complex structures

TL;DR

This paper addresses a gap in the proof of the independence of genus zero Gromov-Witten invariants from divisor choices within the Cieliebak-Mohnke perturbation scheme, ensuring the invariants' well-definedness.

Contribution

It provides a correction to the proof of invariance of genus zero Gromov-Witten invariants by filling a previously identified gap.

Findings

Confirmed the independence of genus zero Gromov-Witten invariants from divisor choices.

Strengthened the theoretical foundation of Gromov-Witten invariants.

Ensured the robustness of the Cieliebak-Mohnke perturbation scheme.

Abstract

We fill a gap pointed out by N. Sheridan in the proof of independence of genus zero Gromov-Witten invariants from the choice of divisor in the Cieliebak-Mohnke perturbation scheme.

Equations

$$J_{\Gamma }:{\overline{\mathcal{U}}}_{\Gamma }\to J_{\tau }(X,\omega ).$$

$$(f^{\prime \prime }{)}^{\ast }{J}_{{\Gamma }^{\prime }}^{\prime },(f^{\prime }{)}^{\ast }{J}_{{\Gamma }^{\prime \prime }}^{\prime \prime }:{\overline{\mathcal{U}}}_{\Gamma }\to \mathcal{J}(X,V^{\prime },V^{\prime \prime })$$

$$J_{V^{\prime },V^{\prime \prime }}\in \bigcap _E{\mathcal{J}}^E(X,V^{\prime },V^{\prime \prime })$$

$$J_{\Gamma }:T({\overline{\mathcal{U}}}_{\Gamma }\times X)\to T({\overline{\mathcal{U}}}_{\Gamma }\times X)$$

$${\mathcal{J}}_{\Gamma }^E(X,V^{\prime },V^{\prime \prime })\subset \mathrm{Map}({\overline{\mathcal{U}}}_{\Gamma },{\mathcal{J}}^E(X,V^{\prime },V^{\prime \prime }))$$

$$f^{\prime }:{\overline{\mathcal{U}}}_{\Gamma }\to {\overline{\mathcal{U}}}_{{\Gamma }^{\prime \prime }},f^{\prime \prime }:{\overline{\mathcal{U}}}_{\Gamma }\to {\overline{\mathcal{U}}}_{{\Gamma }^{\prime }}$$

$$J_{\Gamma }:{\overline{\mathcal{U}}}_{\Gamma }\to \mathcal{J}(X,V^{\prime },V^{\prime \prime })$$

$$f:{\overline{\mathcal{U}}}_{\Gamma }\to {\overline{\mathcal{U}}}_{{\Gamma }^f}$$

$$J_{{\Gamma }_1^f}:{\overline{\mathcal{U}}}_{{\Gamma }_1^f}\to \mathcal{J}(X,V^{\prime },V^{\prime \prime }).$$

Full text

1. 1 Introduction
2. 2 Partly local perturbations
3. 3 Transversality

1. Introduction

A convenient scheme to regularize moduli spaces of genus zero pseudoholomorphic maps was introduced by Cieliebak-Mohnke [1]. In this scheme one chooses a Donaldson hypersurface to stabilize the domains and then a generic domain-dependent almost complex structure to achieve regularity. The scheme leads to a definition of genus zero Gromov-Witten invariants over the rationals that counts some geometric objects, as opposed to the more abstract perturbation schemes in the Kuranishi or polyfold methods. The proof of independence of Gromov-Witten invariants from the choice of Donaldson hypersurfaces in [1, 8.18] depends on the construction of a parametrized moduli space for the following situation: Given a type $\Gamma$ of stable marked curve let $\overline{\mathcal{U}}_{\Gamma}\to\overline{\mathcal{M}}_{\Gamma}$ denote the universal curve over the compactified moduli space $\overline{\mathcal{M}}_{\Gamma}$ of curves of type $\Gamma$. Let $\mathcal{J}_{\tau}(X,\omega)$ denote the space of $\omega$-tamed almost complex structures on the given symplectic manifold $(X,\omega)$ with rational symplectic class $[\omega]\in H^{2}(X,\omega)$. A domain-dependent almost complex structure is a map

[TABLE]

Associated to a coherent collection of sufficiently generic choices $\underline{J}=(J_{\Gamma})$ is a Gromov-Witten pseudocycle $\overline{\mathcal{M}}_{0,n}(X,\beta)\subset X^{n}$ for each number of markings $n$ and each class $\beta\in H_{2}(X)$.

Naturally one wishes to show that the resulting pseudocycle is independent, up to cobordism between pseudocycles, from the choice of Donaldson hypersurface. Suppose that $V^{\prime},V^{\prime\prime}\subset X$ are two Donaldson hypersurfaces and $J^{\prime}=(J^{\prime}_{\Gamma^{\prime}}),J^{\prime\prime}=(J^{\prime\prime}_{\Gamma^{\prime\prime}})$ are two collections of domain dependent almost complex structures depending on the intersection points with $V^{\prime}$ resp. $V^{\prime\prime}$, depending on some combinatorial type $\Gamma^{\prime}$ resp. $\Gamma^{\prime\prime}$. Consider the pullback

[TABLE]

to a common universal curve $\overline{\mathcal{U}}_{\Gamma}$ for some type $\Gamma$ recording both sets of markings (so that if $\Gamma^{\prime}$ resp. $\Gamma^{\prime\prime}$ has $n^{\prime}$ resp. $n^{\prime\prime}$ leaves then $\Gamma$ has $n^{\prime}+n^{\prime\prime}$ leaves). One wishes to construct a homotopy between $(f^{\prime\prime})^{*}J^{\prime}_{\Gamma^{\prime}},\ (f^{\prime})^{*}J^{\prime\prime}_{\Gamma^{\prime\prime}}$ to construct a cobordism between the corresponding pseudocycles $\overline{\mathcal{M}}^{\prime}_{n}(X,\beta)$ and $\overline{\mathcal{M}}_{n}^{\prime\prime}(X,\beta)$ . Unfortunately, as pointed out by Nick Sheridan, the pullbacks $(f^{\prime\prime})^{*}J^{\prime}_{\Gamma^{\prime}},\ (f^{\prime})^{*}J^{\prime\prime}_{\Gamma^{\prime\prime}}$ do not satisfy the locality condition used to show compactness. That is, the restriction of the almost complex structures $(f^{\prime})^{*}J_{\Gamma^{\prime\prime}}^{\prime\prime}$ (or $(f^{\prime\prime})^{*}J_{\Gamma^{\prime}}^{\prime}$) to some irreducible component $C_{v}$ of the domain curve $C$ are not independent of markings on other components $C_{v^{\prime}}\neq C_{v}$, because collapsed components $C_{v}$ may map to non-special points $f^{\prime}(C_{v})=\{w\}\in f^{\prime}(C)$ under the forgetful map $f^{\prime}$.

In this note we modify the definition of the locality on the collapsed components so that one may homotope between the two domain-dependent almost complex structures without losing compactness. Instead of directly homotoping between the given pull-backs, one first homotopes each pullback to an almost complex structure that is equal to a base almost complex structure near any special point.

2. Partly local perturbations

We introduce the following notation for stable maps with two types of markings. Let $\Gamma$ be a combinatorial type of genus zero stable curve with $n=n^{\prime}+n^{\prime\prime}$ markings. Let $V^{\prime},V^{\prime\prime}$ be Donaldson hypersurfaces in the symplectic manifold $(X,\omega)$, that is, symplectic hypersurfaces representing large multiples $k^{\prime}[\omega]$ resp. $k^{\prime\prime}[\omega]$ of the symplectic class $[\omega]\in H^{2}(X,\mathbb{Q})$. Suppose $V^{\prime}$ and $V^{\prime\prime}$ intersect transversely. Let $\mathcal{J}(X,V^{\prime},V^{\prime\prime})$ be the space of $\omega$-tamed almost complex structures on $X$ that make $V^{\prime}$ and $V^{\prime\prime}$ almost complex. Let $\mathcal{J}^{E}(X,V^{\prime},V^{\prime\prime})\subset\mathcal{J}(X,V^{\prime},V^{\prime\prime})$ be some contractible subset of almost complex structures $J:TX\to TX$ preserving $TV^{\prime}$ and $TV^{\prime\prime}$ taming the symplectic form $\omega$ and so that any non-constant pseudoholomorphic $J$-holomorphic map $u:C\to X$ with some given energy bound $E(u)<E$ to $X$ meets $V^{\prime},V^{\prime\prime}$ each in at least three but finitely many distinct points $u^{-1}(V^{\prime}),u^{-1}(V^{\prime\prime})$ in the domain $C$ as in [1, 8.18]. Let

[TABLE]

be a base almost complex structure that satisfies these conditions without restriction on the energy of the map $u:C\to X$.

The universal curve breaks into irreducible components corresponding to the vertices of the combinatorial type. Let $\overline{\mathcal{U}}_{\Gamma}\to\overline{\mathcal{M}}_{\Gamma}$ be the closure of the universal curve of type $\Gamma$. For each vertex $v\in\operatorname{Vert}(\Gamma)$ let $\Gamma(v)$ denote the tree with the single vertex $v$ and edges those of $\Gamma$ meeting $v$. Let $\overline{\mathcal{U}}_{\Gamma,v}\subset\overline{\mathcal{U}}_{\Gamma}$ be the component corresponding to $v$, obtained by pulling back $\overline{\mathcal{U}}_{\Gamma(v)}$ so that $\mathcal{U}_{\Gamma}$ is obtained from the disjoint union of the curves $\mathcal{U}_{\Gamma,v}\to\mathcal{M}_{\Gamma}$ by identifying at nodes.

Cieliebak-Mohnke [1] requires that the almost complex structure is equal to the base almost complex structure near the nodes. This condition is not true for domain-dependent almost complex structures pulled back under forgetful maps, and so must be relaxed as follows. Recall that Knudsen’s (genus zero) universal curve $\overline{\mathcal{U}}_{\Gamma}$ [3] is a smooth projective variety, and in particular a complex manifold. A domain-dependent almost complex structure for type $\Gamma$ of stable genus zero curve is an almost complex structure

[TABLE]

that preserves the splitting of the tangent bundle $T(\overline{\mathcal{U}}_{\Gamma}\times X)$ into factors $T\overline{\mathcal{U}}_{\Gamma}\times TX$ and that is equal to the standard complex structure on the tangent space to the projective variety $\overline{\mathcal{U}}_{\Gamma}$, and gives rise to a map from $\overline{\mathcal{U}}_{\Gamma}$ to $\mathcal{J}(X,V^{\prime},V^{\prime\prime})$ with the same notation $J_{\Gamma}$. Let

[TABLE]

denote the space of such maps taking values in $\mathcal{J}^{E}(X,V^{\prime},V^{\prime\prime})$. With this definition, the standard proof of Gromov convergence applies: Any sequence $u_{\nu}:C_{\nu}\to X$ of $J_{\Gamma}$-holomorphic maps with energy $E(u)<E$ may be viewed as a finite energy sequence of maps to $\overline{\mathcal{U}}_{\Gamma}\times X$. Therefore it has a subsequence with a Gromov limit $u:C\to X$ where the stabilization $C^{s}$ of $C$ is a fiber of $\overline{\mathcal{U}}_{\Gamma}$ and $u$ is pseudoholomorphic for the pull-back of the restriction of $J_{\Gamma}$ to $C^{s}$. If we restrict to sequences of maps $u_{\nu}:C_{\nu}\to X$ sending the markings to $V^{\prime}$ or $V^{\prime\prime}$ then in fact $C^{s}$ is equal to $C$, since non-constant components of $u$ with fewer than three markings are impossible.

We distinguish components of the curve that are collapsed under forgetting the first or second group of markings. Let

[TABLE]

denote the forgetful maps forgetting the first $n^{\prime}$ resp. last $n^{\prime\prime}$ markings and stabilizing. Call a component of $C$ $f^{\prime}$-unstable if it is collapsed by $f^{\prime}$, and $f^{\prime}$-stable otherwise, in which case it corresponds to a component of $f^{\prime}(C)$. $f^{\prime\prime}$-unstable components are defined similarly.

Definition 2.1**.**

(Local and partly local almost complex structures)

1. (a)

A domain-dependent almost complex structure

[TABLE]

is local if and only if for each $v\in\operatorname{Vert}(\Gamma)$ the restriction $J_{\Gamma}|\overline{\mathcal{U}}_{\Gamma,v}$ is local in the sense that $J_{\Gamma}|\overline{\mathcal{U}}_{\Gamma,v}$ is pulled back from some map $J_{\Gamma,v}$ defined on the universal curve $\overline{\mathcal{U}}_{\Gamma(v)}$ and equal to $J_{V^{\prime},V^{\prime\prime}}$ near any special point of $\overline{\mathcal{U}}_{\Gamma,v}$. 2. (b)

A domain-dependent almost complex structure

[TABLE]

is $f^{\prime}$-local if and only if

1. (i)

for each $v\in\operatorname{Vert}(\Gamma)$ such that $\mathcal{U}_{\Gamma,v}$ is $f^{\prime}$-stable (that is, has sufficiently many $V^{\prime\prime}$ markings) then $J_{\Gamma}|\overline{\mathcal{U}}_{\Gamma,v}$ is local in the sense that $J_{\Gamma}|\overline{\mathcal{U}}_{\Gamma,v}$ is pulled back from some map $J_{\Gamma,v}$ defined on the universal curve $\overline{\mathcal{U}}_{\Gamma(v)}$ and equal to $J_{V,V^{\prime}}$ near any point $z\in C$ mapping to a special point $f^{\prime}(z)$ of $f^{\prime}(C)$, and 2. (ii)

for each $v\in\operatorname{Vert}(\Gamma)$ such that $\mathcal{U}_{\Gamma,v}$ is $f^{\prime}$-unstable (that is, does not have sufficiently many $V^{\prime\prime}$ markings) then $J_{\Gamma}|\overline{\mathcal{U}}_{\Gamma,v}$ is constant on each fiber of $\overline{\mathcal{U}}_{\Gamma,v}$.

The definition of $f^{\prime\prime}$-local is similar.

Remark 2.2*.*

Note that $f^{\prime}$-pullbacks $(f^{\prime})^{*}J_{\Gamma^{\prime\prime}}$ are $f^{\prime}$-local, and local almost complex structures are $f^{\prime}$-local. The condition that an almost complex structure be $f^{\prime}$-local is weaker than the condition that it be pulled back under $f^{\prime}$, because the restriction $J_{\Gamma}|\overline{\mathcal{U}}_{\Gamma,v}$ is allowed to depend on special points $z\in\overline{\mathcal{U}}_{\Gamma,v}$ that are forgotten under $f^{\prime}$.

Remark 2.3*.*

One can reformulate the $f^{\prime}$-local condition as a pullback condition for a forgetful map that forgets almost the same markings as those forgotten by $f^{\prime}$. Let $C$ be a curve of type $\Gamma$. Let $C^{{\operatorname{us}}}\subset C$ be the locus collapsed by $f^{\prime}$. For each connected component $C_{i},i=1,\ldots,k$ of $C^{{\operatorname{us}}}$ mapping to a marking of $f^{\prime}(C)$ choose $j(i)$ so that $z_{j(i)}\in C_{i}$. Let $I^{{\operatorname{us}}}\subset\{1,\ldots,n\}$ denote the set of indices $j$ of markings $z_{j}\in C^{{\operatorname{us}}}$ with $z_{j}\neq z_{j(i)},\forall i$. Forgetting the markings with indices in $I^{{\operatorname{us}}}$ and collapsing defines a map $f:C\to f(C)$ such that any collapsed component of $C$ maps to a special point of $f(C)$. Let $\Gamma^{f}$ denote the combinatorial type of $f(C)$. Then $J_{\Gamma}$ is $f^{\prime}$-local if and only if $J_{\Gamma}=f^{*}J_{\Gamma^{f}}$ is pulled back from a local domain-dependent almost complex structure $J_{\Gamma^{f}}:\overline{\mathcal{U}}_{\Gamma^{f}}\to\mathcal{J}(X,V^{\prime},V^{\prime\prime})$. Indeed, the collapsed components under $C\to f(C)$ are the same as those of $f^{\prime}:C\to f^{\prime}(C)$ since adding a single marking $z_{i}$ on the components that collapse $C_{v}$ to markings $f(C_{v})\subset f(C)$ does not stabilize $C_{v}$. So the pull-back condition $J_{\Gamma}=f^{*}J_{\Gamma^{f}}$ requires $J_{\Gamma}$ to be constant on the components $C_{v}$ such that $\dim(f(C_{v}))=0$. On the other hand, any irreducible component of $f(C)$ is isomorphic, as a stable marked curve, to an irreducible component of $C$ not collapsed under $f^{\prime}$.

Remark 2.4*.*

There also exist domain-dependent almost complex structures that are both $f^{\prime}$ and $f^{\prime\prime}$-local. Indeed suppose that $C$ is a curve of type $\Gamma$, and $K\subset\{1,\ldots,n^{\prime}+n^{\prime\prime}\}$ is the set of markings on components collapsed by $f^{\prime}$ or $f^{\prime\prime}$. Forgetting the markings $z_{k},k\in K$ defines a forgetful map $f^{\operatorname{s}\operatorname{s}}:C\to f^{\operatorname{s}\operatorname{s}}(C)$, where $f^{\operatorname{s}\operatorname{s}}(C)$ is of some (possibly empty) type $\Gamma^{\operatorname{s}\operatorname{s}}$. Let $J_{\Gamma^{\operatorname{s}\operatorname{s}}}:\overline{\mathcal{U}}_{\Gamma^{\operatorname{s}\operatorname{s}}}\to\mathcal{J}(X,V^{\prime},V^{\prime\prime})$ be a domain-dependent almost complex structure for type $\Gamma^{\operatorname{s}\operatorname{s}}$. Then $(f^{\operatorname{s}\operatorname{s}})^{*}J_{\Gamma^{\operatorname{s}\operatorname{s}}}$ is both $f^{\prime}$ and $f^{\prime\prime}$-local (taking the constant structure $J_{V^{\prime},V^{\prime\prime}}$ if $\Gamma^{\operatorname{s}\operatorname{s}}$ is empty.)

Lemma 2.5**.**

The space of $f^{\prime}$-local resp. $f^{\prime\prime}$-local resp. $f^{\prime}$ and $f^{\prime\prime}$-local almost complex structures tamed by or compatible with the symplectic form $\omega$ is contractible. Any $f^{\prime}$-local resp. $f^{\prime\prime}$-local resp. $f^{\prime}$ and $f^{\prime\prime}$-local $J_{\Gamma}|\partial\overline{\mathcal{U}}_{\Gamma}$ defined on the boundary $\partial\overline{\mathcal{U}}_{\Gamma}:=\overline{\mathcal{U}}_{\Gamma}|\partial\overline{\mathcal{M}}_{\Gamma}$ extends to a $f^{\prime}$-local resp. $f^{\prime\prime}$-local resp. $f^{\prime}$ and $f^{\prime\prime}$-local structure $J_{\Gamma}$ over an open neighborhood of the boundary $\partial\overline{\mathcal{U}}_{\Gamma}$ in $\overline{\mathcal{U}}_{\Gamma}$.

Proof.

Contractibility follows from the contractibility of tamed or compatible almost complex structures. Since the space of $f^{\prime}$-local tamed almost complex structures is contractible, it suffices to show the existence of an extension of $J_{\Gamma}$ near any stratum $\overline{\mathcal{U}}_{\Gamma_{1}}\subset\overline{\mathcal{U}}_{\Gamma}$ and then patch together the extensions. Local domain-dependent almost complex structures $J_{\Gamma}$ extend by a gluing construction in which open balls $U_{+},U_{-}$ around a node are replaced by a punctured ball $V\cong U_{+}^{\times}\cong U_{-}^{\times}$ on which the almost complex structure is equal to the base almost complex structure $J_{V^{\prime},V^{\prime\prime}}$.

In the partly-local case recall from Remark 2.3 that $J_{\Gamma}$ is the pull-back of a local almost complex structure $J_{\Gamma^{f}}$ near any particular fiber of the universal curve. Define an extension of $J_{\Gamma}$ near curves of type $\Gamma_{1}$ by first extending $J_{\Gamma^{f}}$ and then pulling back. In more detail, let $C$ be such a curve and let $C_{1},\ldots,C_{k}$ denote the connected components of $C$ collapsed by $f^{\prime}$ to a non-special point of $f^{\prime}(C)$. Choose a marking $z_{i}\in C_{i}$ and let $\Gamma^{\operatorname{s}}$ resp. $\Gamma_{1}^{\operatorname{s}}$ denote the type obtained from $\Gamma$ resp. $\Gamma_{1}$ by forgetting all markings on $C_{i}$ except $z_{i}$, for each $i=1,\ldots,k$. Consider the forgetful map

[TABLE]

that forgets all but the marking $z_{i}$ on $C_{i}$. As discussed in Remark 2.3 $J_{\Gamma_{1}}$ is the pullback of a complex structure

[TABLE]

Since the complex structure $J_{\Gamma_{1}^{f}}$ is constant equal to the base almost complex structure $J_{V^{\prime},V^{\prime\prime}}$ near the nodes (which must join non-collapsed components) $J_{\Gamma_{1}^{f}}$ naturally extends to a domain-dependent almost complex structure $J_{\Gamma^{f}}$ on a neighborhood $\mathcal{N}_{\Gamma_{1}^{f}}$ of $\overline{\mathcal{U}}_{\Gamma_{1}^{f}}$ in $\overline{\mathcal{U}}_{\Gamma^{f}}$ by taking $J_{\Gamma^{f}}$ to equal $J_{V^{\prime},V^{\prime\prime}}$ near the nodes. Now take $J_{\Gamma}=f^{*}J_{\Gamma^{f}}$ to obtain an extension of $J_{\Gamma}$ from $\mathcal{U}_{\Gamma_{1}}$ to a neighborhood $f^{-1}(\mathcal{N}_{\Gamma_{1}^{f}})$. The proof for $f^{\prime}$ local or $f^{\prime}$ and $f^{\prime\prime}$-local structures is similar. ∎

3. Transversality

We wish to inductively construct partly-local almost complex structures so that the moduli spaces of stable maps define pseudocycles. Recall that the combinatorial type of a stable map is obtained from the type of stable curve by decorating the vertices with homology classes; we also wish to record the intersection multiplicities with the Donaldson hypersurfaces. More precisely, a type of stable map $u$ from $C$ to $(X,V^{\prime},V^{\prime\prime})$ consists of a type $\Gamma$ the stable curve $C$ (the graph with vertices corresponding to components and edges corresponding to markings and nodes) with the labelling of vertices $v\in\operatorname{Vert}(\Gamma)$ by homology class $d(v)=[u|C_{v}]\in H_{2}(X)$, labelling of the semi-infinite edges $e$ by either $V^{\prime}$ or $V^{\prime\prime}$,111To obtain evaluation maps one should allow additional edges, but here we ignore evaluation maps. and by the intersection multiplicities $m^{\prime}(e),m^{\prime\prime}(e)$ with $V^{\prime}$ and $V^{\prime\prime}$ (possibly zero if the corresponding marking does not map to $V^{\prime}$ or $V^{\prime\prime}$. A stable map is adapted of type $\Gamma$ if each connected component of $u^{-1}(V^{\prime})$ resp. $u^{-1}(V^{\prime\prime})$ contains at least one marking $z_{e}$ corresponding to an edge $e$ labelled $V^{\prime}$ resp. $V^{\prime\prime}$, and each marking $z_{e}$ maps to $V^{\prime}$ or $V^{\prime\prime}$ depending on its label. A stable map is adapted of type $\Gamma$ if

1. (a)

each connected component of $u^{-1}(V^{\prime})$ resp. $u^{-1}(V^{\prime\prime})$ contains at least one marking $z_{e}$ corresponding to an edge $e$ with labelling $m^{\prime}(e)\geq 1$ resp. $m^{\prime\prime}(e)\geq 1$, and 2. (b)

if $m^{\prime}(e)\geq 1$ resp. $m^{\prime\prime}(e)\geq 1$, then the marking $z_{e}$ is mapped to $V^{\prime}$ resp. $V^{\prime\prime}$.

By forgetting the extra data and stabilization one can associate to each type of stable maps to a type of stable curves. In notation we do not distinguish the two notions of types. Given a type of stable map $\Gamma$ choose a domain-dependent almost complex structure $J_{\Gamma}$. Denote by $\mathcal{M}_{\Gamma}(X,J_{\Gamma})$ the moduli space of adapted $J_{\Gamma}$-holomorphic stable maps $u:C\to X$ of type $\Gamma$, such that for each $v\in{\rm Vert}(\Gamma)$ with $d(v)\neq 0$, the image of $u_{v}$ is not contained in $V^{\prime}\cup V^{\prime\prime}$, and for each semi-infinite edge $e$ attached to $v$, the local intersection number of $u_{v}$ with $V^{\prime}$ resp. $V^{\prime\prime}$ at $z_{e}$ is equal to $m^{\prime}(e)$ resp. $m^{\prime\prime}(e)$. The moduli space $\mathcal{M}_{\Gamma}(X,J_{\Gamma})$ is locally cut out by a smooth map of Banach manifolds: Given a local trivialization of the universal curve given by an subset $\mathcal{M}^{i}_{\Gamma}\subset\mathcal{M}_{\Gamma}$ and a trivialization $C\times\mathcal{M}^{i}_{\Gamma}\to\mathcal{U}_{\Gamma}^{i}=\mathcal{U}_{\Gamma}|_{\mathcal{M}^{i}_{\Gamma}}$, we consider the space of maps $\operatorname{Map}(C,X)_{k,p}$ of Sobolev class $k,p$ for $p\geq 2$ satisfying the above constraints and $k$ sufficiently large to the space of $0,1$-forms with values in $TX$ given by the Cauchy-Riemann operator $\overline{\partial}_{J_{\Gamma}}$ associated to $J_{\Gamma}$. The linearization of this operator is denoted $D_{u}$ (or $D_{u,J_{\Gamma}}$ to emphasize dependence on $J_{\Gamma}$) and the map $u$ is called regular if $D_{u}$ is surjective. We call a type $\Gamma$ of stable map $u:C\to X$ crowded if there is a maximal ghost subtree of the domain $C_{1}\subset C$ with more than one marking $z_{e}\in C_{1}$ and uncrowded otherwise. It is not in general possible to achieve transversality for crowded types using the Cieliebak-Mohnke perturbation scheme.

Definition 3.1**.**

We say a domain-dependent almost complex structure $J_{\Gamma}$ is regular for a type of map $\Gamma$ if

1. (a)

if $\Gamma$ is uncrowded then every element of the moduli space $\mathcal{M}_{\Gamma}(X,J_{\Gamma})$ of adapted $J_{\Gamma}$-holomorphic maps is regular; and 2. (b)

If $\Gamma$ is crowded then there exists a regular $J_{\Gamma^{\operatorname{s}}}$ for some uncrowded type $\Gamma^{\operatorname{s}}$ obtained by forgetting all but one marking $z_{e}$ on each maximal ghost component for curves of type $\Gamma$ such that $J_{\Gamma^{\operatorname{s}}}$ is equal to $J_{\Gamma}$ on all non-constant components, that is, all components of $\overline{\mathcal{U}}_{\Gamma}$ on which the maps $u:C\to X$ in $\mathcal{M}_{\Gamma}(X,J_{\Gamma})$ are non-constant.

Recall the construction by Floer [2, Lemma 5.1] of a subspace of smooth functions with a separable Banach space structure. Let $\underline{\epsilon}=(\epsilon_{\ell},\ell\in\mathbb{Z}_{\geq 0})$ be a sequence of constants converging to zero. Let $\mathcal{J}_{\Gamma}(X)_{\underline{\epsilon}}$ denote the space of domain-dependent almost complex structures of finite Floer norm as in [2, Section 5]. In particular, $\mathcal{J}_{\Gamma}(X)_{\underline{\epsilon}}$ allows variations with arbitrarily small support near any point.

Proposition 3.2**.**

1. (a)

For a regular domain-dependent almost complex structure $J_{\Gamma^{\prime\prime}}$ the pull-back $(f^{\prime})^{*}J_{\Gamma^{\prime\prime}}$ is regular, and similarly for the pull-back $(f^{\prime\prime})^{*}J_{\Gamma^{\prime}}$ for regular $J_{\Gamma^{\prime}}$. 2. (b)

Suppose that $J_{\Gamma}|\partial\overline{\mathcal{U}}_{\Gamma}$ is $f^{\prime}$-local and is a regular domain-dependent almost complex structure defined on the boundary $\partial\overline{\mathcal{U}}_{\Gamma}\to\partial\overline{\mathcal{M}}_{\Gamma}$. The set of regular $f^{\prime}$-local extensions is comeager, that is, is the intersection of countably many sets with dense interiors. 3. (c)

Any parametrized-regular homotopy $J_{\Gamma,t}|\partial\overline{\mathcal{U}}_{\Gamma}$ between two regular $f^{\prime}$-local domain-dependent almost complex structures $J_{\Gamma,0},J_{\Gamma,1}$ on the boundary $\partial\overline{\mathcal{U}}_{\Gamma}$ may be extended to a parametrized-regular one-parameter family of $f^{\prime}$-local structures $J_{\Gamma,t}$ equal to $J_{\Gamma,t}$ over $\overline{\mathcal{U}}_{\Gamma}$.

Proof.

Item (a) is immediate from the definition, since any variation of $J_{\Gamma^{\prime\prime}}$ induces a variation of $(f^{\prime})^{*}J_{\Gamma^{\prime\prime}}$. (b) is an application of Sard-Smale applied to a universal moduli space. We sketch the proof which is analogous to that in Cieliebak-Mohnke [1, Chapter 5]. By Lemma 2.5, $J_{\Gamma}|\partial\overline{\mathcal{U}}_{\Gamma}$ has an extension over the interior. For transversality, first consider the case of an uncrowded type $\Gamma$ of stable map. Choose open subsets $L_{\Gamma},N_{\Gamma}\subset\overline{\mathcal{U}}_{\Gamma}$ of the boundary resp. markings and nodes, such that $L_{\Gamma}$ is union of fibers of $\overline{\mathcal{U}}_{\Gamma}$ containing the restriction $\overline{\mathcal{U}}_{\Gamma}|\partial\mathcal{M}_{\Gamma}$ and $N_{\Gamma}$ is sufficiently small so that the intersection of the complement of $N_{\Gamma}$ with each component of each fiber of $\mathcal{U}_{\Gamma}$ not meeting $L_{\Gamma}$ is non-empty. Let $\mathcal{M}^{\operatorname{univ}}_{\Gamma}(X)$ denote the universal moduli space consisting of pairs $(u,J_{\Gamma})$, where $u:C\to X$ is a $J_{\Gamma}$-holomorphic map of some Sobolev class $W^{k,p},kp\geq 3,p\geq 2$ on each component (with $k$ sufficiently large so that the given vanishing order at the Donaldson hypersurfaces $V^{\prime},V^{\prime\prime}$ is well-defined). Let $\mathcal{J}^{E}_{\Gamma}(X,N_{\Gamma},S_{\Gamma})\subset\mathcal{J}^{E}_{\Gamma}(X)$ denote the space of $J_{\Gamma}\in\mathcal{J}_{\Gamma}^{E}(X)_{\underline{\epsilon}}$ that are $f^{\prime}$-local domain-dependent almost complex structures that agree with $J_{V^{\prime},V^{\prime\prime}}$ on the neighborhood $N_{\Gamma}$ of the nodes and markings $z\in\overline{\mathcal{U}}_{\Gamma}$ that map to special points $f^{\prime}(z)\in\overline{\mathcal{U}}_{f^{\prime}(\Gamma)}$ as in Definition 2.1, and equal to the given extension in the neighborhood $L_{\Gamma}$ of the boundary, and constant on the components required by $f^{\prime}$-locality in Definition 2.1. By elliptic regularity, $\mathcal{M}^{\operatorname{univ}}_{\Gamma}(X)$ is independent of the choice of Sobolev exponents.

The universal moduli space is a smooth Banach manifold by an application of the implicit function theorem for Banach manifolds. Let $\mathcal{U}_{\Gamma}^{i}\to\mathcal{M}_{\Gamma}^{i},i=1,\ldots,m$ be a collection of open subsets of the universal curve $\mathcal{U}_{\Gamma}\to\mathcal{M}_{\Gamma}$ on which the universal curve is trivialized via diffeomorphisms $\mathcal{U}_{\Gamma}^{i}\to\mathcal{M}_{\Gamma}^{i}\times C$. The space of pairs $(u:C\to X,J_{\Gamma})$ with $[C]\in\mathcal{M}_{\Gamma}^{i}$, $u$ of type $\Gamma$ of class $W^{k,p}$ on each component, and $J_{\Gamma}\in\mathcal{J}^{E}_{\Gamma}(X,N_{\Gamma},S_{\Gamma})$ is a smooth separable Banach manifold. Since we assume that $J_{\Gamma}$ is regular on the boundary $\partial\mathcal{U}_{\Gamma}$, an argument using Gromov compactness shows that by choosing $L_{\Gamma}$ sufficiently small we may assume that $\tilde{D}_{u,J_{\Gamma}}$ is surjective for $[C]\in L_{\Gamma}$, since regularity is an open condition in the Gromov topology [4, Section 10.7]. Let $\tilde{D}_{u,J_{\Gamma}}$ the linearization of $(u,J_{\Gamma})\mapsto\overline{\partial}_{J_{\Gamma}}u$, and suppose that $\eta$ lies in the cokernel of $\tilde{D}_{u,J_{\Gamma}}$. We have $D_{u}^{*}\eta^{s}=0$ where $D_{u}$ is the usual linearized Cauchy-Riemann operator [4, p. 258] for the map; in the case of vanishing constraints at the Donaldson hypersurfaces see Cieliebak-Mohnke [1, Lemma 6.6]. By variation of the almost complex structure $J_{\Gamma}$ and unique continuation, $\eta$ vanishes on any component on which $u$ is non-constant. On the other hand, for any constant component $u_{v}:C_{v}\to X$, the linearized Cauchy-Riemann operator $D_{u_{v}}$ on a trivial bundle $u_{v}^{*}TX$ is regular with kernel $\operatorname{ker}(D_{u_{v}})$ the space of constant maps $\xi:C_{u}\to(u_{v})^{*}TX$. It follows by a standard inductive argument that the same holds true for a tree $C^{\prime}=\cup_{v\in V}C_{v},{\operatorname{d}}u|_{C^{\prime}}=0$ of constant pseudoholomorphic spheres so the element $\eta$ vanishes on any component $C_{v}\subset C$ on which $u$ is constant. It follows that $\mathcal{M}^{\operatorname{univ},i}_{\Gamma}(X)$ is a smooth Banach manifold. For a comeager subset $\mathcal{J}_{\Gamma}^{{\operatorname{reg}}}(X)\subset\mathcal{J}_{\Gamma}(X)$ of partly almost complex structures in the space above, the moduli spaces $\mathcal{M}^{i}_{\Gamma}(X)=\mathcal{M}_{\Gamma}(X)|_{\mathcal{M}^{i}_{\Gamma}}$ are transversally cut out for each $i=1,\ldots,m$. The transition maps between the local trivializations $\mathcal{M}_{\Gamma}^{i}\cap\mathcal{M}_{\Gamma}^{j}\to\operatorname{Aut}(C)$ induce smooth maps $\mathcal{M}_{\Gamma}^{i}(X)|_{\mathcal{M}_{\Gamma}^{i}\cap\mathcal{M}_{\Gamma}^{j}}\to\mathcal{M}_{\Gamma}^{j}(X)|_{\mathcal{M}_{\Gamma}^{i}\cap\mathcal{M}_{\Gamma}^{j}}$ making $\mathcal{M}_{\Gamma}(X)$ into a smooth manifold.

Next consider a crowded type $\Gamma$. Let $f:\Gamma\to f(\Gamma)$ be a map forgetting all but one marking on each maximal ghost component $C^{\prime}\subset C$ and stabilizing; the multiplicities $m^{\prime}(e),m^{\prime\prime}(e)$ at any marking $z_{e}$ is the sum of the multiplicities of markings in its preimage $f^{-1}(z_{e})$. Define $J_{\Gamma^{f}}$ as follows.

1. (a)

If $\mathcal{U}_{\Gamma^{f},v}\cong\mathcal{U}_{\Gamma,v}$ let $J_{\Gamma^{f}}|\mathcal{U}_{\Gamma^{f},v}$ be equal to $J_{\Gamma}|\mathcal{U}_{\Gamma,v}$. 2. (b)

Otherwise let $J_{\Gamma^{f}}:\mathcal{U}_{\Gamma^{f},v}\to\mathcal{J}^{E}(X,V^{\prime},V^{\prime\prime})$ be constant equal to $J_{V^{\prime},V^{\prime\prime}}$.

The map $J_{\Gamma^{f}}$ is continuous because any non-collapsed ghost component $C_{v}\subset C$ must connect at least two non-ghost components $C_{v_{1}},C_{v_{2}}\subset C$ and the connecting points of the non-ghost components $f^{\prime}(C_{v_{1}}),f^{\prime}(C_{v_{2}})$ is a node of the curve $f^{\prime}(f(C))$ of type $f^{\prime}(\Gamma^{f})$. For a comeager subset of $J_{\Gamma}$ described above, the complex structures $J_{\Gamma^{f}}$ are also regular by the argument for uncrowded types. Item (c) is a parametrized version of (b). ∎

Corollary 3.3**.**

There exists a regular homotopy $J_{\Gamma,t},t\in[-1,1]$ between $(f^{\prime\prime})^{*}J^{\prime}_{\Gamma^{\prime}}$ and $\ (f^{\prime})^{*}J^{\prime\prime}_{\Gamma^{\prime\prime}}$ in the space of maps $\overline{\mathcal{U}}_{\Gamma}\to\mathcal{J}(X,V^{\prime},V^{\prime\prime})$ that are $f^{\prime}$-local for $t\in[-1,0]$ and $f^{\prime\prime}$-local for $t\in[0,1]$

Proof.

Let $\underline{J}=(J_{\Gamma})$ be a collection of regular domain-dependent almost complex structures that are both $f^{\prime}$ and $f^{\prime\prime}$-local, as in Remark 2.4. By part (b) above, for each type $\Gamma$ there exists a regular homotopy from $J_{\Gamma}$ to $(f^{\prime})^{*}J_{\Gamma^{\prime\prime}}$ resp. $(f^{\prime\prime})^{*}J_{\Gamma^{\prime}}$ extending given homotopies on the boundary. The existence of a regular homotopy now follows by induction. ∎