# Deformations of symplectic vortices

Eduardo Gonzalez  Department of Mathematics University of Massachusetts Boston
100 William T. Morrissey Boulevard Boston, MA 02125<EMAIL_ADDRESS>and
Chris Woodward  Mathematics-Hill Center, Rutgers University, 110 Frelinghuysen
Road, Piscataway, NJ 08854-8019, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We prove a gluing theorem for a symplectic vortex on a compact complex curve
and a collection of holomorphic sphere bubbles. Using the theorem we show that
the moduli space of regular stable symplectic vortices on a fixed curve with
varying markings has the structure of a stratified-smooth topological
orbifold. In addition, we show that the moduli space has a non-canonical
$C^{1}$-orbifold structure.

Partially supported by NSF grant DMS060509

###### Contents

  1. 1 Introduction
  2. 2 Deformations of holomorphic curves
  3. 3 Deformations of holomorphic maps from curves
  4. 4 Deformations of symplectic vortices

##  1\. Introduction

In this paper we generalize the following result on existence of universal
deformations for stable (pseudo-)holomorphic maps. Let $(X,\omega)$ be a
compact symplectic manifold equipped with a compatible almost complex
structure $J$, and $({\Sigma},{j})$ a compact nodal complex curve. A map
${u}:{\Sigma}\to X$ is holomorphic if

$\overline{\partial}{u}:=J_{{u}}\circ{\operatorname{d}}{u}-{\operatorname{d}}{u}\circ{j}=0$

on each component of ${\Sigma}$. One naturally has the notion of a stratified-
smooth family of holomorphic maps, and hence the notion of a deformation,
namely the germ of a family around the central fiber together with an
isomorphism of the central fiber with the given map. Recall that a deformation
is universal if any other deformation is obtained from it by pullback, in a
unique way, by a map of parameter spaces. A holomorphic map ${u}:{\Sigma}\to
X$ is regular if the linearized Cauchy-Riemann operator is surjective. The
following theorem is the result of the well-known gluing construction for
holomorphic maps, c.f. Ruan-Tian [21] or the text McDuff-Salamon [15, Chapter
10] in the case of genus zero:

######  Theorem 1.0.1.

A regular holomorphic map ${u}:{\Sigma}\to X$ admits a stratified-smooth
universal deformation iff it is stable.

The construction of the universal deformation proceeds via the implicit
function theorem. For each element in the infinitesimal deformation space of
the stable map one first produces an approximate solution and then applies the
implicit function theorem to find an exact solution. Unfortunately one uses a
different Sobolev space for each “gluing parameter” controlling the domain,
which means that it is rather tricky to show that each nearby stable
holomorphic map occurs only once in the resulting family. A slightly jazzed up
version of the above theorem implies that the gluing construction gives rise
to orbifold charts on the regular locus of the moduli space of stable
holomorphic maps. Uniqueness of the universal deformations implies that the
smooth structures on each stratum are independent of the Sobolev spaces used
in the implicit function theorem. One can make these charts $C^{1}$-compatible
by suitable choices of gluing profiles, that is, coordinates on the local
deformation spaces; however the $C^{1}$-structure on the moduli space is not
canonical. The first part of the paper contains an exposition of the above
theorem, which is rather scattered in the literature.

The main result of the paper is a generalization of the theorem above to
certain gauged (pseudo)holomorphic maps, namely symplectic vortices as
introduced by Mundet [16] and Cieliebak, Gaio and Salamon, see [5]. Let $G$ be
a compact Lie group and $X$ a Hamiltonian $G$-manifold equipped with a moment
map $\Phi:X\to\mathfrak{g}^{*}$ and an invariant almost complex structure $J$.
Let $\Sigma$ be a compact smooth holomorphic curve with complex structure $j$
and equipped with an area form $\operatorname{Vol}_{\Sigma}$. A gauged
holomorphic map with values in $X$ consists of a smooth principal $G$-bundle
$P\to\Sigma$, a connection $A$ on $P$, and a smooth section $u:\Sigma\to
P(X):=P\times_{G}X$ such that $\overline{\partial}_{A}u=0$ where
$\overline{\partial}_{A}$ is defined using the splitting given by the
connection $A$ and the complex structures $J,j$. Let
$F_{A}\in\Omega^{2}(\Sigma,P(\mathfrak{g}))$ denote the curvature of $A$ and
$P(\Phi):P(X)\to P(\mathfrak{g})$ the map induced by $\Phi$. The space of
gauged holomorphic maps admits a formal symplectic structure depending on a
choice of invariant metric on $\mathfrak{g}$ so that the action of the group
of gauge transformations is formally Hamiltonian. A symplectic vortex is a
pair in the zero level set of the moment map: a pair $(A,u)$ such that

$\overline{\partial}_{A}u=0,\quad
F_{A}+u^{*}(P\times_{G}\Phi)\operatorname{Vol}_{\Sigma}=0.$

Thus the moduli space $M(\Sigma,X)$ of symplectic vortices is the symplectic
quotient of the space of gauged maps by the group of gauge transformations. In
certain cases where the moduli spaces are compact Cieliebak, Gaio, Mundet, and
Salamon [4] and Mundet [16] constructed invariants that we will call gauged
Gromov-Witten invariants by integration over these moduli spaces. In general
$M(\Sigma,X)$ admits a compactification $\overline{M}(\Sigma,X)$ consisting of
polystable symplectic vortices given by allowing $u$ to develop holomorphic
sphere bubbles in the fibers of $P\times_{G}X$. A polystable vortex is
strongly stable if the principal component has finite automorphism group, and
regular if a certain linearized operator is surjective, that is, the moduli
space is formally smooth. Our main result is the following:

######  Theorem 1.0.2.

Let $\Sigma,X$ be as above. A regular strongly stable symplectic vortex from
$\Sigma$ to $X$ admits a universal stratified-smooth deformation.

Using the deformations constructed in Theorem 1.0.2 we prove that the moduli
space $\overline{M}^{{\operatorname{reg}}}(\Sigma,X)$ of regular strongly
stable symplectic vortices admits the structure of an oriented stratified-
smooth topological orbifold, and (non-canonically) the structure of a
$C^{1}$-orbifold. The first statement implies that if
$\overline{M}^{\operatorname{reg}}(\Sigma,X)$ is compact then it carries a
rational fundamental class. The second statement implies for example, that if
the target carries a group action then the usual equivariant localization
theorems hold for the induced group action on the moduli space. In the case
that $X$ is a smooth projective variety, algebraic methods explain in [11]
give similar results and provide virtual fundamental classes on the moduli
space. However, the symplectic gluing construction is interesting in its own
right, not in the least because it potentially extends to the case of
Lagrangian boundary conditions. We understand that a forthcoming paper of
Mundet i Riera and Tian gives a gluing construction for two symplectic
vortices, when the structure group is the circle group.

Acknowledgments: We thank Ignasi Mundet i Riera, Melissa Liu, and Robert
Lipshitz for helpful comments and discussions.

##  2\. Deformations of holomorphic curves

The following section is essentially a review of the material that can be
found at the beginning of Siebert [22], with a few additional comments
incorporating terminology of Hofer, Wysocki, and Zehnder [13, Appendix]. In
the first part we review the holomorphic construction of universal
deformations of stable curves. In the second part, we study smooth
deformations of curves.

###  2.1. Holomorphic families of stable curves

A compact, complex nodal curve ${\Sigma}$ is obtained from a collection
$(\Sigma_{1},\ldots,\Sigma_{k})$ of smooth, compact, complex curves by
identifying a collection of distinct nodal points

${w}=\\{\\{w_{1}^{-},w_{1}^{+}\\},\ldots,\\{w_{m}^{-},w_{m}^{+}\\}\\}.$

For $l=1,\ldots,m$, we denote by $\Sigma_{i^{\pm}(l)}$ the components such
that $w_{l}^{\pm}\in\Sigma_{i^{\pm}(l)}$. A point $z\in{\Sigma}$ is smooth if
it is not equal to any of the nodal points. A marked nodal curve is a nodal
curve together with a collection ${z}=(z_{1},\ldots,z_{n})$ of distinct,
smooth points. An isomorphism of marked nodal curves $({\Sigma}_{0},{z}_{0})$
to $({\Sigma}_{1},{z}_{1})$ is an isomorphism
$\phi:{\Sigma}_{0}\to{\Sigma}_{1}$ of nodal curves such that
$\phi(z_{0,i})=z_{1,i}$ for $i=1,\ldots,n$. A marked nodal curve is stable if
it has finite automorphism group, that is, each component contains at least
three marked or nodal points if genus zero, or one special point if genus one.

The combinatorial type $\Gamma({\Sigma})$ of ${\Sigma}$ is the graph whose
vertices are the components and edges are the nodes and markings of
${\Sigma}$. The map ${\Sigma}\mapsto\Gamma({\Sigma})$ extends to a functor
from the category of marked nodal curves to the category of graphs. In
particular, there is a canonical homomorphism
$\operatorname{Aut}({\Sigma})\to\operatorname{Aut}(\Gamma({\Sigma}))$, whose
kernel is the product of the automorphism groups of the components of
${\Sigma}$.

Let $S$ be a complex variety (or scheme). A family of nodal curves over $S$ is
a complex variety ${\Sigma}_{S}$ equipped with a proper flat morphism
$\pi:{\Sigma}_{S}\to S$, such that each fiber ${\Sigma}_{s},s\in S$ is a nodal
curve. A deformation of a marked nodal curve ${\Sigma}$ is a germ of a family
of marked nodal curves ${\Sigma}_{S}$ over a pointed space $(S,0)$ together
with an isomorphism $\varphi:{\Sigma}_{0}\to{\Sigma}$ of the central fiber
${\Sigma}_{0}$ with ${\Sigma}$. A deformation $({\Sigma}_{S},\varphi)$ of
${\Sigma}$ is versal iff any other deformation $({\Sigma}^{\prime}_{S}\to
S^{\prime},\varphi^{\prime})$ is induced from a map $\psi:S^{\prime}\to S$ in
the sense that there exists an isomorphism $\phi$ of ${\Sigma}^{\prime}$ with
the fiber product ${\Sigma_{S}}\times_{S}S^{\prime}$ in a neighborhood of the
central fiber ${\Sigma}_{0}$. A versal deformation is universal if the map
$\phi$ is the unique such map inducing the identity on ${\Sigma}_{0}$. A
deformation has fixed type if the combinatorial type of the fiber is constant.
A universal deformation of fixed type is a deformation of fixed type, which is
universal in the above sense for deformations of fixed type. The space
$\operatorname{Def}({\Sigma})$ of infinitesimal deformations of ${\Sigma}$ is
the tangent space $T_{0}S$ of the base $S$ of a universal deformation, well-
defined up to isomorphism. We write $\operatorname{Def}_{\Gamma}({\Sigma})$
for the space of infinitesimal deformations of fixed type. Let
$\tilde{\Sigma}$ be the normalization of ${\Sigma}$, so that
$\operatorname{Def}_{\Gamma}({\Sigma})$ is isomorphic to the space of
deformations of $\tilde{\Sigma}$ equipped with the additional markings
$w_{1}^{\pm},\ldots,w_{m}^{\pm}$ obtained by lifting the nodes. The general
theory of deformations, see for example [7] in the analytic setting, shows
that any marked nodal curve ${\Sigma}$ admits a versal deformation with smooth
parameter space $S$. ${\Sigma}$ admits a universal deformation
${\Sigma}_{S}\to S$ if and only if ${\Sigma}$ is stable. Furthermore, the
space $\operatorname{Def}({\Sigma})$ of the space of infinitesimal
deformations admits a canonical isomorphism with
$H^{0,1}({\Sigma},T{\Sigma}[-z_{1}-\ldots-z_{n}]),$ where
$T{\Sigma}[-z_{1}-\dots,-z_{n}]$ is the sheaf of vector fields vanishing at
$z_{1},\dots,z_{n}$.

The relationship between the various deformation spaces (in the case with
markings, fixed type, etc.) is given as follows. The space of infinitesimal
automorphisms $\operatorname{aut}({\Sigma},{z})$ of $({\Sigma},{z})$ is the
space
$\operatorname{Vect}({\Sigma},{z})=H^{0}({\Sigma},T{\Sigma}[-z_{1}-\ldots-
z_{n}])$ of holomorphic vector fields vanishing at the marked points. The
short exact sequence of sheaves

$0\to\oplus_{i=1}^{n}T_{z_{i}}{\Sigma}\to T{\Sigma}\to
T{\Sigma}[-z_{1}-\ldots-z_{n}]\to 0$

gives a long exact sequence in cohomology [12, p. 94]

$0\to\operatorname{Vect}({\Sigma},{z})\to\operatorname{Vect}({\Sigma})\to\bigoplus_{i=1}^{n}T_{z_{i}}{\Sigma}\to\operatorname{Def}({\Sigma},{z})\to\operatorname{Def}({\Sigma})\to
0.$

From now on, we omit the markings from the notation, and study deformations of
a nodal marked curve ${\Sigma}=({\Sigma},{z})$. By $T_{w_{i}^{\pm}}{\Sigma}$,
we mean the tangent space in the component of ${\Sigma}$ containing
$w_{i}^{\pm}$. A gluing parameter for the $j$-th node is an element
$\delta_{i}\in T_{w_{i}^{+}}{\Sigma}\otimes T_{w_{i}^{-}}{\Sigma}.$ The
canonical conormal sequence [12, p. 100] gives rise to an exact sequence

(1)
$0\to\operatorname{Def}_{\Gamma}({\Sigma})\to\operatorname{Def}({\Sigma})\to\bigoplus_{i=1}^{m}T_{w_{i}^{+}}{\Sigma}\otimes
T_{w_{i}^{-}}{\Sigma}\to 0.$

After trivialization of the tangent spaces the gluing parameters are
identified with complex numbers.

Universal deformations of a smooth marked curve can be constructed for example
using Teichmüller theory [8] or by Hilbert scheme methods [12, p. 102]. Later
we will need an explicit gluing construction of a universal deformation of a
stable marked curve. This construction seems to be well-known, but the only
proof we could find in the literature is Siebert [22]. The idea is to remove
small neighborhoods of the nodes, and glue the remaining components together.
A local coordinate near a smooth point $z\in{\Sigma}$ is a neighborhood $U$ of
$z$ and a holomorphic isomorphism $\kappa$ of $U$ with a neighborhood of $0$
in the tangent line $T_{z}{\Sigma}$, whose differential $T_{0}U\to
T_{z}\Sigma$ is the identity.

######  Remark 2.1.1.

The space of local coordinates near $z$ is convex, since if
$\kappa_{0},\kappa_{1}$ are local coordinates then any combination
$t\kappa_{0}+(1-t)\kappa_{1}$ is still holomorphic and has the same
differential at $z$, and so by the inverse function theorem is a holomorphic
isomorphism in a neighborhood of $z$.

Any gluing parameter $\delta_{i}$ induces an identification

$T_{w_{i}^{+}}{\Sigma}-\\{0\\}\to T_{w_{i}^{-}}{\Sigma}-\\{0\\},\ \
\lambda_{i}^{+}\mapsto\delta_{i}/\lambda_{i}^{-}.$

Given local coordinates for the nodes of ${\Sigma}$ and a set of gluing
parameters ${\delta}=(\delta_{1},\ldots,\delta_{m})$, define a (possibly
nodal) curve ${\Sigma}^{{\delta}}$ by gluing together small disks around the
node $w_{i}$ by $z\mapsto\delta_{i}/z$, for every gluing parameter
$\delta_{i}$ that is non-zero, where $z$ is the local coordinate given by
$\kappa_{i}$. That is,

(2)
${\Sigma}^{{\delta}}=\bigcup_{i=1}^{k}\Sigma_{i}-\\{w_{1}^{\pm},\ldots,w_{m}^{\pm}\\}/(z\sim(\kappa_{i}^{+})^{-1}(\delta_{i}/\kappa_{i}^{-}(z)),i=1,\ldots,m)$

for pairs of points in the two components such that both coordinates are
defined. In particular, the choice of local coordinates near the nodes defines
a splitting of the sequence (1).

The gluing construction works in families as follows. Let
$I^{i,\pm}_{\Gamma}\to S_{\Gamma}$ resp. ${I}_{\Gamma}\to S_{\Gamma}$ denote
the vector bundle whose fiber at $s\in S_{\Gamma}$ is the tangent line at the
$j$-node resp. tensor product of tangent lines at the nodes,

(3)
$I^{i,\pm}_{\Gamma,s}=T_{w_{i,s}^{\pm}}{\Sigma}_{s},\quad{I}_{\Gamma,s}=\bigoplus_{j=1}^{m}T_{w_{i,s}^{-}}{\Sigma}_{s}\otimes
T_{w_{i,s}^{+}}{\Sigma}_{s}.$

Let ${\Sigma}_{S_{\Gamma}}\to S_{\Gamma}$ be a family of nodal curves of the
same combinatorial type $\Gamma$, with nodal points
$(w_{S_{\Gamma},j}^{\pm})_{i=1}^{m}$. A holomorphic system of local
coordinates for the $i$-th node is a holomorphic map $\kappa_{i}$ from a
neighborhood $U_{i,\pm}$ of the zero section in $I^{i,\pm}_{S}$ to
${\Sigma}_{S}$ which is an isomorphism onto its image and induces the identity
at any point in the zero section. Given a holomorphic system of coordinates
for each node
${\kappa}=(\kappa_{1}^{+},\kappa_{1}^{-},\ldots,\kappa_{m}^{+},\kappa_{m}^{-})$
the gluing construction (2) produces a family ${\Sigma}_{S}\to S$ over an open
neighborhood $S$ of the zero section in the bundle $I\to S_{\Gamma}$ of gluing
parameters.

######  Theorem 2.1.2.

[22, Proposition 2.4] If ${\Sigma}_{\Gamma,S}$ is a family giving a universal
deformation of fixed type, then ${\Sigma}_{S}$ is a universal deformation of
any of its fibers, and in particular is independent up to isomorphism of
deformations of the choice of local coordinates $\kappa$.

The following properties of universal deformations of stable curves will be
used later:

######  Theorem 2.1.3.

[22, Lemma 2.7] For any universal deformation ${\Sigma}_{S}$, the action of
automorphisms $\operatorname{Aut}({\Sigma})$ of ${\Sigma}$ extends to an
action of $\operatorname{Aut}({\Sigma})$ on ${\Sigma}_{S}$, possibly after
shrinking $S$. For any universal deformation, there exists a neighborhood of
the central fiber such that any two fibers ${\Sigma}_{S}$ contained in the
neighborhood are isomorphic, if and only if they are related by an
automorphism of ${\Sigma}$.

If ${\Sigma}$ is not stable, then the above construction produces a minimal
versal deformation of ${\Sigma}$. That is, ${\Sigma}_{S}\to S$ is versal, and
any other versal deformation given by a family
${\Sigma}^{\prime}_{S^{\prime}}\to S^{\prime}$ is obtained by pull-back by a
map $S^{\prime}\to S$.

Algebraic families of connected stable nodal curves with genus $g$ and $n$
markings form the objects of a smooth Deligne-Mumford stack
$\overline{M}_{g,n}$ [6] which admits a coarse moduli space with the structure
of a normal projective variety. The maps
$\operatorname{Def}({\Sigma})\to\overline{M}_{g,n},\ s\mapsto[{\Sigma}_{s}]$
(restricted to a neighborhood of $0$) provide $\overline{M}_{g,n}$ with an
atlas of holomorphic orbifold charts.

###  2.2. Stratified-smooth families of stable curves

We extend the definition of families and deformations to smooth and
stratified-smooth settings. Given a family ${\Sigma}_{S}\to S$ of compact
complex nodal curves, let

$S=\bigcup S_{\Gamma},\quad S_{\Gamma}=\\{s\in S,\ \
\Gamma({\Sigma}_{s})=\Gamma\\}$

denote the stratification by combinatorial type of the fiber. It follows from
the gluing construction of the previous section that if ${\Sigma}_{S}\to S$ is
a family giving a universal deformation, then each $S_{\Gamma}$ is a smooth
manifold, and the restriction ${\Sigma}_{\Gamma,S_{\Gamma}}$ of
${\Sigma}_{S_{\Gamma}}$ to $S_{\Gamma}$ gives a universal deformation of fixed
type $\Gamma$. By a smooth family of curves of fixed type $\Gamma$ we mean a
fiber bundle ${\Sigma}_{\Gamma,S_{\Gamma}}\to S_{\Gamma}$ with fibers of type
$\Gamma$ and smoothly varying complex structure. In the nodal case, it is
obtained from a smooth family of smooth holomorphic curves, identified using a
collection of pairs of smooth sections (nodes).

######  Lemma 2.2.1.

Holomorphic universal deformations of fixed type are also universal in the
category of smooth deformations of ${\Sigma}$. That is, let ${\Sigma}_{S}\to
S,\varphi$ be a universal holomorphic deformation of fixed type of a nodal
curve ${\Sigma}$. Any smooth deformation ${\Sigma}_{S^{\prime}}^{\prime}\to
S^{\prime},\varphi^{\prime}$ of nodal curves of fixed type is obtained by
pull-back ${\Sigma}_{S}\to S$ by a smooth map $S^{\prime}\to S$.

###### Proof.

By the construction of local slices for the action of diffeomorphisms in [8],
[20, Chapter 9]. ∎

Similarly we can define continuous families of holomorphic curves, which
correspond to continuous maps $S^{\prime}\to S$ to the parameter space $S$ for
a universal holomorphic deformation. The following spells out the definition
without reference to the universal holomorphic deformation.

######  Definition 2.2.2.

Let $\Gamma_{0},\Gamma_{1}$ be graphs. A simple contraction $\tau$ is a pair
of maps
$\operatorname{Vert}(\tau):\operatorname{Vert}(\Gamma_{0})\to\operatorname{Vert}(\Gamma_{1})$
and a bijection
$\operatorname{Edge}(\tau):\operatorname{Edge}(\Gamma_{0})\to\operatorname{Edge}(\Gamma_{1})\cup\\{\emptyset\\}$
such that $\Gamma_{1}$ is obtained from $\Gamma_{0}$ identifying the head and
tail of the contracting edge $e$ such that
$\operatorname{Edge}(\tau)(e)=\emptyset$. A contraction is a sequence of
simple contractions.

######  Definition 2.2.3.

A continuous family of nodal holomorphic curves consists of topological spaces
${\Sigma}_{S}$, a surjection ${\Sigma}_{S}\to S$, and a collection of
(possibly nodal) holomorphic structures $j_{{{\Sigma}}_{s}}$ on the fibers
${\Sigma}_{s},s\in S$, which vary continuously in $s$ in the following sense:
for every $s_{0}\in S$ there exists for $s$ in a neighborhood of $s_{0}$ of
some combinatorial type $\Gamma$,

  1. (a)

contractions $\tau_{s}:\Gamma({\Sigma}_{s_{0}})\to\Gamma$, constant in $s\in
S_{\Gamma}$;

  2. (b)

for every node $\\{w_{i}^{\pm}\\}$ collapsed under $\tau_{s}$, a pair of local
coordinates $\kappa^{\pm}_{i}:W^{\pm}_{i}\to\mathbb{C}$

  3. (c)

for every component $\Sigma_{s_{0},i}$ of ${\Sigma}_{s_{0}}$, a real number
$\epsilon_{s}>0$ and maps

$\phi_{i,s}:\Sigma_{s_{0},i}-\cup_{w_{k}^{\pm}\in\Sigma_{s_{0},i},\tau_{s}(w_{k}^{\pm})=\emptyset}(\kappa_{k}^{\pm})^{-1}(B_{\epsilon_{s}})\to{\Sigma}_{s,\tau_{s}(i)}$

such that

  1. (a)

for any $s$, the images of the maps $\phi_{i,s}$ cover ${\Sigma}_{s}$;

  2. (b)

for any nodal point $w_{i}^{\pm}$ of ${\Sigma}_{s}$ joining components
$\Sigma_{s,i^{\pm}(k)}$, there exists a constant
$\lambda_{s}\in\mathbb{C}^{*}$ such that
$(\kappa_{k}^{+}\circ\phi_{s,i^{+}(k)}^{-1}\circ\phi_{s,i^{-}(k)}\circ(\kappa_{k}^{-})^{-1})(z)=\lambda_{s}z$,
if the former is defined, and $\lambda_{s}\to 0$ as $s\to s_{0}$.

  3. (c)

for any $z\in\Sigma_{s_{0},i}$ in the complement of the $W_{k,s}^{\pm}$,
$\lim_{s\to s_{0}}(\phi_{i,s}(z))=z$;

  4. (d)

$\phi_{i,s}^{*}j_{\Sigma_{s,\tau_{s}(i)}}$ converges to $j_{\Sigma_{s_{0},i}}$
uniformly in all derivatives on compact sets;

  5. (e)

if $z_{i}$ is contained in $\Sigma_{s_{0},k}$, then $z_{i}=\lim_{s\to
s_{0}}\phi_{s,k}^{-1}(z_{i,s})$.

A stratified-smooth family of curves is a continuous family ${\Sigma}_{S}\to
S$ over a stratified base $S=\bigcup_{\Gamma}S_{\Gamma}$ such that the
restriction ${\Sigma}_{S_{\Gamma}}$ of ${\Sigma}_{S}$ to $S_{\Gamma}$ is a
smooth family of fixed type $\Gamma$. A stratified-smooth deformation of a
nodal curve ${\Sigma}$ is a germ of a stratified-smooth family of nodal curves
${\Sigma}_{S}$ equipped with an isomorphism of the central fiber
${\Sigma}_{0}$ with ${\Sigma}$. A universal stratified-smooth deformation of
${\Sigma}$ is a deformation with the property that any other stratified-smooth
deformation ${\Sigma}_{S^{\prime}}^{\prime}\to S^{\prime}$ is obtained by
pull-back by maps $\psi:S^{\prime}\to S$,
$\phi:{\Sigma}\times_{S}S^{\prime}\to{\Sigma}^{\prime}$, and any two
isomorphisms $\phi,\phi^{\prime}$ inducing the identity on ${\Sigma}$ are
equal.

Any universal holomorphic deformation is also a universal stratified-smooth
deformation, essentially by Lemma 2.2.1. In the stratified-smooth setting, the
analog of Theorem 2.1.3 fails and we need an additional definition:

######  Definition 2.2.4.

A universal stratified-smooth deformation $(\pi:{\Sigma}_{S}\to S,\phi)$ is
strongly universal if $\pi$ is a universal deformation of any of its fibers,
and two fibers of $\pi$ are isomorphic, if and only if they are related by the
action of $\operatorname{Aut}({\Sigma})$.

The construction of universal deformations extends to the smooth setting as
follows. Let ${\Sigma}_{S_{\Gamma}}\to S_{\Gamma}$ be a smooth family of
curves of fixed type $\Gamma$. A smooth system of local coordinates for the
$i$-th node of ${\Sigma}_{S_{\Gamma}}$ is a smooth map $\kappa_{i}$ from a
neighborhood $U_{i,\pm}$ of the zero section in $I^{i,\pm}$ to
${\Sigma}_{S_{\Gamma}}$ which is an isomorphism onto its image and induces the
identity at zero. Given a universal deformation $({\Sigma}_{S_{\Gamma}}\to
S_{\Gamma},\varphi)$ of fixed type $\Gamma$ and a smooth system of local
coordinates, applying the gluing construction (2) gives a smooth family
${\Sigma}_{S}\to S$ over an open neighborhood $S$ of $0$ in
$\operatorname{Def}({\Sigma})$. We may identify $S$ with
$\operatorname{Def}({\Sigma})$, for simplicity of notation.

######  Theorem 2.2.5.

Let ${\Sigma}$ be a stable curve. The family ${\Sigma}_{S}\to
S\subset\operatorname{Def}({\Sigma})$ constructed by gluing from a family
${\Sigma}_{\Gamma,S}\to S\subset\operatorname{Def}_{\Gamma}({\Sigma})$ of
fixed type, using any smooth family of local coordinates ${\kappa}$ near the
nodes, gives a strongly universal stratified-smooth deformation of ${\Sigma}$.

###### Proof.

Let ${\Sigma}_{S^{{\kappa}}}^{{\kappa}}\to S^{{\kappa}}$ be a family
constructed via gluing using a smooth family of local coordinates ${\kappa}$
as in (2), and ${\Sigma}_{S}\to S$ a universal deformation using a holomorphic
family of local coordinates by the same construction (2). By universality,
there exists a map $\psi:S^{{\kappa}}\to S$ so that
${\Sigma}_{\psi(s)}\cong{\Sigma}_{s}^{{\kappa}}$. It suffices to show that
$\psi$ is a diffeomorphism on each stratum. Consider the canonical map from
$T_{{\delta}}S^{{\kappa}}$ to $\operatorname{Def}({\Sigma}^{{\delta}})$, which
maps an infinitesimal change in the parameter space $S^{{\kappa}}$ to the
corresponding infinitesimal deformation of ${\Sigma}^{{\delta}}$, which we
identify with an element of
$\Omega^{0}({\Sigma},\operatorname{End}(T{\Sigma}^{{\delta}}))$. Let
$U\subset{\Sigma}^{{\delta}}$ denote the gluing region, that is, the image of
the union of domains of the local coordinates. The deformations generated by
the gluing parameters are supported in the gluing region $U$. On the other
hand, linearly independent deformations of fixed type
$\operatorname{Def}_{\Gamma}({\Sigma})$ generate deformations of the glued
curve that are linearly independent on${{\delta}}-U$, for sufficiently small
$U$. (The generated deformations will not vanish on $U$, because of the
varying local coordinates.) Thus the map
$\operatorname{Def}_{\Gamma}({\Sigma})\to\Omega^{0}({\Sigma}-U)$ is injective;
it follows that $TS^{{\kappa}}\to\operatorname{Def}({\Sigma}^{{\delta}})$ is
injective, hence an isomorphism by a dimension count. This shows that the map
$S^{{\kappa}}\to S$ is a covering. Let ${\kappa}_{t}$ be a family of local
coordinates interpolating between ${\kappa}$ and a holomorphic family. The
corresponding family $\psi_{t}$ interpolates between the identity and $\psi$.
Since each $\psi_{t}$ is a covering and $\psi_{0}$ is the identity, each
$\psi_{t}$ is a diffeomorphism. ∎

The strongly universal deformations above defined using smooth families of
local coordinates provide smooth orbifold charts on $\overline{M}_{g,n}$.
Since the space of local coordinates is convex, one can construct the local
coordinates for each stratum compatibly. Namely, let $\Gamma^{\prime}$ be a
combinatorial type degenerating to $\Gamma$. Local coordinates for the nodes
of $M_{g,n,\Gamma}$ induce local coordinates for $M_{g,n,\Gamma^{\prime}}$, in
a neighborhood of $M_{g,n,\Gamma}$, via the gluing construction (2).

######  Definition 2.2.6.

A compatible system of local coordinates for $\overline{M}_{g,n}$ is a system
of local coordinates for the nodes of each stratum $M_{g,n,\Gamma}$, so that
the local coordinates on any stratum $M_{g,n,\Gamma^{\prime}}$ are induced
from those on $M_{g,n,\Gamma}$, in a neighborhood of $M_{g,n,\Gamma}$.

Compatible systems of local coordinates can be constructed by induction on the
dimension of $M_{g,n,\Gamma}$, using convexity on the space of local
coordinates in Remark 2.1.1.

One can modify the gluing construction above by choosing a different smooth
structure on the space of gluing parameters. In the language of Hofer, Wysocki
and Zehnder [13, Appendix],

######  Definition 2.2.7.

A gluing profile is a diffeomorphism $\varphi:(0,1]\to[0,\infty)$. The
diffeomorphism given by $\varphi(\delta)=-1+1/\delta$ will be called the
standard gluing profile; $\varphi(\delta)=e^{1/\delta}-e$ will be called the
exponential gluing profile, and $\varphi(\delta)=-\ln(\delta)$ the logarithmic
gluing profile.

The set of gluing profiles naturally forms a partially ordered set: Write
$\varphi_{1}\geq\varphi_{0}$ and say $\varphi_{1}$ is softer than
$\varphi_{0}$ if $\varphi_{1}^{-1}\varphi_{0}$ extends to a diffeomorphism of
$[0,1]$. Write $\varphi_{1}>\varphi_{0}$ and say that $\varphi_{1}$ is
strictly softer than $\varphi_{0}$ if the derivatives of
$\varphi_{1}^{-1}\varphi_{0}:[0,1]\to[0,1]$ vanish at $0$. The exponential
gluing profile, standard gluing profile, and logarithmic gluing profile form a
decreasingly soft sequence in this partial order.

Fix a gluing profile $\varphi$, and consider once again the gluing
construction.

######  Definition 2.2.8.

Given a nodal curve ${\Sigma}$ with local coordinates ${\kappa}$ near the
nodes, and a collection of gluing parameters
$\delta=(\delta_{1},\ldots,\delta_{m})$, the glued curve
${\Sigma}({\delta},\varphi)$ is defined by gluing together small disks:

(4)
${\Sigma}^{{\delta},\varphi,\kappa}:=\left(\bigcup_{i=1}^{m}\Sigma_{i}-\\{w_{1}^{\pm},\ldots,w_{m}^{\pm}\\}\right)/\sim$

where the equivalence relation $\sim$ is given by

$z\sim(\kappa_{i}^{+})^{-1}(\exp(-\varphi(|\delta_{i}|)-i\arg(\delta_{i}))/\kappa_{i}^{-}(z),\quad,z\in
U_{i}^{-},\quad i=1,\ldots,m.$

More generally, given a family ${\Sigma}_{S_{\Gamma}}\to S_{\Gamma}$ of curves
of constant combinatorial type $\Gamma$ and a system of local coordinates near
the nodes ${\kappa}$, the construction (4) produces a family of curves
${\Sigma}_{S^{{\kappa},\varphi}}^{\varphi,{\kappa}}\to S^{{\kappa},\varphi}$
where $S^{{\kappa},\varphi}$ is the product of $S$ with the space of gluing
parameters.

Let ${\Sigma}$ be a compact, complex nodal curve. For any gluing profile
$\varphi$ and any collection ${\kappa}$ of local coordinates near the nodes,
the family ${\Sigma}_{S^{{\kappa},\varphi}}^{{\kappa},\varphi}\to
S^{{\kappa},\varphi}$ is a stratified-smooth strongly universal deformation,
since it is so for the standard gluing profile. Let
${\kappa}=({\kappa}_{\Gamma})$ be a compatible system of local coordinates
near the nodes, for each combinatorial type $\Gamma$. Each stratified-smooth
universal deformation above defines a classifying map

(5)
$S^{{\kappa},\varphi}/\operatorname{Aut}({\Sigma})\to\overline{M}_{g,n},\quad
s\mapsto[{\Sigma}_{s}]$

which is a homeomorphism onto its image, possibly after shrinking the
parameter space $S^{{\kappa},\varphi}$. (To obtain a precise meaning for
“classifying map” it is necessary to pass to the stacks-theoretic viewpoint,
which we do not discuss here.) The maps (5) provide $\overline{M}_{g,n}$ with
a compatible set of stratified-smooth orbifold charts, since the transition
maps are the identity on the space of gluing parameters by construction, and
smooth on each stratum. We denote by $\overline{M}_{g,n}^{{\kappa},\varphi}$
the smooth structure on $\overline{M}_{g,n}$ defined by the system of local
coordinates ${\kappa}$ near the nodes and the gluing profile $\varphi$; the
use of this smooth structure seems to have been suggested by Hofer. It seems
that these smooth structures might depend on the choice of ${\kappa}$, except
in the case of the logarithmic gluing profile, in which case one has a
canonical smooth structure.

The forgetful maps with respect to these non-standard smooth structures have
regularity properties that are worse than those with respect to the standard
smooth structure. For $2g+n>3$ we have forgetful morphisms
$f_{i}:\overline{M}_{g,n}\to\overline{M}_{g,n-1}$ by forgetting the $i$-th
marking and collapsing unstable components. There are two possibilities: a
genus zero component with one marking and two nodes is replaced by a point; a
genus zero component with two markings and one node is replaced by a single
marking. For any gluing profile, the maps $f_{i}$ are smooth away from the
locus where collapsing occurs. We say a local coordinate on a genus zero curve
is standard if it extends to an isomorphism with the projective line. The
forgetful morphism $f_{i}$ is smooth near the locus of one node, two marking
components if the local coordinates are standard and
$\delta\mapsto\exp(\varphi(\delta))^{-1}$ is smooth, that is, $\varphi$ is at
least as hard as the logarithmic gluing profile. The forgetful morphism
$f_{i}$ is smooth near the locus of curves containing components with two
nodes and one marking if the map
$\delta_{1},\delta_{2}\mapsto\varphi^{-1}(\varphi(\delta_{1})+\varphi(\delta_{2}))$
is smooth. For example, in the logarithmic gluing profile we have
$(\delta_{1},\delta_{2})\mapsto\delta_{1}\delta_{2}$, which is smooth, while
for the standard gluing profile collapsing a component gives the map
$(\delta_{1},\delta_{2})\mapsto\delta_{1}\delta_{2}/(\delta_{1}+\delta_{2})$
in the local gluing parameters, which is not smooth.

##  3\. Deformations of holomorphic maps from curves

This section reviews the construction of a stratified-smooth universal
deformations for stable (pseudo)holomorphic maps. The proof relies on a gluing
theorem, of the sort given by Ruan-Tian [21]; our approach follows that of
McDuff-Salamon [15] who treat the genus zero case. A different set-up for
gluing is described in Fukaya-Oh-Ohta-Ono [9], and explained in more detail in
Abouzaid [1]. The gluing construction gives rise to charts for the moduli
space of regular stable maps.

###  3.1. Stable maps

Let $(X,\omega)$ be a compact symplectic manifold and $\mathcal{J}(X)$ the
space of compatible almost complex structures on $X$. Let
$J\in\mathcal{J}(X)$.

######  Definition 3.1.1.

A marked nodal $J$-holomorphic map to $X$ consists of a nodal curve
${\Sigma}$, a collection ${z}=(z_{1},\ldots,z_{n})$ of distinct, smooth points
on ${\Sigma}$, and a $J$-holomorphic map ${u}:{\Sigma}\to X$. An isomorphism
of marked nodal maps from $({\Sigma}_{0},{z}_{0},{u}_{0})$ to
$({\Sigma}_{1},{z}_{1},{u}_{1})$ is an isomorphism of nodal curves
${\psi}:{\Sigma}_{0}\to{\Sigma}_{1}$ such that ${\psi}(z_{0,i})=z_{1,i}$ for
$i=1,\ldots,n$ and ${u}_{1}\circ{\psi}={u}_{0}$. A marked nodal map
$({\Sigma},{u},{z})$ is stable if it has finite automorphism group or
equivalently each component $\Sigma_{i}$ of genus zero resp. one for which
$u_{i}$ is constant has at least three resp. one special (nodal or marked)
point. The homology class of stable map ${u}:{\Sigma}\to X$ is
${u}_{*}[{\Sigma}]\in H_{2}(X,\mathbb{Z})$.

A _continuous family_ of $J$-holomorphic maps over a topological space $S$ is
a continuous family of nodal curves ${\Sigma}_{S}\to S$ (see Definition 2.2.3)
and a continuous map ${u}:{\Sigma}_{S}\to X$ which is fiberwise holomorphic.
That is, for each $s_{0}\in S$ and each nearby combinatorial type $\Gamma$ we
have

  1. (a)

a sequence of contractions $\tau_{s}:\Gamma({\Sigma}_{s_{0}})\to\Gamma$,
constant in $s\in S_{\Gamma}$;

  2. (b)

for every node $\\{w_{i}^{\pm}\\}$ collapsed under $\tau_{s}$, a pair of local
coordinates $\kappa^{\pm}_{i}:W^{\pm}_{i}\to\mathbb{C}$

  3. (c)

for every component $\Sigma_{s_{0},i}$ of ${\Sigma}_{s_{0}}$, a real number
$\epsilon_{s}>0$ converging to $0$ as $s\to s_{0}$ and maps

$\phi_{i,s}:\Sigma_{s_{0},i}-\cup_{w_{k}^{\pm}\in\Sigma_{s_{0},i},\tau_{s}(w_{k}^{\pm})=\emptyset}(\kappa_{k}^{\pm})^{-1}(B_{\epsilon_{s}})\to{\Sigma}_{s,\tau_{s}(i)}$

such that

  1. (a)

for any $s$, the images of the maps $\phi_{i,s}$ cover ${\Sigma}_{s}$;

  2. (b)

for any nodal point $w_{i}^{\pm}$ of ${\Sigma}_{s}$ joining components
$\Sigma_{s,i^{\pm}(k)}$, there exists a constant
$\lambda_{s}\in\mathbb{C}^{*}$ such that
$(\kappa_{k}^{+}\circ\phi_{s,i^{+}(k)}^{-1}\circ\phi_{s,i^{-}(k)}\circ(\kappa_{k}^{-})^{-1})(z)=\lambda_{s}z$
where defined, and $\lambda_{s}\to 0$ as $s\to s_{0}$.

  3. (c)

for any $z\in\Sigma_{s_{0},i}$ in the complement of the $W_{k,s}^{\pm}$,
$\lim_{s\to s_{0}}(\phi_{i,s}(z))=z$;

  4. (d)

$\phi_{i,s}^{*}j_{\Sigma_{s,\tau_{s}(i)}}$ converges to $j_{\Sigma_{s_{0},i}}$
uniformly in all derivatives on compact sets;

  5. (e)

if $z_{i}$ is contained in $\Sigma_{s_{0},k}$, then $z_{i}=\lim_{s\to
s_{0}}\phi_{s,k}^{-1}(z_{i,s})$.

  6. (f)

$\phi_{i,s}^{*}u_{s}$ converges to $u_{s_{0}}$ uniformly in all derivatives on
compact sets.

######  Remark 3.1.2.

It follows from the assumption that $u_{S}:\Sigma_{S}\to X$ is continuous that
the homology class $u_{s,*}[\Sigma_{s}]$ is locally constant in $s\in S$.
Indeed continuity implies that for $s$ sufficiently close to $s_{0}$, $u_{S}$
is homotopic to a map of the form $v_{S}\circ\gamma_{S}$ where
$\gamma_{s}:\Sigma_{S}\to\Sigma_{s_{0}}$ is a map to the central fiber
$\Sigma_{s_{0}}$ which collapses the gluing regions to the node. Since each
$\gamma_{s}=\gamma_{S}|\Sigma_{s}$ maps $[\Sigma_{s}]$ to $[\Sigma_{s_{0}}]$,
the claim follows.

In particular, taking $S$ to be the topological space given as the closure of
the set $S^{*}$ of rational numbers of the form $1/i,i\in\mathbb{Z}_{>0}$, we
say that a sequence of holomorphic maps $u_{i}:\Sigma_{i}\to X$ Gromov
converges if it extends to a continuous family over $S$. To state the Gromov
compactness theorem, recall that the energy of a map $u:\Sigma\to X$ is

$E(u)=\frac{1}{2}\int|du|^{2}.$

######  Theorem 3.1.3 (Gromov compactness).

Let $X,\omega,J$ be as above. Any sequence $u_{i}:\Sigma_{i}\to X$ of stable
holomorphic maps with bounded energy has a Gromov convergent subsequence.
Furthermore, the limit is unique.

For references and discussion, see for example [14, Theorem 1.8]. The
definition of Gromov convergence passes naturally to equivalence classes of
stable maps. A subset $C$ of $\overline{M}_{g,n}(X,d)$ is Gromov closed if any
sequence in $C$ has a limit point in $C$, and Gromov open if its complement is
closed. The Gromov open sets form a topology for which any convergent sequence
is Gromov convergent, by an argument using [15, Lemma 5.6.5]. Furthermore, any
convergent sequence has a unique limit. Gromov compactness implies that for
any $E>0$, the union of $\overline{M}_{g,n}(X,d)$ over $d\in
H_{2}(X,\mathbb{Z})$ with $(d,[\omega])<E$ is a compact, Hausdorff space.

######  Definition 3.1.4.

Let $X,\omega,J$ be as above. A stratified-smooth family of nodal
$J$-holomorphic maps over a space $S$ is a pair $({\Sigma}_{S},{u}_{S})$ of a
stratified-smooth family of nodal curves ${\Sigma}_{S}\to S$ together with a
continuous map ${u}_{S}:{\Sigma}_{S}\to X$ such that the restriction ${u}_{s}$
of ${u}$ to any fiber ${\Sigma}_{s}$ is holomorphic, and the restriction of
${u}_{S}$ to any stratum ${\Sigma}_{\Gamma}$ is smooth. A stratified-smooth
deformation of a stable $J$-holomorphic map $({\Sigma},{u})$ is a germ of a
stratified-smooth family $({\Sigma}_{S},{u}_{S})$ together with an isomorphism
of nodal maps $\iota:{\Sigma}_{0}\to{\Sigma}$ such that
$\iota^{*}{u}={u}_{0}$. A deformation $({\Sigma}_{S},{u}_{S},\iota)$ of
$({\Sigma},{u})$ is versal if any other (germ of) family of marked, nodal
curves $({\Sigma}^{\prime},{\Sigma}_{0})\to(S^{\prime},0)$ is induced from a
map $\psi:S^{\prime}\to S$ in the sense that there exists an isomorphism
$\phi:{\Sigma}^{\prime}\to{\Sigma}\times_{S}S^{\prime}$ in a neighborhood of
the central fiber ${\Sigma}_{0}$, and ${u}^{\prime}$ is obtained by composing
projection on the first factor with ${u}$. A versal deformation is universal
if the map $\phi$ above is the unique map inducing the identity on
${\Sigma}_{0}$.

###  3.2. Smooth universal deformations of regular stable maps of fixed
combinatorial type.

Let ${u}:{\Sigma}\to X$ be a stable map. For $p>2$ define a fiber bundle
$\mathcal{E}\to\mathcal{B}$ by

$\mathcal{B}=\mathcal{J}({\Sigma})\times\operatorname{Map}({\Sigma},X)_{1,p},\quad\mathcal{E}_{\zeta,{u}}=\Omega^{0,1}({\Sigma},{u}^{*}TX)_{0,p},$

where the latter is the space of $(0,1)$-forms with respect to the pair
$({j}(\zeta),J)$. Consider the Cauchy-Riemann section,

$\overline{\partial}:\mathcal{B}\to\mathcal{E},\ \ \
(j,{u})\mapsto\overline{\partial}_{j}{u},\ \
\overline{\partial}_{j}{u}={\frac{1}{2}}({\operatorname{d}}{u}\circ{j}(\zeta)-J_{u}\circ{\operatorname{d}}{u}).$

Let

$\operatorname{ev}:\mathcal{B}\to X^{2m},\ \ \
{u}\mapsto({u}(w_{1}^{-}),{u}(w_{1}^{+}),\ldots,{u}(w_{m}^{-}),{u}(w_{m}^{+}))$

denote the map evaluating at the nodal points. The space of stable maps of
type $\Gamma$ is given as $(\overline{\partial},\operatorname{ev})^{-1}(0)$.
To obtain a Fredholm map, we quotient by diffeomorphisms of $\Sigma$, or
equivalently, restrict to a minimal versal deformation ${\Sigma}_{S}\to S$ of
${\Sigma}$ of fixed type. This means that for each
$\zeta\in\operatorname{Def}_{\Gamma}({\Sigma})$ near $0$ we have a complex
structure ${j}(\zeta)$ on ${\Sigma}$, which we may assume agrees with
${j}={j}(0)$ near the nodes. Then the Cauchy-Riemann section induces a map

$\operatorname{Def}_{\Gamma}(\Sigma)\times\Omega^{0}(\Sigma,u^{*}TX)\to\mathcal{E}.$

Linearizing the Cauchy-Riemann section, together with the differences at the
nodes, gives rise to a Fredholm operator

$\tilde{D}_{{u}}:\operatorname{Def}_{\Gamma}({\Sigma})\times\Omega^{0}({\Sigma},{u}^{*}TX)\to\Omega^{0,1}({\Sigma},{u}^{*}TX)\oplus\bigoplus_{i=1}^{m}{u}(w_{i}^{\pm})^{*}TX$
(6)
$\tilde{D}_{{u}}({\zeta},{\xi}):=\left(\pi^{0,1}_{{\Sigma}}(\nabla{\xi}-{\frac{1}{2}}J({u}){\operatorname{d}}{u}Dj({\zeta})-{\frac{1}{2}}J_{{u}}(\nabla_{{\xi}}J)_{{u}}\partial{u})),({\xi}(w_{i}^{+})-{\xi}(w_{i}^{-}))_{i=1}^{m}\right)$

given by the linearized Cauchy-Riemann operator on each component, and the
difference of the values of the section at the nodes
$w^{\pm}_{1},\ldots,w^{\pm}_{m}$. The map ${u}=({\Sigma},{u},{z})$ is regular
if $\tilde{D}_{{u}}$ is surjective. This is independent of the choice of
representatives $j(\zeta)$: any two such choices
${j}^{\prime}(\zeta),{j}(\zeta)$ are related by a diffeomorphism of
${\Sigma}$. The space of infinitesimal deformations of ${u}$ of fixed type is

$\operatorname{Def}_{\Gamma}({u})=\operatorname{ker}(\tilde{D}_{{u}})/\operatorname{aut}({\Sigma}).$

The space of infinitesimal deformations of ${u}$ is

$\operatorname{Def}({u})=\operatorname{Def}_{\Gamma}({u})\oplus\bigoplus_{i=1}^{m}T_{w_{j}^{+}}{\Sigma}\otimes
T_{w_{j}^{-}}{\Sigma}$

where $\Gamma$ is the type of ${u}$.

######  Theorem 3.2.1.

Let $X,\omega,J$ be as above. A regular marked nodal $J$-holomorphic map
${u}=({\Sigma},{u},{z})$ admits a strongly universal deformation
$({\Sigma}_{S},{u}_{S},{z}_{S})$ with parameter space
$S\subset\operatorname{Def}_{\Gamma}({u})$ of fixed type if and only if ${u}$
is stable.

###### Proof.

Let $({\Sigma},{u})$ be a stable map to $X$ and ${\Sigma}_{S}\to
S\subset\operatorname{Def}_{\Gamma}({\Sigma})$ a minimal versal deformation of
${\Sigma}$ of fixed type constructed in (2). We may write any map
$C^{0}$-close to ${u}$ as $\exp_{{u}}({\xi})$ for some
${\xi}\in\Omega^{0}({\Sigma},{u}^{*}TX)$. Let
$\Psi_{{u}}({\xi}):{u}^{*}TX\to\exp_{{u}}({\xi})^{*}TX$ denote parallel
transport along geodesics with respect to the Hermitian connection
$\tilde{\nabla}=\nabla-{\frac{1}{2}}J(\nabla J);$ here $\nabla$ is the Levi-
Civita connection, see [15, Chapter 2]. This defines an isomorphism

(7)
$\Psi_{{u}}({\xi})^{-1}:\Omega^{0,1}_{{j}}({\Sigma},\exp_{{u}}({\xi})^{*}TX)\to\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX).$

where subscript $j$ denotes the space of $0,1$-forms taken with respect to the
complex structure $j$ on ${\Sigma}$. There is an isomorphism of
$\Omega^{0,1}_{{j}(\zeta)}({\Sigma},{u}^{*}TX)$ with
$\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX)$ given by composing the inclusion

$\Omega^{0,1}_{{j}(\zeta)}({\Sigma},{u}^{*}TX)\to\Omega^{1}({\Sigma},{u}^{*}TX)_{\mathbb{C}}=\Omega^{1}({\Sigma};{u}^{*}TX)\otimes_{\mathbb{R}}\mathbb{C}$

with the projection
$\Omega^{1}({\Sigma};{u}^{*}TX)\otimes_{\mathbb{R}}\mathbb{C}\to\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX).$
We denote by

$\Psi_{j}(\zeta):\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX)\to\Omega^{0,1}_{{j}(\zeta)}({\Sigma},{u}^{*}TX)$

the resulting map; one can think of this as a connection over the space of
complex structures on ${\Sigma}$ on the bundle whose fiber is the space of
$0,1$-forms with respect to $j(\zeta)$. By composing $\Psi_{{u}}({\xi})^{-1}$
and $\Psi_{j}(\zeta)^{-1}$ we obtain an identification

(8)
$\Psi_{j,{u}}(\zeta,\xi)^{-1}:\Omega^{0,1}_{{j}(\zeta)}({\Sigma},\exp_{{u}}({\xi})^{*}TX)\to\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX).$

Define

$\mathcal{F}_{{u}}:\operatorname{Def}_{\Gamma}({\Sigma})\times\Omega^{0}({\Sigma},{u}^{*}TX)\to\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX)$
$(\zeta,{\xi})\mapsto\Psi_{j,{u}}(\zeta,{\xi})^{-1}(\overline{\partial}_{{j}(\zeta)}(\exp_{{u}}({\xi}))).$

The operator $\tilde{D}_{{u}}$ is the linearization of $\mathcal{F}_{{u}}$.
The implicit function theorem implies that if ${u}$ is regular then the zero
set of $\mathcal{F}_{{u}}$ is modelled locally on a neighborhood of $0$ in
$\operatorname{ker}(\tilde{D}_{{u}})$. Furthermore, by elliptic regularity the
zero set consists entirely of smooth $J$-holomorphic maps [15, Section B.4].
Thus we obtain a smooth family of stable maps in a neighborhood of $0$ in
$\operatorname{ker}(\tilde{D}_{{u}})$. The action of $\operatorname{Aut}({u})$
on the space of stable maps with domain ${\Sigma}$ induces an inclusion of the
Lie algebra $\operatorname{aut}({u})$ into
$\operatorname{ker}(\tilde{D}_{{u}})$. Restricting to
$\operatorname{Def}_{\Gamma}({u})$, identified with a complement of
$\operatorname{aut}({\Sigma})$ (that is, a slice for the
$\operatorname{Aut}({u})$ action) gives a family $({\Sigma}_{S},{u}_{S})\to
S\subset\operatorname{Def}_{\Gamma}({u})$ of fixed type. The family
$({\Sigma}_{S},{u}_{S})$, together with the canonical identification $\iota$
of the central fiber with ${\Sigma}$, is a universal smooth deformation of
fixed type. Indeed, another smooth family
$({\Sigma}_{S^{\prime}},{u}_{S^{\prime}})$ over a base $S^{\prime}$ is in
particular a deformation of the underlying curve. After shrinking
$S^{\prime}$, each fiber of
$({\Sigma}_{s^{\prime}},{u}_{s^{\prime}}^{\prime})$ corresponds to a zero of
$\mathcal{F}_{{u}}$, and so lies in the image of the map given by the implicit
function theorem. The uniqueness part of the implicit function theorem gives a
smooth map $\psi:S^{\prime}\to\operatorname{Def}_{\Gamma}({u})$ and an
identification ${\Sigma}_{S^{\prime}}\to\psi^{*}{\Sigma}_{S}$. Any two such
maps inducing the same map on the central fiber are close in a neighborhood of
the central fiber. Since the automorphism groups of the central fibre are
discrete, any automorphism group is discrete. Thus any two such automorphisms
defined in a neighborhood of the central fiber, and equal on the central fiber
must be equal in a neighborhood of the central fiber. This shows that the
identification is unique, so that the deformation given by the gluing
construction is universal.

If ${u}$ is not stable, then it has no universal deformation since the
identification with the central fiber is unique only up to a continuous family
of automorphisms. ∎

Let $M_{g,n,\Gamma}^{{\operatorname{reg}}}(X,d)$ denote the moduli space of
regular stable maps of combinatorial type $\Gamma$. A family ${u}_{S}$ over
$S\subset\operatorname{Def}_{\Gamma}({u})$ induces a map

(9) $S\to M_{g,n,\Gamma}^{{\operatorname{reg}}}(X,d),\ s\mapsto[{u}_{s}]$

where $[{u}_{s}]$ denotes the isomorphism class of ${u}_{s}:{\Sigma}_{s}\to
X$.

######  Theorem 3.2.2.

For any $g,n,d$ and combinatorial type $\Gamma$ with $m$ nodes,
$M_{g,n,\Gamma}^{{\operatorname{reg}}}(X,d)$ has the structure of a smooth
orbifold of dimension $(1-g)(\dim(X)-6)+2(c_{1}(TX),d)-2m+2n$, with tangent
space at $[{u}]$ isomorphic to $\operatorname{Def}_{\Gamma}({u})$.

###### Proof.

By Theorem 3.2.1, the maps (9) for families giving universal deformations are
homeomorphisms onto their image and provide compatible charts. The dimension
formula follows from Riemann-Roch: The index of $\tilde{D}_{{u}}$ may be
deformed to a complex linear operator by homotoping the zero-th order terms
(which define a compact operator) to zero. ∎

###  3.3. Constructing stratified-smooth deformations of varying type

The main result of this section is Theorem 1.0.1, which is probably well-
known, cf. [19], [21], but for which we could not find an explicit reference.
The theorem itself will not be used, but the estimates involved in the proof
will be needed later for the corresponding result for vortices. The proof uses
a gluing construction for holomorphic maps, which produces from a smooth
family of holomorphic maps of fixed type, a stratified-smooth family of maps
of varying type.

Step 1: Approximate Solution

######  Definition 3.3.1.

Let ${\Sigma}$ be a compact, complex nodal curve. A gluing datum for
${\Sigma}$ consists of

  1. (a)

a collection of gluing parameters ${\delta}=(\delta_{1},\ldots,\delta_{m})$ in
the bundle ${I}$ of (3);

  2. (b)

local coordinates $\kappa_{j}^{\pm}$ near the nodes $w_{j}^{\pm}$ for
$j=1,\ldots,m$;

  3. (c)

a parameter $\rho$ which describes the width of the annulus on which the
gluing of maps is performed;

  4. (d)

a gluing profile $\varphi$, see Definition 2.2.7;

  5. (e)

a smooth cutoff function

(10) $\alpha:\mathbb{C}\to[0,1],\ \ \
\alpha(z)=\left\\{\begin{array}[]{ll}0&|z|\leq 1\\\ 1&|z|\geq
2\end{array}\right\\}.$

We first treat the case that $\varphi$ is the standard gluing profile. Let a
gluing datum be given, and ${\Sigma}^{{\delta}}$ denote the glued curve from
(2). Let ${u}:{\Sigma}\to X$ be a holomorphic map. Near each node $w_{k}$ let
$i^{\pm}(k)$ denote the components on either side of $w_{k}$. In the
neighborhoods $U_{k}^{\pm}$ (assuming they have been chosen sufficiently
small) define maps

$\xi_{k}^{\pm}:U_{k}^{\pm}\to T_{x_{k}}X,\ \ \
u_{i^{\pm}(k)}(z)=\exp_{x_{k}}(\xi_{k}^{\pm}(z))$

where $x_{k}=u(w_{k})$ and $\exp_{x_{k}}:T_{x_{k}}X\to X$ denotes geodesic
exponentiation. Given a holomorphic map ${u}:{\Sigma}\to X$, and a gluing
datum $({\delta},{\kappa},\rho,\varphi,\alpha)$ define the pre-glued map by
interpolating between the maps on the various components using the given
cutoff function and local coordinates: ${u}^{{\delta}}={u}(z)$ for
$z\notin\cup_{k}U_{k}^{\pm}$ and otherwise

(11)
${u}^{{\delta}}(z)=\exp_{x_{k}}(\alpha(\kappa_{k}^{\pm}(z)/\rho|\delta_{k}|^{1/2})\xi_{k}^{\pm}(z))\quad
z\in U_{k}^{\pm}.$

######  Remark 3.3.2.

The same formula but with domain ${\Sigma}$ (not the glued curve) defines an
intermediate map ${u}_{0}^{{\delta}}:{\Sigma}\to X$ which is constant near the
nodes. The right inverse of $\tilde{D}_{{u}_{0}^{{\delta}}}$ will be used in
the gluing construction.

First we estimate the failure of ${u}^{{\delta}}$ to satisfy the Cauchy-
Riemann equation. Define on ${\Sigma}^{{\delta}}$ the $C^{0}$-metric $g$ by
the identification

(12)
${\Sigma}^{{\delta}}={\Sigma}-\bigcup_{k,\pm}\kappa_{k}^{\pm}(B_{|\delta_{k}|^{1/2}}(0))/\left(\kappa^{+}_{k}(\partial
B_{|\delta_{k}|^{1/2}}(0))\sim\kappa^{-}_{k}(\partial
B_{|\delta_{k}|^{1/2}}(0))\right)$

using a Kähler metric on $\Sigma$, see Figure 1.

Figure 1. Continuous metric on a glued curve

The generalized Sobolev spaces $W^{l,p}$ with respect to this metric are
defined for $p\geq 1$ and integers $l\in\\{0,1\\}$, see [2] or [3]. For any
vector bundle $E$ we denote by $\Omega({\Sigma}^{{\delta}},E)_{l,p,{\delta}}$
the space of $W^{l,p}$ forms with values in $E$. If $p=\infty$ the norm is
independent of ${\delta}$ and we drop it from the notation. Let
$\|\cdot\|_{k,p,{\delta}}$ denote the Sobolev $W^{k,p}$-norm on
$\Omega^{0}({u}^{*}TX)$ defined using the ${\delta}$-dependent metric (12).

######  Proposition 3.3.3.

Suppose that ${u}:{\Sigma}\to X$ is a stable map, and
${u}^{{\delta}}:{\Sigma}^{{\delta}}\to X$ is the pre-glued map defined in
(11), defined for ${\delta}$ sufficiently small. There is a constant $c$ and
an $\epsilon>0$ such that if $\|{\delta}\|<\epsilon,\rho>1/\epsilon,$ and
$|\delta_{k}|^{2}\rho<\epsilon,\ \ \ k=1,\ldots,m$ then

$\|\overline{\partial}{u}^{{\delta}}\|_{0,p,{\delta}}^{p}\leq
c\sum_{k=1}^{m}(|\delta_{k}|^{1/2}\rho)^{2}.$

###### Proof.

Compare with McDuff-Salamon [15, Chapter 10]. The error term
$\overline{\partial}{u}({\delta})$ can be estimated by terms of two types;
those involving derivatives of the cutoff functions and those involving
derivatives of the map $\xi_{k}$. The derivative of $\exp_{x_{k}}$ is
approximately the identity near the node. The derivatives of $\alpha$ grow
like $1/\rho|\delta_{k}|^{1/2}$, while the norm of $\xi_{k}^{\pm}$ is bounded
by a constant times $\rho|\delta_{k}|^{1/2}$ on the gluing region. The term
involving the derivatives of $\alpha$ is bounded and supported on a region of
area less than $\pi\rho^{2}|\delta_{k}|$ for each node. The derivatives of
$\xi_{k}^{\pm}$ are also uniformly bounded, and the area bound gives the
required estimate. ∎

Let ${\Sigma}_{S}\to S$ with $S\subset\operatorname{Def}_{\Gamma}({\Sigma})$
be a family giving a minimal versal deformation of ${\Sigma}$ of fixed type,
and ${\Sigma}_{S_{{\delta}}({\delta})}\to
S_{{\delta}}\subset\operatorname{Def}({\Sigma}^{{\delta}})$ a family giving a
minimal versal deformation of ${\Sigma}^{{\delta}}$. The gluing construction
(2) applied to the family ${\Sigma}_{S}$ produces a map

(13)
$\operatorname{Def}_{\Gamma}({\Sigma})\to\mathcal{J}({\Sigma}^{{\delta}}),\quad\zeta\mapsto
j^{{\delta}}(\zeta)$

which maps any deformation of the original curve to the corresponding
deformation of the glued curve. In other words, any variation of complex
structure on ${\Sigma}$ of fixed type induces a variation of complex structure
on ${\Sigma}^{{\delta}}$. Similarly, for any
$\xi\in\Omega^{0}({\Sigma},{u}^{*}TX)$ we obtain an element
$\xi^{{\delta}}\in\Omega^{0}({\Sigma}^{{\delta}},{u}^{*}TX)$.

######  Proposition 3.3.4.

Suppose that ${u},{u}^{{\delta}}$ are as above, and
$(\zeta,\xi)\in\operatorname{Def}_{\Gamma}(u)$. There is a constant $c$ and an
$\epsilon>0$ such that if $\|{\delta}\|<\epsilon,\rho>1/\epsilon$,
$\|\zeta\|+\|{\xi}\|_{1,p}\leq\epsilon$, and $|\delta_{k}|^{2}\rho<\epsilon$
for $k=1,\ldots,m$ then

$\|\overline{\partial}_{{j}^{{\delta}}(\zeta)}\exp_{{u}^{{\delta}}}({\xi}^{{\delta}})\|_{0,p,{\delta}}^{p}\leq
c\sum_{k=1}^{m}(|\delta_{k}|^{1/2}\rho)^{2}.$

Step 2: Uniformly bounded right inverse

We wish to show that the map in Proposition 3.3.4 can be corrected to obtain a
holomorphic map. Define

(14)
$\mathcal{F}_{{u}}^{{\delta}}:\operatorname{Def}_{\Gamma}({\Sigma})\times\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)\to\Omega^{0,1}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)$
$(\zeta,\xi)\mapsto\Psi_{j,{u}^{{\delta}}}(\zeta,\xi)^{-1}(\overline{\partial}_{{j}^{{\delta}}(\zeta)}(\exp_{{u}^{{\delta}}}(\xi))).$

Here the operator $\Psi_{j,{u}^{\delta}}$ is as in (8). Let
$\tilde{D}_{{u}}^{{\delta}}(\xi)$ be the associated linear operator, that is,
the linearization of (14) at $\xi$. This operator naturally extends to a map
from Sobolev $1,p$-completion of the second factor of the domain to the
$0,p$-completion of the codomain. We denote by
$\tilde{D}_{{u}}^{{\delta}}:=\tilde{D}_{{u}}^{{\delta}}(0)$. We will construct
an approximate inverse

(15)
$T_{{\delta}}:\Omega^{0,1}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)\to\operatorname{Def}_{\Gamma}({\Sigma})\oplus\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)$

to $\tilde{D}_{{u}}^{{\delta}}$. The construction depends on a carefully
chosen cutoff function:

######  Lemma 3.3.5.

[15, Section 10.3] For any $\delta>0,\rho>1$ there exists a function
$\beta_{\rho,\delta}:\mathbb{R}^{2}\to[0,1]$ that satisfies

$\beta_{\rho,\delta}(z)=\begin{cases}0&|z|\leq\sqrt{\delta/\rho}\\\
1&|z|\geq\sqrt{\delta\rho}\end{cases}$

and for all $\xi\in W^{1,p}(B_{\rho|\delta_{k}|})$ satisfying $\xi(0)=0$

(16) $\|(\nabla\beta_{2;\rho,\delta})\xi\|_{0,p}\leq
c\log(\rho^{2})^{-1+1/p}\|\xi\|_{1,p},\quad\|\beta_{2;\rho,\delta}\|_{1,2}\leq
C\log(\rho^{2})^{-1/2}.$

Recall the map ${u}_{0}$ from Remark 3.3.2.

######  Lemma 3.3.6.

For sufficiently small ${\delta}$ there exists a right inverse
$Q_{{u}_{0}^{{\delta}}}$ of $\tilde{D}_{{u}_{0}^{{\delta}}}$ with image the
$L^{2}$-perpendicular of the kernel of $\tilde{D}_{{u}_{0}^{{\delta}}}$.

###### Proof.

Consider the maps defined by parallel transport using the modified Levi-Civita
connection,
$\Pi_{{u}_{0}^{{\delta}}}^{{u}}:\Omega^{0}({\Sigma},({u}_{0}^{{\delta}})^{*}TX)\to\Omega^{0}({\Sigma},({u}^{{\delta}})^{*}TX)$
$\Psi_{{u}_{0}^{{\delta}}}^{{u}}:\Omega^{0,1}({\Sigma},({u}_{0}^{{\delta}})^{*}TX)\to\Omega^{0,1}({\Sigma},({u}^{{\delta}})^{*}TX).$
The operator
$\Psi_{{u}_{0}^{{\delta}}}^{{u}}\tilde{D}_{{\Sigma},{u}_{0}^{{\delta}}}(\Pi_{{u}_{0}^{{\delta}}}^{{u}})^{-1}$
approaches the operator $\tilde{D}_{{u}}$ as ${\delta}\to 0$, c.f. [15, Remark
10.2.2]. The statement of the lemma follows. ∎

Define the approximate right inverse for $\tilde{D}_{{u}}^{{{\delta}}}$ by
composing the right inverse $Q_{{u}_{0}^{{\delta}}}$ with a cutoff and
extension operator:
$T_{{\delta}}:=P_{{\delta}}Q_{{u}_{0}^{{\delta}}}K_{{\delta}},$ defined as
follows. The cutoff operator

$K_{{\delta}}:\Omega^{0,1}({u}^{{\delta}}*TX)_{0,p,{\delta}}\to\Omega^{0,1}({u}_{0}^{{\delta},*}TX)_{0,p}$

is defined by

$(K_{{\delta}}(\eta))(z)=\begin{cases}\eta(z)&z\notin\bigcup_{k,\pm}B_{|\delta_{k}(z)|^{1/2}}(w_{k}^{\pm})\\\
0&\text{otherwise}\end{cases}.$

We have $\|K_{{\delta}}\eta\|_{0,p}\leq\|\eta\|_{0,p,{\delta}}$ by definition
of the $0,p,{\delta}$ norm. The extension operator

$P_{{\delta}}:\operatorname{Def}_{\Gamma}({\Sigma})\oplus\Omega^{0}({\Sigma},{u}_{0}^{{\delta}}*TX)_{1,p,{\delta}}\to\operatorname{Def}({\Sigma}^{{\delta}})\oplus\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{1,p,{\delta}}$

is defined as follows. For each component $\Sigma_{i}$ let $\Sigma_{i}^{*}$
denote the complements of small balls around the nodes

$\Sigma_{i}^{*}=\Sigma_{i}-\bigcup_{l,w_{l}^{\pm}\in\Sigma_{i}}B_{|\delta_{l}(z)|^{1/2}/\rho}(w_{l}^{\pm})$

and the inclusion $\pi_{i}:\Sigma_{i}^{*}\to{\Sigma}^{{\delta}}$ induces a map
$\pi_{i,*}:\Omega^{0}(\Sigma_{i}^{*},u_{i}^{*}TX)\to\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{0,p}.$
Define

$P_{{\delta}}(\zeta,\xi)=(\zeta^{{\delta}},\xi^{{\delta}})$

where $\zeta^{{\delta}}$ is the image of $\zeta$ under the gluing map (13) and
$\xi^{{\delta}}$ is obtained by patching together the sections $\xi$; on the
gluing region arising from gluing the $k$-th node $w_{k}$ the section
$\xi^{{\delta}}$ is given by the sum

$\pi_{i^{+}(k),*}\beta_{\rho,\delta_{k}}(\xi_{i^{+}(k)}-\xi(w_{k}))\\\
+\pi_{i^{-}(k),*}\beta_{\rho,\delta_{k}}(\xi_{i^{-}(k)}-\xi(w_{k}))+\xi(w_{k}).$

Fix a metric $\|\cdot\|$ on the finite-dimensional space
$\operatorname{Def}_{\Gamma}({\Sigma})$ and define

$\|(\zeta,\xi)\|_{1,p,{\delta}}=\left(\|\zeta\|^{p}+\|\xi\|_{1,p,{\delta}}^{p}\right)^{1/p}.$

######  Proposition 3.3.7.

Let ${u}:{\Sigma}\to X$ be a stable map. There exist constants $c,C>0$ such
that if $\|{\delta}\|<c$ then the approximate inverse $T_{{\delta}}$ of (15)
satisfies

$\|(\tilde{D}_{{u}}^{{\delta}}T_{{\delta}}-I)\eta\|_{0,p,{\delta}}\leq{\frac{1}{2}}\|\eta\|_{0,p,{\delta}},\quad\|T_{{\delta}}\|<C.$

###### Proof.

By construction $T_{{\delta}}$ is an exact right inverse for
$\tilde{D}_{{u}}^{{\delta}}$ away from gluing region. In the gluing region the
variation of complex structure on the curve vanishes and
$D_{{u}}^{{\delta}}=D_{x_{k}}$, the standard Cauchy-Riemann operator with
values in $T_{x_{k}}X$. So

$\displaystyle\tilde{D}_{{u}}^{{\delta}}T_{{\delta}}\eta-\eta$
$\displaystyle=$ $\displaystyle\sum
D_{x_{k}}\beta_{2;\rho,\delta}(z)(\xi_{i^{\pm}(k)}(z)-\xi_{i^{\pm}(k)}(w_{k}))$
$\displaystyle=$
$\displaystyle\sum(D_{x_{k}}\beta_{2;\rho,\delta}(z))\xi_{i^{\pm}(k)}(z)$

since $K_{{\delta}}\eta=0$ on $B_{|\delta_{k}|^{1/2}}(0)$ in the components
adjacent to the node. Since $p>2$, the $0,p,{\delta}$-norm of the right hand
side is controlled by the ordinary $L^{p}$ norm. By (16) we have

$\|\tilde{D}_{{u}}^{{\delta}}T_{{\delta}}\eta-\eta\|_{0,p,{\delta}}\leq\sum_{k}c|\log(\rho)|^{2/p-2}\|\xi_{i^{\pm}(k)}-\xi(w_{k})\|_{1,p}.$

The last factor is bounded by $\|K_{{\delta}}\eta\|_{0,p}$, by the uniform
bound on $Q_{{\delta}}$, and hence $\|\eta\|_{0,p,{\delta}}$, by the uniform
bound on $K_{{\delta}}$. ∎

Define a right inverse $Q_{{\delta}}$ to $\tilde{D}_{{u}}^{{\delta}}$ by the
formula

$Q_{{\delta}}=T_{{\delta}}(\tilde{D}_{{u}}^{{\delta}}T_{{\delta}})^{-1}=\sum_{k\geq
0}T_{{\delta}}(\tilde{D}_{{u}}^{{\delta}}T_{{\delta}}-I)^{k}.$

The uniform bound on $T_{{\delta}}$ from Lemma 3.3.7 implies a uniform bound
on $Q_{{\delta}}$.

Step 3: Uniform quadratic estimate

######  Proposition 3.3.8.

Let ${u}:{\Sigma}\to X$ be a stable map, ${\delta}$ a collection of gluing
parameters, and ${u}^{{\delta}}:{\Sigma}^{{\delta}}\to X$ the approximate
solution defined above. For every constant $c>0$ there exist constants
$c_{0},\delta_{0}>0$ such that if
${u}\in\operatorname{Map}({\Sigma},X)_{1,p},\xi\in\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{1,p}$

$\|{\operatorname{d}}{u}\|_{0,p}\leq c_{0},\ \ \ \|\xi\|_{L_{\infty}}\leq
c_{0}\ \ \ \ \|\zeta\|\leq c_{0},\ \ \ \ \|{\delta}\|<\delta_{0}$

then

$\|(D\mathcal{F}_{{u}}^{{\delta}}(\zeta,\xi,\zeta_{1},\xi_{1})-\tilde{D}_{{u}}^{{{\delta}}})(\zeta_{1},\xi_{1})\|_{0,p,{\delta}}\leq
c\|\zeta,\xi\|_{1,p}(\|\zeta_{1},\xi_{1}\|_{1,p}.$

Here $D\mathcal{F}_{{u}}^{{\delta}}(\zeta,\xi,\zeta_{1},\xi_{1})$ denotes the
derivative evaluated at $\zeta,\xi$, applied to $\zeta_{1},\xi_{1}$. We use
similar notation throughout the discussion. The proof uses uniform estimates
on Sobolev embedding:

######  Lemma 3.3.9.

There exists a constant $c>0$ independent of $\delta$ such that the embedding

$\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{{1,p},{\delta}}\to\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{0,\infty}$

has norm less than $c$.

###### Proof.

One writes the Sobolev norms as a contribution from each component of the
curve ${\Sigma}$. Then on each piece, the metric near the boundary is
uniformly comparable with the flat metric. The claim then follows from [2,
Chapter 4] which shows that the constants in the Sobolev embeddings depend
only on the dimensions of the cone in the cone condition. ∎

###### Proof of Proposition.

For simplicity we assume a single gluing parameter $\delta$. Let
$\Psi_{{u}}^{{\delta},x}(\zeta,\xi):\Lambda^{0,1}T_{z}^{*}{\Sigma}^{\delta}\otimes
T_{x}X\to\Lambda^{0,1}_{{j}^{{\delta}}(\zeta)}T_{z}^{*}{\Sigma}\otimes
T_{\exp_{x}(\xi)}X$ denote pointwise parallel transport as in (7) using the
parallel transport using the modified Levi-Civita connection, and projecting
onto the $0,1$-part of the form defined using the complex structure
${j}^{{\delta}}(\zeta)$ obtained from gluing ${j}(\zeta)$, see (13). Let

$\Theta_{{u}}^{{\delta},x}(\zeta,\xi,\zeta_{1},\xi_{1};\eta)=\tilde{\nabla}_{t}\Psi_{{u}}^{{\delta},x}(\zeta+t\zeta_{1},\xi+t\xi_{1})\eta.$

For $\xi,\eta$ sufficiently small there exists a constant $c$ such that

(17) $|\Theta_{{u}}^{\delta,x}(\zeta,\xi,\zeta_{1},\xi_{1};\eta)|\leq
c\|\xi,\zeta\|\|\xi_{1},\zeta_{1}\|\|\eta\|$

where the norms on the right-hand side are any norms on the finite dimensional
vector spaces $T_{{\Sigma}}M_{g,n,\Gamma}$ and $T_{x}X$. This estimate is
uniform in $\delta$, for the variation in complex structure vanishes in a
neighborhood of the nodes. Differentiate the equation
$\Psi_{{u}}^{\delta}(\zeta,\xi)\mathcal{F}_{{u}}^{\delta}(\zeta,\xi)=\overline{\partial}_{{j}^{\delta}(\zeta)}(\exp_{{u}^{\delta}}(\xi)))$
with respect to $(\zeta_{1},\xi_{1})$ to obtain

(18)
$\Theta_{{u}^{\delta}}(\zeta,\xi,\zeta_{1},\xi_{1},\mathcal{F}_{{u}}^{\delta}(\zeta,\xi))+\Psi_{{u}}^{\delta}(\zeta,\xi)(D\mathcal{F}_{{u}}^{\delta}(\zeta,\xi,\zeta_{1},\xi_{1}))=\\\
\tilde{D}_{{u}}^{\delta}(\xi,Dj^{\delta}(\zeta,\zeta_{1}),D\exp_{{u}^{\delta}}(\xi,\xi_{1})).$

Using the pointwise inequality

$|\mathcal{F}_{{u}}^{\delta}(\zeta,\xi)|<c|{\operatorname{d}}{\exp_{{u}^{\delta}}(\xi)}|<c(|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)$

for $\zeta,\xi$ sufficiently small, the estimate (17) on $\Phi$ produces a
pointwise estimate

$|(\Psi_{{u}}^{\delta})^{-1}(\xi)\Theta_{{u}}^{\delta}(\zeta,\xi,\zeta_{1},\xi_{1},\mathcal{F}_{{u}}^{\delta}(\zeta,\xi))|\leq
c(|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)\,|(\xi,\zeta)|\,|(\xi_{1},\zeta_{1})|.$

Hence

(19)
$\|\Psi_{{u}^{\delta}}^{-1}(\xi)\Theta_{{u}}^{\delta}(\zeta,\xi,\zeta_{1},\xi_{1},\mathcal{F}^{\delta}_{{u}}(\zeta,\xi))\|_{0,p}\\\
\leq
c(1+\|{\operatorname{d}}{u}^{\delta}\|_{0,p}+\|\nabla\xi\|_{0,p})\|(\xi,\zeta)\|_{0,\infty}\|(\xi_{1},\zeta_{1})\|_{0,\infty}.$

It follows that

(20)
$\|\Psi_{{u}}^{\delta}(\xi)^{-1}\Theta_{{u}}^{\delta}(\zeta,\xi,\zeta_{1},\xi_{1},\mathcal{F}_{{u}}^{\delta}(\zeta,\xi))\|_{0,p}\leq
c\|(\xi,\zeta)\|_{1,p}\|(\xi_{1},\zeta_{1})\|_{1,p}$

since the $W^{1,p}$ norm controls the $L^{\infty}$ norm by Lemma 3.3.9.

We next show that there exists a constant $c>0$ such that uniformly in
$\delta$,

(21)
$\|\Psi_{{u}^{\delta}}(\xi)^{-1}\tilde{D}_{{u}}^{\delta}(\xi,D{j}^{{\delta}}(\zeta,\zeta_{1}),D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))-\tilde{D}_{{u}}^{\delta}(\zeta_{1},\xi_{1})\|_{0,p}\leq
c\|\zeta,\xi\|_{1,p}\|\zeta_{1},\xi_{1}\|_{1,p}.$

Indeed differentiate (14) to obtain

(22)
$\tilde{D}_{{u}}^{\delta}(\xi,D{j}^{\delta}(\zeta,\zeta_{1}),D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))=\nabla^{0,1}_{j^{\delta}(\zeta)}D\exp_{{{u}^{\delta}}}(\xi,\xi_{1})-\\\
{\frac{1}{2}}\pi^{0,1}_{j^{\delta}(\zeta)}J_{\exp_{{u}^{\delta}}(\xi)}{\operatorname{d}}\exp_{{u}^{\delta}}(\xi)D{j}^{\delta}(\zeta,\zeta_{1})$$-J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{{u}}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial\exp_{{u}^{\delta}}(\xi).$

Hence

$\tilde{D}_{{u}}^{\delta}(\xi,D{j}^{\delta}(\zeta,\zeta_{1}),D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))-\Psi_{{u}^{\delta}}(\xi)\tilde{D}_{{u}}^{\delta}(\zeta_{1},\xi_{1})=\Pi_{1}+\Pi_{2}+\Pi_{3}$

where the three terms $\Pi_{1},\Pi_{2},\Pi_{3}$ are

$\Pi_{1}=\nabla^{0,1}_{{j}^{\delta}(\zeta)}D\exp_{{{u}^{\delta}}}(\xi,\xi_{1})-\Psi_{{u}^{\delta}}(\xi)\nabla^{0,1}_{{j}^{\delta}(0)}\xi_{1}$
$\Pi_{2}=-{\frac{1}{2}}\pi^{0,1}_{{j}^{\delta}(\zeta)}J_{\exp_{{u}}(\xi)}{\operatorname{d}}\exp_{{u}^{\delta}}(\xi)D{j}^{\delta}(\zeta,\zeta_{1})+\Psi_{{u}^{\delta}}(\xi)\pi^{0,1}_{{j}^{\delta}(0)}{\frac{1}{2}}J_{{u}^{\delta}}{\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1})$
$\Pi_{3}=-{\frac{1}{2}}J_{\exp_{{u}}(\xi)}(\nabla^{0,1}_{D\exp_{{{u}}^{\delta}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(\zeta)}\exp_{{u}^{\delta}}(\xi)+{\frac{1}{2}}\Psi_{{u}^{\delta}}(\xi)J_{{u}^{\delta}}(\nabla_{\xi_{1}}J_{{u}})\partial_{{j}^{\delta}(0)}{u}^{\delta}.$

The first difference has norm given by

(23)
$|\pi_{{j}^{\delta}(\zeta)}^{0,1}(\nabla(D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))-\Psi_{{u}^{\delta}}(\xi)\nabla\xi_{1})|\\\
\leq|\pi_{{j}^{\delta}(\zeta)}^{0,1}(\nabla(D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))-D\exp_{{{u}^{\delta}}}(\xi,\nabla\xi_{1}))|+|\pi^{0,1}_{{j}^{\delta}(\zeta)}(D\exp_{{{u}^{\delta}}}(\xi,\nabla\xi_{1})-\Psi_{{u}^{\delta}}(\xi)\nabla\xi_{1})|\\\
\leq
c|\nabla\xi||\xi_{1}|+c(|\zeta|+|\xi|)|\nabla\xi_{1}|+c|{\operatorname{d}}{u}^{\delta}||\xi||\xi_{1}|.$

We write for the second difference

(24)
$|\pi^{0,1}_{{j}^{\delta}(\zeta)}(J_{\exp_{{u}^{\delta}}(\xi)}{\operatorname{d}}\exp_{{u}^{\delta}}(\xi)D{j}^{\delta}(\zeta,\zeta_{1})-\Psi_{{u}^{\delta}}(\xi)J_{{u}^{\delta}}{\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1}))|\\\
\leq|\pi^{0,1}_{{j}^{\delta}(\zeta)}(J_{\exp_{{u}^{\delta}}(\xi)}{\operatorname{d}}\exp_{{u}^{\delta}}(\xi)D{j}^{\delta}(\zeta,\zeta_{1})-J_{\exp_{{u}^{\delta}}(\xi)}\Psi_{{u}^{\delta}}(\xi){\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1}))|+\\\
|\pi^{0,1}_{{j}^{\delta}(\zeta)}(J_{\exp_{{u}^{\delta}}(\xi)}\Psi_{{u}^{\delta}}(\xi){\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1})-\Psi_{{u}^{\delta}}(\xi)J_{{u}}{\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1}))|\\\
\leq
c(|\zeta|+|\xi|+|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)||\zeta_{1}|.$

The third term can be estimated pointwise by

$|J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{u}^{\delta}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(\zeta)}\exp_{{u}^{\delta}}(\xi)-\Psi_{{u}^{\delta}}(\xi)J_{{u}^{\delta}}(\nabla_{\xi_{1}}J_{{u}^{\delta}})\partial_{{j}^{\delta}(0)}{u}^{\delta}|$
$\leq|J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{{u}}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(\zeta)}\exp_{{u}^{\delta}}(\xi)-J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{u}^{\delta}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(0)}\exp_{{u}^{\delta}}(\xi)|$
$+|J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{{u}}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(0)}\exp_{{u}^{\delta}}(\xi)-\Psi_{{u}^{\delta}}(\xi)J_{{u}^{\delta}}(\nabla_{\xi_{1}}J_{{u}^{\delta}})\partial_{{j}^{\delta}(0)}{u}^{\delta}|$
$\leq
c|\zeta|(|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)|\xi_{1}|+c(|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)(|\xi_{1}|)$

for $\xi$ sufficiently small. Combining these estimates and integrating, using
the $0,p,\delta$-norms on ${\operatorname{d}}{u}$, $\nabla\xi,\nabla\xi_{1}$
and the $L^{\infty}$ norms on the other factors and Lemma 3.3.9, completes the
proof. ∎

Step 4: Implicit Function Theorem

For any $(\zeta_{0},\xi_{0})\in\operatorname{Def}_{\Gamma}({u})$ we denote by
$\zeta_{0}^{{\delta}}$ the deformation of ${\Sigma}^{{\delta}}$ defined in
(13) and by $\xi_{0}^{{\delta}}$ the section of ${u}^{{\delta},*}TX$ defined
as in (11).

######  Theorem 3.3.10.

Let ${u}:{\Sigma}\to X$ be a stable map. There exist constants
$\epsilon_{0},\epsilon_{1}>0$ such that for any
$(\zeta_{0},\xi_{0},{\delta})\in\operatorname{ker}\tilde{D}_{{u}}\times\mathbb{R}^{m}$
of norm at most $\epsilon_{0}$, there is a unique
$(\zeta_{1},\xi_{1})=(\tilde{D}_{{u}}^{{\delta}})^{*}\eta_{1}$ of norm at most
$\epsilon_{1}$ such that the map
$\exp_{{u}^{{\delta}}}(\xi_{0}^{{\delta}}+\xi)$ is
${j}^{\delta}(\zeta_{0}+\zeta_{1})$-holomorphic, and depends smoothly on
$\zeta_{0},\xi_{0}$.

###### Proof.

The first claim is an application of the quantitative version of the implicit
function theorem (see for example [15, Appendix A.3]) using the uniform error
bound from Proposition 3.3.3, uniformly bounded right inverse from Proposition
3.3.7, and uniform quadratic estimate from Proposition 3.3.8. ∎

Step 5: Rigidification

In the previous step we have constructed a family of stable maps which we will
show eventually gives rise to a parametrization of all nearby stable maps. A
more natural way of parametrizing nearby stable maps involves examining the
intersections with a family of codimension two submanifolds. For example, this
construction of charts is that given in the algebraic geometry approach of
Fulton-Pandharipande [10]. In order to carry this out in the symplectic
approach, we study the differentiability of the evaluation maps. Let
$u_{S}:\Sigma_{S}\to X$ over a parameter space
$S\subset\operatorname{Def}({u})$ be the family of maps defined in the
previous step. The following is similar to [9, Lemma A1.59].

######  Theorem 3.3.11.

If ${u}_{S}$ is constructed using the exponential gluing profile $\varphi$ and
$U\subset{\Sigma}$ is an open neighborhood of the nodes then the map
$(z,s)\mapsto{u}_{s}(z)$ is differentiable on a neighborhood of
$({\Sigma}-U)\times\\{0\\}$.

###### Proof.

For simplicity, we assume that there is a single gluing parameter $\delta$.
Differentiability for $\delta$ is studied in McDuff-Salamon [15, Section
10.6]. The discussion in our case is somewhat easier, because we use a fixed
right inverse in the gluing construction. Given
$(\zeta,\xi)\in\operatorname{Def}_{\Gamma}({\Sigma})\times\Omega^{0}({\Sigma}^{\delta},({u}^{\delta})^{*}TX)$,
we constructed a unique correction $(\zeta_{1},\xi_{1})$ in the image of the
right inverse such that
$\overline{\partial}_{{j}^{\delta}(\zeta_{0}+\zeta_{1})}\exp_{{u}^{\delta}}(\xi_{0}+\xi_{1})=0.$
For $\delta$ fixed, $(\zeta_{1},\xi_{1})$ depends smoothly on
$(\zeta_{0},\xi_{0})$, by the implicit function theorem. Hence the evaluation
at $z\in{\Sigma}-U$ also depends smoothly on $(\zeta_{0},\xi_{0})$.

The computation of the derivative with respect to the gluing parameter is
complicated by the fact that for each $\delta$ a different implicit function
theorem is applied to obtain the correction. Let
$\tilde{D}_{\delta}=D\mathcal{F}_{{u}^{\delta}}$. Differentiating the equation
$\mathcal{F}_{{u}^{\delta}}(\zeta_{0}^{\delta}+\zeta_{1},\xi_{0}^{\delta}+\xi_{1})=0$
with respect to $\delta$ gives

$\tilde{D}_{\delta}\left(\frac{d}{d\delta}\zeta_{1},D\exp_{{u}^{\delta}}(\xi_{0}^{\delta}+\xi_{1},0,\frac{d}{d\delta}\xi_{1})\right)=-\tilde{D}_{\delta}\left(0,D\exp_{{u}^{\delta}}(\xi_{0}^{\delta}+\xi_{1},\frac{d}{d\delta}{u}^{\delta},0)\right).$

From (11) we have in the gluing region,

$\displaystyle\frac{d}{d\delta}\overline{\partial}{u}^{\delta}$
$\displaystyle=$
$\displaystyle\frac{d}{d\delta}\overline{\partial}\exp_{x}(\left(\alpha(\rho\varphi^{-1/2}|z|)\xi(z))\right)$
$\displaystyle=$ $\displaystyle
D\exp_{x}\left(\alpha(\rho\varphi^{-1/2}|z|)\xi(z),\alpha^{\prime}(\rho\varphi^{-1/2}|z|)|z|\xi(z)\frac{d}{d\delta}\varphi^{-1/2}\rho\right)^{0,1}.$

Hence there exists a constant $C$ depending on $\rho,\alpha$ but not on
$\delta$ such that

(25) $\left|\frac{d}{d\delta}\overline{\partial}{u}^{\delta}\right|\leq
C\left|\frac{d}{d\delta}\varphi^{-1/2}\right|.$

Now ${\operatorname{d}}\varphi$ is given by

${\operatorname{d}}(e^{1/\delta}-e)^{-1/2}=(1/2)(e^{1/\delta}-e)^{-3/2}e^{1/\delta}\delta^{-2}{\operatorname{d}}\delta=(1/2)(e^{1/3\delta}-e^{-2/3\delta+1})^{-3/2}\delta^{-2}{\operatorname{d}}\delta.$

For $\delta$ small, this is less than
${\frac{1}{2}}e^{-1/2\delta}\delta^{-2}$. Integrating and using the pointwise
estimate (25) we obtain for some constant $C>0$,

$\left\|\frac{d}{d\delta}\overline{\partial}{u}^{\delta}_{s}\right\|_{0,p}\leq
Ce^{-1/2\delta}\delta^{-2}\leq Ce^{-1/\delta}$

for sufficiently small $\delta$. Now the uniform quadratic estimates imply
that $\tilde{D}_{\delta}=D\mathcal{F}_{{u}^{\delta}}(\zeta,\xi)$ is uniformly
bounded from below on the right inverse of
$\tilde{D}_{{u}}^{\delta}=D\mathcal{F}_{{u}}^{\delta}(0,0)$, for
$(\zeta_{0},\xi_{0})$ sufficiently small. It follows that

$\left\|(\frac{d}{d\delta}\zeta_{1},\frac{d}{d\delta}\xi_{1})\right\|_{1,p}\leq
Ce^{-1/\delta}$

for $\zeta_{0},\xi_{0},\delta$ sufficiently small as well. Hence the same is
true for the evaluation $\frac{d}{d\delta}\xi_{1}(z)$ for $z\in{\Sigma}-U$. In
particular, $\lim_{\delta\to
0}(\partial_{\delta}\exp_{{u}^{\delta}}(\xi_{0}^{\delta}+\xi_{1}))(z)=0.$ It
follows that the differential of the evaluation map has a continuous limit as
${\delta}\to 0$, which completes the proof of the Theorem. ∎

Using the evaluation maps in the previous step, we construct embeddings of the
families constructed above into suitable moduli spaces of stable marked
curves, given by adding additional marked points which map to fixed
submanifolds in $X$. A codimension two submanifold $Y\subset X$ is transverse
to ${u}:{\Sigma}\to X$ if ${u}$ meets $Y$ transversally in a single point
${u}(z)$.

######  Definition 3.3.12.

Let ${u}:{\Sigma}\to X$ be a stable map. Given any family
${Y}=(Y_{1},\ldots,Y_{\ell})$ of codimension two submanifolds transverse to
${u}$ and a family ${\Sigma}_{S},{u}_{S},{z}_{S}$ with parameter space $S$ of
an $n$-marked stable map $({\Sigma},{u},{z})$, the rigidified family of
$n+\ell$-marked nodal surfaces is defined by

(26) ${\Sigma}_{S}^{{Y},{u}}:=({\Sigma}_{S},(z_{1,S},\ldots,z_{n+\ell,S}))\to
S,\ \ \ \ {u}_{s}(z_{n+i,s})\in Y_{i}.$

######  Proposition 3.3.13.

Let ${u}_{S}$ be a family of stable maps over a parameter space
$S\subset\operatorname{Def}({u})$ given by the gluing construction using a
gluing profile $\varphi$ and system of coordinates ${\kappa}$. Suppose that
the evaluation map $\operatorname{ev}:({\Sigma}-U)\times S\to X$ is $C^{1}$,
and that the rigidified family has stable underlying curves. Then the
rigidified family of curves ${\Sigma}_{S}^{{Y},{u}}$ is $C^{1}$ with respect
to the gluing profile and local coordinates, that is, the map
$S\mapsto\overline{M}_{g,n+l},\quad s\mapsto{\Sigma}_{s}^{{Y},{u}}$ is $C^{1}$
with respect to the smooth structure defined by $\varphi,{\kappa}$.

###### Proof.

By the implicit function theorem for $C^{1}$ maps and differentiability of
evaluation maps from the previous subsection. ∎

######  Definition 3.3.14.

Let ${Y},{u}$ be as above. The pair $({Y},{u})$ is compatible if

  1. (a)

each $Y_{j}$ intersects ${u}$ transversally in a single point
$z_{j}\in{\Sigma}$;

  2. (b)

if $\xi\in\operatorname{ker}(\tilde{D}_{{u}})$ satisfies $\xi(z_{n+j})\in
T_{{u}(z_{n+j})}Y_{j}$ for $j=1,\ldots,l$ then $\xi=0$;

  3. (c)

the curve ${\Sigma}$ marked with the additional points
$z_{n+1},\ldots,z_{n+\ell}$ is stable;

  4. (d)

if some automorphism of $({\Sigma},{u})$ maps $z_{i}$ to $z_{j}$ then $Y_{i}$
is equal to $Y_{j}$.

The second condition says that there are no infinitesimal deformations which
do not change the positions of the extra markings.

######  Proposition 3.3.15.

Let ${u}$ be a parametrized regular stable map, and ${u}_{S}$ the stratified-
smooth universal deformation constructed in Theorem 3.3.10 with base
$S\subset\operatorname{Def}({u})$. There exists a collection ${Y}$ compatible
with ${u}$. Furthermore if the evaluation map is $C^{1}$ as in Proposition
3.3.13 then ${\Sigma}^{{Y},{u}}_{S}$ defines an $C^{1}$-immersion of $S$ into
$\operatorname{Def}({\Sigma}^{{Y},{u}})$.

###### Proof.

First we show the existence of a compatible collection. Given a regular stable
map $({\Sigma},{z}=(z_{1},\ldots,z_{n}),{u}:{\Sigma}\to X)$, choose
$Y_{1},\ldots,Y_{k}$ transverse ${u}$ on the unstable components of
${\Sigma}$, so that ${\Sigma}_{1}=({\Sigma},(z_{1},\ldots,z_{n+k}))$ is a
stable curve. Let ${\Sigma}_{S_{1},1}\to S_{1}$ denote a universal deformation
of ${\Sigma}_{1}$. By universality, the family ${\Sigma}_{S}^{{Y},{u}}$ is
induced by a map $\psi:S\to S_{1}$. We successively add marked points until
$\psi$ is an immersion: Suppose that $\psi$ is not an immersion. Then we may
choose an additional marked point $z_{n+k+1}\in{\Sigma}$ such that
${\operatorname{d}}\operatorname{ev}_{n+k+1}$ is non-trivial on
$\operatorname{ker}D\psi$. Since ${u}$ is holomorphic,
${\operatorname{d}}{u}(z_{n+k+1})$ is rank two at $z_{n+k+1}$. Let
$Y_{n+k+1}\subset X$ be a codimension two submanifold containing
${u}(z_{n+k+1})$ such that ${u}$ is transverse to $Y_{n+k+1}$ at $z_{n+k+1}$,
and $Y_{n+k+1}$ is transversal to $\operatorname{ev}_{n+k+1}$ at
${\Sigma},{u}$. Suppose $z_{n+n^{\prime}+1}$ has orbit
$z_{n+k+1},z_{n+k+2},\ldots,z_{n+l}$ under the group
$\operatorname{Aut}({u})$. Repeating the same submanifold for each marking
related by automorphisms gives a collection invariant under the action of
automorphisms. The map $\psi_{1}$ for the new family has property that the
dimension of $\operatorname{ker}(D\psi_{1})$ has dimension at least two less
than that of $\operatorname{ker}(D\psi)$. It follows that the procedure
terminates after adding a finite number of markings. The last claim follows
from the second condition in Definition 3.3.14. ∎

Step 6: Surjectivity

In this section, we show that the family constructed above contains a Gromov
neighborhood of the central fiber. First we show:

######  Proposition 3.3.16.

There exists a constant $\epsilon>0$ such that any stable map
$({\Sigma}_{1},{u}_{1})$ with complex structure on ${\Sigma}_{1}$ given by
$j^{\delta}(\zeta)$ for some
$\zeta\in\operatorname{Def}({\Sigma}^{{\delta}})$, and
${u}_{1}:=\exp_{{u}^{{\delta}}}({\xi})$ with
$\|\zeta\|^{2}+\|\xi\|^{2}_{1,p,{\delta}}<\epsilon$ is of the form in Theorem
3.3.10 for some
$(\xi_{1},\zeta_{1})\in\operatorname{Im}(\tilde{D}_{{u}}^{{{\delta}}})^{*}$.

###### Proof.

Compare with [15, Section 10.7.3]. Let $(\zeta,{\xi})$ be as in the statement
of the Proposition. We claim that
$(\zeta,{\xi})=(\zeta_{0}^{{\delta}},{\xi}_{0}^{{\delta}})+(\zeta_{1},{\xi}_{1})$
for some $(\zeta_{0},{\xi}_{0})\in\operatorname{ker}(\tilde{D}_{{u}})$ and
$(\zeta_{1},{\xi}_{1})\in\operatorname{Im}((D_{{u}}^{{{\delta}}})^{*})$ with
small norm. It then follows by the implicit function theorem that
$(\zeta_{1},{\xi}_{1})$ is the solution given in Theorem 3.3.10. Now
$\zeta=\zeta_{0}^{{\delta}}$ for some
$\zeta_{0}\in\operatorname{Def}_{\Gamma}({\Sigma})$ and gluing parameters
${\delta}$, because $\operatorname{Def}({\Sigma}^{{\delta}})$ is the direct
sum of the image of $\operatorname{Def}_{\Gamma}({\Sigma})$ and
$\mathbb{C}^{m}$. By the gluing theorem for indices (see e.g. [23, Theorem
2.4.1]), the image of $\operatorname{Def}_{\Gamma}({u})$ under the gluing map
projects isomorphically onto $\operatorname{ker}(\tilde{D}_{{u}}^{{\delta}})$
for ${\delta}$ sufficiently small, and so $\operatorname{Def}_{\Gamma}({u})$
is transverse to $\operatorname{Im}(\tilde{D}_{{u}}^{{\delta}})^{*}$, for
${\delta}$ sufficiently small. The claim then follows from the inverse
function theorem. ∎

Given a regular stable ${u}$ with stable domain, consider the family of
$J$-holomorphic maps ${u}_{S}$ produced by Theorem 3.3.10 with parameter space
a neighborhood $S$ of $0$ in $\operatorname{Def}({u}),$ equipped with a
canonical identification $\iota$ of the central fiber with the original map
${u}$. In the case that the domain ${\Sigma}$ is not a stable (marked) curve,
we choose codimension two submanifolds ${Y}=(Y_{1},\ldots,Y_{l})$ meeting
${u}$ transversally so that ${\Sigma}$ with the additional marked points is
stable. Applying this to the family ${u}_{S}$ gives a family of marked stable
maps ${u}_{S}^{{Y}}$ with $n+l$ marked points over a parameter space
$S\subset\operatorname{Def}({u}^{{Y}})$ in the deformation space of the map
with the additional marked points. Now
$\operatorname{Def}({u}^{{Y}})\cong\operatorname{Def}({u})\oplus\bigoplus_{i=1}^{l}T_{z_{i}}{\Sigma}$
includes the deformations of the markings, but these are fixed by requiring
that the additional marked points map to the given collection ${Y}$.
Forgetting the additional marked points gives a family ${u}_{S}$ of stable
maps with $n$ marked points over a neighborhood of $0$ in
$\operatorname{Def}({u})$.

######  Proposition 3.3.17.

$({u}_{S},\iota)$ is a versal stratified-smooth deformation of ${u}$, and in
fact ${u}_{S}$ gives a versal stratified-smooth deformation of any of its
fibers.

###### Proof.

First suppose that ${\Sigma}$ is stable. Let $({u}_{S^{1}}^{1},\iota^{1})$ be
another stratified-smooth deformation of ${u}$ with parameter space $S^{1}$.
Let ${\Sigma}_{S}\to S\subset\operatorname{Def}({\Sigma})$ be a minimal versal
deformation of ${\Sigma}$. The family ${\Sigma}_{S^{1}}^{1}$ is obtained by
pull-back of ${\Sigma}_{S}$ by a stratified-smooth map $\psi:S^{1}\to S$. By
definition the map ${u}_{s}^{1}$ converges to the central fiber in the Gromov
topology as $s$ converges to the base point $0\in S^{1}$. The exponential
decay estimate of [15, Lemma 4.7.3] for holomorphic cylinders of small energy
imply that for $s$ sufficiently close to $0$, ${\Sigma}_{s}^{1},{u}_{s}^{1}$
is given by exponentiation, ${u}_{s}^{1}=\exp_{{u}^{{\delta}}}({\xi})$ for
some ${\xi}\in\Omega^{0}({u}^{{\delta},*}TX)$ with
$\|{\xi}\|_{1,p}<\epsilon_{1}$, for $s$ sufficiently close to $0$. Proposition
3.3.16 produces a stratified-smooth map $\psi:S^{1}\to\operatorname{Def}({u})$
such that ${u}_{S^{1}}^{1}$ is the pull-back of $\psi$. To show that the
deformation $({u}_{S},\iota)$ is universal, let
$\phi_{j}:{\Sigma}_{S^{1}}^{1}\to\psi_{j}^{*}{\Sigma}_{S},j=0,1$ be
isomorphisms of families inducing the identity on the central fiber. The
difference between the two automorphisms is an automorphism of the family
${\Sigma}_{S^{1}}^{1}$ inducing the identity on the central fiber; since the
automorphism group of the central fiber is discrete, the automorphism must be
the identity. In the case that ${\Sigma}$ is not stable, after adding marked
points passing through $Y_{1},\ldots,Y_{l}$, we obtain a family
${u}_{S^{1}}^{1,{Y}}$ of stable maps with $n+l$ marked points. By the case
with stable domain, this family is obtained by pull-back of ${u}_{S}^{{Y}}$ by
some map $S^{1}\to S$. Hence ${u}_{S^{1}}^{1}$ is obtained by pull-back by the
same map. The argument for an arbitrary fiber is similar and left to the
reader. ∎

######  Remark 3.3.18.

In the case that ${\Sigma}$ is unstable, it seems likely that restricting the
family of Theorem 3.3.10 to $\operatorname{Def}({u})$ (that is, the
perpendicular of $\operatorname{aut}({\Sigma})$) also gives a universal
deformation, but we do not know how to prove this. The problem is that in this
case, several different gluing parameters give the same curve, and we do not
have an implicit function theorem for varying gluing parameter.

Step 7: Injectivity

By injectivity, we mean that the family constructed above contains each nearby
stable map exactly once, up to the action of $\operatorname{Aut}(u)$. This is
part of what we called “strongly universal” in Definition 2.2.4.

######  Theorem 3.3.19.

The versal deformations constructed in Step 6 above are strongly universal.

###### Proof.

Let ${u}_{S}$ be a deformation constructed as in Step 6, using the exponential
gluing profile. Let ${\Sigma}_{1,S^{1}}\to S^{1}$ be a family giving a
universal deformation of the curve ${\Sigma}^{{Y},{u}}$ obtained by adding the
additional markings mapping to the given submanifolds. By Definition 3.3.14,
the family ${\Sigma}_{S}^{{Y},{u}}$ induces a map $\phi:S\to S_{1}$ whose
differential is injective in a neighborhood of $0$. By the inverse function
theorem for $C^{1}$ maps, $\phi$ induces a homeomorphism onto its image. In
particular, any two distinct fibers of ${\Sigma}_{S}^{{Y},{u}}$ are non-
isomorphic, and so two fibers of ${\Sigma}_{S}$ are isomorphic if and only if
they are related by a permutation of the markings. After shrinking $S$, this
happens only if the permutation is induced by an automorphism of ${u}$. Given
another family ${u}_{S^{\prime}}^{\prime}:{\Sigma}_{S^{\prime}}\to S^{\prime}$
corresponding to a deformation of a fiber of ${u}_{S}\to S$, by the uniqueness
part of the implicit function theorem, a map $\phi^{\prime}:S^{\prime}\to
S_{1}$ so that ${u}_{S^{\prime}}^{\prime}$ is obtained by pull-back from
${u}_{S}$, and this map is unique by the injectivity just proved. This shows
that ${u}_{S}$ gives a stratified-smooth universal deformation of any of its
fibers, and so is strongly universal. ∎

The Theorem implies that the families in the universal deformations
constructed above define stratified-smooth-compatible charts for the moduli
space $\overline{M}_{g,n}(X,d)$. That is, for any stratum
$M_{g,n,\Gamma}(X,d)$, the restriction of the charts given by the universal
deformation of some map of type $\Gamma$ to $M_{g,n,\Gamma}(X,d)$ are smoothly
compatible.

######  Corollary 3.3.20.

Let $X,J$ be as above. For any $g\geq 0,n\geq 0$, the strongly universal
stratified-smooth deformations of parametrized regular stable maps provide
$\overline{M}_{g,n}^{{\operatorname{reg}}}(X)$ with the structure of a
stratified-smooth topological orbifold.

In order to apply localization one needs to know that the fixed point sets
admit tubular neighborhoods. For this it is helpful to know that
$\overline{M}_{g,n}^{\operatorname{reg}}(X,d)$ admits a $C^{1}$ structure. In
order to obtain compatible charts, we construct the local coordinates
inductively as in Definition 2.2.6, starting with the strata of highest
codimension.

######  Proposition 3.3.21.

Let $X,J$ be as above. For any compatible system of local coordinates near the
nodes, the strongly universal deformations constructed using the exponential
gluing profile equip $\overline{M}_{g,n}^{\operatorname{reg}}(X)$ with the
structure of a $C^{1}$-orbifold.

###### Proof.

We claim that the charts induced by the universal deformations are
$C^{1}$-compatible, assuming they are constructed from the same system of
local coordinates near the nodes. Given two sets of submanifolds
${Y}_{1},{Y}_{2}$, define ${Y}={Y}_{1}\cup{Y}_{2}$. The family
${\Sigma}^{{Y},{u}}$ admits proper étale forgetful maps
${\Sigma}^{{Y},{u}}_{S}\to{\Sigma}_{S}^{{Y}_{j},{u}},\quad j=1,2.$ The fiber
consists of reorderings of the additional marked points induced by the action
of $\operatorname{Aut}({\Sigma},{u})$, and the diagram provided by
${\Sigma}^{{Y},{u}}$ expresses the composition as a smooth $C^{1}$-morphism of
orbifolds. ∎

######  Remark 3.3.22.

Any compact $C^{1}$ orbifold admits a compatible $C^{\infty}$ structure, in
analogy with the situation with manifolds. Indeed, as is well known any
orbifold admits a presentation as the quotient of a manifold (namely its
orthogonal frame bundle) by a locally free group action, and so the orbifold
case follows from the equivariant case proved in Palais [18]. Hence
$\overline{M}^{{\operatorname{reg}}}_{g,n}(X,d)$ if compact admits a (non-
canonical) smooth structure. Presumably the compactness assumption may be
removed but we have not proved that this is so. See however the construction
of smoothly compatible Kuranishi charts in [9, Appendix].

##  4\. Deformations of symplectic vortices

We begin by reviewing the theory of symplectic vortices introduced by Mundet i
Riera [16] and Salamon and collaborators [5]. Let $\Sigma$ be a compact
complex curve, $G$ a compact Lie group, and $\pi:P\to\Sigma$ a smooth
principal $G$-bundle. Given any left $G$-manifold $F$ we have a left action of
$G$ on $P\times F$ given by $g(p,f)=(pg^{-1},gf)$ and we denote by
$P(F)=(P\times F)/G$ the quotient, that is, the associated fiber bundle with
fiber $F$. Let $X$ be a compact Hamiltonian $G$-manifold with symplectic form
$\omega$ and moment map $\Phi:X\to\mathfrak{g}^{*}$. The action of $G$ on $X$
induces an action on $\mathcal{J}(X)$; and we denote by $\mathcal{J}(X)^{G}$
the invariant subspace. Let $\psi:\Sigma\to BG$ be a classifying map for $P$,
so that $P\cong\psi^{*}EG$ and
$P(X)\cong\psi^{*}EG\times_{G}X\cong\psi^{*}X_{G}$ where
$X_{G}=EG\times_{G}X$. Continuous sections $u:\Sigma\to P(X)$ are in one-to-
one correspondence with lifts of $\psi$ to $X_{G}$. The homology class
$\deg(u)$ of the section $u$ is defined to be the homology class $\deg(u)\in
H_{2}^{G}(X,\mathbb{Z})$ of the corresponding lift. Let $\mathcal{A}(P)$ be
the space of smooth connections on $P$, and $P(\mathfrak{g})$ the adjoint
bundle. For any $A\in\mathcal{A}(P)$, let
$F_{A}\in\Omega^{2}(\Sigma,P(\mathfrak{g}))$ the curvature of $A$. Any
connection $A\in\mathcal{A}(P)$ induces a map of spaces of almost complex
structures

$\mathcal{J}(X)^{G}\to\mathcal{J}(P(X)),\ \ J\mapsto J_{A}$

by combining the almost complex structures on $X$ and $\Sigma$ using the
splitting defined by the connection. Let $\Gamma(\Sigma,P(X))$ denote the
space of smooth sections of $P(X)$. Consider the vector bundle

(27)
$\bigcup_{u\in\Gamma(\Sigma,P(X))}\Omega^{0,1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))\to\Gamma(\Sigma,P(X)).$

We denote by $\overline{\partial}_{A}$ the section given by the Cauchy-Riemann
operator defined by $J_{A}$. A gauged map from $\Sigma$ to $X$ is a datum
$(P,A,u)$ where $A\in\mathcal{A}(P)$ and $u:\Sigma\to P(X)$ is a section. A
gauged holomorphic map is a gauged map $(P,A,u)$ such that
$\overline{\partial}_{A}u=0$. Let $\mathcal{H}(P,X)$ be the space of gauged
holomorphic maps with underlying bundle $P$. Let $\mathcal{G}(P)$ denote the
group of gauge transformations

$\mathcal{G}(P)=\\{a:P\to P,a(pg)=a(p)g,\ \ \ \pi\circ a=\pi\\}.$

The Lie algebra of $\mathcal{G}(P)$ is the space of sections
$\Omega^{0}(\Sigma,P(\mathfrak{g}))$ of the adjoint bundle
$P(\mathfrak{g})=P\times_{G}\mathfrak{g}$. We identify
$\mathfrak{g}\to\mathfrak{g}^{*}$, and hence $P(\mathfrak{g})\to
P(\mathfrak{g}^{*})$, using an invariant metric on $\mathfrak{g}$. Let
$P(\Phi):P(X)\to P(\mathfrak{g})$ denote the map induced by the equivariant
map $\Phi:X\to\mathfrak{g}$.

######  Definition 4.0.1.

A gauged holomorphic map $(A,u)\in\mathcal{H}(P,X)$ is a symplectic vortex (or
vortex for short) if it satisfies

$F_{A}+\operatorname{Vol}_{\Sigma}u^{*}P(\Phi)=0.$

An $n$-marked symplectic vortex is a vortex $(A,u)$ together with $n$-tuple
${z}=(z_{1},\ldots,z_{n})$ of distinct points on $\Sigma$. A marked vortex
$(A,u,{z})$ is stable if it has finite automorphism group.

The equation in the definition can be interpreted as the zero level set
condition for a formal moment map for the action of the group of gauge
transformations, see [16], [5]. The energy of a gauged holomorphic map $(A,u)$
is given by

$E(A,u)={\frac{1}{2}}\int_{\Sigma}\left(|{\operatorname{d}}_{A}u|^{2}+|F_{A}|^{2}+|u^{*}P(\Phi)|^{2}\right)\operatorname{Vol}_{\Sigma}.$

The _equivariant symplectic area_ of a pair $(A,u)$ is the pairing of the
homology class $\deg(u)$ with the class $[\omega_{G}=\omega+\Phi]\in
H^{2}(X_{G})$,

$D(A,u)=(\deg(u),[\omega_{G}])=([\Sigma],u^{*}[\omega_{G}]).$

######  Lemma 4.0.2.

Suppose $\operatorname{Vol}_{\Sigma}$ is the Kähler form for the metric on
$\Sigma$. The energy and equivariant area are related by

(28)
$E(A,u)=D(A,u)+\int_{\Sigma}\left(|\overline{\partial}_{A}u|^{2}+{\frac{1}{2}}|F_{A}+\operatorname{Vol}_{\Sigma}u^{*}P(\Phi)|^{2}\right)\operatorname{Vol}_{\Sigma}.$

###### Proof.

See [4, Proposition 2.2]. ∎

In particular, for any symplectic vortex the energy and action are equal. Let
$M(P,X)$ denote the moduli space of vortices

$M(P,X):=\mathcal{H}(P,X)/\kern-3.01385pt/\mathcal{G}(P)=\\{F_{A}+\operatorname{Vol}_{\Sigma}u^{*}P(\Phi)=0\\}/\mathcal{G}(P).$

Let $M_{n}(P,X)$ denote the moduli space of $n$-marked vortices, up to gauge
transformation, and $M_{n}(\Sigma,X)=\bigcup_{P\to\Sigma}M_{n}(P,X)$ the union
over types of bundles $P$. Clearly, $M_{n}(\Sigma,X)$ is homeomorphic to the
product $M(\Sigma,X)\times M_{n}(\Sigma)$ where $M(\Sigma,X):=M_{0}(\Sigma)$
and $M_{n}(\Sigma)$ denotes the configuration space of $n$-tuples of distinct
points on $\Sigma$.

We wish to study families and deformations of symplectic vortices. For
families with smooth domain, the definitions are straightforward:

######  Definition 4.0.3.

A smooth family of vortices on a principal $G$-bundle $P$ on $\Sigma$ over a
parameter space $S$ consists of a family of connections depending smoothly on
$s\in S$, that is, a smooth map $A_{S}:S\times P\to T^{*}P\otimes\mathfrak{g}$
on $P$ such that the restriction $A_{s}$ of $A_{S}$ to any $\\{s\\}\times P$
is a connection, together with a smooth family of (pseudo)holomorphic sections
$u_{S}=(u_{s})_{s\in S}$, such that each pair $(A_{s},u_{s}),s\in S$ is a
symplectic vortex. A deformation of $(A,u)$ is a germ of a smooth family
$(A_{S},u_{S})$ together with an isomorphism (gauge transformation) relating
$(A_{0},u_{0})$ with $(A,u)$. A deformation is universal if it satisfies the
condition as in Definition 3.1.4, and strongly universal if it satisfies the
conditions in Definition 2.2.4.

We define a linearized operator associated to a vortex as follows. Define

(29)
${\operatorname{d}}_{A,u}:\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\Sigma,u^{*}TP(X))\to\Omega^{2}(\Sigma,P(\mathfrak{g}))$
${\operatorname{d}}_{A,u}(a,\xi):={\operatorname{d}}_{A}a+\operatorname{Vol}_{\Sigma}u^{*}L_{\xi}P(\Phi).$

Here $L_{\xi}P(\Phi)$ denotes the derivative of $P(\Phi)$ with respect to the
vector field generated by $\xi$, and $u^{*}L_{\xi}P(\Phi)$ its evaluation at
$u$. Define an operator

(30)
${\operatorname{d}}_{A,u}^{*}:\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\Sigma,u^{*}TP(X))\to\Omega^{0}(\Sigma,P(\mathfrak{g}))$
${\operatorname{d}}_{A,u}^{*}(a,\xi)={\operatorname{d}}_{A}^{*}a+u^{*}L_{J\xi}P(\Phi).$

(This is not the adjoint of operator in (32), but rather defined by analogy
with the case $X$ trivial.) It is shown in [5, Section 4] that if $(A,u)$ is
stable then the set

$W_{A,u}=\\{(A+a,\exp_{u}(\xi)),(a,\xi)\in\operatorname{ker}{\operatorname{d}}_{A,u}^{*}\\}$

is a slice for the gauge group action near $(A,u)$. Define

(31)
$\mathcal{F}_{A,u}:\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))\\\
\to(\Omega^{0}\oplus\Omega^{2})(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0,1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))\\\
(a,\xi)\mapsto\left(F_{A+a}+\operatorname{Vol}_{\Sigma}\exp_{u}(\xi)^{*}P(\Phi),{\operatorname{d}}_{A,u}^{*}(a,\xi),\Psi_{u}(\xi)^{-1}\overline{\partial}_{A+a}\exp_{u}(\xi)\right).$

Let

$\Omega^{1}(\Sigma,P(\mathfrak{g}))\to\Omega^{1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X)),\
\ \ a\mapsto a_{X}$

denote the map induced by the infinitesimal action. The linearization of the
last component (31) is

$D_{A,u}(a,\xi)=(\nabla_{A}\xi)^{0,1}+{\frac{1}{2}}J_{u}(\nabla_{\xi}J)_{u}\partial_{A}u+a_{X}^{0,1}.$

Here $0,1$ denotes projection on the $0,1$-component. The linearized operator
for a vortex $(A,u)$ is the operator

(32)
$\tilde{D}_{A,u}=({\operatorname{d}}_{A,u},{\operatorname{d}}_{A,u}^{*},D_{A,u}):\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))\\\
\to(\Omega^{0}\oplus\Omega^{2})(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0,1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))$

A vortex $(A,u)$ is regular if the operator $\tilde{D}_{A,u}$ is surjective. A
marked vortex $(A,u,{z})$ is regular if the underlying unmarked vortex is
regular. The space of infinitesimal deformations of $(A,u)$ is
$\operatorname{Def}(A,u):=\operatorname{ker}(\tilde{D}_{A,u}).$

######  Theorem 4.0.4.

Any regular vortex with smooth domain $(A,u)$ has a strongly universal smooth
deformation if and only if it is stable.

###### Proof.

Give the spaces of connections and sections the structure of Banach manifolds
by taking completions with respect to Sobolev norms $1,p$ for $1$-forms, and
$0,p$ for $0$ and $2$-forms. For $p>2$, the map $\mathcal{F}_{A,u}$ is a
smooth map of Banach spaces.

(33)
$\mathcal{F}_{A,u}:\Omega^{1}(\Sigma,P(\mathfrak{g}))_{1,p}\oplus\Omega^{0}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))_{1,p}\\\
\to(\Omega^{0}\oplus\Omega^{2})(\Sigma,P(\mathfrak{g}))_{0,p}\oplus\Omega^{0,1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))_{0,p}$

equivariant for the action of the group $\mathcal{G}(P)_{2,p}$ of gauge
transformations of class $2,p$. Suppose that $(A,u)$ is regular and stable. By
the implicit function theorem, there is a local homeomorphism

$\operatorname{ker}(\tilde{D}_{A,u})\to\left\\{\begin{array}[]{c}F_{A+a}+\operatorname{Vol}_{\Sigma}(\exp_{u}(\xi))^{*}P(\Phi)=0\\\
\overline{\partial}_{A+a}(\exp_{u}(\xi))=0\\\ \
{\operatorname{d}}_{A,u}^{*}(a,\xi)=0\end{array}\right\\}.$

This gives rise to a family $(A_{S},u_{S})\to S$ over a neighborhood $S$ of
$0$ in $\operatorname{ker}(\tilde{D}_{A,u})$. By [5, Theorem 3.1],
$(A_{S},u_{S})$ is a smooth family, assuming $(A,u)$ is smooth. Given any
other family $(A_{S^{\prime}}^{\prime},u_{S^{\prime}}^{\prime})\to S^{\prime}$
of stable vortices with $(A^{\prime}_{0},u_{0}^{\prime})=(A,u)$, the implicit
function theorem provides a smooth map $S^{\prime}\to S$ so that
$(A^{\prime}_{S^{\prime}},u^{\prime}_{S^{\prime}})$ is obtained from $(A,u)$
by pull-back. The first property of the universal deformation is a consequence
of the slice condition; the second property follows from the fact that the
projection
$\operatorname{ker}(\tilde{D}_{A,u})\to\operatorname{ker}(\tilde{D}_{A_{s},u_{s}})$
is an isomorphism for sufficiently small $s$. ∎

Let $M^{\operatorname{reg}}_{n}(\Sigma,X)$ denote the moduli space of regular,
stable $n$-marked symplectic vortices from $\Sigma$ to $X$. We denote by
$(c_{1}^{G}(TX),d)$ the pairing of $d$ with the first Chern class
$c_{1}^{G}(P(TX)\to P(X))$

######  Theorem 4.0.5.

Let $\Sigma,X,J$ be as above. $M^{\operatorname{reg}}_{n}(\Sigma,X)$ has the
structure of a smooth orbifold with tangent space at $[A,u]$ isomorphic to
$\operatorname{Def}(A,u)$, and dimension of the component of homology class
$d\in H_{2}^{G}(X)$ is given by

$\dim(M^{\operatorname{reg}}_{n}(\Sigma,X,d))=(1-g)(\dim(X)-2\dim(G))+2((c_{1}^{G}(TX),d)+n).$

###### Proof.

Charts for $M^{{\operatorname{reg}}}_{n}(\Sigma,X)$ are provided by the
strongly universal deformations. The dimension of the tangent space at $[A,u]$
is given by the index of the linearized operator $\tilde{D}_{A,u}$, which
deforms via Fredholm operators to the sum of the operator
${\operatorname{d}}_{A}\oplus{\operatorname{d}}_{A}^{*}$ for the connection,
which has index $2\dim(G)(g-1)$, and the linearized Cauchy-Riemann operator on
the nodal curve, which has index $(1-g)\dim(X)+2n+2(c_{1}^{G}(TX),d)$ by
Riemann-Roch, if $(A,u)$ has equivariant homology class $d$ (which determines
the first Chern class of $P$ by projection.) ∎

###  4.1. Polystable vortices

The moduli space of symplectic vortices admits a compactification which allows
bubbling of the section in the fibers.

######  Definition 4.1.1.

A nodal gauged marked holomorphic map from $\Sigma$ to $X$ consists of a datum
$(\hat{\Sigma},P,A,{u},{z})$ where $P\to\Sigma$ is a principal $G$-bundle,
$A\in A(P)$ is a connection, $\hat{\Sigma}$ is a marked nodal curve,
$v:\hat{\Sigma}\to\Sigma$ is a holomorphic map of degree $[\Sigma]$, and
${u}:\hat{\Sigma}\to P(X)$ is a $J_{A}$-holomorphic map from a nodal curve
$\hat{\Sigma}$ such that $\pi\circ{u}$ has class $[\Sigma]$. In other words,

  1. (a)

$\hat{\Sigma}$ is a connected nodal complex curve consisting of a principal
component $\Sigma_{0}$ equipped with an isomorphism with $\Sigma$ together
with a number of projective lines $\Sigma_{1},\ldots,\Sigma_{k}$. We denote by
$w_{1}^{\pm},\ldots,w_{m}^{\pm}$ the nodes. For each $i=1,\ldots,k$, we denote
by $w_{i}^{0}\in\Sigma_{0}$ the attaching point to the principal component.

  2. (b)

$(A,u)\in\mathcal{H}(P,X)$ is a gauged holomorphic map from $\Sigma$ to $X$;

  3. (c)

for each non-principal component $\Sigma_{i}$, a holomorphic map
$u_{i}:\Sigma_{i}\to P(X)_{w_{i}^{0}}$;

  4. (d)

${z}=(z_{1},\ldots,z_{n})\in\hat{\Sigma}$ are distinct, smooth points of
$\hat{\Sigma}$.

Let $\mathcal{H}(\hat{\Sigma},P(X))$ denote the space of nodal gauged
holomorphic sections with domain $\hat{\Sigma}$ and bundle $P$. The group of
gauge transformations $\mathcal{G}(P)$ acts on
$\mathcal{H}(\hat{\Sigma},P(X))$ by $g(A,{u})=(g^{*}A,g\circ{u}).$ The
generating vector field for $\zeta\in\Omega^{0}(\Sigma,P(\mathfrak{g}))$
acting on $\mathcal{H}(\hat{\Sigma},P(X))$ at $(\hat{\Sigma},A,{u})$ is the
tuple given by

(34)
$\zeta_{\mathcal{H}(\hat{\Sigma},P(X))}(\hat{\Sigma},A,{u})=({\operatorname{d}}_{A}\zeta,u_{0}^{*}P(\zeta_{X}),(u_{i}^{*}P(\zeta_{X}(w_{i}^{0})))_{i=1}^{k})$

in
$\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\hat{\Sigma},{u}^{*}T^{\operatorname{vert}}P(X))$.
Here $P(\zeta_{X})\in\Omega^{0}(\Sigma,P(\operatorname{Vect}(X)))$ is the
fiber-wise vector field generated by $\zeta$ and $u_{0}^{*}P(\zeta)\in
T^{\operatorname{vert}}P(X)$ is the evaluation at $u_{0}$. Similarly for the
bubble components $u_{1},\ldots,u_{k}$ in the fibers
$P(X)_{w^{0}_{1}},\ldots,P(X)_{w^{0}_{k}}$. A slice is given by taking the
perpendicular to the tangent spaces to the $\mathcal{G}(P)$-orbits. We will
assume for simplicity that the stabilizer of the $\mathcal{G}(P)$ action on
the principal component is finite, so that a slice is given locally by the
kernel of ${\operatorname{d}}_{A,u_{0}}^{*}$, that is, the Coulomb gauge
condition on the principal component. The implicit function theorem shows that
any nearby pair $(A_{1},{u}_{1})$ is complex gauge equivalent to a pair of the
form $(A+a,\exp_{{u}}({v}))$ with
$(a,{v})\in\operatorname{ker}{\operatorname{d}}_{A,u_{0}}^{*}$.

######  Definition 4.1.2.

A nodal vortex is a stable nodal gauged holomorphic map such that the
principal component is an vortex. A nodal vortex $(\hat{\Sigma},A,{u},{z})$ is
polystable if each sphere bubble $\Sigma_{i}$ on which $u_{i}$ is constant has
at least three marked or singular points, and stable if it has finite
automorphism group. An isomorphism of nodal vortices
$(\hat{\Sigma},A,{u},{z}),(\hat{\Sigma}^{\prime},A^{\prime},{u}^{\prime},{z}^{\prime})$
consists of an automorphism of the domain, acting trivially on the principal
component, and a corresponding automorphism of the principal bundle mapping
$(A,{u})$ to $(A^{\prime},{u}^{\prime})$ and mapping the markings ${z}$ to
${z}^{\prime}$. For any nodal section ${u}:\hat{\Sigma}\to P(X)$, the homology
class of ${u}$ is defined as the sum of the homology class $d_{0}\in
H_{2}^{G}(X,\mathbb{Z})$ of the principal component $u_{0}$ and the homology
classes $d_{i}\in H_{2}(X,\mathbb{Z}),i=1,\ldots,k$ of the sphere bubbles,
using the inclusion $H_{2}(X,\mathbb{Z})\to H_{2}^{G}(X,\mathbb{Z})$ given by
equivariant formality. The combinatorial type $\Gamma(\hat{\Sigma},A,{u},{z})$
of a gauged nodal map is a rooted graph whose vertices represent the
components of $\hat{\Sigma}$, whose finite edges represent the nodes, semi-
infinite edges represent the markings, and whose root vertex represents the
principal component.

Note that there is no condition for points on the principal component. In
particular, nodal gauged holomorphic maps with no markings can be polystable.
The term polystable is borrowed from the vector bundle case. In that
situation, a bundle is stable if it is flat and has only central automorphisms
and polystable if it is a direct sum of stable bundles of the same slope. Any
flat bundle is automatically polystable; a bundle is semistable if it is grade
equivalent to a polystable bundle. In particular, the moduli space of stable
bundles is definitely not compact, and we feel that the vortex terminology
should include this fact as a special case.

From now on, we fix the bundle $P$.

######  Definition 4.1.3.

Let $X$ as above. A smooth family of fixed type of nodal vortices to $X$
consists of a smooth family $\hat{Sigma}_{S}\to S$ of nodal curves of fixed
type, a smooth family of holomorphic maps $v_{S}:\hat{\Sigma}_{S}\to\Sigma$ of
class $[\Sigma]$, a smooth family ${u}_{S}:\hat{\Sigma}_{S}\to P(X)$ of maps,
and a smooth family $A_{S}:S\times P\to T^{*}P$ of connections over $S$. A
smooth deformation of a nodal vortex $(A,\hat{\Sigma},{u},{z})$ of fixed type
consists of a germ of a smooth family
$(A_{S},\hat{\Sigma}_{S},{u}_{S},{z}_{S})$ of nodal vortices of fixed type
together with an identification $\iota$ of of the central fiber with
$(A,\hat{\Sigma},{u},{z})$. A stratified-smooth family of marked nodal
symplectic vortices is a datum $(\hat{\Sigma}_{S},A_{S},{u}_{S},{z}_{S})$
consisting of a stratified-smooth family $\hat{\Sigma}_{S}\to S$ of nodal
curves, a stratified-smooth family of holomorphic maps
$v:\hat{\Sigma}_{S}\to\Sigma$ of class $[\Sigma]$, a stratified-smooth family
$A_{S}$ of connections on $P$, a stratified-smooth family of maps
${u}_{S}:\hat{\Sigma}_{S}\to P(X)$; such that each triple
$(\hat{\Sigma}_{s},A_{s},{u}_{s},{z}_{s})$ is a marked nodal symplectic
vortex. A family of polystable symplectic vortices is a family of marked nodal
symplectic vortices such that any fiber is polystable.

A smooth vector bundle ${E}\to\hat{\Sigma}$ is a collection of smooth vector
bundles $E_{i}$ over the components $\Sigma_{i}$ of $\hat{\Sigma}$, equipped
with identifications of the fibers at nodal points $E_{i^{+}(w^{+}_{k})}\to
E_{i^{-}(w^{-}_{k})},k=1,\ldots,m$. We denote by $\Omega(\hat{\Sigma},{E})$
the sum over components,
$\Omega(\hat{\Sigma},{E})=\bigoplus_{i=1}^{k}\Omega(\Sigma_{i},E_{i})$ where
$E_{i}={E}|\Sigma_{i}.$

######  Definition 4.1.4.

For a polystable vortex $(\hat{\Sigma},A,{u})$, let $\tilde{D}_{A,{u}}$ denote
the linearized operator

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$\Omega^{1}({\Sigma},P(\mathfrak{g}))\oplus\Omega^{0}(\hat{\Sigma},{u}^{*}T^{\operatorname{vert}}P(X))\\\
\to(\Omega^{0}\oplus\Omega^{2})({\Sigma},P(\mathfrak{g}))\oplus\Omega^{0,1}(\hat{\Sigma},{u}^{*}T^{\operatorname{vert}}P(X))\oplus\bigoplus_{k=1}^{m}T_{{u}(w_{k}^{\pm})}^{\operatorname{vert}}P(X)$

given by the linearized vortex operator
$({\operatorname{d}}_{A,u_{0}},D_{A,u_{0}})$ on the principal component, the
linearized Cauchy-Riemann operator $\tilde{D}_{u_{i}}$ on the bubbles, the
slice operator ${\operatorname{d}}_{A,u_{0}}^{*}$, and the difference operator
on the fibers over the nodes

$\Omega^{0}(\hat{\Sigma},{u}^{*}T^{\operatorname{vert}}P(X))\to\bigoplus_{i=1}^{m}T_{{u}(w_{i}^{\pm})}^{\operatorname{vert}}P(X),\quad{\xi}\mapsto({\xi}(w_{i}^{+})-{\xi}(w_{i}^{-}))_{i=1}^{m}.$

$(A,{u})$ is regular if $\tilde{D}_{A,{u}}$ is surjective. The space of
infinitesimal deformations of $A,{u}$ of fixed type is
$\operatorname{Def}_{\Gamma}(A,{u}):=\operatorname{ker}\tilde{D}_{A,{u}}/\operatorname{aut}(\hat{\Sigma})$
where $\operatorname{aut}(\hat{\Sigma})$ denotes the group of infinitesimal
automorphisms acting trivially on the principal component. The space of
infinitesimal deformations of $A,{u}$ is

$\operatorname{Def}(A,{u}):=\operatorname{Def}_{\Gamma}(A,{u})\oplus\bigoplus_{i=1}^{m}T_{w_{j}^{+}}\hat{\Sigma}\otimes
T_{w_{j}^{-}}\hat{\Sigma}$

consisting of a deformation of fixed type together with a collection of gluing
parameters.

###  4.2. Constructing deformations of symplectic vortices

First we construct deformations of fixed type.

######  Theorem 4.2.1.

A regular polystable vortex has a strongly universal smooth deformation of
fixed type if and only if it is stable.

The proof is by the implicit function theorem applied to the map

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$\mathcal{F}_{A,{u}}(a,{\xi})=(F_{A+a}+\operatorname{Vol}_{\Sigma}(\exp_{u_{0}}(\xi_{0}))^{*}P(\Phi),{\operatorname{d}}_{A,u_{0}}^{*}(a,{\xi}),\\\
\Psi_{u_{0}}(\xi_{0})^{-1}\overline{\partial}_{A}u_{0},(\Psi_{u_{i}}(\xi_{i})^{-1}\overline{\partial}u_{i})_{i=1}^{k},({\xi}(w_{i}^{+})-{\xi}(w_{i}^{-}))_{i=1}^{m})$

whose linearization is $\tilde{D}_{A,{u}}$. The proof is left to the reader.
We denote by $M_{n,\Gamma}(\Sigma,X,d)$ of the moduli space of isomorphism
classes of polystable vortices of combinatorial type $\Gamma$ of homology
class $d\in H_{2}^{G}(X,\mathbb{Z})$, and
$M^{{\operatorname{reg}}}_{n,\Gamma}(\Sigma,X,d)$ the regular locus.

######  Corollary 4.2.2.

$M_{n,\Gamma}^{\operatorname{reg}}(\Sigma,X,d)$ has the structure of a smooth
orbifold of dimension given, if $\Sigma$ is connected, by
$(1-g)(\dim(X)-2\dim(G))+2(n+(c_{1}^{G}(TX),d)-m)$ where $m$ is the number of
nodes.

We now prove that a regular stable symplectic vortex from $\Sigma$ to $X$
admits a strongly universal stratified-smooth deformation if it is strongly
stable, that is, Theorem 1.0.2. We explain the construction for a single
bubble only, so that $\hat{\Sigma}$ is the union of a principal component
$\Sigma_{+}=\Sigma$ and a holomorphic sphere $\Sigma_{-}$, attached by a
single pair $w_{\pm}$ of nodes. We denote by $(A,u_{+})$ the restriction to
the principal component and by $u_{-}$ the bubble, so that
$x:=u_{+}(w_{+})=u_{-}(w_{-})$ and ${u}=(u_{+},u_{-})$. We choose a local
coordinate near $w$, equivariant for the action of the automorphism group
$\operatorname{Aut}(A,{u})$ in the sense that $\operatorname{Aut}(A,{u})$ acts
on the local coordinate by multiplication by roots of unity. The construction
depends on the following choices:

######  Definition 4.2.3.

A gluing datum for $(\hat{\Sigma},A,{u})$ consists of

  1. (a)

neighborhoods $U_{\pm}$ of the nodes $w_{\pm}$;

  2. (b)

local coordinates ${\kappa}=(\kappa_{+},\kappa_{-})$ on $U_{\pm}$;

  3. (c)

a trivialization $P|_{U_{0}}\to G\times U_{0}$ on $U_{0}$;

  4. (d)

a gluing parameter $\delta$;

  5. (e)

an annulus parameter $\rho$ ;

  6. (f)

a cutoff function $\alpha$ as in (10).

Step 1: Approximate Solution

Given a nodal vortex $(A,u)$ as above and a gluing datum we wish to define an
approximate solution to the vortex equations $(A,u^{\delta})$. Let
$\exp_{x}:T_{x}X\to X$ denote the exponential map defined by the metric on
$X$. Define sections

$\xi_{\pm}:U_{\pm}\to T_{x}X,\quad u(z)=\exp_{x}(\xi_{\pm}(z)).$

Let $\hat{\Sigma}^{\delta}$ denote the surface obtained by gluing; since the
bubble is genus zero, this surface is isomorphic to $\Sigma$ but not
canonically. Define the pre-glued section
${u}^{\delta}:\hat{\Sigma}^{\delta}\to P(X)$,

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${u}^{\delta}(z)=\exp_{x}\left(\alpha(|\kappa_{+}(z)|/\rho|\delta|^{1/2})(\xi_{+}(z)-\xi_{\pm}(w^{\pm}))\right.+\\\
\left.\alpha(|\kappa_{-}(z)|/\rho|\delta|^{1/2})(\xi_{-}(z)-\xi_{\pm}(w^{\pm}))+\xi_{\pm}(w^{\pm})\right)$

for $|\kappa_{\pm}(z)|\leq 2|\delta|^{1/2}\rho^{2}$; elsewhere let
${u}^{\delta}(z)={u}(z)$, using the identification of $\Sigma$ with
$\hat{\Sigma}$ away from the gluing region. We do not modify $A$ in the bubble
region; this is because after re-scaling the connection on the bubble is
already close to the trivial connection.

The pair $(A,{u}^{\delta})$ is an approximate solution to the vortex equations
in the following Sobolev norms. Let
$\Omega(\Sigma^{\delta},P^{\delta}(\mathfrak{g}))_{k,p}$ denote the
$W^{k,p}$-completion using the standard metric. Let $g^{\delta}$ denote the
$C^{0}$ metric on the glued surface in (12). Let
$\Omega^{0,1}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{k,p,\delta}$.
Let

$\mathcal{H}_{\delta}:=\Omega^{2}(\Sigma,P(\mathfrak{g}))_{{0,3}}\oplus\Omega^{0}(\Sigma,P(\mathfrak{g}))_{0,3}\oplus\Omega^{0,1}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{{0,3},\delta}$

with norm

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$\|(\phi,\psi,\eta)\|_{{\delta}}^{2}=\|\phi\|_{{0,3}}^{2}+\|\psi\|_{0,3}^{2}+\|\eta\|_{0,3,\delta}^{2}.$

Let

$\mathcal{I}_{\delta}:=\Omega^{1}(\Sigma,P(\mathfrak{g}))_{{1,3}}\oplus\Omega^{0}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta}$

with norm

$\|(a,\xi)\|_{{\delta}}^{2}=\|a\|_{{1,3}}^{2}+\|\eta\|_{{1,3},\delta}^{2}.$

Locally the moduli space of polystable vortices is in bijection with the zero
set of the map

(39)
$\mathcal{F}_{A,{u}}^{{\delta}}:\mathcal{I}_{{\delta}}\to\mathcal{H}_{{\delta}}\\\
(a,\xi)\mapsto\left(F_{A+a}+\operatorname{Vol}_{\Sigma}\exp_{{u}^{{\delta}}}(\xi)^{*}P(\Phi),{\operatorname{d}}_{A,u_{+}}^{*}(a,\xi),\Psi_{{u}^{{\delta}}}(\xi)^{-1}\overline{\partial}_{A+a}\exp_{{u}^{{\delta}}}(\xi)\right).$

Here the second map enforces a slice condition. That
$\mathcal{F}_{A,{u}}^{{\delta}}$ is well-defined follows from Sobolev
embedding: In particular, there is a $\delta$-uniform embedding

(40)
$\Omega^{0}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta}\to\Omega^{0}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{0,\infty},$

since the dimensions of the cone in the cone condition [2, Chapter 4] are
uniformly bounded in $\delta$, and the metric uniformly comparable to the flat
metric.

######  Lemma 4.2.4.

Let $(A,{u})$ be a symplectic vortex on a nodal curve with a single node
$w=(w^{+},w^{-})$. There exist constants $c_{0},c_{1}>0$ such that if
$|\delta|<c_{1},\rho>1/c_{1}$ and$|\delta|\rho^{4}<c_{1}$ then the pair
$(A,{u}^{\delta})\in\mathcal{A}(P)\times\Gamma(P(X))$ satisfies

(41) $\|\mathcal{F}_{A,{u}}^{{\delta}}(0,0)\|\leq
c_{0}|\delta|^{1/3}\rho^{2/3}.$

###### Proof.

The expression $\overline{\partial}_{A}{u}^{\delta}$ can be expressed as a sum
of terms involving derivatives of the cutoff function $\alpha$, terms
involving derivatives of $\xi_{j}$, and terms involving the connection $A$ on
the bubble region. The derivative of $\alpha$ is bounded by
$C/\rho|\delta|^{1/2}$, while the norm of $\xi_{j}$ is bounded by
$C\rho|\delta|^{1/2}$ on the gluing region. Hence the term involving the
derivative of $\alpha$ is bounded and supported on a region of area less than
$C|\delta|\rho^{2}$. In the given trivialization we have

$\overline{\partial}_{A}u=\overline{\partial}u+A_{X}^{0,1}(u)$

where $A^{0,1}$ is the $0,1$-form defined by
$A\in\Omega^{1}(B_{R},\mathfrak{g})$ is the connection $1$-form in the local
trivialization and $A_{X}^{0,1}(u)$ is the corresponding form with values in
$T^{\operatorname{vert}}P(X)\otimes_{\mathbb{R}}\mathbb{C}$. We have

$\|A_{X}^{0,1}(u)\|_{0,3,\delta}\leq\|A_{X}^{0,1}(u)\|_{0,3}$

since $p\geq 2$; for $p=2$ the $W^{0,3,\delta}$ and $W^{0,3}$ norms are the
same, by conformal invariance; for $p>2$ the $0,3$-norm is strictly greater.
Hence

$\|\overline{\partial}_{A}u^{\delta}\|_{0,3,\delta}\leq
C\max(|\delta|^{1/3}\rho^{2/3},|\delta|).$

The moment map term
$F_{A}+({u}^{\delta})^{*}P(\Phi)\operatorname{Vol}_{\Sigma}$ vanishes except
on $|\kappa_{+}|\leq\rho|\delta^{1/2}|$, where it is uniformly bounded. Hence
for $\delta$ small

$\|F_{A}+({u}^{\delta})^{*}P(\Phi)\operatorname{Vol}_{\Sigma}\|_{0,3}\leq
C\rho^{2/3}|\delta|^{1/3}.$

The statement of the lemma follows. ∎

We also wish to perform the gluing construction in families, that is, for each
nearby vortex and gluing parameter we wish to find a solution to the vortex
equations on $\Sigma^{\delta}$. Define

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$\mathcal{F}_{A,{u}}^{D,\delta}:\operatorname{Def}_{\Gamma}(A,{u})\times\mathcal{I}_{{\delta}}\to\mathcal{H}_{{\delta}},\quad(a,{\xi},a_{1},\xi_{1})\mapsto\\\
\left(F_{A}+a_{0}+a_{1}+\operatorname{Vol}_{\Sigma}\exp_{{u}^{{\delta}}}({\xi}_{0}^{\delta}+\xi_{1})^{*}P(\Phi),{\operatorname{d}}_{A,{u}^{\delta}}^{*}(a_{0}+a_{1},{\xi}_{0}^{\delta}+\xi),\right.\\\
\left.\Psi_{{u}^{{\delta}}}({\xi}_{0}^{\delta}+\xi_{1})^{-1}\overline{\partial}_{A+a_{0}+a_{1}}\exp_{{u}^{{\delta}}}({\xi}_{0}^{\delta}+\xi_{1})\right)$

The following is proved in the same way as Lemma 4.2.4 and left to the reader:

######  Lemma 4.2.5.

Let $(A,{u})$ be as above. There exist constants $c_{0},c_{1}>0$ such that if
$|\delta|<c_{1},\rho>1/c_{1}$, $\|(a,\xi)\|\leq c_{1}$ and
$|\delta|\rho^{4}<c_{1}$ then

(43) $\|\mathcal{F}_{A,{u}}^{D,{\delta}}(a,\xi,0,0)\|\leq
c_{0}|\delta|^{1/3}\rho^{2/3}.$

Step 2: Uniformly bounded right inverse

In preparation for the construction of the uniformly bounded right inverse of
$\tilde{D}_{\delta}$ we define the intermediate family $(A,{u}_{0}^{\delta})$
of gauged holomorphic maps on the nodal curve $\hat{\Sigma}$ is the family
defined by the equations (4.2.4), using the identification of $\hat{\Sigma}$
and $\hat{\Sigma}^{\delta}$ away from the gluing region. Thus $u_{0}^{\delta}$
is constant in a neighborhood of the node $w^{\pm}$. We identify
$({u}_{0}^{\delta})^{*}T^{\operatorname{vert}}P(X)$ with
${u}^{*}T^{\operatorname{vert}}P(X)$ by geodesic parallel transport.

######  Lemma 4.2.6.

The operator $\tilde{D}_{A,{u}_{0}^{\delta}}$ converges in the operator norm
to $\tilde{D}_{A,{u}}$ as $\delta\to 0,\rho\to\infty,\delta\rho^{4}\to 0$.

###### Proof.

The section ${u}_{0}^{\delta}$ converges in the $W^{1,3}$ norm to $u$ as
$\rho^{2}|\delta|^{1/2}\to 0$. It follows that the operator
$\xi\mapsto\operatorname{Vol}_{\Sigma}(u^{\delta}_{0})^{*}L_{\xi}P(\Phi)$
converges to $\xi\mapsto\operatorname{Vol}_{\Sigma}u^{*}L_{\xi}P(\Phi)$. Hence
${\operatorname{d}}_{A,u_{0}^{\delta},\epsilon}$ converges to
${\operatorname{d}}_{A,u,\epsilon}$, and similarly for
${\operatorname{d}}_{A,u_{0},\epsilon}^{*}$. The operator
$D_{A,{u}_{0}^{\delta}}$ converges to $D_{A,{u}}$, as in Lemma 3.3.6. ∎

######  Proposition 4.2.7.

Let $(A,{u})$ be a nodal vortex, and $(A,{u}^{\delta})$ the approximate
solution constructed above. There exist constants $c,C>0$ such that if
$|\delta|<c$ then there exists an approximate right inverse $T_{\delta}$ of
the parametrized linear operator
$\tilde{D}_{\delta}:=\tilde{D}_{A,{u}^{\delta}}$ that is, a map
$T_{\delta}:\mathcal{H}_{\delta}\to\mathcal{I}_{\delta}$ such that

$\|(\tilde{D}_{\delta}T_{\delta}-I)\eta\|_{\delta}\leq{\frac{1}{2}}\|\eta\|_{\delta},\quad\|T_{{\delta}}\eta\|_{\delta}\leq
C\|\eta\|_{\delta}.$

Given such an approximate inverse, we obtain a uniformly bounded right inverse
$Q_{\delta}$ to $\tilde{D}_{\delta}$ by the formula

$Q_{\delta}=T_{\delta}(\tilde{D}_{\delta}T_{\delta})^{-1}=\sum_{k\geq
0}T_{\delta}(\tilde{D}_{\delta}T_{\delta}-I)^{k}.$

###### Proof of 4.2.7.

By the regularity assumption, $\tilde{D}_{A,{u}}$ is surjective when
restricted to the space of vectors $(a,{\xi})$ such that
$\xi_{0}(0)=\xi_{1}(\infty)$. By Lemma 4.2.6,
$\tilde{D}^{0}_{\delta}:=\tilde{D}_{A,{u}_{0}^{\delta}}$ is surjective for
sufficiently small $\rho,\delta$, when restricted to the same space. The
approximate right inverse is constructed by composing a cutoff operator
$K_{\delta}$, right inverse $Q_{\delta}$, and gluing operator $R_{\delta}$, as
follows. (In other words, the right inverse is defined by truncating the given
functions, applying the right inverse for the linearized operator on the nodal
curve, and then gluing back together using cutoff functions again.) For
simplicity we assume that there is a single node. Define the cutoff operator

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$K_{\delta}:\Omega^{0,1}(\hat{\Sigma}^{\delta},{u}^{\delta,*}T^{\operatorname{vert}}P(X))_{0,3,\delta}\to\Omega^{0,1}(\hat{\Sigma},({u}^{\delta}_{0})^{*}T^{\operatorname{vert}}P(X))_{0,3}\\\
K_{\delta}(\eta)=\begin{cases}0&\kappa_{\pm}(z)\in
B_{|\delta|^{{\frac{1}{2}}}}(0)\\\ \eta&\text{otherwise}\end{cases}.$

Then $\|K_{\delta}(\eta)\|_{0,3}\leq\|\eta\|_{0,3,\delta}.$ Define the gluing
operator

$R_{\delta}:\Omega^{0}(\hat{\Sigma},({u}^{\delta}_{0})^{*}TX)_{{1,3}}\to\Omega^{0}(\hat{\Sigma}^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta}$

as follows. Let $\hat{\Sigma}_{\pm}^{*}$ denote the complements of small balls
around the nodes
$\hat{\Sigma}_{\pm}^{*}=\Sigma_{\pm}-B_{\rho^{2}|\delta|^{1/2}}(w^{\pm})$. Let
$\pi_{\pm}:\hat{\Sigma}_{\pm}^{*}\to\hat{\Sigma}^{\delta}$ denote the
inclusions. These induce maps of sections with compact support in
$\hat{\Sigma}_{\pm}^{*}$,
$\pi_{\pm,*}:\,\Omega^{0}_{c}(\hat{\Sigma}_{\pm}^{*},u_{\pm}^{*}TX)\to\Omega^{0}(\hat{\Sigma}^{\delta},u^{\delta,*}TX).$
Define $R_{\delta}(\xi)=\xi^{\delta}$ where

$\xi^{\delta}=\pi_{+,*}\beta_{\rho,\delta}(\xi_{+}-\xi_{+}(w_{+}))+\pi_{-,*}\beta_{\rho,\delta}(\xi_{-}-\xi_{-}(w_{-}))+\xi(w)$

for $\kappa_{\pm}(z)\in B_{|\delta|^{1/2}\rho^{2}}(0)$ and $\xi^{\delta}=\xi$
otherwise. Here $x=\xi^{\pm}(w^{\pm})$ is the value of $\xi$ at the node.
Define $T_{\delta}:=(I\times R_{\delta})Q_{\delta}(I\times K_{\delta}).$ That
is, if $(a,\xi)=Q_{\delta}K_{\delta}(\phi,\psi,\eta)$ then
$T_{\delta}(\phi,\psi,\eta)=(a,\xi^{\delta}).$ The map $T_{\delta}$ is the
required approximate right inverse. The difference
$(\tilde{D}_{\delta}T_{\delta}-I)(\phi,\psi,\eta)$ is the sum of terms

${\operatorname{d}}_{A,u^{\delta}}(a,\xi^{\delta})-\phi,\quad{\operatorname{d}}_{A,u_{0}}^{*}(a,\xi^{\delta})-\psi,\quad
D_{A,u^{\delta}}(a,\xi^{\delta})-\eta.$

By definition ${\operatorname{d}}_{A,u_{0}^{\delta}}(a,\xi)=\phi$, so the
first difference has contributions involving the difference between
$u_{0}^{\delta}$ and $u^{\delta}$, and between $\xi^{\delta}$ and $\xi$. But
since
${\operatorname{d}}_{A,u}(a,\xi)={\operatorname{d}}_{A}a+u^{*}L_{\xi_{X}}P(\Phi)$,
these terms are zeroth order in $u,\xi$,

$\|(u^{\delta})^{*}L_{\xi^{\delta}_{X}}P(\Phi)-u_{0}^{\delta}L_{\xi_{X}}P(\Phi)\|_{0,3}\leq
C|\delta|\|\xi\|_{C^{0}};$

that is, a constant times the area of $\pi_{-,*}(\hat{\Sigma}_{-}^{*})$, which
goes to zero as $\delta$ does. A similar discussion holds for the second
difference. The third difference has terms arising from the cutoff function
and the term $a_{X}^{0,1}(u)$ on the bubble region
$|\kappa_{+}(z)|\leq|\delta|^{1/2}$. We have

$\|a_{X}^{0,1}(u)\|_{0,3,\delta}\leq\|a_{X}^{0,1}(u)\|_{0,3}\leq C|\delta|.$

since $p\geq 2$; for $p=2$ the $W^{0,3,\delta}$ and $W^{0,3}$ norms are the
same, by conformal invariance; for $p>2$ the $0,3$-norm is strictly greater.
The term involving the derivative of the cutoff function satisfies

$\|{\operatorname{d}}\beta_{\rho,\delta}(\xi-\xi(w))\|_{0,3,\delta}\leq
c\log(\rho^{2})^{-2/3}\|(\xi-\xi(w))\|_{1,3}$

by Lemma 3.3.5, and so vanishes in the limit $\rho\to\infty$. Using the
uniform bound on $Q_{\delta}$, the total difference is bounded by
$C(\log(\rho^{2})^{-2/3}+|\delta|)\|(\phi,\psi,\eta)\|$, and so vanishes in
the limit $\delta\to 0,\rho\to\infty,|\delta|\rho^{2}\to 0$. The uniform bound
on $T_{\delta}$ follows from the uniform bound on $Q_{\delta}$ and the cutoff
and extension operators. ∎

Step 3: Uniform quadratic estimate

######  Proposition 4.2.8.

There exist constants $c,C>0$ such that if $\|\xi\|_{1,3,\delta}<c$,
$|\delta|<c,\rho>1/c$ and $|\delta|\rho^{4}<c$ then the map
$\mathcal{F}^{\delta}_{A,{u}}$ satisfies a quadratic bound

$\|D\mathcal{F}_{A,{u}}^{\delta}(a_{1},\xi_{1})-\tilde{D}_{A,{u}^{\delta}}(a_{1},\xi_{1})\|_{\delta}\leq
C\|a,\xi\|_{\delta}\|a_{1},\xi_{1}\|_{\delta}.$

###### Proof.

The norm of the non-linear part of the curvature $\|[a,a_{1}]\|_{{0,3}}$ is
bounded by Sobolev multiplication. The other term appearing in the first
vortex equation satisfies

$\|\exp_{{u}^{\delta}}(\xi_{0})^{*}L_{\xi_{1,X}}P(\Phi)-({u}^{\delta})^{*}L_{\xi_{1,X}}P(\Phi)\|_{{0,3}}\leq
C\|\xi_{0}\|_{1,3,\delta}\|\xi_{1}\|_{1,3,\delta}$

for some constant $C$ independent of $\delta$, using that $W^{1,3,\delta}$
norm controls the $W^{0,3}$ norm uniformly. The non-linear terms in the
Cauchy-Riemann equation are estimated as in Theorem 4.2.8 and [15, Section
3.5, Lemma 10.3.1]; note that we are fixing the complex structure on $\Sigma$,
which avoids the more complicated analysis we gave in the previous section.
The second vortex equation also involves a term of mixed type
$\Psi_{u}(\xi+\xi_{1})^{-1}(a_{1})_{X}^{0,1}(\exp_{u}(\xi+\xi_{1}))-\Psi_{u}(\xi)^{-1}(a_{1})_{X}^{0,1}(\exp_{u}(\xi)).$
It follows from uniform Sobolev embedding that this difference has
$0,3,\delta$-norm bounded by $C\|a_{1}\|_{{1,3}}\|\xi_{1}\|_{1,3,\delta}$ for
some constant $C$ independent of $\delta$. ∎

Step 4: Implicit Function Theorem

######  Theorem 4.2.9.

Let $(A,{u})$ be a regular stable nodal vortex of combinatorial type $\Gamma$.
There exist constants $\epsilon_{0},\epsilon_{1}>0$ such that for every
$(a,{\xi},\delta)\in\operatorname{Def}_{\Gamma}(A,{u})$ with norm less than
$\epsilon_{0}$, there exists a unique $(\phi,\psi,\eta)$ of norm less than
$\epsilon_{1}$ such that if $(a_{1},\xi_{1})=Q_{\delta}(\phi,\psi,\eta)$ then
$(A+a_{0}+a_{1},\exp_{{u}^{\delta}}({\xi}_{0}^{\delta}+\xi_{1}))$ is a
symplectic vortex in Coulomb gauge with respect to $(A,{u}^{\delta})$. The
solution depends smoothly on $a_{0},{\xi}_{0}$, and transforms equivariantly
the action of $\mathcal{G}(P)_{A,{u}}$ on
$\operatorname{Def}_{\Gamma}(A,{u})$.

###### Proof.

Uniform error and quadratic estimates are those for
$\mathcal{F}_{A,{u}}^{\delta}$ in Lemmas 4.2.4, 4.2.7, and 4.2.8, in a
uniformly bounded neighborhood of $0$ in $\operatorname{Def}_{\Gamma}(A,{u})$.
Then the first claim is an application of the quantitative version of the
implicit function theorem (see for example [15, Appendix A.3]). Equivariance
follows from uniqueness of the solution given by the implicit function
theorem, since the map $\mathcal{F}^{D,\delta}_{A,{u}}$ is equivariant for the
action of $\mathcal{G}(P)_{A,{u}}$. ∎

Step 5: Rigidification

As in the case of holomorphic maps in the previous section, there is a more
natural way of parametrizing nearby symplectic vortices which involves
examining the intersections of the sections with submanifolds of $P(X)$, and
framings induced by parallel transport. First we study the differentiability
of the evaluation maps. The gluing construction of the previous step gives
rise to a deformation $(A_{S},{u}_{S})$ of $(A,{u})$ with parameter space a
neighborhood $S$ of $0$ in $\operatorname{Def}(A,{u})$, and so a map
$S\to\overline{M}_{n}(\Sigma,X),\ s\mapsto(\hat{\Sigma}_{s},A_{s},{u}_{s})$
Consider the map

(45) $\operatorname{ev}:(\hat{\Sigma}-U)\times S\to
P(X),\quad(z,s)\mapsto{u}_{s}(z).$

######  Proposition 4.2.10.

The map $\operatorname{ev}$ of (45) is $C^{1}$ for the family constructed by
gluing in Theorem 4.2.9 using the exponential gluing profile.

###### Proof.