# Moduli stacks of Higgs bundles on stable curves

Oren Ben-Bassat, Sourav Das, Tony Pantev

## Abstract

In this article, we construct a flat degeneration of the derived moduli stack
of Higgs bundles on smooth curves using the stack of expanded degenerations of
Jun Li. We show that there is an intrinsic relative log-symplectic form on the
degeneration and we compare it with the one constructed by the second author.
We show that the Hitchin map of the degeneration we construct has complete
fibers. Furthermore, we show that the Hitchin map is flat and that a suitable
open subset of the smooth locus of the reduced nilpotent cone is Lagrangian.
We also extend the construction of the moduli of Higgs bundles along with the
relative log-symplectic form over the universal moduli stack of stable curves.

###### Contents

  1. 1 Introduction
    1. 1.1 Notations and Conventions
  2. 2 Acknowledgements
  3. 3 Preliminaries
    1. 3.1 Space of bounded Expanded degenerations
      1. 3.1.1 Degeneration of curves.
      2. 3.1.2 Space of expanded degenerations
      3. 3.1.3 Stack of bounded Expanded degenerations
      4. 3.1.4 An alternative construction of the stack $\mathfrak{M}$
    2. 3.2 Construction of the family of curves
    3. 3.3 Log structures on Derived Artin stacks
      1. 3.3.1 Log structures on Artin stacks
      2. 3.3.2 Locally free Log structures on Derived Artin stacks
      3. 3.3.3 Relative logarithmic cotangent complex
    4. 3.4 Relative shifted log-symplectic forms
  4. 4 Logarithmic Dolbeault Moduli stack and shifted log-symplectic form
  5. 5 Completeness of the Hitchin map
  6. 6 Flatness of the Hitchin map
  7. 7 On the relative logarithmic Dolbeault moduli over $\overline{\mathcal{M}_{g}}$
    1. 7.1 Log structures on ${\mathcal{M}^{ss}_{g}}$
      1. 7.1.1 Versal deformation space of ${\mathcal{M}^{ss}_{g}}$ and the versal picture of the map $\pi:\mathcal{M}^{ss}_{g}\longrightarrow\overline{\mathcal{M}_{g}}$
    2. 7.2 Relative log-cotangent complex of the map ${\mathcal{M}^{ss}_{g}}\longrightarrow\overline{\mathcal{M}_{g}}$
    3. 7.3 Relative logarithmic Dolbeault shape and shifted symplectic forms
  8. 8 Appendix: Classical Artin stack of Gieseker-Higgs bundles and its local properties
    1. 8.1 The stacks of Gieseker vector bundles and Gieseker Higgs bundles
      1. 8.1.1 Stack of torsion-free Hitchin pairs
      2. 8.1.2 Classical Artin Stack of Gieseker-Higgs bundles
      3. 8.1.3 Construction of an atlas for $N_{Gie}$
      4. 8.1.4 Construction of an atlas for $M^{{}^{cl}}_{Gie}$
    2. 8.2 Dimension and local properties of $M^{cl}_{Gie,0}$
      1. 8.2.1 Relative Log-Symplectic reduction

##  1 Introduction

It is well known that moduli stacks of vector bundles on nodal curves are not
complete. Often their completions involve adding torsion free sheaves as
points of the boundary. Because we want to use the mapping space techniques of
shifted symplectic geometry, we instead adopt the alternative approach of
completing the moduli by adding boundary points which are bundles on a
bubbling of the node. This approach goes back to the classical works of David
Gieseker [11] and Jun Li [14]. In our setting we will implement the approach
by utilizing Jun Li’s stack of expanded degenerations. This allows us to use
only vector bundles as opposed to torsion-free coherent sheaves and apply
techniques of shifted symplectic geometry [28] obtaining a relative symplectic
structure on a complete moduli stack. We extend the Hitchin map to Higgs
bundles on the bubbled curves parametrized by the stack of expanded
degenerations and then use the completeness of this moduli stack to prove that
our extended Hitchin map is complete.

The derived moduli stack of Higgs bundles on a smooth curve $C$ has virtual
dimension
$-\chi_{\text{Dol}}(C,\mathcal{E}nd(E,\phi))=n^{2}\deg(c_{1}(T_{C}))=2n^{2}(g-1)$
where $g$ is the genus of $C$ while the derived moduli stack of vector bundles
(or any other Lagrangian in the derived moduli stack of Higgs bundle) has
virtual dimension
$-\chi(Y,\mathcal{E}nd(E))=\int_{C}\text{ch}(\mathcal{E}nd(E))\text{td}(C)=n^{2}(g-1)$.
These moduli stacks are homotopically locally of finite presentation and so
exhibit the “hidden smoothness” envisioned in derived deformation theory. In
view of this it is natural to study other geometric properties of these
moduli. This article investigates $0$-shifted symplectic structures when $C$
degenerates to a nodal curve. After writing this article we realized there are
a lot of relations with the articles [2], [3] and significant overlap with a
forthcoming article [10] of Ron Donagi and Andres Fernandez Herrero. In their
work they give a construction of a good moduli space for the semistable locus,
and semistable reduction relative to the Hitchin fibration (so the moduli
space is proper over the Hitchin base in families). This uses the “infinite
dimensional GIT” picture developed with Dan Halpern-Leistner. They also showed
flatness of the Hitchin morphism, and that things are syntomic (as classical
stacks), including the symplectic leaves. They are pursuing the story with
punctures: thinking about the log Poisson reduction picture for the relative
log cotangent stack of framed Gieseker vector bundles.

To start with, we fix a family of projective curves $\mathcal{X}$ over the
spectrum of a discrete valuation ring $S$ such that the generic fibre is
smooth, the closed fibre is an irreducible nodal curve with a single node and
the total space $\mathcal{X}$ is smooth. We also fix a rank $n$ and degree $d$
for the vector bundles in our moduli problem. In Section §2, we recall the
construction of the stack $\mathfrak{M}$ of bounded expanded degenerations
(bounded by the integer $n$) of the family of curves $\mathcal{X}/S$ following
[14] and [37]. One of the main results in this subsection is Lemma 3.9, which
we use to prove the following proposition.

Proposition 3.15 The morphism $\mathfrak{M}\longrightarrow S$ of ordinary
Artin stacks is a log-smooth map. Moreover, the relative log-cotangent complex
$\mathbb{L}^{log}_{\mathfrak{M}/S}=0$.

In subsection §2.3, we define shifted log-symplectic structure on a quasi-
smooth derived Artin stack (Definition 3.18) equipped with a locally free log
structure. In subsection §2.4, we recall the definition of relative shifted
symplectic forms for a quasi-smooth morphism of derived Artin stacks. We
define relative shifted log-symplectic forms for certain logarithmic morphisms
of derived Artin stacks equipped with locally-free log structures.

In Section §3, we construct a logarithmic version of the relative Dolbeault
moduli stack for the universal expanded family of curves
$\mathcal{\mathcal{missing}}X_{\mathfrak{M}}\rightarrow\mathfrak{M}$. We show
that the relative logarithmic Dolbeault moduli stack has a relative
$0$-shifted log-symplectic form over $S$. Moreover, we show that the relative
log-symplectic form is an extension of Hitchin’s symplectic form on the
generic fibre of the moduli stack over $S$. This was proved for moduli schemes
in [9]. The main result of this section are the following.

Theorem 4.11 $\mathcal{X}_{Dol}$ is $\mathcal{O}$-compact and
$\mathcal{O}$-oriented over $S$. Hence,
$\mathsf{Map}_{\mathcal{S}}(\mathcal{X}_{Dol},BGL_{n}\times\mathcal{S})$ has a
$0$-shifted relative symplectic structure over $S$.

Theorem 4.13 The derived Artin stack
$\mathsf{Map}_{{}_{\mathfrak{M}}}(\mathcal{X}_{\mathfrak{M},Dol},BGL_{n}\times\mathfrak{M})$
has a natural relative $0$-shifted log-symplectic structure over $S$.

In Section §4, we define the Hitchin map on the classical Artin stack of
Gieseker-Higgs bundles $M^{cl}_{Gie}$ (see 8.1.2). We prove that the Hitchin
map is complete.

Theorem 5.3 The morphism $h:M^{cl}_{Gie}\longrightarrow B$ is complete.

This result was proved in [1] for the Hitchin map on the moduli scheme in the
case where rank and degree are co-prime. We prove it here for the moduli stack
and the argument does not require us to assume that the rank and degree are
coprime.

In section §5, we study the reduced global nilpotent cone of $M^{cl}_{Gie}$,
which is the reduction of the scheme theoretic fibre over the point $0\in B$.
We prove that every irreducible component of of the reduced nilpotent cone has
an open subset (denoted by $\mathcal{N}ilp^{sm,gen}$) which is an isotropic
substack of $M$ (the derived stack of Higgs bundles) with respect to its log-
symplectic form. We use this to compute the dimension of the reduced nilpotent
cone and to show that the Hitchin map is flat. The main theorem in this
section is the following.

Theorem 6.9

  1. 1.

The Hitchin map $h:M^{cl}_{Gie}\longrightarrow B$ is surjective.

  2. 2.

The sub-stack $\mathcal{N}ilp^{sm,gen}$ is relatively isotropic over $S$.

  3. 3.

The Hitchin map $h:M^{cl}_{Gie}\longrightarrow B$ is flat.

In Section §6, we construct the Gieseker-like derived moduli stack of Higgs
bundles $\mathcal{M}^{Dol}_{g}$ over the moduli stack of stable curves of
genus $g\geq 2$. We prove the following theorem.

Theorem 7.9 There is a $0$-shifted relative log-symplectic form on
$\mathcal{M}^{Dol}_{g}$ (relative to the moduli stack of stable curves
$\overline{\mathcal{M}_{g}}$).

In the Appendix, we construct the relative classical Artin stack of Gieseker-
Higgs bundles and study its local properties. The main results of the appendix
are Proposition 8.9 and Theorem 8.10. In the first lemma we prove that the
stack of Gieseker vector bundles is an almost very good stack. We use this in
Lemma 8.10 to show that the classical stack of Gieseker-Higgs bundles is an
irreducible, local complete intersection.

Proposition 8.9 The closed fibre $N^{cl}_{Gie,0}$ is an irreducible,
equidimensional, almost very good stack ([32, Definition 2.1.2]) with normal
crossing singularities.

Theorem 8.10 The stack $M^{cl}_{Gie,0}$ is an irreducible local complete
intersection of pure dimension $2\dim N_{Gie,0}+1$.

###  1.1 Notations and Conventions

  1. $\bullet$

$\mathbb{k}$ be an algebraically closed field of characteristic zero.

  2. $\bullet$

$S:=\\{\eta,\eta_{0}\\}$ denotes the spectrum of a complete discrete valuation
ring, where $\eta$ denotes the generic point, and $\eta_{0}$ denotes the
closed point.

  3. $\bullet$

$\mathcal{X}\to S$ denotes a flat family of curves whose generic fibre is
smooth projective, and the closed fibre is a nodal curve with a single node.
We denote the nodal curve by $X_{0}$ and the node by $x$. We denote its
normalisation by $q:\widetilde{X}_{0}\longrightarrow X_{0}$ and the two pre-
images of the node $x$ by $\\{x^{+},x^{-}\\}$.

  4. $\bullet$

$dg_{\mathbb{k}}$ is the category of dg-modules over $\mathbb{k}$ (i.e. of
complexes of $\mathbb{k}$-modules). By convention, the differential of an
object in $dg_{\mathbb{k}}$ _increases_ degrees.

  5. $\bullet$

$cdga_{\mathbb{k}}$ is the category of commutative dg-algebras over
$\mathbb{k}$, and $cdga_{\mathbb{k}}^{\leq 0}$ its full subcategory of non-
positively graded commutative dg-algebras.

  6. $\bullet$

$dg_{\mathbb{k}}$, $cdga_{\mathbb{k}}$ (respectively $cdga_{\mathbb{k}}^{\leq
0}$) are endowed with their natural model structures for which equivalences
are quasi-isomorphisms, and fibrations are epimorphisms (respectively
epimorphisms in strictly negative degrees).

  7. $\bullet$

${\operatorname{\mathit{dAff}}}_{\mathbb{k}}:=(cdga_{\mathbb{k}}^{\leq
0})^{op}$ is the category of derived affine $\mathbb{k}$-schemes.

  8. $\bullet$

The $\infty$-categories associated with the model categories
$dg_{\mathbb{k}},cdga_{\leq 0},\operatorname{\mathit{dAff}}_{\mathbb{k}}$ are
denoted by $\textbf{dg}_{\mathbb{k}},\textbf{cdga}_{\leq
0},\textbf{dAff}_{\mathbb{k}}$.

  9. $\bullet$

The $\infty$-category of simplicial sets is denoted by $\mathbb{S}$. It is
also called the $\infty$-category of spaces, and space will be used to mean
simplicial set.

  10. $\bullet$

The $\infty$-category of derived stacks over $\mathbb{k}$, for the étale
topology, is denoted by $\textbf{dSt}_{\mathbb{k}}$. If $X$ is a derived
stack, the $\infty$-category of derived stacks over $X$ is denoted by
$\textbf{dSt}_{X}$. The truncation of a derived stack $X$ is denoted by
$h^{0}(X)$.

  11. $\bullet$

For a family of semistable curves $\mathcal{C}$ over a scheme $T$,
$\mathcal{C}_{lDol}$ denotes the relative logarithmic Dolbeault shape of
$\mathcal{C}/T$.

${\mathcal{C}_{lDol}}$${\mathcal{C}}$${T}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{r}$

##  2 Acknowledgements

T.P. (University of Pennsylvania) was partially supported by NSF FRG grant
DMS- 2244978, NSF/BSF grant DMS-2200914, NSF grant DMS-1901876, and Simons
Collaboration grant number 347070. O.B. (University of Haifa) and S.D. would
like to acknowledge NSF-BSF grant 2021717 for supporting S.D. as a postdoc.
S.D. would like to thank Professor Alek Vainshtein of the University of Haifa,
Israel for the financial support from his Israel Science Foundation grant
number 876/20 during a postdoc position at the University of Haifa. S.D. would
also like to thank Professor Vikraman Balaji for the financial support from
his SERB Core Research Grant during a postdoc position at Chennai Mathematical
Institute.

##  3 Preliminaries

###  3.1 Space of bounded Expanded degenerations

In this subsection, we will recall a construction by Jun Li [14] called the
stack of expanded degenerations. We will use this stack to construct our
degeneration. The construction of Jun Li starts with a one-parameter family of
varieties (of any dimension) such that the total space of the family is
smooth, the generic fibre is smooth and the special fibre is a normal crossing
divisor in the total space of the family. For our purpose, we only need to
consider the expanded degenerations of such a family for curves. We will make
use of the standard fact that for any nodal curve one can always construct a
smoothing over the spectrum of a discrete valuation ring which also has a
smooth total space. The precise setting we consider can be spelled out as
follows.

####  3.1.1 Degeneration of curves.

Start with a flat family of projective curves $\mathcal{X}\longrightarrow S$,
such that

  1. 1.

the generic fibre $\mathcal{X}_{\eta}$ is a smooth curve of genus $g\geq 2$,

  2. 2.

the closed fibre is a nodal curve $X_{0}$ with a single node $x\in X_{0}$, and

  3. 3.

the total space $\mathcal{X}$ is regular over ${\rm Spec}\,\mathbb{k}$.

Let us denote the relative dualising sheaf by $\omega_{\mathcal{X}/S}$. From
here onwards, we assume that $S$ is a neighbourhood of the origin in
$\mathbb{A}^{1}:=Spec\leavevmode\nobreak\ \leavevmode\nobreak\ \mathbb{k}[t]$.

######  Definition 3.1.

(Gieseker curve/Expanded degeneration/Modification) Let $X_{0}$ be a nodal
curve with a single node $x\in X_{0}$, and let $x^{\pm}$ label the two
preimages of $x$ in the normalization $\widetilde{X}_{0}$ of $X_{0}$. Let $r$
be a positive integer.

  1. 1.

A _chain of $r$ projective lines_ is a scheme $R[r]$ of the form
$\cup_{i=1}^{{}^{r}}R[r]_{i}$ such that

    1. (a)

$R[r]_{i}\cong\mathbb{P}^{{}^{1}}$,

    2. (b)

for any $i<j$, $R[r]_{i}\cap R[r]_{j}$ consists of a single point $p_{j}$ if
$j=i+1$ and empty otherwise.

We call $r$ the length of the chain $R[r]$. Let us choose and fix two smooth
points $p_{1}$ and $p_{r+1}$ on $R[r]_{1}$ and $R[r]_{r}$, respectively.

$p_{1}$$p_{r+1}$

  2. 2.

A _Gieseker curve_ $X_{r}$ is the categorical quotient of the disjoint union
of the curves $\tilde{X}_{0}$ and $R[r]$ obtained by identifying $x^{+}$ with
$p_{1}$ and $x^{-}$ with $p_{r+1}$. We will also refer to these curves as
_expanded degenerations_ or _bubblings_ of the nodal curve $X_{0}$.

######  Definition 3.2.

(Family of Gieseker curves/Expanded degenerations/Modifications:) For every
$S$-scheme $T$, a _modification of $\mathcal{X}$_ relative to $T$ is a
commutative diagram

${\mathcal{X}^{mod}_{T}}$${\mathcal{X}_{T}:=\mathcal{X}\times_{S}T}$${T}$$\scriptstyle{p_{T}}$$\scriptstyle{\pi_{T}}$
(3.1)

such that

  1. 1.

$p_{T}:\mathcal{X}^{mod}_{T}\longrightarrow T$ is flat;

  2. 2.

the horizontal morphism is finitely presented and is an isomorphism over
smooth fibers $(\mathcal{X}_{T})_{t}$ of $\mathcal{X}_{T}\to T$;

  3. 3.

over each closed point $t\in T$ which maps to $\eta_{0}\in S$, we have
$(\mathcal{X}^{mod}_{T})_{t}\cong X_{r}$ for some integer $r$ and the
horizontal morphism restricts to the morphism $X_{r}\to X_{0}$ which contracts
the $\mathbb{P}^{1}$’s in $X_{r}$ to the node $x\in X_{0}$.

######  Definition 3.3.

(Morphisms of families of expanded degenerations:) Let $T\to S$ be a scheme
mapping to $S$, and let $\mathcal{X}^{mod}_{T}$ and
${\mathcal{X}^{\prime}}^{mod}_{T}$ be two $T$-relative modifications of
$\mathcal{X}\to S$.

We call $\mathcal{X}^{mod}_{T}$ and $\mathcal{X}^{\prime mod}_{T}$
_isomorphic_ if there exists an isomorphism
$\sigma_{T}:\mathcal{X}^{mod}_{T}\longrightarrow\mathcal{X}^{\prime mod}_{T}$
such that the following diagram commutes

${\mathcal{X}^{mod}_{T}}$${\mathcal{X}^{\prime
mod}_{T}}$${\mathcal{X}_{T}}$$\scriptstyle{\pi_{T}}$$\scriptstyle{\sigma_{T}}$$\scriptstyle{\pi^{\prime}_{T}}$
(3.2)

####  3.1.2 Space of expanded degenerations

As before we will choose an uniformizer on $S$ so that $S$ is identified with
a neighborhood of zero in $\mathbb{A}^{1}$. For any positive integer $n$, we
set

$S[n]:=S\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ (3.3)

where the map $\mathbb{A}^{n+1}\longrightarrow\mathbb{A}^{1}$ is given by
$(t_{1},\dots,t_{n+1})\mapsto t_{1}\cdots t_{n+1}$.

Consider the group

$G[n]:=\mathbb{G}_{m}^{n}$ (3.4)

acting on $\mathbb{A}^{n+1}$ by

$(\sigma_{1},\ldots,\sigma_{n})\cdot(t_{1},\ldots,t_{n+1}):=(\sigma_{1}\cdot
t_{1},\ldots,\sigma_{i-1}^{-1}\cdot\sigma_{i}\cdot
t_{i},\ldots,\sigma_{n}^{-1}\cdot t_{n+1}).$ (3.5)

The map $\mathbb{A}^{n+1}\to\mathbb{A}^{1}$, $(t_{1},\ldots,t_{n+1})\mapsto
t_{1}t_{2}\cdots t_{n+1}$ intertwines this action with the trivial action on
$\mathbb{A}^{1}$ and hence (3.5) induces an action of $G[n]$ on $S[n]$.

In [14, Section 1.1], Jun Li constructed a $G[n]$-equivariant family of
expanded degenerations $W[n]$ over $S[n]$. It is constructed by several
birational transformations of the family $\mathcal{X}\times_{S}S[n]$. The
curves occurring in the family are all possible expanded degenerations of the
family $\mathcal{X}/S$ for which the length of rational chains is bounded by
$n$.

####  3.1.3 Stack of bounded Expanded degenerations

The significance of the family $W[n]\to S[n]$ is that it can be used as an
atlas for the universal family of expanded degenerations with bubbling of
length $\leq n$.

Indeed, recall from [14, Definition 1.9, Proposition 1.10] the following

######  Definition 3.4.

The _stack of expanded degenerations_ of $\mathcal{X}/S$ is the stack

$\mathfrak{M}:{Sch}/S\longrightarrow{Groupoids}$

given by the assignment

$T\mapsto\left\\{\text{ \begin{minipage}[c]{303.53377pt} Pairs $(W_{T},\pi)$
where $T$ is a scheme, $W_{T}\to T$ is a family of projective curves, and
$\pi:W_{T}\to\mathcal{X}$ is a morphism over $S$ such that there exists an
\'{e}tale cover $\widetilde{T}\to T$ and a map $\widetilde{T}\to S[n]$ so that
$W_{T}\times_{T}\widetilde{T}\cong W[n]\times_{S[n]}\widetilde{T}$
\end{minipage}}\right\\}$ (3.6)

Two such families $W_{T}$ and $W^{\prime}_{T}$ are isomorphic if there is a
$T$-isomorphism $f:W_{T}\longrightarrow W^{\prime}_{S}$ compatible to the
tautological projections $W_{T}\longrightarrow W$ and
$W^{\prime}_{T}\longrightarrow W$.

By construction, the stack of expanded degenerations has the following
properties

  1. 1.

$\mathfrak{M}$ is a smooth Artin stack of finite type.

  2. 2.

The projection map $\mathfrak{M}\longrightarrow S$ is generically an
isomorphism.

  3. 3.

The closed fibre of $\mathfrak{M}\to S$ is a normal crossing divisor in
$\mathfrak{M}$.

Here we use the following

######  Definition 3.5.

Let $\mathcal{Y}$ be a smooth Artin stack and $\mathcal{D}$ be a closed sub-
stack of co-dimension one. We say that $\mathcal{D}$ is a _normal crossing
divisor_ in $\mathcal{Y}$ if for any smooth morphism
$f:T\rightarrow\mathcal{Y}$ from a smooth scheme $T$, the pull-back of the
divisor $\mathcal{D}\times_{\mathcal{Y}}T$ is a normal crossing divisor in
$T$.

####  3.1.4 An alternative construction of the stack $\mathfrak{M}$

In [37, Definition 2.22], Zijun Zhou gave an alternative construction of the
stack $\mathfrak{M}$ which will be useful for our purpose. Here we briefly
recall his construction.

First let us introduce some useful notation and terminology. For any integer
$n$, we set $[n]:=\\{1,\dots,n+1\\}$. It will be useful to have short names
for the natural maps between the spaces $S[k]$ for various valuse of $k$.

######  Definition 3.6.

Given any subset $I\subseteq[n]$ of cardinality $k+1$, we define

  * •

The _Standard Embedding_ corresponding to $I$ is the embedding

$\gamma_{I}:S[k]\hookrightarrow S[n]$ (3.7)

given by $\gamma_{I}(t_{1},\dots,t_{k+1})=t_{i}$ if $i\in I$ and
$\gamma_{I}(t_{1},\dots,t_{k+1})=1$ otherwise.

  * •

The _Standard Open Embedding_ corresponding to $I$ is the open embedding

$\tau_{I}:S[k]\times\mathbb{G}_{m}^{n-k}\hookrightarrow S[n]$ (3.8)

where $S[k]\times\mathbb{G}_{m}^{n-k}$ denotes the open subset swept by $S[k]$
under the free action of the subgroup $\mathbb{G}_{m}^{n-k}\subset G[n]$ whose
$j$-th component is $1$ if $j\notin I$.

######  Definition 3.7.

(i)

Given two subsets $I$ and $I^{\prime}$ of $[n]$ both of order $k+1$, define an
equivalence relation $R_{I,I^{\prime}}$ on $S[n]$ by setting

$R_{I,I^{\prime}}=S[k]\times\mathbb{G}_{m}^{n-k}\rightrightarrows S[n]$ (3.9)

where the two maps are $\tau_{I}$ and $\tau_{I^{\prime}}$.

(ii)

For every $k\leq 0$ define a discrete equivalence relation
$R_{k}\rightrightarrows S[n]$ as

$R_{k}:=\coprod_{|I|=|I^{\prime}|=k+1,I,I\subseteq[n]}R_{I,I^{\prime}}\
\rightrightarrows\ S[n].$ (3.10)

(iii)

Define the amalgamated discrete equivalence relation $R_{dis}$ on $S[n]$ by
setting

$R_{dis}:=\coprod_{k\leq n}R_{k}\ \rightrightarrows\ S[n]$ (3.11)

(iv)

Finaly we define the total smooth groupoid

$R_{tot}:=\mathbb{G}_{m}^{n}\times R_{dis}\rightrightarrows S[n]$

which is generated by $R_{dis}$ and the action of $\mathbb{G}_{m}^{n}$ on
$S[n]$.

$R_{tot}:=\mathbb{G}_{m}^{n}\times R_{dis}$ (3.12)

which is generated by $R_{dis}$ and the action of $\mathbb{G}_{m}^{n}$ on
$S[n]$.

######  Definition 3.8.

Let

$\mathfrak{M}_{n}:=[R_{tot}\rightrightarrows S[n]]$ (3.13)

the quotient stack of $S[n]$ by the smooth groupoid $R_{tot}$.

From [37, Remark 2.23], it follows that the stack $\mathfrak{M}_{n}$ is the
stack of expanded degenerations of the family $\mathcal{X}/S$ bounded by the
integer $n$.

Notation: From here onwards, we will only work with $\mathfrak{M}_{n}$ and we
drop the subscript $"n"$ from the notation and denote it simply by
$\mathfrak{M}$.

######  Lemma 3.9.

The morphism between quotient stacks

$[S[n]/G[n]]\longrightarrow\mathfrak{M}$ (3.14)

is étale.

###### Proof.

Let us first show that the relation $R_{dis}$ descends to an equivalence
relation on the quotient stack $[S[n]/G[n]]$). Recall that
$R_{dis}:=\coprod_{k\leq n}R_{k}$ and
$R_{k}:=\coprod_{|I|=|I^{\prime}|=k+1,I,I\subseteq[n]}R_{I,I^{\prime}}$, where
the equivalence relations $R_{I,I^{\prime}}$ are given by

$S[k]\times\mathbb{G}_{m}^{n-k}\rightrightarrows S[n]$ (3.15)

Recall that the map $R_{I}:S[k]\times\mathbb{G}_{m}^{n-k}\longrightarrow S[n]$
is given by the standard open embedding.

For simplicity, let us assume that $S=\mathbb{A}^{1}$. Then
$S[n]=\mathbb{A}^{n+1}$. Notice that $G[n]:=\mathbb{G}_{m}^{n}$. But in fact,
there is an action of a bigger group $\mathbb{G}_{m}^{n+1}$ on
$\mathbb{A}^{n+1}$ given by the coordinatewise multiplication. Moreover, the
group $\mathbb{G}_{m}^{n}$ can be identified as a subgroup of
$\mathbb{G}_{m}^{n+1}$ consisting of elements of the form
$(t_{1},\dots,t_{n+1})$ such that $\prod^{n+1}_{i=1}t_{i}=1$. Let’s consider
the action of the bigger group $G[n+1]$.

Let us first define a natural action of $G[n+1]$ on
$R_{I}:=\mathbb{A}^{k+1}\times\mathbb{G}_{m}^{n-k}$. Notice that the
cardinality of the set $I$ equals $k+1$. It is an ordered subset of $[n+1]$.
Therefore, the complement $J:=[n+1]\setminus I$ is also an ordered set of
cardinality $n-k$, where the order is the induced order from $[n+1]$. Now we
see that
$\mathbb{G}_{m}^{n+1}\cong\mathbb{G}_{m}^{I}\times\mathbb{G}_{m}^{J}$.
Finally, we define the action of $\mathbb{G}_{m}^{n+1}$ on
$R_{I}:=\mathbb{A}^{I}\times\mathbb{G}_{m}^{J}$ by the obvious action of
$\mathbb{G}_{m}^{I}\times\mathbb{G}_{m}^{J}$. It is easy to check that the map
$R_{I}:\mathbb{A}^{k+1}\times(\mathbb{k}^{*})^{n-k}\longrightarrow\mathbb{A}^{n+1}$
is $\mathbb{G}_{m}^{n+1}$-equivariant. Hence, it is also equivariant under the
action of the smaller subgroup $\mathbb{G}_{m}^{n}$. Therefore, the
equivalence relations $R_{I,I^{\prime}}$ descends to the quotient stack
$[S[n]/G[n]]$.

Now we will show that the relation $\underline{R}_{dis}$ is an étale
equivalence relation. First of all, notice that the maps
$R_{I}:S[k]\times\mathbb{G}_{m}^{n-k}\longrightarrow S[n]$ are open
immersions. Therefore,
$R_{k}:=\coprod_{|I|=|I^{\prime}|=k+1,I,I\subseteq[n]}R_{I,I^{\prime}}\rightrightarrows\mathbb{A}^{k+1}$
is an étale equivalence relation because both the projections are Zariski open
immersions. Therefore, $R_{dis}$ on $S[n]$ and $[R_{dis}/G[n]]$ on
$[S[n]/G[n]]$ defines an étale equivalence relation. ∎

###  3.2 Construction of the family of curves

In this subsection, we will briefly recall from [14, 37] the fact that there
is a family of expanded degeneration (whose rational chain is bounded by
length $n$). Let us choose an étale neighbourhood $U_{p}$ of the node
$p\in\mathcal{X}$ such that the nodal curve $U_{p}\times_{\mathcal{X}}X_{0}$
is a reducible nodal curve with two smooth connected components intersecting
transversally at the node $p$. We choose a étale covering
$\mathcal{X}=U_{p}\cup V$ such that $p\notin V$. We have the following pushout
diagram.

${U_{p}\times_{\mathcal{X}}V}$${V}$${U_{p}}$${\mathcal{X}}$ (3.16)

Then notice that $U_{p}\longrightarrow S$ is a simple degeneration required to
construct the expanded degeneration [14, 37]. We denote by
$U_{p}[n]\longrightarrow S[n]$ the family of expanded degenerations. Also,
$V[n]:=V\times_{S}S[n]$. Now we construct the family of expanded
degenerations, denoted by $\mathcal{X}[n]$ over $S[n]$ by the following push-
out diagram.

${U_{p}[n]\times_{(\mathcal{X}\times_{S}S[n])}V[n]}$${V[n]}$${U_{p}[n]}$${\mathcal{X}[n]}$
(3.17)

Since it is an étale gluing, the total space $\mathcal{X}[n]$, in general, is
an algebraic space. The family of curves $\mathcal{X}[n]\longrightarrow S[n]$
is the desired family of expanded degeneration of the original family
$\mathcal{X}\longrightarrow S$. We refer to [37, Proposition 2.13] for the
interesting properties of the family of expanded degenerations. we should
remark that the properties (3) and (4) of [37, Proposition 2.13] imply that
the family of expanded degenerations descends to the stack of expanded
degenerations $\mathfrak{M}$. We denote this family of expanded degenerations
by $\mathcal{X}_{\mathfrak{M}}$.

###  3.3 Log structures on Derived Artin stacks

In this subsection, we define shifted log-symplectic structure on a quasi-
smooth derived Artin stack (Definition 3.18) equipped with a locally free log
structure. We refer to [22, 23] and [25] for the prerequisite material on log-
structure/locally free log-structure on Artin/derived Artin stacks. We now
recall few necessary definitions.

####  3.3.1 Log structures on Artin stacks

Let $\mathfrak{L}^{0}$ denote the algebraic stack, which classifies fine log
structures, and $\mathfrak{L}^{1}$ denote the algebraic stack, which
classifies morphisms of fine log structures. We denote by
$\mathfrak{L}^{0}_{f}$ and $\mathfrak{L}^{1}_{f}$ the classifying substack of
locally free log structures and morphisms between locally free log structures,
respectively. Given a logarithmic scheme $(S,M,\alpha)$, the stack
$\mathfrak{L}^{0}_{S,f}:=\mathfrak{L}^{1}_{f}\times_{\mathfrak{L}^{0}_{f}}S$
is the classifying stack of logarithmic morphisms from a scheme with locally
free log structure to $(S,M,\alpha)$.

######  Definition 3.10.

A locally free log structure on an algebraic stack $\mathfrak{X}$ is a
morphism of stacks $\mathfrak{X}\longrightarrow\mathfrak{L}^{0}_{f}$.

######  Remrk 3.11.

By [25, Proposition 1.7], we know that the algebraic stack
$\mathcal{L}^{0}_{f}$ and $\mathcal{L}^{1}_{f}$ are smooth.

####  3.3.2 Locally free Log structures on Derived Artin stacks

######  Definition 3.12.

A locally free log structure on a derived algebraic stack $\mathfrak{X}$ is a
morphism $\mathfrak{X}\longrightarrow\mathcal{L}^{0}_{f}$ between stacks.

######  Definition 3.13.

A morphism of locally-free derived log stacks
$\mathfrak{X}\longrightarrow\mathfrak{Y}$ is a commutative diagram of stacks

${\mathfrak{X}}$${\mathcal{L}^{1}_{f}}$${\mathfrak{Y}}$${\mathcal{L}^{0}_{f}}$
(3.18)

####  3.3.3 Relative logarithmic cotangent complex

######  Definition 3.14.

Given a morphism between derived stacks with locally free log structures
$f:\mathfrak{X}\longrightarrow\mathfrak{Y}$, we define the relative log-
cotangent complex by

${\mathbb{L}}^{log}_{\mathfrak{X}/\mathfrak{Y}}:=\mathbb{L}_{\mathfrak{X}/\mathcal{L}^{0}_{f}}$
(3.19)

Here $\mathbb{L}_{\mathfrak{X}/\mathcal{L}^{0}_{f}}$ denote the relative
cotangent complex for the morphism of stacks
$\mathfrak{X}\longrightarrow\mathcal{L}^{0}_{f}$.

######  Proposition 3.15.

The morphism $\mathfrak{M}\longrightarrow S$ of ordinary Artin stacks is a
log-smooth map. Moreover, the relative log-cotangent complex
$\mathbb{L}^{log}_{\mathfrak{M}/S}=0$.

###### Proof.

From Lemma 3.9, it follows that the map
$[S[n]/G[n]]\longrightarrow\mathfrak{M}$ is étale. Notice that the varieties
$S[n]$ and $S$ are log smooth varieties with the log structure coming from the
divisor given by the preimage of the closed point of $S$ under the map
$S[n]\longrightarrow S$. Since the map is basically the same as the map
$\mathbb{A}^{n+1}\longrightarrow\mathbb{A}^{1}$ given by
$(t_{1},\dots,t_{n+1})\mapsto t_{1}\cdots t_{n+1}$, the map is a log-smooth
morphism. Therefore, it follows that the map $[S[n]/G[n]]\longrightarrow S$ is
also a log-smooth morphism of log-smooth Artin stacks.

The explicit description of the relative log-cotangent complex is the
following. The pull-back (to $S[n]$) of the log-cotangent complex of the stack
$[S[n]/G[n]]$ is given by

$\big{[}\Omega_{S[n]}(\log\leavevmode\nobreak\ \partial
S[n])\longrightarrow\mathcal{O}_{S[n]}^{\oplus n}\big{]}$ (3.20)

where the rank $(n+1)$ locally free sheaf
$\Omega_{S[n]}(\log\leavevmode\nobreak\ \partial S[n])$ sitting in degree $0$.
Since the action of $G[n]$ on $S[n]$ respects the normal crossing divisor of
$S[n]$, the action lifts to an action on the log-cotangent bundle of $S[n]$
such that the natural log-symplectic structure is $G[n]$-equivariant. The
morphism (3.20) is given by the moment map for the action of $G[n]$ on the
log-cotangent bundle of $S[n]$. We recall the action of $G[n]$ on $S[n]$

$(t_{1},\dots,t_{n+1})\cdot(x_{1},\dots,x_{n+1})=(t_{1}x_{1},t_{2}x_{2},\dots,t_{n+1}x_{n+1})$
(3.21)

where $t_{1}\cdots t_{n+1}=1$. Also remember the map $S[n]\longrightarrow S$
is given by $(t_{1},\dots,t_{n+1})\mapsto\prod^{n+1}_{i=1}t_{i}$. Therefore,
we see that the composite map

$\Omega_{S}(\log\leavevmode\nobreak\ \leavevmode\nobreak\ \partial
S)\hookrightarrow\Omega_{S[n]}(\log\leavevmode\nobreak\ \leavevmode\nobreak\
\partial S[n])\longrightarrow\mathcal{O}_{S[n]}^{\oplus n}$ (3.22)

is $0$. Therefore, we have a complex

$\big{[}\Omega_{S[n]/S}(\log\leavevmode\nobreak\ \leavevmode\nobreak\ \partial
S[n])\longrightarrow\mathcal{O}_{S[n]}^{\oplus n}\big{]}$ (3.23)

This is precisely the relative log cotangent complex of the morphism
$\big{[}S[n]/G[n]\big{]}\longrightarrow S$ pulled back to $S[n]$. But notice
that the morphism $\Omega_{S[n]/S}(\log\leavevmode\nobreak\
\leavevmode\nobreak\ \partial S[n])\longrightarrow\mathcal{O}_{S[n]}^{\oplus
n}$ is an isomorphism. Therefore, the relative log-cotangent complex of the
morphism $\big{[}S[n]/G[n]\big{]}\longrightarrow S$ is equivalent to $0$. ∎

###  3.4 Relative shifted log-symplectic forms

In this subsection, we will recall the definition of relative shifted
symplectic forms for a quasi-smooth morphism of derived Artin stacks. We will
define relative shifted log-symplectic forms for certain logarithmic morphisms
of derived Artin stacks equipped with locally-free log structures.

Let $\pi:M\longrightarrow S$ be a relatively derived Artin stack with $S$ a
scheme. Suppose we also assume that the stack $M$ and the scheme $S$ are
equipped with locally free log structures such that the map $\pi$ is a
morphism of log structures. The above data is equivalent to a map of stacks
$\pi_{log}:M\longrightarrow\mathcal{L}^{0}_{S}$, where $\mathcal{L}^{0}_{S}$
is the classifying log stack for $S$. We also denote by $\mathcal{L}^{0}_{S}$
the associated derived stack. We further assume that the map $\pi_{log}$ is a
quasi-smooth morphism of derived stacks (defined below 3.18).

######  Definition 3.16.

(Locally of finite presentation morphism)[29, Def. 2.16]

  1. 1.

(classical) A map $R\longrightarrow S$ of discrete commutative $k$-algebras is
finitely presented if $\mathsf{Hom}_{R/CAlg(k)}(S,-)$ commutes with filtered
colimits.

  2. 2.

(derived) A map $A\longrightarrow B$ in $cdga^{\leq 0}$ is derived finitely
presented if $\mathsf{Map}_{A/cdga^{\leq 0}_{k}}(B,-)$ commutes with
(homotopy) filtered colimits.

######  Remrk 3.17.

A map $A\longrightarrow B$ in $cdga^{\leq 0}$ is finitely presented if and
only if

  1. 1.

$H^{0}(A)\longrightarrow H^{0}(B)$ is classically finitely presented, and

  2. 2.

$\mathbb{L}_{B/A}$ is a perfect $B$-dg module (i.e., a dualizable object in
$(dgmod(B),\otimes)$). ([30, 2.2.] and [19, Theorem 7.4.3.18])

######  Definition 3.18.

(Quasi-smooth morphism of Derived stacks) A map of derived Artin stacks
$f:\mathcal{X}\longrightarrow\mathcal{Y}$ is called a quasi-smooth morphism if
it is locally of finite presentation and the relative cotangent complex
$\mathbb{L}_{f}$ has Tor amplitude in $[-1,1]$.

######  Definition 3.19.

[25, Definition 2.12] We define the relative log cotangent complex
${\mathbb{L}}^{log}_{M/S}:=\mathbb{L}_{\pi_{log}}$. Here
$\mathbb{L}_{\pi_{log}}$ denotes the relative cotangnet complex for the
morphism $\pi_{log}$.

######  Remrk 3.20.

Since the map $\pi_{log}:M\longrightarrow\mathcal{L}^{0}_{S}$ is assumed to be
quasi-smooth, therefore ${\mathbb{L}}^{log}_{M/S}$ is a perfect complex over
$M$ and is of Tor-amplitude $[-1,1]$.

Now we recall the definitions of relative shifted symplectic forms for a
finitely presented map of derived Artin stacks $M\longrightarrow N$ from [28],
[8, Definition 1.4.1]. The finitely presented condition implies that the
relative cotangent and relative tangent complexes are perfect.

######  Definition 3.21.

We define the relative de Rham algebra for a finitely presented morphism of
derived Artin stacks $M:=\operatorname{Spec}A\longrightarrow N$, for a cdga
$A$,

$DR(M/N):=\operatorname{Sym}_{A}(\mathbb{L}_{M/N}[1])),$ (3.24)

where $\mathbb{L}_{M/N}$ is the relative cotangent complex. Notice that
$Sym_{A}(\mathbb{L}_{M/N}[1])\cong\oplus^{\infty}_{p=0}\wedge^{p}\mathbb{L}_{M/N}[p]$
as $A$ modules and therefore
$Sym_{A}(\mathbb{L}_{M/N}[1])\cong\oplus^{\infty}_{p=0}\wedge^{p}\mathbb{L}_{M/N}[p]$,
as $\mathbb{k}$-vector spaces. A cohomological differential is induced from
the internal differential of $A$, which we denote by $d$. The differential $d$
makes it a commutative differential graded algebra (cdga). We also have the de
Rham differential
$d_{DR}:\wedge^{p}\mathbb{L}_{M/N}\longrightarrow\wedge^{p+1}\mathbb{L}_{M/N}$.
The de Rham differential defines a mixed structure on $DR(M/N)$ with
$\epsilon:=d_{DR}$. This makes $DR(M/N)$ a mixed cdga. There are two types of
gradings: namely, the cohomological grading (induced by the internal
differential $d$) and a “weight grading” defined as
$DR(M/N)(p):=\wedge^{p}\mathbb{\mathbb{missing}}L_{M/N}[p]$. This graded mixed
structure on $DR(M/N)$ is also compatible with the multiplicative structure
and makes it into a graded mixed cdga over $\mathbb{k}$. The degree and weight
of the internal differential $d$ are $1$ and $0$, respectively. The degree and
weight of $\epsilon$ are $-1$ and $1$, respectively.

The assignment $(\operatorname{Spec}A\longrightarrow N)\mapsto
DR(\operatorname{Spec}A/N)$ defines a functor

$cdga^{\leq 0}_{N}\longrightarrow\epsilon-dg^{gr}_{\mathbb{k}}$

We can derive this functor on the left, by precomposing it with cofibrant
replacement functor on $cdga^{\leq 0}_{N}$, to obtain

$LDR(-/N):cdga^{\leq 0}_{N}\longrightarrow\epsilon-dg^{gr}_{\mathbb{k}}$

which preserves quasi-isomorphisms. Therefore, it induces a well-defined
$\infty$-functor
$\mathbf{DR}(-/N):\mathbf{cdga}_{N}\longrightarrow\epsilon-\mathbf{dg}^{gr}$.

Given a cdga $E\in cdga_{\leq 0}$, we get a simplicial set $|E|$ given by the
Dold-Kan correspondence.

######  Definition 3.22.

For $A\in cdga_{\leq 0}$ with a map $M:=\operatorname{Spec}A\longrightarrow N$
of derived stacks and two integers $p\geq 0$ and $n\in\mathbb{Z}$, we define

  1. 1.

$\mathcal{A}^{p}_{N}(M,n):=|\wedge^{p}\mathbb{L}_{M/N}[n]|$, a simplicial set.

  2. 2.

$\mathcal{A}^{p,cl}_{N}(M,n):=|NC^{w}(DR(M/N))[n-p](p)|$, where for a cdga
$E$, $NC^{w}(E):=NC(E)(p)$ and $NC^{n}(E)(p):=\prod_{i\geq 0}E^{n-2i}(p+i).$

We have two $\infty$-functors $\boldsymbol{\mathcal{A}^{p}_{N}(-,n)}$ and
$\boldsymbol{\mathcal{A}^{p,cl}_{N}(-,n)}$ from
$\mathbf{cdga}_{N}\longrightarrow\mathbb{S}$, where $\mathbf{cdga}_{N}$ denote
the $\infty$-category of cdga’s over a fixed derived stack $N$ and
$\mathbb{S}$ denotes the $\infty$-category of simplicial sets.

######  Definition 3.23.

For $A\in cdga_{N}$ ($M:=SpecA$) , the simplicial set
$\mathcal{A}^{p}_{N}(M,n)$ (respectively $\mathcal{A}^{p,cl}_{N}(M,n)$) is
called the space of $p$-forms of degree $n$ on the derived stack $M$, relative
to $N$ (respectively the space of closed $p$-forms of degree $n$ on the
derived stack $M$, relative to $N$).

The two $\infty$-functors $\boldsymbol{\mathcal{A}^{p}_{N}(-,n)}$ and
$\boldsymbol{\mathcal{A}^{p,cl}_{N}(-,n)}$ can be viewed as derived prestacks
$\mathbf{dAff}^{op}_{N}\longrightarrow\mathbb{S}$. Following the arguments of
[28, Proposition 1.11], one can show that the derived pre-stacks
$\boldsymbol{\mathcal{A}^{p}_{N}(-,n)}$ and
$\boldsymbol{\mathcal{A}^{p,cl}_{N}(-,n)}$ are derived stacks for the big
étale site over $N$. For the definitions of an isotropic fibration structure
and Lagrangian fibration structure, we refer to [7, Definition 1.13].

Now, in order to define relative shifted log $p$-forms for a logarithmic map
of derived Artin stacks equipped with locally free log structures
$M\longrightarrow S$, we put $N:=\mathcal{L}^{0}_{S}$. We denote the space of
closed relative log $p-$forms of degree $n$ over $S$ by
$\mathcal{A}^{p,cl}_{S,log}(-,n)=\mathcal{A}^{p,cl}_{\mathcal{L}^{0}_{S}}(-,n)$.

######  Definition 3.24.

  1. 1.

We define the space of $S$ relative logarithmic $n$-shifted $p$-forms on
$M\in\mathbf{dSt}_{S,log}$ by
$\mathcal{A}_{log}^{p}(M/S,n):=\mathsf{Map}_{{}_{\textbf{dSt}_{S,log}}}(M,\mathcal{A}^{p}_{\mathcal{L}^{0}_{S}}(-,n))$.

  2. 2.

We define the space of closed relative logarithmic $n$-shifted $p$-forms
$\mathcal{A}_{log}^{p,cl}(M/S,n):=\mathsf{Map}_{{}_{\textbf{dSt}_{S,log}}}(M,\mathcal{A}^{p,cl}_{\mathcal{L}^{0}_{S}}(-,n))$.

  3. 3.

A $2$-form $\omega\in\mathcal{A}_{log}^{2}(M/S,n)$ is non-degenerate if the
induced map in $D_{qcoh}(M)$

$\Theta_{\omega}:\mathbb{T}^{log}_{M/S}\longrightarrow\mathbb{L}^{log}_{M/S}[n]$
(3.25)

is a quasi-isomorphism. We denote by $\mathcal{A}_{log}^{2}(M/S,n)^{nd}$ the
full subspace of $\mathcal{A}_{log}^{2}(M/S,n)$ which is the union of all the
connected components consisting of non-degenerate relative logarithmic
$2$-forms of degree $n$ on $M$.

  4. 4.

Finally we define the space of $n$-shifted relative log symplectic forms as
the homotopy fibre product

$Symp_{log}(M/S,n):=\mathcal{A}_{log}^{2}(M/S,n)^{nd}\times^{h}_{\mathcal{A}_{log}^{2}(M/S,n)}\mathcal{A}_{log}^{2,cl}(M/S,n).$

##  4 Logarithmic Dolbeault Moduli stack and shifted log-symplectic form

In this section, we will construct a logarithmic version of relative Dolbeault
moduli stack for the family of curves
$\mathcal{X}_{\mathfrak{M}}\rightarrow\mathfrak{M}$. We will show that the
relative logarithmic Dolbeault moduli stack has a relative 0-shifted log-
symplectic form over the spectrum of a discrete valuation ring $S$. Moreover,
we will show that the relative log-symplectic form is an extension of
Hitchin’s symplectic form on the generic fibre of the moduli stack over the
spectrum of a discrete valuation ring. This was proved for moduli schemes in
[9].

######  Definition 4.1.

Let $\mathcal{X}\longrightarrow S$ be a morphism of classical Artin stacks of
relative dimension $1$. We call it a semi-stable family of curves if every
geometric fibre of $\mathcal{X}\longrightarrow S$ is isomorphic to a semi-
stable curve i.e., it satisfies the following properties

  1. 1.

every geometric fibre is a connected projective nodal curve,

  2. 2.

for every sub-curve $E$ isomorphic to $\mathbb{P}^{1}$ it intersects with its
complement at exactly two smooth points.

######  Definition 4.2.

let $C$ be a nodal curve and $D$ denote the set of nodes. Let
$q:\tilde{C}\longrightarrow C$ denote the normalisation and $\tilde{D}$ denote
the pre-image $q^{-1}(D)$. Then the dualising sheaf $\omega_{C}$ is the kernel
of the map

$q_{*}\Omega_{\tilde{C}}(\tilde{D})\longrightarrow\oplus_{x\in
D}\mathbb{k}_{x},$ (4.1)

where

  1. 1.

$\mathbb{k}_{x}$ denotes the sky-scraper sheaf at the point $x$.

  2. 2.

the map $q_{*}\Omega_{\tilde{C}}(x^{+}+x^{-})\longrightarrow\mathbb{k}_{x}$ is
given by

$s\mapsto Res(s;x^{+})+Res(s;x^{-})$ (4.2)

We denote it by $\omega_{C}$. Here, $Res(s;x)$ denotes the residue of a form
$s$ at a point $x$. It is straightforward to check that it is a locally free
sheaf.

######  Remrk 4.3.

For the definition and existence of the relative dualising sheaf for a family
of semi-stable/ pre-stable curves we refer to [34, 109.19 The relative
dualizing sheaf].

######  Definition 4.4.

Let $\mathcal{X}\longrightarrow\mathcal{S}$ be a morphism of derived Artin
stacks of relative dimension $1$. We call it a semi-stable family of curves if
the induced map between underlying classical Artin stacks is family of semi-
stable curves.

######  Remrk 4.5.

The above definition makes sense because the space of deformations of a
projective curve $C$ over Spec $\mathbb{k}[\epsilon]$, (where
$\mathbb{k}[\epsilon]$ denote the free cdga generated by $\epsilon$ with deg
$\epsilon=-1$) can be interpreted as $H^{2}(C,T_{C})$, which is equal to $0$
because dim $C=1$. Here $T_{C}$ denote the tangent complex of the curve $C$,
which is a perfect complex because $C$ is a local complete intersection. We
refer to [29, Proposition 6.3] for the precise statement.

######  Definition 4.6.

A Higgs bundle over a family of semistable curves $\mathcal{X}/\mathcal{S}$ is
a locally free $\mathcal{O}_{\mathcal{X}}$ module $\mathcal{E}$ with a Higgs
field
$\phi:\mathcal{E}\longrightarrow\mathcal{E}\otimes\omega_{\mathcal{X}/\mathcal{S}}$,
where $\omega_{\mathcal{X}/\mathcal{S}}$ denotes the relative dualising sheaf
for the family $\mathcal{X}/\mathcal{S}$.

We define
$Tot(\omega^{\vee}_{\mathcal{X}/\mathcal{S}}):=\underline{Spec}_{\mathcal{X}}Sym_{\mathcal{O}_{\mathcal{X}}}(\omega_{\mathcal{X}/\mathcal{S}})$;
it is the total space of the dual of the relative dualising sheaf of the
family of curves $\mathcal{X}\longrightarrow\mathcal{S}$.

######  Definition 4.7.

Let $\widehat{\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})}$
denote the formal completion of
$\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})$ along the zero
section. Notice that It is a formal group scheme over $\mathcal{X}$, whereas
$\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})$ is an abelian
group scheme over $\mathcal{X}$. The logarithmic Dolbeault shape of
$\mathcal{X}/\mathcal{S}$ is defined as the following quotient stack.

$\mathcal{X}_{lDol/\mathcal{S}}:=[\widehat{\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})}\times_{S}\mathcal{X}\rightrightarrows\mathcal{X}]$
(4.3)

The nilpotent logarithmic Dolbeault shape of $\mathcal{X}/\mathcal{S}$ is
defined as follows.

$\mathcal{X}^{nil}_{lDol/\mathcal{S}}:=[\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})\times_{\mathcal{S}}\mathcal{X}\rightrightarrows\mathcal{X}]$
(4.4)

######  Definition 4.8.

Let $X$ be a Gorenstein scheme of dimension $1$. Let us denote the dualising
sheaf by $\omega_{X}$. Let $(\mathcal{E},\phi)$ be a Higgs bundle i.e.,
$\mathcal{E}$ is a locally free sheaf on $X$ and
$\phi:\mathcal{E}\longrightarrow\mathcal{E}\otimes\omega_{X}$ is any sheaf
homomorphism (called Higgs field).

######  Lemma 4.9.

  1. 1.

$Qcoh(\mathcal{X}_{lDol})\cong
Mod_{Sym_{\mathcal{O}_{\mathcal{X}}}\omega^{\vee}_{\mathcal{X}/\mathcal{S}}}(QCoh(X))$

  2. 2.

$H^{*}_{S}(\mathcal{X}_{lDol},E)\cong H^{*}_{lDol,S}(\mathcal{X},\mathcal{E})$

###### Proof.

The proof follows verbatim [27, Proposition 5.1.2] after replacing the
relative cotangent complex $\mathbb{L}_{\mathcal{X}/\mathcal{S}}$ with the
relative dualising complex $\omega_{\mathcal{X}/\mathcal{S}}$ . ∎

######  Remrk 4.10.

Notice that $\mathcal{X}_{lDol/\mathcal{S}}$ is the classifying stack
$B\widehat{\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})}$ of
the formal group scheme
$\widehat{\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})}$ over
$\mathcal{X}$. The nilpotent logarithmic Dolbeault shape
$\mathcal{X}^{nil}_{lDol/\mathcal{S}}$ is the classifying stack of the abelian
group scheme $Tot(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})$ over
$\mathcal{X}$. Therefore we have the following equivalence of catgeories.

$\left\\{\begin{array}[]{@{}ll@{}}\text{Quasi-coherent sheaves}\\\
\text{over}\leavevmode\nobreak\
B\widehat{\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})}\end{array}\right\\}\cong\left\\{\begin{array}[]{@{}ll@{}}\text{Quasi-
coherent sheaves over}\leavevmode\nobreak\ \mathcal{X}\leavevmode\nobreak\
\text{with}\\\ \text{an action of the sheaf of algebras}\\\
\operatorname{Sym}(\omega^{\vee}_{\mathcal{X}/\mathcal{S}})\end{array}\right\\}$
(4.5)

The quadratic dual algebra $(Sym\leavevmode\nobreak\
\omega^{\vee}_{\mathcal{X}/\mathcal{S}})^{!}$ is isomorphic to the dg-algebra
with zero differential $\operatorname{Sym}\leavevmode\nobreak\
(\omega_{\mathcal{X}/\mathcal{S}}[-1])$. Moreover, we have the following
equivalence of categories ([26]).

$\left\\{\begin{array}[]{@{}ll@{}}\text{Quasi-coherent sheaves
over}\leavevmode\nobreak\ \mathcal{X}\leavevmode\nobreak\ \text{with}\\\
\text{an action of the sheaf of algebras}\\\
\operatorname{Sym}(\omega^{\vee}_{\mathcal{X}/\mathcal{S}})\end{array}\right\\}\cong\left\\{\begin{array}[]{@{}ll@{}}\text{Quasi-
coherent sheaves over}\\\ \operatorname{Spec}\leavevmode\nobreak\
\operatorname{Sym}\leavevmode\nobreak\
(\omega_{\mathcal{X}/\mathcal{S}}[-1])\end{array}\right\\}$ (4.6)

From (4.5) and (4.6), we have the following.

$B\widehat{\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})}\cong\operatorname{Spec}\leavevmode\nobreak\
\operatorname{Sym}\leavevmode\nobreak\
(\omega_{\mathcal{X}/\mathcal{S}}[-1]).$ (4.7)

Similarly, we have the following equivalence.

$\left\\{\begin{array}[]{@{}ll@{}}\text{Vector bundle}\\\
\text{over}\leavevmode\nobreak\
B{\operatorname{Tot}(\omega_{\mathcal{X}/\mathcal{S}}^{\vee})}\end{array}\right\\}\cong\left\\{\begin{array}[]{@{}ll@{}}\text{Vector
bundle over}\leavevmode\nobreak\ \mathcal{X}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{with an}\\\ \text{ action of the sheaf of
algebras}\\\
({\operatorname{Sym}(\omega_{\mathcal{X}/\mathcal{S}}}))^{\vee}\cong\widehat{\operatorname{Sym}(\omega^{\vee}_{\mathcal{X}/\mathcal{S}}})\end{array}\right\\}\cong\left\\{\begin{array}[]{@{}ll@{}}\text{nilpotent
Higgs}\\\ \text{bundle over}\leavevmode\nobreak\
\mathcal{X}\end{array}\right\\}$ (4.8)

######  Theorem 4.11.

$\mathcal{X}_{lDol}$ is $\mathcal{O}$-compact and $\mathcal{O}$-oriented over
$\mathcal{S}$. Hence,
$\mathsf{Map}_{\mathcal{S}}(\mathcal{X}_{lDol},BGL_{n}\times\mathcal{S})$ has
a $0$-shifted relative symplectic structure over $\mathcal{S}$.

###### Proof.

Proof of $\mathcal{O}$-compactness: Let us begin with recalling the notations
for the following maps from Section 1.1.

${\mathcal{X}_{lDol}}$${\mathcal{X}}$${\mathcal{S}}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{r}$
(4.9)

From (4.7), we have

$Rq_{*}\mathcal{O}_{\mathcal{X}_{lDol}}\cong\mathcal{O}_{\mathcal{X}}\oplus\omega_{\mathcal{X}/\mathcal{S}}[-1],$
(4.10)

which is clearly a perfect complex. Therefore
$\mathcal{O}_{\mathcal{X}_{{}Dol}}$ is $\mathcal{O}$-compact.

Proof of $\mathcal{O}$-orientation: Using (4.10) we have

$(Rp_{*}\mathcal{O}_{\mathcal{X}_{{l}Dol}})^{\vee}[-2]\cong(Rr_{*}\mathcal{O}_{\mathcal{X}}\oplus
Rr_{*}\omega_{\mathcal{X}/\mathcal{S}}[-1])^{\vee}[-2]\cong(Rr_{*}\mathcal{O}_{\mathcal{X}})^{\vee}[-2]\oplus(Rr_{*}\omega_{\mathcal{X}/\mathcal{S}}[-1])^{\vee}[-2].$
(4.11)

By Serre duality we have

$\displaystyle
Rr_{*}(\omega_{\mathcal{X}/\mathcal{S}}[-1])\cong(Rr_{*}((\omega_{\mathcal{X}/\mathcal{S}}[-1])^{\vee}\otimes\omega_{\mathcal{X}/\mathcal{S}}[1]))^{\vee}\cong(Rr_{*}(\mathcal{O}_{\mathcal{X}}[2]))^{\vee}\cong(Rr_{*}\mathcal{O}_{\mathcal{X}})^{\vee}[-2]$
$\displaystyle\implies(Rr_{*}(\omega_{\mathcal{X}/\mathcal{S}}[-1]))^{\vee}[-2]\cong
Rr_{*}(\mathcal{O}_{\mathcal{X}}[2])[-2]\cong
Rr_{*}\mathcal{O}_{\mathcal{X}}\hskip 60.00009pt$

We see that
$H^{0}(Rr_{*}\mathcal{O}_{\mathcal{X}})\cong\mathcal{O}_{\mathcal{S}}$. We
choose an isomorphism once and for all and denote it by $\eta$. This defines
an element

$\eta\in(Rp_{*}\mathcal{O}_{\mathcal{X}_{{l}Dol}})^{\vee}[-2]=\mathsf{Hom}(Rp_{*}\mathcal{O}_{\mathcal{X}_{lDol}},\mathcal{O}_{\mathcal{S}}[-2]).$
(4.12)

We want to show that it is an $\mathcal{O}$ orientation.

Let $\mathcal{E}$ be a perfect complex on
$\mathcal{X}_{{l}Dol}\times_{\mathcal{S}}\textbf{Spec}A$, for an
$A\in\textbf{cdga}^{\leq 0}_{{\mathcal{S}}}$. We have to show that the
following map

$(-\cap\eta_{A}):R{p_{A}}_{*}\mathcal{E}\longrightarrow(R{p_{A}}_{*}(\mathcal{E}^{\vee}))^{\vee}[-2]=(R{p_{A}}_{*}(\mathcal{E}^{\vee})[2])^{\vee}$
(4.13)

is a quasi-isomorphism of $A$-dg-modules, where
$\eta_{A}:Rp_{A*}\mathcal{O}_{\mathcal{X}_{lDol,A}}\longrightarrow A[-2]$ is
the derived pullback of
$\eta:Rp_{*}\mathcal{O}_{\mathcal{X}_{lDol}}\longrightarrow\mathcal{O}_{\mathcal{S}}[-2]$
under the map ${\rm Spec}\,\leavevmode\nobreak\ A\longrightarrow\mathcal{S}$
and $(R{p_{A}}_{*}(\mathcal{E}^{\vee}))^{\vee}$ is the derived $A$-dual of
$R{p_{A}}_{*}(\mathcal{E}^{\vee})$. Here
$p_{A}:\mathcal{X}_{{l}Dol}\times_{\mathcal{S}}\textbf{Spec}A\longrightarrow\textbf{Spec}\leavevmode\nobreak\
A$ is the projection. The LHS is isormorphic to
$R{r_{A}}_{*}\mathcal{D}_{(E_{A},\phi_{A})}$, where

$\displaystyle\mathcal{D}_{(E_{A},\phi_{A})}:=[E_{A}\xrightarrow{\phi_{A}}E_{A}\otimes\omega_{\mathcal{X}_{A}/\operatorname{Spec}A}]$
$\displaystyle 0\hskip 113.81102pt1$

corresponding to $R{q_{A}}_{*}\mathcal{E}_{A}$ (here $E_{A}$ is sitting in
degree $0$). The RHS is isomorphic to
$(R{r_{A}}_{*}(\mathcal{D}^{\boldsymbol{\vee}}_{(E_{A},\phi_{A})}))^{\vee}$,
where

$\displaystyle\mathcal{D}^{\boldsymbol{\vee}}_{(E_{A},\phi_{A})}:=[E^{\vee}_{A}\xrightarrow{-\phi^{\vee}_{A}}E^{\vee}_{A}\otimes\omega_{\mathcal{X}_{A}/\operatorname{Spec}\leavevmode\nobreak\
A}]$ (4.14) $\displaystyle-2\hskip 99.58464pt-1$ (4.15)

where $\phi^{\vee}_{A}$ is the image of $\phi_{A}$ under the natural
isomorphism $\mathcal{E}nd\leavevmode\nobreak\
E_{A}\cong\mathcal{E}nd\leavevmode\nobreak\ E^{\vee}_{A}$. Now since the
Grothendieck-Serre dual of the complex $\mathcal{D}_{(E_{A},\phi_{A})}$ is the
same as $\mathcal{D}^{\boldsymbol{\vee}}_{(E_{A},\phi_{A})}$, the morphism
(4.13) is quasi-isomorphism. (Notice that the bold-font dual
$\mathcal{D}^{\boldsymbol{\vee}}_{(E_{A},\phi_{A})}$ in (4.14) is to remind
that it is not the ordinary dual)

Therefore, we have shown that $\mathcal{X}_{lDol}$ is $\mathcal{O}$-compact
and $\mathcal{O}$-oriented over ${\mathcal{S}}$ and from [28, Theorem 2.5], it
follows that $\mathsf{Map}_{\mathcal{S}}(\mathcal{X}_{lDol},BGL_{n})$ has a
$0$-shifted relative symplectic structure over ${\mathcal{S}}$. ∎

Let us fix an integer $n$, which will represent rank in the moduli problem.
Let $S$ be a spectrum of a discrete valuation ring over $\mathbb{k}$. We start
with the set-up as in 3.1.1. Let us denote the relative dualising sheaf by
$\omega_{\mathcal{X}/S}$. Let $\mathfrak{M}$ denote the stack of expanded
degeneration bounded by the integer $n$. By abuse of notation, we denote the
universal curve over $\mathfrak{M}$ by $\mathcal{X}_{\mathfrak{M}}$. We denote
the relative logarithmic Dolbeault shape by
$\mathcal{X}_{\mathfrak{M},{l}Dol}$.

######  Proposition 4.12.

The morphism
$\mathsf{Map}_{{}_{\mathfrak{M}}}(\mathcal{X}_{\mathfrak{M},{l}Dol},BGL_{n}\times\mathfrak{M})\longrightarrow\mathfrak{M}$
is a quasi-smooth morphism of derived Artin stacks.

###### Proof.

Let us denote
$\mathsf{Map}_{{}_{\mathfrak{M}}}(\mathcal{X}_{\mathfrak{M},{l}Dol},BGL_{n}\times\mathfrak{M})$
by $M$, for simplicity of notation. The relative tangent complex of the
morphism $\pi:M\longrightarrow\mathfrak{M}$ is given by

$\mathbb{T}_{M/\mathfrak{M}}\cong Rp_{*}\mathcal{E}nd\leavevmode\nobreak\
\mathcal{E}[1]\cong Rr_{*}\mathcal{C}(E,\phi)[1]$ (4.16)

where $\mathcal{C}(E,\phi)$ denotes the following complex

$[\mathcal{E}nd\leavevmode\nobreak\
E\xrightarrow{[-,\phi]}\mathcal{E}nd\leavevmode\nobreak\
E\otimes\omega_{\mathcal{X}_{M}/M}]$ (4.17)

with $\mathcal{E}nd\leavevmode\nobreak\ E$ sitting in the degree $0$. Here
$\mathcal{E}$ denotes the universal sheaf on $\mathcal{X}_{M,{l}Dol}$ and
$(E,\phi)$ denotes the corresponding universal Higgs complex on $M$. From [35,
Theorem 0.3] it follows that, $\mathbb{L}_{M/\mathfrak{M}}$ is a perfect
complex because the projection map $\mathcal{X}_{M,lDol}\longrightarrow M$ is
a proper locally complete intersection morphism. Also, using Grothendieck-
Serre duality we can see that the complex $Rr_{*}\mathcal{C}(E,\phi)[1]$ has
Tor-amplitude in $[-1,1]$. Therefore we conclude that the morphism
$\mathsf{Map}_{{}_{\mathfrak{M}}}(\mathcal{X}_{\mathfrak{M},{l}Dol},BGL_{n}\times\mathfrak{M})\longrightarrow\mathfrak{M}$
is a quasi-smooth morphism of derived Artin stacks. ∎

######  Theorem 4.13.

The derived Artin stack
$\mathsf{Map}_{{}_{\mathfrak{M}}}(\mathcal{X}_{\mathfrak{M},lDol},BGL_{n}\times\mathfrak{M})$
has a natural relative $0$-shifted log-symplectic structure over $S$.

###### Proof.

Consider the composite morphism
$M:=\mathsf{Map}_{{}_{\mathfrak{M}}}(\mathcal{X}_{\mathfrak{M},lDol},BGL_{n}\times\mathfrak{M})\longrightarrow\mathfrak{M}\longrightarrow\mathcal{L}^{0}_{S}$.
The composite morphism induces a log structure on $M$, which is the pull-back
of the log structure from $\mathfrak{M}$. The first map is quasi-smooth and
the second map is smooth with
$\mathbb{L}_{\mathfrak{M}/\mathcal{L}^{0}_{S}}\cong 0$. Therefore it follows
that $\mathbb{L}_{M/\mathcal{L}^{0}_{S}}\cong\mathbb{L}_{M/\mathfrak{M}}\cong
Rr_{*}\mathcal{C}(E,\phi)[1]$. So, the relative log-cotangent complex of the
map $M\longrightarrow S$ is isomorphic to the relative cotangent complex of
the map $M\longrightarrow\mathfrak{M}$. From 4.11, it follows that the stack
$M$ has a $0$-shifted symplectic structure relative to the stack
$\mathfrak{M}$, which translates into the fact that there is a $0$-shifted
log-symplectic structure on $M$ relative to $S$.

The log-symplectic pairing can be described as follows. The stack $BGL_{n}$
has a $2$-shifted sympletcic form given by the trace pairing.

$T_{BGL_{n}}\wedge T_{BGL_{n}}=\mathfrak{g}l^{*}_{n}[1]\wedge
gl^{*}_{n}[1]\xrightarrow{(A,B)\mapsto Trace(AB)}\mathbb{k}[2]$ (4.18)

Now given a map $f_{A}:\mathcal{X}_{lDol,A}\longrightarrow
BGL_{n}\times\operatorname{Spec}A$, It corresponds to a Principal $GL_{n}$
bundle $P$ on $\mathcal{X}_{lDol,A}$ and a Higgs bundle
$(E\xrightarrow{\phi_{A}}E\otimes\omega_{\mathcal{X}_{A}/A})$. We have the
following induced pairing by pulling back (4.18).

$f^{*}_{A}T_{BGL_{n}}\otimes
f^{*}_{A}T_{BGL_{n}}=\text{ad}P[1]\otimes\text{ad}P[1]\xrightarrow{Tr:=Trace}\mathcal{O}_{\mathcal{X}_{lDol,A}}[2]$
(4.19)

Now by pushing it forward by the map $q_{A}$, we get

${{q_{A}}_{*}(\text{ad}P[1])\otimes{q_{A}}_{*}(\text{ad}P[1])}$${(\mathcal{O}_{\mathcal{X}_{A}}\oplus\omega_{\mathcal{X}_{A}/A}[-1])[2]}$
(4.20)

The complex ${q_{A}}_{*}(\text{ad}P[1])=[\mathcal{E}nd\leavevmode\nobreak\
E\xrightarrow{[-,\phi_{A}]}\mathcal{E}nd\leavevmode\nobreak\
E\otimes\omega_{\mathcal{X}_{A}/A}]$ and the pairing in (4.20) can be
described by the following map of complexes.

${-2}$${-1}$${0}$${(\mathcal{E}nd\leavevmode\nobreak\ E)^{\otimes
2}}$${\mathcal{E}nd\leavevmode\nobreak\
E\otimes(\mathcal{E}nd\leavevmode\nobreak\
E\otimes\omega_{\mathcal{X}_{A}/A})\oplus(\mathcal{E}nd\leavevmode\nobreak\
E\otimes\omega_{\mathcal{X}_{A}/A})\otimes\mathcal{E}nd\leavevmode\nobreak\
E}$${(\mathcal{E}nd\leavevmode\nobreak\
E\otimes\omega_{\mathcal{X}_{A}/A})^{\otimes
2}}$${\mathcal{O}_{\mathcal{X}_{A}}}$${\omega_{\mathcal{X}_{A}/A}}$${0}$$\scriptstyle{d_{1}}$$\scriptstyle{Tr}$$\scriptstyle{d_{2}}$$\scriptstyle{Tr}$$\scriptstyle{0}$
(4.21)

where the map $d_{1}:=([-,\phi],[-,\phi])$ and
$d_{2}:=[-,\phi]\otimes\mathbb{1}+\mathbb{1}\otimes[-,\phi]$. Finally, by
pushing forward (4.20) by the map $r_{A}$ and then composing with the
orientation $\eta_{A}$ (see (4.13)), we get our log-symplectic pairing. From
the above description it is evident that this log-symplectic form coincides
with the log-symplectic form discussed in [9]. ∎

##  5 Completeness of the Hitchin map

In this section, we define the Hitchin map on the classical Artin stack of
Gieseker-Higgs bundles $M^{cl}_{Gie}$ (see 8.1.2). We prove that the Hitchin
map is complete. This result was proved in [1] for the Hitchin map on the
moduli scheme in the case where rank and degree are co-prime. We prove it here
for the moduli stack and the argument does not require us to assume that the
rank and degree are coprime. For various definitions related to Gieseker-Higgs
bundles, classical Artins stack of Gieseker-Higgs bundles, and its local
structure, and dimension refer to the appendix.

######  Definition 5.1.

[33, p7] A morphism of Artin stacks $f:\mathcal{X}\longrightarrow\mathcal{Y}$
is called complete if given any discrete valuation ring $A$ with function
field $K$ and any commutative diagram with solid arrows of the following type

${\operatorname{Spec}\leavevmode\nobreak\
K}$${\mathcal{X}}$${\operatorname{Spec}\leavevmode\nobreak\
A}$${\mathcal{Y}}$$\scriptstyle{f}$ (5.1)

$\exists$ a finite field extension $K\longrightarrow K^{\prime}$ with
$A^{\prime}$ the integral closure of $A$ in $K^{\prime}$ and a lift (the
dotted arrow) $Spec\leavevmode\nobreak\ A^{\prime}\longrightarrow\mathcal{X}$
making all the triangles in the diagram commute.

${\operatorname{Spec}\leavevmode\nobreak\
K^{\prime}}$${\operatorname{Spec}\leavevmode\nobreak\
K}$${\mathcal{X}}$${\operatorname{Spec}\leavevmode\nobreak\
A^{\prime}}$${\operatorname{Spec}\leavevmode\nobreak\
A}$${\mathcal{Y}}$$\scriptstyle{f}$ (5.2)

A morphism of derived Artin stacks is called complete if the underlying
morphism of ordinary Artin stacks is complete.

######  Definition 5.2.

(Hitchin map) We recall the following two well-known definitions

  1. 1.

$B:=\oplus^{n}_{i=1}R^{0}\pi_{*}\omega_{\mathcal{X}/S}$. It is a vector bundle
over the spectrum of a discrete valuation ring $S$. It is well known as the
Hitchin Base.

  2. 2.

There is a natural map $h:M\longrightarrow B$, which sends a family of Higgs
bundles $(\mathcal{E},\phi)$ to
$(-Trace(\phi),Trace(\wedge^{2}\phi),...,(-1)^{i}Trace(\wedge^{i}\phi),...,(-1)^{n}Trace(\wedge^{n}\phi))$.

######  Theorem 5.3.

The morphism $h:M^{cl}_{Gie}\longrightarrow B$ is complete.

###### Proof.

Let $T:=Spec\leavevmode\nobreak\ A$ be a spectrum of a discrete valuation ring
with function field $K$. Let us denote $T^{o}:=Spec\leavevmode\nobreak\ K$.
Suppose we are given a commutative diagram

${T^{o}}$${M^{cl}_{Gie}}$${T}$${B}$$\scriptstyle{h}$ (5.3)

For the proof of completeness, we need to recall the defintion and few facts
about of the Artin stack (and the construction of their atlas) of torsion-free
sheaves/ torsion-free Higgs pairs on the origianal family of curves
$\mathcal{X}/S$ (see Appendix 8.1.1). There is a morphism
$\theta:M^{cl}_{Gie}\rightarrow TFH(\mathcal{X}/S)$ given by the pushforward
of the Hitchin map under the modification map of Gieseker curves
$\mathcal{X}^{mod}\rightarrow\mathcal{X}$. Therefore, from (5.3) we get the
following commutative diagram (without the dotted arrow)

${T^{o}}$${TFH(\mathcal{X}/S)}$${T}$ (5.4)

Using [12, Proposition 6] and [20, Lemma 6.5], we can get the dotted
arrow/extension. First of all, notice that the surface
$\mathcal{X}\times_{S}T$ is either a normal surface with an isolated
singularity of type $\frac{\mathbb{k}[[x,y,t]]}{xy-t^{d}}$ or isomorphic to
the product $X_{0}\times T$ depending on whether the spectrum of a discrete
valuation ring $T$ is faithfully flat over $S$ or not.

In the first case, it is a routine checking that [12, Proposition 6] holds for
the normal surface, i.e., any vector bundle $\mathcal{E}_{K}$ over the generic
fibre $\mathcal{X}\times_{S}T^{o}$ can be extended to the surface
$\mathcal{X}\times_{S}T$ as a family of torsion-free sheaves (Let us denote it
by $\mathcal{F}$). Using [20, Lemma 6.5] we also extend the Higgs field over
the complement of the node in the surface $\mathcal{X}\times_{S}T$. Let
$i:\mathcal{X}\times_{S}T^{o}\hookrightarrow\mathcal{X}\times_{S}T$ denote the
inclusion.

${0}$${\mathcal{F}}$${i_{*}\mathcal{E}_{K}}$${\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}}}$${0}$${0}$${\mathcal{F}\otimes\omega_{\mathcal{X}_{T}/T}}$${i_{*}\mathcal{E}_{K}\otimes\omega_{\mathcal{X}_{T}/T}}$${\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}}\otimes\omega_{\mathcal{X}_{T}/T}}$${0}$
(5.5)

Consider the composite map
$\mathcal{F}\longrightarrow\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}}\otimes\omega_{\mathcal{X}_{T}/T}$.
Since we have extended the Higgs field on $\mathcal{F}$ except at the node,
therefore this composite map vanishes at the generic point of the closed fibre
$X_{0}$. Now if we show that the sheaf
$\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}}$ is torsion-free, then it will prove
that the Higgs field extends everywhere. To see this we restrict the upper
exact sequence to the closed fibre $X_{0}$. We get

$0\longrightarrow
Tor^{1}_{{}_{\mathcal{X}_{T}}}(\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}},\mathcal{O}_{X_{0}})\longrightarrow\mathcal{F}|_{X_{0}}\longrightarrow(i_{*}\mathcal{E}_{K})|_{X_{0}}\longrightarrow\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}}\longrightarrow
0$ (5.6)

One can easily check that
$Tor^{1}_{{}_{\mathcal{X}_{T}}}(\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}},\mathcal{O}_{X_{0}})\cong\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}}\otimes\mathcal{O}_{{}_{\mathcal{X}_{T}}}(-X_{0})$.
Since $\mathcal{O}_{{}_{\mathcal{X}_{T}}}(-X_{0})$ is a locally-free sheaf,
$\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}}$ is torsion-free if and only if
$Tor^{1}_{{}_{\mathcal{X}_{T}}}(\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}},\mathcal{O}_{X_{0}})$
is torsion-free. But since
$Tor^{1}_{{}_{\mathcal{X}_{T}}}(\frac{i_{*}\mathcal{E}_{K}}{\mathcal{F}},\mathcal{O}_{X_{0}})$
is a subsheaf of $\mathcal{F}|_{X_{0}}$, therefore it is torsion-free.

In the second case, the surface $\mathcal{X}_{T}\cong X_{0}\times T$.
Therefore the normalisation of $\mathcal{X}_{T}$ is isomorphic to
$\tilde{X_{0}}\times T$. Let us denote the normalisation morphism by $n$. From
[DI, Proposition 3.2], it follows that we can extend the pullback of the
generic Higgs bundle $(n^{*}\mathcal{E}_{K},n^{*}\phi_{K})$ as a good
Generalised parabolic Hitchin pair which in turn gives us the desired torsion-
free Hitchin pair on the surface $\mathcal{X}_{T}$.

So now since we have extended the generic Higgs bundle as a torsion-free
Hitchin pair, we have the following commutative diagram

${T^{o}}$${M^{cl}_{Gie}}$${T}$${TFH(\mathcal{X}/S)}$$\scriptstyle{\theta}$
(5.7)

Let $R^{\Lambda}_{S}\xrightarrow{smooth}TFH(\mathcal{X}/S)$ denote a suitable
Quot scheme (we omit the integer $m$ from the notation) so that the image of
the map $T\longrightarrow TFH(\mathcal{X}/S)$ lies inside the image of the
smooth map $R^{\Lambda}_{S}\longrightarrow TFH(\mathcal{X}/S)$ (see 8.1.1).
Therefore, there exists a finite cover
$T^{\prime}:=\operatorname{Spec}A^{\prime}$ of $T$ such that the composite map
$T^{\prime}\longrightarrow TFH(\mathcal{X}/S)$ lifts to a map
$T^{\prime}\longrightarrow R^{\Lambda}_{S}$. We denote by
$(T^{\prime})^{o}:=SpecK^{\prime}$, where $K^{\prime}$ is the function field
of $A^{\prime}$ ($K^{\prime}$ is a finite extension of $K$). Let
$\mathcal{Y}^{H}_{S}$ denote the Quot scheme for Gieseker-Hitchin pairs
constructed over the Quot scheme $R^{\Lambda}_{S}$ (see 8.1.4). Let us denote
the projection map by $\tilde{\theta}:\mathcal{Y}^{H}_{S}\longrightarrow
R^{\Lambda}_{S}$. Then there exists a lift
$(T^{\prime})^{o}\longrightarrow\mathcal{Y}^{H}_{S}$ of the map
$(T^{\prime})^{o}\longrightarrow M^{cl}_{Gie}$. Therefore, we get a following
commutative diagram

${(T^{\prime})^{o}}$${\mathcal{Y}^{H}_{S}}$${T^{\prime}}$${R^{\Lambda}_{S}}$$\scriptstyle{\tilde{\theta}}$
(5.8)

From [1, Proposition 5.11], it follows that the map
$\mathcal{Y}^{H}_{S}\longrightarrow R^{\Lambda}_{S}$ is proper (see Remarks
5.4 and 5.5 for an outline of the argument). So there exists a lift
$T^{\prime}\longrightarrow\mathcal{Y}^{H}_{S}$. The composite map
$T^{\prime}\longrightarrow\mathcal{Y}^{H}_{S}\longrightarrow M^{cl}_{Gie}$ is
our desired extension. ∎

######  Remrk 5.4.

Let $Y\longrightarrow R$ be a quasi-projective morphism of separated schemes
over a spectrum of a discrete valuation ring $S$ and isomorphic over the
generic point of $S$. Suppose any map $T\longrightarrow R$ flat over $S$ can
be lifted to a map $T\longrightarrow Y$. Then we claim that the map
$Y\longrightarrow R$ is proper. To see this, take the closure of $Y$ inside
some relative projective space (relative over $R$) to get a projective
morphism $\overline{Y}\longrightarrow R$. Take any element
$y\in\overline{Y}\setminus Y$. Then there exists a map
$T\longrightarrow\overline{Y}$ (here $T$ is a d.v.r) passing through $y$ and
flat over $S$. Therefore we get a map $T\longrightarrow R$. But then this map
can be lifted to a map $T\longrightarrow Y$. Let $y^{\prime}\in Y$ be the
image of the closed point of $T$ under this map. But, by separatedness of the
$\overline{Y}$, we must have $y=y^{\prime}$ and therefore $\overline{Y}=Y$ and
the map $Y\longrightarrow R$ is proper.

######  Remrk 5.5.

Let $\mathcal{X}\longrightarrow S$ be the original family of curves and let
$(\mathcal{F},\phi)$ be a flat family of torsion-free Higgs pair on
$\mathcal{X}_{T}\longrightarrow T$, where $T\longrightarrow S$ is a surjective
map of spectrum of a discrete valuation rings. Notice that the surface
$\mathcal{X}_{T}$ may have a singularity of type
$\frac{\mathbb{k}[|x,y,t|]}{xy-t^{n}}$. Let
$r:\mathcal{X}^{{res}}_{T}\longrightarrow\mathcal{X}_{T}$ be the minimal
resolution of singularities. Then from [15, Proposition 6.5] it follows that
the vector bundle $\mathcal{E}:=\frac{r^{*}\mathcal{F}}{Torsion}$ has the
property that $r_{*}\mathcal{E}\cong\mathcal{F}$. Notice also that by
construction and by the property that $r_{*}\mathcal{E}\cong\mathcal{F}$, it
follows that the natural map $r^{*}r_{*}\mathcal{E}\longrightarrow\mathcal{E}$
is a surjective map of sheaves, which means $\mathcal{E}|_{R}$ is globally
generated. Here $R$ denotes the chain of rational curves in
$\mathcal{X}^{{res}}_{T}$. Hence, by definition (Def. 8.1 and Remark 8.2), the
vector bundle $\mathcal{E}$ is a Gieseker vector bundle. The pullback of the
Higgs field $\phi$ defines a Higgs field on $\mathcal{E}$.

##  6 Flatness of the Hitchin map

In this section, we study the reduced global nilpotent cone of $M^{cl}_{Gie}$
(see 8.1.2), which is the ordinary scheme theoretic fibre over the point $0\in
B$ equipped with the reduced induced structure. We prove that every
irreducible component of of the reduced nilpotent cone has an open subset
which is an isotropic substack of $M$ (the derived stack of Higgs bundles)
with respect to its log-symplectic form. We use this to compute the dimension
of the reduced nilpotent cone and to show that the Hitchin map is flat.

######  Definition 6.1.

  1. 1.

The nilpotent cone is the Hitchin fibre over the zero section $0_{S}$ of
$B\longrightarrow S$, i.e., the following fibre product (fibre product of
classical Artin stacks)

${\mathcal{N}ilp:=0_{S}\times_{B}M^{cl}_{Gie}}$${M^{cl}_{Gie}}$${0_{S}}$${B}$
(6.1)

The fibre product is, in general, non-reduced.

  2. 2.

We can put the reduced induced structure on $\mathcal{N}ilp$. We refer to the
resulting stack as the reduced nilpotent cone and denote it by
$\mathcal{N}ilp^{red}$. It is a stack over the spectrum of a discrete
valuation ring $S$. We denote the closed fibre by $\mathcal{N}ilp^{red}_{0}$.

######  Remrk 6.2.

First, we notice that $\mathcal{N}ilp^{red}_{0}$ may have several irreducible
components. Since it is reduced, the generic points are smooth points. A
priori, a generic point is a tuple $(X_{s},\mathcal{E},\phi)$, where $X_{s}$
is a Gieseker curve with $s$ many $\mathbb{P}^{1}$’s $(0\leq s\leq n)$,
$\mathcal{E}$ is a Gieseker vector bundle and $\phi$ is a nilpotent Higgs
field, i.e., $\phi^{n}=0$. Given a nilpotent Higgs field, we get a canonical
filtration by saturated torsion-free sub-sheaves

$\mathcal{E}^{0}:=0\subsetneq\mathcal{E}^{1}:=Ker\phi\subsetneq\mathcal{E}^{2}:=Ker\phi^{2}\subsetneq\dots\subsetneq\mathcal{E}^{k}:=Ker\phi^{k}\subsetneq\mathcal{E}^{k+1}:=\mathcal{E}$
(6.2)

for some integer $k$ such that $\phi^{k+1}=0$.

######  Definition 6.3.

(Type of a nilpotent torsion-free Higgs pair on the nodal curve $X_{0}$) Let
$(\mathcal{F},\psi)$ be a torsion-free Higgs pair on $X_{0}$, where
$\mathcal{F}$ is a torsion-free coherent sheaf on $X_{0}$ and
$\psi:\mathcal{F}\longrightarrow\mathcal{F}\otimes\omega_{X_{0}}$ is a map of
coherent sheaves. Suppose that the Higgs field is nilpotent i.e.,
$\psi^{n}=0$, where $n$ denotes the rank of the torsion-free sheaf
$\mathcal{F}$. Then as above we get a natural flag of saturated subsheaves of
$\mathcal{F}$.

$\mathcal{F}^{0}:=0\subsetneq\mathcal{F}^{1}:=Ker\psi\subsetneq\mathcal{F}^{2}:=Ker\psi^{2}\subsetneq\dots\subsetneq\mathcal{F}^{k}:=Ker\psi^{k}\subsetneq\mathcal{F}^{k+1}:=\mathcal{F}$
(6.3)

For every $1\leq i\leq k+1$, we define
$n_{i}:=\textsf{rank}\leavevmode\nobreak\
\Large(\frac{\mathcal{F}^{i}}{\mathcal{F}^{i-1}}\Large)$ and
$d_{i}:=\deg\leavevmode\nobreak\
\Large(\frac{\mathcal{F}^{i}}{\mathcal{F}^{i-1}}\Large)$. We say that the
nilpotent Higgs bundle $(\mathcal{F},\psi)$ is of type
$\\{(n_{i},d_{i})\\}^{i=k+1}_{i=1}$. We denote it by
$\textsf{Type}\leavevmode\nobreak\ \leavevmode\nobreak\ (\mathcal{F},\psi)$.

######  Definition 6.4.

(Type of a nilpotent Higgs bundle on a Gieseker curve $X_{s}$ for some
$s\in[0,n]$) Let $s\in[0,n]$ be an integer. Let $\pi_{s}:X_{s}\longrightarrow
X_{0}$ be the Gieseker curve with exactly $s$ many $\mathbb{P}^{1}$’s. Let
$(\mathcal{E},\phi)$ be a Gieseker-Higgs bundle on $X_{s}$. Then, by
definition, $((\pi_{s})_{*}\mathcal{E},(\pi_{s})_{*}\phi)$ is a nilpotent
torsion-free Higgs pair on $X_{0}$. We define
$\textsf{Type}\leavevmode\nobreak\ \leavevmode\nobreak\
(X_{s},\mathcal{E},\phi):=\textsf{Type}\leavevmode\nobreak\
\leavevmode\nobreak\ ((\pi_{s})_{*}\mathcal{E},(\pi_{s})_{*}\phi)$.

######  Lemma 6.5.

Let $(\pi_{s}:\mathcal{X}_{s}\rightarrow X_{0},\mathcal{E},\phi)$ be a
nilpotent Gieseker-Higgs bundle. Then we have the following induced filtration
as in (6.2).

$\mathcal{E}^{0}:=0\subsetneq\mathcal{E}^{1}:=Ker\phi\subsetneq\mathcal{E}^{2}:=Ker\phi^{2}\subsetneq\dots\subsetneq\mathcal{E}^{k}:=Ker\phi^{k}\subsetneq\mathcal{E}^{k+1}:=\mathcal{E}$
(6.4)

The induced nilpotent torsion-free Higgs pairs
$(\mathcal{F}:=(\pi_{s})_{*}\mathcal{E},\psi:=(\pi_{s})_{*}\phi)$ also has a
natural filtration as in (6.3).

$\mathcal{F}^{0}:=0\subsetneq\mathcal{F}^{1}:=Ker\psi\subsetneq\mathcal{F}^{2}:=Ker\psi^{2}\subsetneq\dots\subsetneq\mathcal{F}^{k}:=Ker\psi^{k}\subsetneq\mathcal{F}^{k+1}:=\mathcal{F}$
(6.5)

Then $(\pi_{s})_{*}\mathcal{E}^{i}\cong\mathcal{F}^{i}$ for every $1\leq i\leq
k+1$.

###### Proof.

For every $i\in[1,k+1]$, we have morphisms
$\phi^{i}:\mathcal{E}\longrightarrow\mathcal{E}\otimes\omega^{\otimes
i}_{X_{s}}$ and
$\psi^{i}:\mathcal{F}\longrightarrow\mathcal{F}\otimes\omega^{\otimes
i}_{X_{0}}$. Moreover, $\textsf{Ker}(\phi^{i})=\mathcal{E}^{i}$ and
$\textsf{Ker}(\psi^{i})=\mathcal{F}^{i}$. Let $\sigma$ be a local section of
$(\pi_{s})_{*}\mathcal{E}^{i}$ on a neighbourhood $U$ of the node of $X_{0}$.
It is an element of $\mathcal{E}^{i}((\pi_{s})^{-1}(U))$. Therefore the
section $\phi^{i}(\sigma)=0$ on the open set
$(\pi_{s})^{-1}(U)\cap\tilde{X_{0}}=U^{o}$, where $U^{o}$ denotes the
complement of the node of $X_{0}$. Therefore $\psi^{i}(\sigma)=0$ on $U^{o}$,
because $\psi=\phi$ on $U^{o}$. Since $U^{o}$ is dense in $U$, therefore
$\psi^{i}(\sigma)=0$ on $U$ and $\sigma\in\mathcal{F}^{i}$. Therefore,
$(\pi_{s})_{*}\mathcal{E}^{i}\subseteq\mathcal{F}^{i}$ for all $i\in[1,k+1]$.

For the converse, let $\sigma\in\mathcal{F}^{i}(U)$. Since
$\mathcal{F}^{i}\subseteq\mathcal{F}=(\pi_{s})_{*}\mathcal{E}$, we have
$\sigma\in\mathcal{E}((\pi_{s})^{-1}(U))$. Since $\psi^{i}(\sigma)=0$,
therefore $\phi^{i}(\sigma)=0$ on $(\pi_{s})^{-1}(U^{o})$ and hence, by
continuity, on $(\pi_{s})^{-1}(U)\cap\tilde{X_{0}}$. Now notice that
$\phi^{i}(\sigma)$ is a section of $\mathcal{E}\otimes\omega_{X_{s}}$. The
bundle $\mathcal{E}\otimes\omega_{X_{s}}$ is a Gieseker-vector bundle becase
$\omega_{X_{s}}|_{R}\cong\mathcal{O}_{R}$, where $R$ the chain of
$\mathbb{P}^{1}$’s in $X_{s}$. Therefore since the section vanishes at the two
points $\tilde{X_{0}}\cap R$, therefore it must vanish everywhere. This
implies that
$\sigma\in\mathcal{E}^{i}((\pi_{s})^{-1}(U))=((\pi_{s})_{*}\mathcal{E}^{i})(U)$.
Therefore, $\pi_{s*}(\mathcal{E}^{i})=\mathcal{F}^{i}$ for all $i\in[1,k+1]$.
∎

######  Definition 6.6.

We define $\mathcal{N}ilp^{sm,gen}_{0}$ to be the open substack of
$\mathcal{N}ilp^{red}_{0}$ consisiting of nilpotent Higgs bundles
$(X_{s},\mathcal{E},\phi)$ which satisfies the following two conditions.

  1. 1.

the $\textsf{Type}(X_{s},\mathcal{E},\phi)$ is of general type i.e., it is the
same as the Type of one of the generic points of the reduced nilpotent cone.

  2. 2.

$(X_{s},\mathcal{E},\phi)$ is a smooth point of $\mathcal{N}ilp^{red}_{0}$.

######  Lemma 6.7.

Let $(\mathcal{X}^{mod},\mathcal{E},\phi)$ denote the restriction of the
universal modification, universal vector bundle and the universal Higgs field
to $\mathcal{N}ilp^{sm,gen}_{0}$. Consider the filtration 6.2 of $\mathcal{E}$
induced by the Higgs field $\phi$. Then the sheaves $\mathcal{E}^{i}$’s in the
filtration are all flat over $\mathcal{N}ilp^{sm,gen}_{0}$.

###### Proof.

To see this consider any connected component $C$ of
$\mathcal{N}ilp^{sm,gen}_{0}$. Then for any element $c:{\rm
Spec}\,\mathbb{k}\longrightarrow C$, we have the following exact sequence.

$0\longrightarrow\textsf{K}_{c}^{i}\longrightarrow
c^{*}\mathcal{E}\xrightarrow{(c^{*}\phi)^{i}}c^{*}\mathcal{E}\otimes\omega_{c^{*}\mathcal{X}^{mod}}\longrightarrow\textsf{CK}^{i}_{c}\longrightarrow
0$

Here $\textsf{K}_{c}^{i}$ and $\textsf{CK}^{i}_{c}$ denote the Kernel and
Cokernel of the map $(c^{*}\phi)^{i}$, respectively. We also have the
following exact sequence.

$0\longrightarrow\underline{\textsf{K}}^{i}_{c}\longrightarrow(\pi_{s})_{*}(c^{*}\mathcal{E})\xrightarrow{(\pi_{s})_{*}((c^{*}\phi)^{i})=((\pi_{s})_{*}(c^{*}\phi))^{i}}(\pi_{s})_{*}(c^{*}\mathcal{E}\otimes\omega_{c^{*}\mathcal{X}^{mod}})\longrightarrow\underline{\textsf{CK}}^{i}_{c}\longrightarrow
0$

Here $\underline{\textsf{K}}^{i}_{c}$ and $\underline{\textsf{CK}}^{i}_{c}$
are Kernel and Cokernel, respectively. Notice from Lemma 6.5 it follows that
$\underline{\textsf{K}}^{i}_{c}=(\pi_{s})_{*}(\textsf{K}^{i}_{c})$. Since the
$\textsf{Type}\leavevmode\nobreak\ \leavevmode\nobreak\
(c^{*}{\mathcal{X}^{mod}},c^{*}\mathcal{E},c^{*}\phi)$ is constant over $C$,
therefore the Hilbert polynomial of $\underline{\textsf{K}}^{i}_{c}$ does not
depend on $c\in C$. Therefore, the Hilbert Polynomial of
$\underline{\textsf{CK}}^{i}_{c}$ does not depend on $c\in C$. Therefore,
$\underline{\textsf{CK}}^{i}$ is flat over $C$. Therefore,
$\underline{\textsf{K}}^{i}$ is flat over $C$ and therefore $\textsf{K}^{i}$
is flat over $C$. ∎

######  Proposition 6.8.

The tangent complex of $\mathcal{N}ilp^{red}_{0}$ at a point
$(X_{s},\mathcal{E},\phi)\in\mathcal{N}ilp^{sm,gen}_{0}$ is given by
$R\Gamma(\mathcal{SC}{(\mathcal{E},\phi)})$, where
$\mathcal{SC}{(\mathcal{E},\phi)}$ is the following complex of sheaves on
$X_{s}$.

$\mathcal{SC}{(\mathcal{E},\phi)}:=[SC(\mathcal{E},\phi)\xrightarrow{[-,\phi]}SC(\mathcal{E},\phi)\otimes\omega_{X_{s}}]$
(6.6)

where $SC(\mathcal{E},\phi)\subseteq\mathcal{E}nd\mathcal{E}$ is the sheaf of
local sections $s$ of $\mathcal{E}nd\mathcal{E}$ such that
$s(\mathcal{E}^{i})\subseteq\mathcal{E}^{i-1}$ for $i=1,\dots,k+1$.

###### Proof.

We choose a trivialisation $X_{k}=\cup_{i\in I}V_{i}$ of the vector bundle
$\mathcal{E}$ and the line bundle $\omega_{X_{s}}$. Then a first-order
infinitesimal deformation (as a Higgs bundle) of $(\mathcal{E},\phi)$ can be
described as a pair $(s_{ij},t_{i})$, where
$s_{ij}\in\Gamma(V_{ij},\mathcal{E}nd\mathcal{E})$ and
$t_{i}\in\Gamma(V_{i},\mathcal{E}nd\mathcal{E}\otimes\omega_{X_{s}})$
satisfying $s_{ij}+s_{jk}=s_{ik}$ and $t_{i}-t_{j}=[s_{ij},\phi]$ (see [5,
Theorem 2.3] and [6, Proposition 3.1.2]).

Let $\operatorname{Spec}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{k}[\epsilon]\longrightarrow\mathcal{N}ilp^{sm,gen}_{0}$ be a map such
that the image of the closed point is given by the Higgs bundle
$(X_{s},\mathcal{E},\phi)$. Let us denote by the corresponding first order
infinitesimal deformation of the nilpotent Higgs bundle $(\mathcal{E},\phi)$
by $(\mathcal{E}[\epsilon],\phi[\epsilon])$. We assume that
$(\mathcal{E}[\epsilon],\phi[\epsilon])$ is a nilpotent Higgs bundle. We
define $\mathcal{E}^{i}[\epsilon]:=Ker\leavevmode\nobreak\
(\phi[\epsilon])^{i}$ for $i\in[1,k+1]$. Since the induced flag
$\mathcal{E}^{\bullet}$ is flat over $\mathcal{N}ilp^{sm,gen}_{0}$ (see Lemma
6.7), therefore $\mathcal{E}^{\bullet}[\epsilon]$ is flat over
$\operatorname{Spec}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{k}[\epsilon]$. Since $\mathcal{E}^{i}[\epsilon]$ is flat over
$\operatorname{Spec}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{k}[\epsilon]$, therefore it is an extension of $\mathcal{E}^{i}$ by
$\mathcal{E}^{i}$.

${0}$${\mathcal{E}}$${\mathcal{E}[\epsilon]}$${\mathcal{E}}$${0}$${0}$${\mathcal{E}^{i}}$${\mathcal{E}^{i}[\epsilon]}$${\mathcal{E}^{i}}$${0}$
(6.7)

Now it is straightforward to check that for $(s_{ij},t_{i})$ to be an
infinitesimal deformation of $(\mathcal{E},\phi)$ as a nilpotent Higgs field,
it has to satisfy the extra condition that
$s_{ij}(\mathcal{E}^{\bullet})\subseteq\mathcal{E}^{\bullet-1}$, where
$\mathcal{E}^{\bullet}$ is the flag (6.2). This means that
$s_{ij}\in\Gamma(V_{ij},SC(\mathcal{E},\phi))$ for all $i,j$ and therefore,
the Proposition follows. ∎

######  Theorem 6.9.

  1. 1.

The Hitchin map $h:M^{cl}_{Gie}\longrightarrow B$ is surjective.

  2. 2.

The sub-stack $\mathcal{N}ilp^{sm,gen}$ is relatively isotropic over the
spectrum of a discrete valuation ring $S$.

  3. 3.

The Hitchin map $h:M^{cl}_{Gie}\longrightarrow B$ is flat.

###### Proof.

Proof of (1): To prove the surjectivity we use the spectral correspondence [1,
Lemma 2.4]. Given any point $a_{\bullet}:=(a_{1},\dots,a_{n})\in B$, we
consider the function
$s(a_{\bullet}):=t^{n}+a_{1}t^{n-1}+\dots+a_{n-1}t+a_{n}$ on the total space
$Tot(\omega_{X_{0}})$. Here $t$ denotes the canonical section of
$f^{*}\omega_{X_{0}}$, where $f:Tot(\omega_{X_{0}})\longrightarrow X_{0}$ is
the projection map. Then the vanishing locus $V(s(a_{\bullet}))$ defines a
closed sub-scheme of $Tot(\omega_{X_{0}})$. Notice that $Tot(\omega_{X_{0}})$
is only quasi-projective and it is an open subscheme of
$Z:=\mathbb{P}(\omega_{X_{0}}^{*}\oplus\mathcal{O}_{X_{0}})$. But since
$s(a_{\bullet})$ is a monic polynomial, therefore the closure of
$V(s(a_{\bullet}))$ in $Z$ is $V(s(a_{\bullet}))$ itself. In particular,
$V(s(a_{\bullet}))$ is a closed subscheme in $Z$ such that
$V(s(a_{\bullet}))\cap D_{\infty}=\emptyset$, where $D_{\infty}:=Z\setminus
Tot(\omega_{X_{0}})$. Therefore, by spectral correspondence, it follows that
any rank $1$ locally free sheaf on $V(s(a_{\bullet}))$ corresponds to a Higgs
bundle $(\mathcal{E},\phi)$ on the nodal curve $X_{0}$ whose characteristic
polynomial is given by $a_{\bullet}:=(a_{1},\dots,a_{n})\in B$. Therefore, the
Hitchin map is surjective.

Proof of (2): We want to show that $\mathcal{N}ilp^{sm,gen}$ is relatively
isotropic over $S$. First, we notice that the smooth locus of the generic
fibre of $\mathcal{N}ilp^{red}\longrightarrow S$ is isotropic, and its
dimension is equal to $n^{2}(g-1)$, which is the same as the dimension of the
stack of rank $n$ vector bundles on a smooth projective curve of genus $g$.
Therefore, we have dim $\mathcal{N}ilp^{red}_{0}\geq n^{2}(g-1)$. We want to
show that the $\mathcal{N}ilp^{red,sm}_{0}$ is an isotropic substack in
$M^{cl}_{Gie,0}$. The tangent complex at a point of
$\mathcal{N}ilp^{red,sm}_{0}$ is given by (6.6). We want to show that the
following composite morphism is quasi-isomorphic to $0$.

$R\Gamma(\mathcal{SC}(\mathcal{E},\phi))[1]\otimes
R\Gamma(\mathcal{SC}(\mathcal{E},\phi))[1]\longrightarrow
R\Gamma(\mathcal{C}(\mathcal{E},\phi))[1]\otimes
R\Gamma(\mathcal{C}(\mathcal{E},\phi))[1]\longrightarrow
R\Gamma(\omega_{X_{s}}[1])$ (6.8)

Notice that the morphism

$R\Gamma(\mathcal{C}(\mathcal{E},\phi))[1]\otimes
R\Gamma(\mathcal{C}(\mathcal{E},\phi))[1]\longrightarrow
R\Gamma(\omega_{X_{s}}[1])$

is given by the log-symplectic pairing (Theorem 4.13). If we show that the
composite morphism 6.8 is quasi-isomorphic to $0$, this will imply that the
$\mathcal{N}ilp^{red,sm}_{0}$ is an isotropic substack. To show this, we
choose a trivialisation $X_{s}=\cup_{i\in I}V_{i}$ of the vector bundle
$\mathcal{E}$ and of the line bundle $\omega_{X_{s}}$. With respect to this
cover, the complex $R\Gamma(\mathcal{SC}(\mathcal{E},\phi))[1]$ is quasi-
isomorphic to the following Čech complex

$\prod SC(\mathcal{E},\phi)(V_{i})\longrightarrow\prod
SC(\mathcal{E},\phi)(V_{ij})\oplus\prod(SC(\mathcal{E},\phi)\otimes\omega)({V_{i}})\longrightarrow\prod(SC(\mathcal{E},\phi)\otimes\omega)(V_{ij})\oplus\prod
SC(\mathcal{E},\phi)(V_{ijk})$,

(6.9)

where the term $\prod SC(\mathcal{E},\phi)({V_{i}})$ is in degree $-1$.

For convenience, let us write (6.8) in the following way.

${R\Gamma(\mathcal{SC}(\mathcal{E},\phi))[1]}$${R\Gamma(\mathcal{C}(\mathcal{E},\phi))[1]}$${(R\Gamma(\mathcal{C}(\mathcal{E},\phi))[1])^{\vee}}$${(R\Gamma(\mathcal{SC}(\mathcal{E},\phi))[1])^{\vee}}$$\scriptstyle{\omega^{\flat}}$
(6.10)

The arrow in the middle is given by the log-symplectic pairing (4.13).

Notice that the composite map can only be non-nozero on three cohomologies,
namely in the degree $-1$, $0$ and $1$.

Case 1: the induced map $H^{0}(\omega^{\flat})$: An element of $\prod
SC(\mathcal{E},\phi)(V_{ij})\oplus\prod(SC(\mathcal{E},\phi)\otimes\omega)({V_{i}})$
is given by $\\{(s_{ij},t_{i})\\}_{i,j\in I}$, where $(s_{ij},t_{i})$, where
$s_{ij}\in\Gamma(V_{ij},\mathcal{E}nd\mathcal{E})$ and
$t_{i}\in\Gamma(V_{i},\mathcal{E}nd\mathcal{E}\otimes\omega_{X_{s}})$ such
that $s_{ij}(\mathcal{E}^{\bullet})\subseteq\mathcal{E}^{\bullet-1}$, where
$\mathcal{E}^{\bullet}$ is the flag (6.2). Then from (6.8) it follows that the
pairing of two elements $\\{(s_{ij},t_{i})\\}$ and
$\\{(s^{\prime}_{ij},t^{\prime}_{i})\\}$ is given by
$\\{T_{ij}:=Trace(s_{ij}\circ t^{\prime}_{j}-t_{i}\circ
s^{\prime}_{ij})\in\omega_{X_{s}}(V_{ij})\\}_{i,j\in I}$.

Claim: The section $T_{ij}$ vanishes for all $i,j\in I$.

Proof of the claim: We first observe that, we can choose the trivialisation
$X_{s}=\cup_{i\in I}V_{i}$ of $\mathcal{E}$ and $\omega_{X_{s}}$ in such a way
that

  1. 1.

$V_{i}$ contains at most one node for each $i\in I$,

  2. 2.

$V_{i}$’s are connected.

We notice that $V_{ij}$’s also contain at most one node. Suppose, $V_{ij}$
contains a node $p$. Then $V_{ij}=V^{1}_{ij}\coprod V^{2}_{ij}$, the union of
two smooth irreducible components of $V_{ij}$. We denote by
$V^{o}_{ij}:=V_{ij}\setminus p$, $V^{1,o}_{ij}:=V^{1}_{ij}\setminus p$ and
$V^{2,o}_{ij}:=V^{2}_{ij}\setminus p$. We consider the restriction of the
section $T_{ij}$ to the two open subsets $V^{1,o}_{ij}$ and $V^{2,o}_{ij}$.
Notice that the restrictions of the flag (6.2) to these two open subsets are
all sub-bundles. We notice that $SC(\mathcal{E}|_{V^{1,o}_{ij}},\phi)$ is the
nilpotent part of the parabolic sub-algebra of
$\mathcal{E}nd\mathcal{(}E|_{V^{1,o}_{ij}})$ given by the natural flag of sub-
bundles induced by $\phi$. Therefore it is clear that the trace pairing is $0$
i.e., $(T_{ij})|_{V^{1,o}_{ij}}=0$. Similarly, $(T_{ij})|_{V^{2,o}_{ij}}=0$.
Since $V^{o}_{ij}$ is a dense open subset of $V_{ij}$, therefore, by
continuity, $T_{ij}=0$.

If a $V_{ij}$ does not contain any node then the proof is similar.

Case 2: the induced map $H^{-1}(\omega^{\flat})$: In this case, we look at the
pairing of $\\{t_{i}\in SC(\mathcal{E},\phi)\otimes\omega_{X_{s}}\\}$ and
$\\{s_{ij}\in SC(\mathcal{E},\phi)\\}$. The pairing is given by
$\\{Trace(t_{i}\circ s_{ij})\\}$. By similar arguments as above, we can show
that $Trace(t_{i}\circ s_{ij})=0\leavevmode\nobreak\ \leavevmode\nobreak\
\forall i,j\in I$.

Case 3: the induced map $H^{1}(\omega^{\flat})$:In this case, we look at the
pairing of
$\\{t_{ij}\in(SC(\mathcal{E},\phi)\otimes\omega_{X_{s}})(V_{ij})\\}$ and
$\\{s_{j}\in SC(\mathcal{E},\phi)(V_{j})\\}$. The pairing is given by
$\\{Trace(t_{ij}\circ s_{j})\\}$. By similar arguments as above, we can show
that $Trace(t_{ij}\circ s_{j})=0\leavevmode\nobreak\ \leavevmode\nobreak\
\forall i,j\in I$.

Therefore, we conclude that the smooth locus of this particular component is
isotropic.

Proof of (3) From (2), it follows that the dimension of
$\mathcal{N}ilp^{red}_{0}=n^{2}(g-1)$. Now from Lemma 8.10, it follows that
the dimension of $M^{cl}_{Gie}$ is equal to $2n^{2}(g-1)+2$. Once we know the
dimension, it is easy to see that $M^{cl}_{Gie}$ is a local complete
intersection (Remark 8.12). Since $M^{cl}_{Gie}$ is a local complete
intersection, and all the fibres of the Hitchin map
$h:M^{cl}_{Gie}\longrightarrow B$ are of dimension equal to $n^{2}(g-1)$,
therefore by Miraculous flatness criterion, it follows that the Hitchin map is
flat. ∎

##  7 On the relative logarithmic Dolbeault moduli over
$\overline{\mathcal{M}_{g}}$

In this section, we will construct the Gieseker-like derived moduli stack of
Higgs bundles over the moduli stack of stable curves of genus $g\geq 2$. We
will also show that there is a relative $0$-shifted log-symplectic forms on
this moduli stack relative to the moduli stack of stabel curves. Let us begin
with a few definitions.

######  Definition 7.1.

(The moduli stack of stable curves $\overline{\mathcal{M}_{g}}$) A projective
variety $C$ of dim $1$ is called a stable curve if it is either smooth or has
nodal singularities, and the automorphism group $Aut(C)$ is a finite group.

The moduli stack of stable curves is a functor.

$\overline{\mathcal{M}_{g}}:Sch/\mathbb{k}\longrightarrow Groupoids$
$T\mapsto\left\\{\begin{array}[]{@{}ll@{}}\text{families of stable curves}\\\
\text{ of genus}\leavevmode\nobreak\ g\leavevmode\nobreak\
\text{over}\leavevmode\nobreak\ T\end{array}\right\\}$ (7.1)

Here, morphisms in the groupoid $\overline{\mathcal{M}_{g}}(T)$ are just
isomorphisms of stable curves over $T$.

######  Remrk 7.2.

It is well known that if $g\geq 2$, the functor $\overline{\mathcal{M}_{g}}$
is a smooth Deligne-Mumford stack. The locus of nodal curves forms a normal
crossing divisor in $\overline{\mathcal{M}_{g}}$. Therefore, it induces a log
structure on $\overline{\mathcal{M}_{g}}$. From [18, Lemma 4.4], it follows
that the universal curve $\mathcal{C}_{g}$ over $\overline{\mathcal{M}_{g}}$
also induces a log structure on it and these two log structures are
isomorphic.

######  Definition 7.3.

(The moduli stack of semi-stable curves $\mathcal{M}^{ss}_{g}$) A connected
projective variety $C$ of dim $1$ is called a semi-stable curve if the
following properties are satisfied

  1. 1.

it is either smooth or has nodal singularities,

  2. 2.

every rational irreducible component $R$ is smooth, and

  3. 3.

the intersection $R\cdot\overline{C\setminus R}=2$.

The moduli stack of semi-stable curves is a functor

${\mathcal{M}^{ss}_{g}}:Sch/\mathbb{k}\longrightarrow Groupoids$
$T\mapsto\left\\{\begin{array}[]{@{}ll@{}}\text{families of semi-stable
curves}\\\ \text{of genus}\leavevmode\nobreak\ g\leavevmode\nobreak\
\text{over}\leavevmode\nobreak\ T\end{array}\right\\}$ (7.2)

Morphisms in the groupoid ${\mathcal{M}^{ss}_{g}}(T)$ are just isomorphisms of
families of semi-stable curves over $T$.

######  Remrk 7.4.

It is well known that if $g\geq 2$, the functor ${\mathcal{M}^{ss}_{g}}$ is a
smooth Artin stack [34, 0E72, Lemma 21.5].

###  7.1 Log structures on ${\mathcal{M}^{ss}_{g}}$

The locus of singular curves in ${\mathcal{M}^{ss}_{g}}$ forms a normal
crossing divisor, and hence induces a log structure on
${\mathcal{M}^{ss}_{g}}$. We denote this divisor by
$\partial\mathcal{M}^{ss}_{g}$. Let us denote the universal curve over
${\mathcal{M}^{ss}_{g}}$ by $\mathcal{D}_{g}$. There is the “stabilization
morphism” $\pi:\mathcal{M}^{ss}_{g}\longrightarrow\overline{\mathcal{M}_{g}}$
such that the induced morphism
$\mathcal{D}_{g}\longrightarrow\pi^{*}\mathcal{C}_{g}$ is the universal
modification morphism over $\mathcal{M}^{ss}_{g}$.

####  7.1.1 Versal deformation space of ${\mathcal{M}^{ss}_{g}}$ and the
versal picture of the map
$\pi:\mathcal{M}^{ss}_{g}\longrightarrow\overline{\mathcal{M}_{g}}$

Let $\mathcal{X}$ be a stable curve of genus $g$ and $\mathcal{X}^{mod}$ be a
semi-stable curve whose stable model is $\mathcal{X}$. Let
$\\{c_{i}\\}^{l}_{i=1}$ be the nodes of the curve $\mathcal{X}$. Let
$\\{d_{ij}\\}^{\iota_{i}}_{i=1}$ be the nodes of $\mathcal{X}^{mod}$ over the
node $c_{i}$ for every $i=1,\dots,l$. From [30, Proposition 3.3.2], it follows
that

  1. 1.

there exists a versal deformation space of the nodal curve $\mathcal{X}$ which
is formally smooth to $\mathbb{k}[|z_{1},\dots,z_{l}|]$ i.e., the versal space
is isomorphic to $\mathbb{k}[|z_{1},\dots,z_{l}|][|z_{l+1},\dots,z_{N}|]$,
where $N:=3g-3$ (here $g$ is the genus of the curves) and $z_{i}$ is the
equation of the $i$-th node of $\mathcal{X}$ for $i=1,\dots,l$,and

  2. 2.

there exits a versal deformation space of the nodal curve $\mathcal{X}^{mod}$
which is isomorphic to
$\mathbb{k}[|\\{\\{z_{ij}\\}^{l}_{i=1}\\}^{\iota_{i}}_{j=1}|][|z_{l+1},\dots,z_{N}|]$,
where $z_{ij}=0$ is the equation of the node $d_{ij}$ for $i=1,\dots,l$ and
$j=1,\dots,\iota_{i}$.

  3. 3.

Moreover, there is a morphism between the versal deformation spaces, which is
as follows.

$\displaystyle\mathbb{k}[|z_{1},\dots,z_{l}|][|z_{l+1},\dots,z_{N}|]\longrightarrow\mathbb{k}[|\\{\\{z_{ij}\\}^{l}_{i=1}\\}^{\iota_{i}}_{j=1}|][|z_{l+1},\dots,z_{N}|]$
$\displaystyle z_{i}\mapsto z_{i1}\cdots z_{i\iota_{i}}\hskip
30.00005pt\forall i=1,\dots,l\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{,and}$ $\displaystyle
z_{j}\mapsto z_{j}\hskip 30.00005pt\forall j=l+1,\dots,N$

######  Proposition 7.5.

  1. 1.

The curves $\mathcal{C}_{g}$ and $\mathcal{D}_{g}$ induce locally-free log
structures on $\overline{\mathcal{M}_{g}}$ and $\mathcal{M}^{ss}_{g}$,
respectively. These log structures concide with the log structures induced by
the boundary divisors $\partial\overline{\mathcal{M}_{g}}$ and
$\partial\mathcal{M}^{ss}_{g}$, respectively.

  2. 2.

The morphism $\mathcal{M}^{ss}_{g}\longrightarrow\overline{\mathcal{M}_{g}}$
is a log-smooth morphism of locally free log structures.

###### Proof.

The proof follows from the following general lemma.

######  Lemma 7.6.

Let $T$ be a scheme and $\mathcal{X}\longrightarrow T$ be a family of stable
curves. Let $\mathcal{X}^{mod}\longrightarrow\mathcal{X}$ be a modification
over $T$. Let us denote the log structures on $T$ induced by $\mathcal{X}$ and
$\mathcal{X}^{mod}$ by $(T,P,\alpha:P\longrightarrow\mathcal{O}_{T})$ and
$(T,Q,\beta:Q\longrightarrow\mathcal{O}_{T})$, respectively. Here $P$ and $Q$
are the étale sheaves of monoids and $\alpha$ and $\beta$ are maps of sheaves
such that $\alpha^{-1}(\mathcal{O}^{\times}_{T})\cong P^{\times}$ and
$\beta^{-1}(\mathcal{O}^{\times}_{T})\cong Q^{\times}$. Then there is a
natural map of logarithmic schemes $(T,Q,\beta)\longrightarrow(T,P,\alpha)$.

###### Proof.

From [18, Lemma 4.4], it follows that the curves $\mathcal{X}$ and
$\mathcal{X}^{mod}$ induce two log structures on $T$. We denote these two log
structures by $(T,P,\alpha:P\longrightarrow\mathcal{O}_{T})$ and
$(T,Q,\beta:Q\longrightarrow\mathcal{O}_{T})$, respectively. We want to show
that there is a morphism between these two log structures.

Let $t\in T$ be a closed point. Let us denote the maximal ideal of
$\mathcal{O}^{h}_{T,t}$ by $m_{T,t}$. Let $\\{c_{i}\\}^{l}_{i=1}$ be the nodes
of the curve $\mathcal{X}_{t}$. Let $\\{d_{ij}\\}^{\iota_{i}}_{i=1}$ be the
nodes of $\mathcal{X}^{mod}_{t}$ over the node $c_{i}$ for every
$i=1,\dots,l$. The Henselian local ring of $\mathcal{X}$ at the node $c_{i}$
is isomorphic to the Henselisation of $\frac{\mathcal{O}^{h}_{T,t}[|x,y|]}{xy-
t_{i}}$ at the ideal $(x,y,m_{T,t})$, for some $t_{i}\in m_{T,t}$ for every
$i=1,\dots,l$. Here $(x,y,m_{T,t})$ denote the ideal generated by $x$,$y$ and
$m_{T,t}$. Similarly, the Henselian local ring of $\mathcal{X}^{mod}$ at the
node $d_{ij}$ is isomorphic to the Henselisation of
$\frac{\mathcal{O}^{h}_{T,t}[|x,y|]}{xy-t_{ij}}$ at the ideal $(x,y,m_{T,t})$,
for some $t_{ij}\in m_{T,t}$ for every $i=1,\dots,l$ and
$j=1,\dots,\iota_{i}$. From [18, Lemma 2.1 and 2.2], it follows that the
elements $t_{i}$’s and $t_{ij}$’s are uniquely determined in
$\widehat{\mathcal{O}}_{T,t}$ and in $\mathcal{O}^{h}_{T,t}$.

Claim:

$t_{i}=t_{i1}\cdots t_{i\iota_{i}}\hskip 30.00005pt\forall i=1,\dots,l$ (7.3)

Proof of the claim: Since $t_{i}$’s and $t_{ij}$’s are uniquely determined in
$\widehat{\mathcal{O}}_{T,t}$ and in $\mathcal{O}^{h}_{T,t}$, it is enough to
prove the above identities in $\widehat{\mathcal{O}}_{T,t}$.

By versality property of the family described in subsubsection 7.1.1), we have
a commutative diagram

${\widehat{\mathcal{O}}_{T,t}}$${\mathbb{k}[|\\{\\{z_{ij}\\}^{l}_{i=1}\\}^{\iota_{i}}_{j=1}|][|z_{l+1},\dots,z_{N}|]}$${\mathbb{k}[|z_{1},\dots,z_{l}|][|z_{l+1},\dots,z_{N}|]}$
(7.4)

given by the family of curves $\mathcal{X}$ and $\mathcal{X}^{mod}$ over $T$.
By the uniqueness of the equations of the nodes in
$\widehat{\mathcal{O}}_{T,t}$, we have that $t_{i}=z_{i}$ and $t_{ij}=z_{ij}$
for every $i=1,\dots,l$ and $j=1,\dots,\iota_{i}$. Since $z_{i}=z_{i1}\cdots
z_{i\iota_{i}}$ for every $i=1,\dots,l$, we have $t_{i}=t_{i1}\cdots
t_{i\iota_{i}}\hskip 5.0pt\forall i=1,\dots,l$ in
$\widehat{\mathcal{O}}_{T,t}$. This completes the proof of the claim.

Proof of (1): Again since the elements $t_{i}$’s and $t_{ij}$’s are uniquely
determined in $\widehat{\mathcal{O}}_{T,t}$ and in $\mathcal{O}^{h}_{T,t}$,
using the versal maps
$\mathbb{k}[|\\{\\{z_{ij}\\}^{l}_{i=1}\\}^{\iota_{i}}_{j=1}|][|z_{l+1},\dots,z_{N}|]\longrightarrow\widehat{\mathcal{O}}_{T,t}$
and
$\mathbb{k}[|z_{1},\dots,z_{l}|][|z_{l+1},\dots,z_{N}|]\longrightarrow\widehat{\mathcal{O}}_{T,t}$
one can easily check (1).

Proof of (2): Now the morphism of log structures is induced by the following
diagram of pre-log structures.

${e_{i}}$${\sum^{\iota_{i}}_{j=1}e_{ij}}$${e_{i}}$${\mathbb{N}^{\oplus
l}}$${\oplus^{l}_{i=1}(\mathbb{N}^{\oplus\iota_{i}})}$${e_{ij}}$${t_{i}}$${\mathcal{O}^{h}_{T,t}}$${\mathcal{O}^{h}_{T,t}}$${t_{ij}}$$\scriptstyle{=}$$\scriptstyle{t_{i}=t_{i1}\cdots
t_{i\iota_{i}}}$ (7.5)

It is straightforward to check that the induced morphism is log-smooth. ∎

∎

###  7.2 Relative log-cotangent complex of the map
${\mathcal{M}^{ss}_{g}}\longrightarrow\overline{\mathcal{M}_{g}}$

######  Proposition 7.7.

The relative logarithmic cotangent complex $\mathbb{L}^{log}_{f}\cong 0$.

###### Proof.

We have a commutative diagram (not a cartesian square)

${\mathcal{D}_{g}}$${\mathcal{C}_{g}}$${\mathcal{M}^{ss}_{g}}$${\overline{\mathcal{M}_{g}}}$$\scriptstyle{\tilde{p}}$$\scriptstyle{\tilde{f}}$$\scriptstyle{p}$$\scriptstyle{f}$
(7.6)

As discussed above, the family of semi-stable curves induces log structures on
the base and the family of curves, making the projection morphism a log-smooth
morphism. Therefore, the maps $\tilde{p}$ and $p$ are log-smooth morphisms
with the induced log structures. The space of infinitesimal first-order log-
deformations of the curves is isomorphic to
$H^{1}(\mathcal{C}_{g},\omega_{\mathcal{C}_{g}/{\overline{\mathcal{M}_{g}}}}^{\vee})$
and
$H^{1}(\mathcal{D}_{g},\omega_{\mathcal{D}_{g}/{{\mathcal{M}^{ss}_{g}}}}^{\vee})$,
respectively. Therefore the log-tangent complex of
$\overline{\mathcal{M}_{g}}$ is isomorphic to
$Rp_{*}(\omega^{\vee}_{\mathcal{C}_{g}/\overline{\mathcal{M}_{g}}})[1]$ and
the log-tangent complex of $\mathcal{M}^{ss}_{g}$ is isomorphic to
$R\tilde{p}_{*}(\omega^{\vee}_{\mathcal{D}_{g}/{\mathcal{M}^{ss}_{g}}})[1]$.
Now notice that

$Rf_{*}\circ
R\tilde{p}_{*}(\omega^{\vee}_{\mathcal{D}_{g}/{\mathcal{M}^{ss}_{g}}})[1]=Rp_{*}R\tilde{f}_{*}(\omega^{\vee}_{\mathcal{D}_{g}/{\mathcal{M}^{ss}_{g}}})[1]\cong
Rp_{*}(\omega^{\vee}_{\mathcal{C}_{g}/\overline{\mathcal{M}_{g}}})[1].$

Therefore, we conclude that the natural map between the two log-tangent
complexes ( and the map between log-cotangent complexes) is an isomorphism.
Using the distinguished triangle of log-cotangent complexes of a map, we have
the following traingles.

$f^{*}\mathbb{L}^{log}_{\overline{\mathcal{M}_{g}}}\longrightarrow\mathbb{L}^{log}_{{\mathcal{M}^{ss}_{g}}}\longrightarrow\mathbb{L}^{log}_{f}\longrightarrow
f^{*}\mathbb{L}^{log}_{\overline{\mathcal{M}_{g}}}[1]$ (7.7)
$\mathbb{L}^{log}_{{\mathcal{M}^{ss}_{g}}}\longrightarrow\mathbb{L}^{log}_{f}\longrightarrow
f^{*}\mathbb{L}^{log}_{\overline{\mathcal{M}_{g}}}[1]\longrightarrow\mathbb{L}^{log}_{{\mathcal{M}^{ss}_{g}}}[1]$
(7.8)

Since, the map
$f^{*}\mathbb{L}^{log}_{\overline{\mathcal{M}_{g}}}\longrightarrow\mathbb{L}^{log}_{{\mathcal{M}^{ss}_{g}}}$
is an equivalence, therefore we conclude that $\mathbb{L}^{log}_{f}\cong 0$. ∎

###  7.3 Relative logarithmic Dolbeault shape and shifted symplectic forms

Let $\mathcal{D}^{lDol}_{g}$ denote the relative logarithmic Dolbeault moduli
stack for the family of curves
$\mathcal{D}_{g}\longrightarrow\mathcal{M}^{ss}_{g}$. Then
$\mathcal{M}^{lDol}_{g}:=\mathsf{Map}_{\mathcal{M}^{ss}_{g}}(\mathcal{D}^{lDol}_{g},BGL_{n}\times\mathcal{M}^{ss}_{g})$
is the relative derived moduli stack of Gieseker-Higgs bundles viewed over
$\overline{\mathcal{M}_{g}}$.

######  Proposition 7.8.

The morphism $\mathcal{M}^{lDol}_{g}\longrightarrow\mathcal{M}^{ss}_{g}$ is a
quasi-smooth morphism of derived Artin stacks.

###### Proof.

Same as the proof of Proposition 4.12. ∎

We equip $\mathcal{M}^{lDol}_{g}$ with the locally free log-structure pulled
back from $\mathcal{M}^{ss}_{g}$ via the morphism
$\mathcal{M}^{lDol}_{g}\longrightarrow\mathcal{M}^{ss}_{g}$.

######  Theorem 7.9.

There is a $0$-shifted relative log-symplectic form on
$\mathcal{M}^{lDol}_{g}$ (relative to the moduli stack of stable curves
$\overline{\mathcal{M}_{g}}$).

###### Proof.

Consider the composite morphism
$\mathcal{M}^{lDol}_{g}\xrightarrow{\pi}\mathcal{M}^{ss}_{g}\xrightarrow{f}\overline{\mathcal{M}_{g}}$.
We have a distinguished triangle

$\pi^{*}\mathbb{L}^{log}_{\mathcal{M}^{ss}_{g}/\overline{\mathcal{M}_{g}}}\longrightarrow\mathbb{L}^{log}_{\mathcal{M}^{lDol}_{g}/\overline{\mathcal{M}_{g}}}\longrightarrow\mathbb{L}^{log}_{\mathcal{M}^{lDol}_{g}/\overline{\mathcal{M}_{g}}}\longrightarrow\pi^{*}\mathbb{L}^{log}_{\mathcal{M}^{ss}_{g}/\overline{\mathcal{M}_{g}}}[1].$
(7.9)

But since
$\mathbb{L}^{log}_{\mathcal{M}^{ss}_{g}/\overline{\mathcal{M}_{g}}}\cong 0$
(Proposition 7.7), therefore the relative log cotangent complex of the
composite morphism is isomorphic to the relative log cotangent complex of the
morphism $\pi$. But notice that the log structure of $\mathcal{M}^{lDol}_{g}$
is pulled back of the log structure of $\mathcal{M}^{ss}_{g}$ via the map
$\pi$. Therefore the relative log-cotangent complex of the morphism $\pi$ is
isomorphic to the relative cotangent complex of the morphism $\pi$. Now since
the relative logarithmic Dolbeault stack over the moduli stacks
$\mathcal{M}^{lDol}_{g}$ is $\mathcal{O}$-compact and $\mathcal{O}$-oriented
(Theorem 4.11); therefore $\mathcal{M}^{lDol}_{g}$ has a $0$-shifted relative
symplectic form over $\mathcal{M}^{ss}_{g}$, which is a $0$-shifted relative
log-symplectic form viewed over the moduli stack of stable curves
$\overline{\mathcal{M}_{g}}$. ∎

##  8 Appendix: Classical Artin stack of Gieseker-Higgs bundles and its local
properties

In this section we will define and construct the relative classical Artin
stack of Gieseker-Higgs bundles and study its local properties. The main
results of the appendix are Lemma 8.9 and Lemma 8.10. In the first lemma we
prove that the stack of Gieseker vector bundles is an almost very good stack.
We use this in Lemma 8.10 to show that the classical stack of Gieseker-Higgs
bundles is an irreducible, local complete intersection.

###  8.1 The stacks of Gieseker vector bundles and Gieseker Higgs bundles

######  Definition 8.1.

(Gieseker vector bundle) A vector bundle $\mathcal{E}$ of rank $n$ on $X_{r}$
with $r\geq 1$ is called a Gieseker vector bundle if

  1. 1.

$\mathcal{E}|_{R[r]}$ is a strictly standard vector bundle on $R[r]\subset
X_{r}$, i.e., for each $i=1,\dots,r$, $\exists$ non-negative integers $a_{i}$
and $b_{i}$ such that $\mathcal{E}|_{R[r]_{i}}\cong\mathcal{O}^{\oplus
a_{i}}\oplus\mathcal{O}(1)^{\oplus b_{i}}$, and

  2. 2.

the direct image $\pi_{r*}(\mathcal{E})$ is a torsion-free
$\mathcal{O}_{X_{0}}$-module.

Any vector bundle on $X_{0}$ is called a Gieseker vector bundle. In the
literature, a Gieseker vector bundle is also called an admissible vector
bundle.

A Gieseker vector bundle $(X_{r},\mathcal{E})$ is called a stable Gieseker
vector bundle if $\pi_{r*}\mathcal{E}$ is a stable torsion-free sheaf on the
irreducible nodal curve $X_{0}$, where $\pi_{r}:X_{r}\longrightarrow X_{0}$ is
the natural contraction map.

A (stable) Gieseker vector bundle on a modification $\mathcal{X}^{mod}_{T}$ is
a vector bundle such that its restriction to each
$(\mathcal{X}^{mod}_{T})_{t}$ is a (stable) Gieseker vector bundle.

######  Remrk 8.2.

A Gieseker vector bundle can also be defined as a vector bundle
$(X_{r},\mathcal{E})$ on a Gieseker curve $X_{r}$ satisfying the following two
conditions

  1. 1.

$\mathcal{E}|_{R[r]}$ is a globally generated vector bundle on $R[r]$,

  2. 2.

the direct image $\pi_{r*}(\mathcal{E})$ is a torsion-free
$\mathcal{O}_{X_{0}}$-module.

######  Definition 8.3.

(Gieseker-Higgs bundle) A Gieseker-Higgs bundle on $\mathcal{X}^{mod}_{T}$ is
a pair $(\mathcal{E}_{T},\phi_{T})$, where $\mathcal{E}_{T}$ is a vector
bundle on $\mathcal{X}^{mod}_{T}$, and
$\phi_{T}:\mathcal{E}_{T}\longrightarrow\mathcal{E}_{T}\otimes\omega_{\mathcal{X}^{mod}_{T}/T}$
is an $\mathcal{O}_{\mathcal{X}^{mod}_{T}}$ -module homomorphism satisfying
the following

  1. 1.

$\mathcal{E}_{T}$ is a Gieseker vector bundle on $\mathcal{X}^{mod}_{T}$,

  2. 2.

for each closed point $t\in T$ over $\eta_{0}\in S$, the direct image
$(\pi_{t})_{*}(\mathcal{E}_{t})$ is a torsion-free sheaf on $X_{0}$ and
$(\pi_{t})_{*}\phi_{t}:(\pi_{t})_{*}(\mathcal{E}_{t})\longrightarrow(\pi_{t})_{*}(\mathcal{E}_{t})\otimes\omega_{X_{0}}$
is an $\mathcal{O}_{X_{0}}$-module homomorphism. We refer to such a pair
$((\pi_{t})_{*}(\mathcal{E}_{t}),(\pi_{t})_{*}\phi_{t})$ as a torsion-free
Higgs pair on the nodal curve $X_{0}$.

######  Remrk 8.4.

A Gieseker-Higgs bundle can also be defined as a Higgs bundle
$(X_{r},\mathcal{E},\phi)$ on a Gieseker curve $X_{r}$ satisfying the
following two conditions

  1. 1.

$\mathcal{E}|_{R[r]}$ is a globally generated vector bundle on $R[r]$,

  2. 2.

the direct image $(\pi_{r})_{*}(\mathcal{E})$ is a torsion-free
$\mathcal{O}_{X_{0}}$-module.

######  Remrk 8.5.

From [21, Definition-Notation 1, Lemma 2, and Proposition 5], it follows that
that for moduli problem of vector bundles (Higgs bundles) of rank $n$, we have
to consider Gieseker curves $X_{r}$, where $r=0,\dots,n$.

We fix the rank and degree to be $n$ and $d$, respectively.

####  8.1.1 Stack of torsion-free Hitchin pairs

We define the moduli stack of torsion-free Hitchin pairs

$TFH(\mathcal{X}/S):{Sch}/S\rightarrow{Sets}$

$T\mapsto\left\\{\begin{array}[]{@{}ll@{}}\text{Families of torsion-free
Hitchin pairs}\\\
(\mathcal{F},\phi:\mathcal{F}\longrightarrow\mathcal{F}\otimes\omega_{\mathcal{X}/S})\\\
\text{of rank}\leavevmode\nobreak\ n\leavevmode\nobreak\ \text{and
degree}\leavevmode\nobreak\ d\leavevmode\nobreak\ \text{over the original}\\\
\text{family of curves}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{X}/S\leavevmode\nobreak\ \leavevmode\nobreak\
(\ref{DegeCourbes})\end{array}\right\\}$ (8.1)

It is an Artin stack. For the construction of an atlas for
$TFH(\mathcal{X}/S)$, we refer to [1, 5.0.7. The total family construction.].
Following the notation from [1], we denote it by $\coprod_{m\geq
m_{0}}R^{\Lambda,m}_{S}$. The superscript ”$\Lambda$” is because for the
construction of the Quot scheme of torsion-free Hitchin pairs one views them
as $\Lambda$-modules, where $\Lambda=\operatorname{Sym}\leavevmode\nobreak\
(\omega^{\vee}_{\mathcal{X}/S})$ [31]. We denote the stack of families of
torsion-free sheaves over $\mathcal{X}/S$ of rank $n$ and degree $d$ by
$TF(\mathcal{X}/S)$. It is a reduced and irreducible Artin stack.

####  8.1.2 Classical Artin Stack of Gieseker-Higgs bundles

Given any derived Artin stack
$F:\textsf{cdga}_{\mathbb{k}}\longrightarrow\mathbb{S}$ one can define a
classical Artin stack by considering the composition functor
$F^{cl}:\textsf{alg}_{\mathbb{k}}\longrightarrow\textsf{cdga}_{\mathbb{k}}\longrightarrow\mathbb{S}$.
We call $F^{cl}$ as the underlying classical Artin stack of $F$. Following
this notation, we denote the underlying classical Artin stack of the derived
stack $M$ of Higgs bundles over the family of curves
$\mathcal{X}_{\mathfrak{M}}/\mathfrak{M}$ by $M^{cl}$.

Let us denote by $Coh(\mathcal{X}/S)$ the Artin stack of coherent
$\mathcal{O}_{\mathcal{X}}-$ modules which are flat over $S$. There is a
natural map $\theta:M^{cl}\longrightarrow Coh(\mathcal{X}/S)$ which is given
by the pushforward of the underlying bundle $\pi_{*}\mathcal{E}$, where
$\pi:X_{k}\longrightarrow X_{0}$ is the modification morphism. Consider the
open sub-stack $TF(\mathcal{X}/S)\subset Coh(\mathcal{X}/S)$ consisting of
torsion-free sheaves. Therefore the stack $\theta^{-1}(TF(\mathcal{X}/S))$ is
open in $M^{cl}$. Let us consider the natural map
$\pi^{*}\pi_{*}\mathcal{E}\longrightarrow\mathcal{E}$ over the universal curve
$\mathcal{X}^{mod}$ over $M^{cl}$. Consider the sub-stack of $M^{cl}$ where
the map is surjective. It is again an open sub-stack. Let us denote it by
$M^{gg}$ (the superscript ”gg” stands for globally generated). Then we define
a open sub-stack $M^{cl}_{Gie}:=\theta^{-1}(TF(\mathcal{X}/S))\cap M^{gg}$. It
is an open sub-stack and it consists of Gieseker-Higgs bundles of rank $n$ and
degree $d$. There is a natural map $M^{cl}_{Gie}\longrightarrow
TFH(\mathcal{X}/S)$ which is given by
$(\mathcal{X}^{mod},\mathcal{E},\phi)\mapsto(\pi_{*}\mathcal{E},\pi_{*}\phi)$,
where $\pi:\mathcal{X}^{mod}\longrightarrow\mathcal{X}$ is the modification
map.

We denote by $M^{{}^{cl}}_{Gie}$ the underlying classical stack of Gieseker-
Higgs bundles of rank $n$ and degree $d$. We denote by $M^{{}^{cl}}_{Gie,0}$
the closed fibre of the map $M^{{}^{cl}}_{Gie}\longrightarrow S$. Now we
recall the construction of an atlas for the stack $M^{{}^{cl}}_{Gie}$ as well
as the classical stack of Gieseker vector bundles. Let us denote by $N_{Gie}$
the classical stack of Gieseker vector bundles.

####  8.1.3 Construction of an atlas for $N_{Gie}$

Let $\mathcal{O}_{\mathcal{X}/S}(1)$ be a relatively ample line bundle for the
family of curves $\mathcal{X}/S$. The set of all flat families of stable
torsion-free sheaves (Higgs pairs) of degree $d$ and rank $n$ over
$\mathcal{X}$ forms a bounded family. Therefore we can choose a large integer
$m_{0}$ such that given any family of stable torsion-free Higgs pairs
$(\mathcal{F}_{S},\phi_{S})$, the sheaf
$\mathcal{F}_{s}\otimes\mathcal{O}_{\mathcal{X}_{s}}(m_{0})$ is generated by
global sections and
$H^{1}(\mathcal{X}_{s},\mathcal{F}_{s}\otimes\mathcal{O}_{\mathcal{X}_{s}}(m_{0}))=0$
for every geometric point $s\in S$. Set
$N(m_{0}):=H^{0}(\mathcal{X}_{s},\mathcal{F}_{s}\otimes\mathcal{O}_{\mathcal{X}_{s}}(m_{0}))$
for any geometric point $s\in S$. We denote by $Grass(N(m_{0}),n)$ the
Grassmannian of $n$ dimensional quotient vector spaces of
$\mathbb{k}^{N(m_{0})}$.

######  Definition 8.6.

Let $\mathcal{G}^{m_{0}}_{S}:{Sch}/S\longrightarrow{Sets}$ be the functor
defined as follows:

$\mathcal{G}^{m_{0}}_{S}(T)=\\{(\Delta_{T},V_{T})\\},$ (8.2)

where

$\Delta_{T}\subset\mathcal{X}\times_{S}T\times{Grass}(N(m_{0}),n)$ (8.3)

is a closed subscheme, and $V_{T}$ is a vector bundle on $\Delta_{T}$ such
that

  1. 1.

the projection $j:\Delta_{T}\longrightarrow T\times{Grass}(N(m_{0}),n)$ is a
closed immersion,

  2. 2.

the projection $\Delta_{T}\longrightarrow\mathcal{X}\times_{S}T$ is a
modification,

  3. 3.

the projection $p_{T}:\Delta_{T}\longrightarrow T$ is a flat family of
Gieseker curves,

  4. 4.

Let $\mathcal{V}$ be the tautological quotient bundle of rank $n$ on
${Grass}(N(m_{0}),n)$ and $\mathcal{V}_{T}$ its pullback to
$T\times{Grass}(N(m_{0}),n)$. Then

$V_{T}:=j^{*}(\mathcal{V}_{T})$ (8.4)

be such that $V_{T}$ is a Gieseker vector bundle on the modification
$\Delta_{T}$ of rank $n$ and degree $d(m_{0})^{\prime}:=N(m_{0})+n(g-1)$.

  5. 5.

for each $t\in T$, the quotient
$\mathcal{O}_{\Delta_{t}}^{N}(m_{0})\longrightarrow V_{t}$ induces an
isomorphism

$H^{0}(\Delta_{t},\mathcal{O}_{\Delta_{t}}^{N(m_{0})})\cong
H^{0}(\Delta_{t},V_{t})$ (8.5)

and $H^{1}(\Delta_{t},V_{t})=0$.

We denote by $P(m_{0})$ the Hilbert polynomial of the closed subscheme
$\Delta_{s}$ of $\mathcal{X}_{s}\times Grass(N(m_{0}),n)$ for any geometric
point $s\in S$ with respect to the polarisation
$\mathcal{O}_{\mathcal{X}_{s}}(1)\boxtimes\mathcal{O}_{Grass(N(m_{0}),n)}(1)$,
where $\mathcal{O}_{Grass(N(m_{0}),n)}(1)$ is the line bundle
$\det\leavevmode\nobreak\ \leavevmode\nobreak\ \mathcal{V}$.

It is shown in [21, Proposition 8] that the functor $\mathcal{G}^{m_{0}}_{S}$
is represented by a $PGL(N(m_{0}))$-invariant open subscheme
$\mathcal{Y}^{m_{0}}_{S}$ of the Hilbert scheme
$\mathcal{H}_{S}:={Hilb}^{P(m_{0})}(\mathcal{X}\times{Grass}(N(m_{0}),n))$.

Finally, the disjoint union $\mathcal{Y}_{S}:=(\coprod_{{}_{N(m),m\geq
m_{0}}}\mathcal{Y}^{m}_{S})\longrightarrow N_{Gie}$ is an atlas. The varieties
$\mathcal{Y}^{m}_{S}$ are all smooth varieties and the projection map to the
stack is also smooth.

####  8.1.4 Construction of an atlas for $M^{{}^{cl}}_{Gie}$

######  Definition 8.7.

We define a functor

$\mathcal{G}^{H,m}_{S}:{Sch}/\mathcal{Y}_{S}\longrightarrow{Groups}$ (8.6)

which maps

$T\longrightarrow H^{0}(T,(p_{T})_{*}(\mathcal{E}nd\leavevmode\nobreak\
\leavevmode\nobreak\ \mathcal{V}_{T}\otimes\omega_{\Delta_{T}/T})),$

where
$p_{T}:\Delta_{T}:=\Delta_{{}_{\mathcal{Y}_{S}}}\times_{{}_{\mathcal{Y}_{S}}}T\longrightarrow
T$ is the projection, and $\omega_{\Delta_{T}/T}$ denotes the relative
dualising sheaf of the family of curves $p_{T}$.

Since $\mathcal{Y}^{m}_{S}$ is a reduced scheme, the functor
$\mathcal{G}^{H,m}_{S}$ is representable, i.e., there exists a linear
$\mathcal{Y}^{m}_{S}$\- scheme $\mathcal{Y}^{H,m}_{S}$, which represents it.
For a $S$-scheme $T$, a point in $\mathcal{G}^{H,m}_{S}(T)$ is given by
$(V_{T},\phi_{T})$, where

  1. 1.

$V_{T}\in\mathcal{G}^{m}_{S}(T)$, and

  2. 2.

$(V_{T},\phi_{T})$ is a Gieseker-Higgs bundle.

Finally, the disjoint union $\mathcal{Y}^{H}_{S}:=(\coprod_{{}_{N(m),m\geq
m_{0}}}\mathcal{Y}^{H,m}_{S})\longrightarrow M^{{}^{cl}}_{Gie}$ is an atlas.
The projection map to the stack is smooth.

###  8.2 Dimension and local properties of $M^{cl}_{Gie,0}$

In this subsection, we will compute the dimension of $M^{cl}_{Gie,0}$ and show
that it is a local complete intersection.

####  8.2.1 Relative Log-Symplectic reduction

We will do this by a log-symplectic reduction on every atlas of $N_{Gie,0}$.
Let us pick one of the atlas $\mathcal{Y}^{m}_{S}$ of $M^{cl}_{Gie}$. The
group $GL_{N(m)}$ acts on $\mathcal{Y}^{m}_{S}$ and the quotient stack
$[\mathcal{Y}^{m}_{S}/GL_{N(m)}]$ is an open sub-stack of $N_{Gie}$. We
consider the relative log-cotangent bundle
$\Omega^{log}_{\mathcal{Y}^{m}_{S}/S}$. Since the $GL_{N(m)}$ action preserves
the normal crossing divisor $\mathcal{Y}^{m}_{0}$ (the closed fibre of
$\mathcal{Y}^{m}_{S}\longrightarrow S$), the action of $GL_{N(m)}$ lifts to an
action on $\Omega^{log}_{\mathcal{Y}^{m}_{S}/S}$ with a moment map
$\mu_{log}:\Omega^{log}_{\mathcal{Y}^{m}_{S}/S}\longrightarrow\mathfrak{gl}_{N(m)}^{*}$.
Notice that the action of $GL_{N(m)}$ has a generic stabiliser
$\mathbb{G}_{m}$; therefore the map $\mu_{log}$ actually factors through
$\Omega^{log}_{\mathcal{Y}^{m}_{S}/S}\longrightarrow\mathfrak{pgl}_{N(m)}^{*}$.
Since we know that there exists an open subset of $\mathcal{Y}^{m}_{S}$ where
the action of $GL_{N(m)}$ has stabiliser isomorphic to $\mathbb{G}_{m}$
(namely, the locus of stable vector bundles), therefore the map
$\Omega^{log}_{\mathcal{Y}^{m}_{S}/S}\longrightarrow\mathfrak{pgl}_{N(m)}^{*}$
is surjective. Then one can show that
$\mathcal{Y}^{H,m}_{S}=\mu_{log}^{-1}(0)$. To see this, notice that
$\mathcal{Y}^{m}_{S}$ is a principal $GL_{N(m)}$-bundle over an open subset of
the stack $N_{Gie}$. Let $a$ denote the projection
$a:\mathcal{Y}^{m}_{S}\longrightarrow N_{Gie}$. Let us write the cotangent
sequence.

$0\longrightarrow a^{*}\Omega^{log,*}_{N_{Gie}/S}\longrightarrow
T^{log,*}_{\mathcal{Y}^{m}_{S}/S}\longrightarrow\mathfrak{g}l^{*}_{N(m)}\otimes\mathcal{O}_{\mathcal{Y}^{m}_{S}}$
(8.7)

Notice that
$[T^{log,*}_{\mathcal{Y}^{m}_{S}/S}\longrightarrow\mathfrak{g}l^{*}_{N(m)}\otimes\mathcal{O}_{\mathcal{Y}^{m}_{S}}]$
is the cotangent complex of the stack $N_{Gie}$. Therefore the $0$-th
cohomology of the complex at a point
$[\mathcal{O}^{N(m)}\rightarrow\mathcal{E}]\in\mathcal{Y}^{m}_{S}$ is
isomorphic to
$H^{1}(\mathcal{E}nd\mathcal{E})^{\vee}\cong\mathsf{Hom}(\mathcal{E},\mathcal{E}\otimes\omega)$.
Therefore by definition 8.1.4, $\mathcal{Y}^{H,m}_{S}=\mu_{log}^{-1}(0)$.
Therefore $[\mu_{log}^{-1}(0)/GL_{N(m)}]$ is an open subset of $M^{cl}_{Gie}$.
Therefore, we will compute the dimension of $\mu_{log}^{-1}(0)$ and that it is
a local complete intersection. We recall a result on the dimension of the
image of a cotangent fibre under the moment map.

######  Lemma 8.8.

dim $\mu_{log}(\Omega^{log}_{\mathcal{Y}^{m}_{S}/S,y})\geq\leavevmode\nobreak\
dim\leavevmode\nobreak\
\big{(}\frac{\mathfrak{gl}_{N(m)}}{\mathfrak{gl}_{N(m),y}}\big{)}^{*}$, where
$\mathfrak{gl}_{N(m),y}$ is the Lie algebra of the stabilizer of
$y\in\mathcal{Y}^{m}_{S}$ under the action of $GL_{N}(m)$.

###### Proof.

Consider the diagram

${0}$${\mathfrak{gl}_{N(m),y}}$${\mathfrak{gl}_{N(m)}}$${\frac{\mathfrak{gl}_{N(m)}}{\mathfrak{gl}_{N(m),y}}}$${0}$${0}$${K}$${T^{log}_{\mathcal{Y}^{m}_{S}/S,y}}$${T_{\mathcal{Y}^{m}_{S}/S,y}}$
(8.8)

Here $K$ denotes the Kernel of the natural map
$T^{log}_{\mathcal{Y}^{m}_{S}/S,y}\longrightarrow
T_{\mathcal{Y}^{m}_{S}/S,y}$. Notice that the map
$\mathfrak{gl}_{N(m)}\longrightarrow T_{\mathcal{Y}^{m}_{S}/S,y}$ is the
differential of the orbit map, and it factors through
$T^{log}_{\mathcal{Y}^{m}_{S}/S,y}$ because the action of $GL_{N(m)}$
preserves the normal crossing divisor. It is well known that the rightmost
vertical map is injective [32, Lemma 2.4.1]. Therefore we can complete the
diagram.

${0}$${\mathfrak{gl}_{N(m),y}}$${\mathfrak{gl}_{N(m)}}$${\frac{\mathfrak{gl}_{N(m)}}{\mathfrak{gl}_{N(m),y}}}$${0}$${0}$${K}$${T^{log}_{\mathcal{Y}^{m}_{S}/S,y}}$${T_{\mathcal{Y}^{m}_{S}/S,y}}$$\scriptstyle{\iota}$$\scriptstyle{\iota_{log}}$
(8.9)

Therefore we see that $Kernel\leavevmode\nobreak\ (\iota_{log})\subset
Kernel\leavevmode\nobreak\ (\iota)=\mathfrak{gl}_{N(m),y}$. Since the moment
and log-moment maps are the dual of the maps $\iota$ and $\iota_{log}$,
therefore $dim\leavevmode\nobreak\
\mu_{log}(\Omega^{log}_{\mathcal{Y}^{m}_{S}/S,y})\geq\leavevmode\nobreak\
dim\leavevmode\nobreak\
\big{(}\frac{\mathfrak{gl}_{N(m)}}{\mathfrak{gl}_{N(m),y}}\big{)}^{*}$. ∎

######  Proposition 8.9.

The closed fibre $N^{cl}_{Gie,0}$ is an irreducible, equidimensional, almost
very good stack ([32, Definition 2.1.2]) with normal crossing singularities.

###### Proof.

From [16, Theorem 9.5],it follows that the normalisation of $N^{cl}_{Gie,0}$
is a $KGL_{n}$ bundle over the stack of vector bundles $Bun(\tilde{X}_{0})$ of
rank $n$ and degree $d$ over the normalisation $\tilde{X}_{0}$ of the nodal
curve $X_{0}$. Here $KGL_{n}$ denotes the compactification of $GL_{n}$
constructed by Kausz [17]. More precisely, let $E$ is the universal $GL_{n}$
bundle over $\tilde{X_{0}}\times Bun(\tilde{X_{0}})$. Consider the
$GL_{n}\times GL_{n}$ bundle $E_{x_{1}}\times_{Bun(\tilde{X_{0}})}E_{x_{2}}$
over $Bun(\tilde{X_{0}})$. Then the associated $KGL_{n}$ fibration
$(E_{x_{1}}\times_{Bun(\tilde{X_{0}})}E_{x_{2}})\times_{GL_{n}\times
GL_{n}}KGL_{n}\cong\widetilde{N^{cl}_{Gie,0}}$. This is the stack of Gieseker
vector bundle data ([16, Definition 4.7]) which is equivalent to ([DI, Lemma
5.6]) the stack of marked Gieseker vector bundles (i.e., a Gieseker vector
bundle with a marked node ([DI, Definition 5.1])). It is obvious that the
automorphism group of a marked Gieseker vector bundle is isomorphic to the
automorphism group of the corresponding Gieseker vector bundle. Therefore it
is enough to show that the stack of Gieseker vector bundle data is smooth,
equidimensional, irreducible and almost very good.

Now let us recall that the map
$\tilde{\pi}:\widetilde{N^{cl}_{Gie,0}}\longrightarrow Bun(\tilde{X}_{0})$ is
given by $(X^{m,n},s_{1},s_{2},\tilde{\mathcal{E}},\phi)\mapsto
h_{*}(\mathcal{E}(-s_{1}-s_{2}))(p_{1}+p_{2})$, where
$h:X^{m,n}\longrightarrow\tilde{X}_{0}$ is the modification map ([16, Lemma
9.3]). Here $(X^{m,n},s_{1},s_{2},\tilde{\mathcal{E}},\phi)$ is a Gieseker
vector bundle data and here $\phi$ is not a Higgs bundle but an identification
between the fibres
$\tilde{\mathcal{E}}_{s_{1}}\xrightarrow{\cong}\tilde{\mathcal{E}}_{s_{2}}$
([16, Definition 4.7]). It is straightforward to check that the induced map
$Aut(X^{m,n},s_{1},s_{2},\tilde{\mathcal{E}},\phi)\longrightarrow
Aut(h_{*}(\mathcal{E}(-s_{1}-s_{2}))(p_{1}+p_{2}))$ is injective. For
notational convenience, let us denote $Y:=\widetilde{N^{cl}_{Gie,0}}$ and
$Z:=Bun(\tilde{X}_{0})$.

From [4, Proposition 2.1.2] and [32, Definition 2.1.2, Remark 2.1.3], it
follows that $Z$ is smooth, irreducible, almost very good stack. More
precisely, the $\operatorname{codim}_{Z}(Z_{k})>k\leavevmode\nobreak\
\leavevmode\nobreak\ \forall k\geq 1$, where $Z_{k}:=\\{z\in
Z\leavevmode\nobreak\ |\leavevmode\nobreak\ \text{dim}\leavevmode\nobreak\
Aut\leavevmode\nobreak\ (z)=1+k\\}$. From the fact that
$Aut(X^{m,n},s_{1},s_{2},\tilde{\mathcal{E}},\phi)\subset
Aut(h_{*}(\mathcal{E}(-s_{1}-s_{2}))(p_{1}+p_{2}))$, it follows that
$\tilde{\pi}(Y_{k})\subset Z_{k}$. Therefore,
$Y_{k}\subset\tilde{\pi}^{-1}(Z_{k})$ and $\dim\leavevmode\nobreak\
Y_{k}\leq\dim\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{\pi}^{-1}(Z_{k})=\dim\leavevmode\nobreak\ \leavevmode\nobreak\
Z_{k}+\dim\leavevmode\nobreak\ \leavevmode\nobreak\ KGL_{n}$. Now
$\operatorname{codim}_{Y}(Y_{k})=\dim\leavevmode\nobreak\ \leavevmode\nobreak\
Y-\dim\leavevmode\nobreak\ \leavevmode\nobreak\ Y_{k}\geq\leavevmode\nobreak\
\leavevmode\nobreak\ \dim\leavevmode\nobreak\ \leavevmode\nobreak\
Y-\dim\leavevmode\nobreak\ \leavevmode\nobreak\ Z_{k}-\dim\leavevmode\nobreak\
\leavevmode\nobreak\ KGL_{n}=\dim\leavevmode\nobreak\ \leavevmode\nobreak\
Z-\dim\leavevmode\nobreak\ \leavevmode\nobreak\
Z_{k}=\operatorname{codim}_{Z}(Z_{k})>k\leavevmode\nobreak\
\leavevmode\nobreak\ \forall k>0$. Therefore, $\widetilde{N^{cl}_{Gie,0}}$ is
smooth irreducible, almost very good stack and $N^{cl}_{Gie,0}$ is an
irreducible, almost very good stack. ∎

######  Theorem 8.10.

The stack $M^{cl}_{Gie,0}$ is an irreducible local complete intersection of
pure dimension $2\cdot\text{dim}\leavevmode\nobreak\ \leavevmode\nobreak\
N_{Gie,0}+1$.

###### Proof.

It is enough to show that $\mu_{log,0}^{-1}(0)=\mathcal{Y}^{H,m}_{0}$ is a
local complete intersection of dimension $2\cdot\text{dim}\leavevmode\nobreak\
\leavevmode\nobreak\ N_{Gie,0}+1+\dim\leavevmode\nobreak\ GL_{N(m)}$. Here
$\mu_{log,0}$ denotes the restriction of $\mu_{log}$ to the closed fibre of
$\Omega^{log}_{\mathcal{Y}^{m}_{S}/S}\longrightarrow S$. Since the map
$\mu_{log,0}:\Omega^{log}_{\mathcal{Y}^{m}_{0}}\longrightarrow\mathfrak{pgl}^{*}_{N(m)}$
is surjective, the dimension of the generic fibre is equal to
$\dim\leavevmode\nobreak\ \mathcal{Y}^{m}_{0}-\dim\leavevmode\nobreak\
\mathfrak{gl}^{*}_{N(M)}+1$. Therefore for every irreducible component $I$ of
$\mu^{-1}_{log,0}(0)$, we have

$\dim\leavevmode\nobreak\ I\geq\dim\leavevmode\nobreak\
\mathcal{Y}^{m}_{0}-\dim\leavevmode\nobreak\ \mathfrak{gl}^{*}_{N(M)}+1.$
(8.10)

Suppose that there exists a component $I\subseteq\mu^{-1}_{log,0}(0)$ which
does not dominates $N_{Gie,0}$. In that case $p(I)\subseteq N_{k}$ for some
$k\geq 1$, where $N_{k}:=\\{x\in N|\dim\leavevmode\nobreak\ Stab(x)=k+1\\}$. A
priori, the generic point of $p(I)$ may belong to the singular locus of $N$.
But, in any case, using Lemma 8.8, we have

$\displaystyle\dim\leavevmode\nobreak\ I-\dim\leavevmode\nobreak\
p(I)\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leq\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \dim\leavevmode\nobreak\
\mathcal{Y}^{m}_{0}-\dim\leavevmode\nobreak\
\big{(}\frac{\mathfrak{gl}_{N(m)}}{\mathfrak{gl}_{N(m),y}}\big{)}^{*}\hskip
40.00006pt(\text{using Lemma}\leavevmode\nobreak\ \leavevmode\nobreak\
\ref{Moment})$ $\displaystyle\implies\dim\leavevmode\nobreak\
I\leq\dim\leavevmode\nobreak\ p(I)+\dim\leavevmode\nobreak\
\mathcal{Y}^{m}_{0}-\dim\leavevmode\nobreak\
\big{(}\frac{\mathfrak{gl}_{N(m)}}{\mathfrak{gl}_{N(m),y}}\big{)}^{*}\hskip
125.00018pt$ $\displaystyle<(\dim\leavevmode\nobreak\
\mathcal{Y}^{m}_{0}-k)+\dim\leavevmode\nobreak\
\mathcal{Y}^{m}_{0}-\dim\leavevmode\nobreak\
\big{(}\frac{\mathfrak{gl}_{N(m)}}{\mathfrak{gl}_{N(m),y}}\big{)}^{*}\hskip
100.00015pt$ $\displaystyle(\text{because}\leavevmode\nobreak\
N_{Gie,0}\leavevmode\nobreak\ \text{ is a almost very good stack})$
$\displaystyle=2\cdot\dim\leavevmode\nobreak\
\mathcal{Y}^{m}_{0}-\dim\leavevmode\nobreak\
\mathfrak{gl}^{*}_{N(m)}+(\dim\leavevmode\nobreak\
\mathfrak{gl}^{*}_{N(m),y}-(k+1))+1*\hskip 50.00008pt$
$\displaystyle=2\cdot\dim\leavevmode\nobreak\
\mathcal{Y}^{m}_{0}-\dim\leavevmode\nobreak\ \mathfrak{gl}^{*}_{N(m)}+1\hskip
160.00024pt$ $\displaystyle(\text{because}\leavevmode\nobreak\
\leavevmode\nobreak\ \dim\leavevmode\nobreak\
\mathfrak{gl}^{*}_{N(m),y}-(k+1),\text{by definition})$
$\displaystyle\leq\dim\leavevmode\nobreak\ \leavevmode\nobreak\ I\hskip
200.0003pt(\text{from}\leavevmode\nobreak\ \leavevmode\nobreak\ \eqref{geq})$

where $y$ denotes the generic point of $p(I)$. But this is a contradiction.
Therefore, $k$ must be equal to $0$, which implies that every component $I$
dominates $\mathcal{Y}^{m}_{0}$ and $\mu^{-1}_{log,0}(0)$ is an equi-
dimensional of dimension $2\cdot\dim\leavevmode\nobreak\
\mathcal{Y}^{m}_{0}-\dim\leavevmode\nobreak\ \mathfrak{gl}^{*}_{N(m)}+1$. Once
we know the dimension, it is easy to see that $\mu^{-1}_{log,0}(0)$ is a local
complete intersection. Therefore, $M^{cl}_{Gie,0}$ is an equi-dimensional
local complete intersection stack.

We define

$M^{cl,sm}_{Gie,0}:=\\{(\mathcal{E},\phi)\in M^{cl}_{Gie,0}\ \ |\ \
\mathbb{H}^{2}(\mathcal{C}(\mathcal{E},\phi))\cong\mathbb{k}\\},$

where $\mathcal{C}(\mathcal{E},\phi))$ denote the complex
$[\mathcal{E}nd\mathcal{E}\xrightarrow{[-,\phi]}\mathcal{E}nd\mathcal{E}\otimes\omega]$.
This substack is obviously an open subset of $M^{cl}_{Gie,0}$ because the
minimum dimension of $\mathbb{H}^{2}(\mathcal{C}(\mathcal{E},\phi))$ for any
Gieseker-Higgs bundle is $1$. Using the fact that $N^{cl}_{Gie,0}$ is an
irreducible, almost very good stack with a generic stabiliser of dimension $1$
and the fact that $M^{cl}_{Gie,0}$ is equi-dimensional, it is not difficult to
see that $M^{cl,sm}_{Gie,0}$ is also dense in $M^{cl}_{Gie,0}$.

But notice that $M^{cl,sm}_{Gie,0}$ is a vector bundle over $N_{Gie,0}$ with
fibres $H^{0}(\mathcal{E}nd\mathcal{E}\otimes\omega)$ (which remains constant
in this locus). Since $N_{Gie,0}$ is irreducible, $M^{cl,sm}_{Gie,0}$ is
irreducible. Therefore, $M^{cl}_{Gie,0}$ is also irreducible because
$M^{cl,sm}_{Gie,0}$ is dense in $M^{cl}_{Gie,0}$. ∎

######  Remrk 8.11.

Since the geometric fibres of the map $M^{cl}_{Gie}\longrightarrow S$ are of
dimension equal to $2n^{2}(g-1)+1$, therefore $\dim
M^{cl}_{Gie}=2n^{2}(g-1)+1+\dim S=2n^{2}(g-1)+2$.

######  Remrk 8.12.

As seen in subsubsection 8.2.1, the stack $M^{cl}_{Gie}$ can be covered by
open substacks of the form $[\mu_{log}^{-1}(0)/GL_{N(m)}]$, where
$\mu_{log}:\Omega^{log}_{\mathcal{Y}^{m}_{S}/S}\longrightarrow\mathfrak{pgl}_{N(m)}^{*}$
is the moment map and $\coprod_{m\geq m_{0}}\mathcal{Y}^{m}_{S}$ is a suitable
atlas of the stack $N_{Gie}$. Now notice that the dimension of
$\Omega^{log}_{\mathcal{Y}^{m}_{S}/S}$ is equal to $2n^{2}(g-1)+2N(m)^{2}+\dim
S$. Therefore the expected dimension of $\mu_{log}^{-1}(0)$ is equal to

$2n^{2}(g-1)+N(m)^{2}+1+\dim S.$

From the previous remark, we know that the dimension of $M^{cl}_{Gie}$ is