# The Algebraic and Analytic Compactifications
of the Hitchin Moduli Space

Siqi He  Morningside Center of Mathematics, Chinese Academy of Sciences,
Beijing, 100190 China<EMAIL_ADDRESS>,  Rafe Mazzeo  Department of
Mathematics, Stanford University, Stanford, CA, 94305<EMAIL_ADDRESS>,
Xuesen Na  Department of Mathematics, University of Illinois, Champaign, IL
61820, USA<EMAIL_ADDRESS>and  Richard A. Wentworth  Department of
Mathematics, University of Maryland, College Park, MD 20742, USA<EMAIL_ADDRESS>
###### Abstract.

Following the work of Mazzeo-Swoboda-Weiss-Witt [30] and Mochizuki [33], there
is a map $\overline{\Xi}$ between the algebraic compactification of the
Dolbeault moduli space of $\mathrm{SL}(2,\mathbb{C})$ Higgs bundles on a
smooth projective curve coming from the $\mathbb{C}^{\ast}$ action, and the
analytic compactification of Hitchin’s moduli space of solutions to the
$\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by
adding solutions to the decoupled equations, known as “limiting
configurations”. This map extends the classical Kobayashi-Hitchin
correspondence. The main result of this paper is that $\overline{\Xi}$ fails
to be continuous at the boundary over a certain subset of the discriminant
locus of the Hitchin fibration. This suggests the possibility of a third,
refined compactification which dominates both.

##  1\. Introduction

Let $\Sigma$ be a closed Riemann surface of genus $g\geq 2$. The coarse
Dolbeault moduli space of $\mathrm{SL}(2,\mathbb{C})$ semistable Higgs bundles
on $\Sigma$, denoted by $\mathcal{M}_{\mathrm{Dol}}$, and Hitchin’s moduli
space of solutions to the $\mathsf{SU}(2)$ self-duality equations on $\Sigma$,
denoted by $\mathcal{M}_{\mathrm{Hit}}$, have been extensively studied since
their introduction over 35 years ago. The Kobayashi-Hitchin correspondence,
proved in [21], gives a homeomorphism between these two moduli spaces:

$\Xi:\mathcal{M}_{\mathrm{Dol}}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\mathcal{M}_{\mathrm{Hit}}\
.$

The space $\mathcal{M}_{\mathrm{Dol}}$ is naturally a quasiprojective variety
[35, 45], and similarly to monopole moduli spaces,
$\mathcal{M}_{\mathrm{Hit}}$ fails to be compact. Recently, there has been
interest from several directions on natural compactifications of these two
spaces. A key feature on the Dolbeault side is the existence of a
$\mathbb{C}^{\ast}$ action with the Białynicki-Birula property, and this may
be used to define a completion of $\mathcal{M}_{\mathrm{Dol}}$ as a projective
variety [19, 7, 11]. The ideal points are identified with the $\mathbb{C}^{*}$
orbits in the complement of the nilpotent cone of
$\mathcal{M}_{\mathrm{Dol}}$. The Hitchin moduli space also admits a more
recently introduced compactification, $\overline{\mathcal{M}}_{\mathrm{Hit}}$,
based on the work of several authors (see [30, 33, 47]). The boundary of
$\overline{\mathcal{M}}_{\mathrm{Hit}}$ is given by gauge equivalence classes
of _limiting configurations_. This compactification is relevant to many
aspects of Hitchin’s moduli space. For more details, we refer the reader to
[10, 29, 14, 15, 36, 27, 3], and the references therein.

By the work of [30, 33], there is a natural extension

$\overline{\Xi}:\overline{\mathcal{M}}_{\mathrm{Dol}}\longrightarrow\overline{\mathcal{M}}_{\mathrm{Hit}}$

of the Kobayashi-Hitchin correspondence to the two compactifications described
above, and it is of interest to study the geometry of this map. This involves
another key feature of Hitchin’s moduli space; namely, spectral curves.
Spectral curves and spectral data [22] play a central role in the realization
of the Dolbeault moduli space as an algebraically complete integrable system
$\mathcal{H}:\mathcal{M}_{\mathrm{Dol}}\to\mathcal{B}$. In the case of
$\mathrm{SL}(2,\mathbb{C})$, the base $\mathcal{B}$ is the space of
holomorphic quadratic differentials on $\Sigma$. Given $q\in H^{0}(K^{2})$,
one obtains a (scheme theoretic) spectral curve $S_{q}$. This curve is reduced
if $q\neq 0$, irreducible if $q$ is not the square of an abelian differential,
and smooth if $q$ has simple zeros. We let $\mathcal{B}^{\rm
reg}\subset\mathcal{B}$ denote the open cone of quadratic differentials with
simple zeros.

The ideal points of both compactifications
$\overline{\mathcal{M}}_{\mathrm{Dol}}$ and
$\overline{\mathcal{M}}_{\mathrm{Hit}}$ have associated nonzero quadratic
differentials, and therefore spectral curves. We write
$\overline{\mathcal{M}}_{\mathrm{Dol}}^{\mathrm{reg}}$ for the elements in
$\overline{\mathcal{M}}_{\mathrm{Dol}}$ with smooth spectral curves, and
$\overline{\mathcal{M}}_{\mathrm{Dol}}^{\mathrm{sing}}=\overline{\mathcal{M}}_{\mathrm{Dol}}\setminus\overline{\mathcal{M}}_{\mathrm{Dol}}^{\mathrm{reg}}$
for those with singular spectral curves; similarly for
$\overline{\mathcal{M}}_{\mathrm{Hit}}^{\mathrm{reg}}$ and
$\overline{\mathcal{M}}_{\mathrm{Hit}}^{\mathrm{sing}}$. We then have the
following result.

######  Theorem 1.1.

The restriction of the compactified Kobayashi-Hitchin map
$\overline{\Xi}:\overline{\mathcal{M}}_{\mathrm{Dol}}\to\overline{\mathcal{M}}_{\mathrm{Hit}}$
to the locus with smooth associated spectral curves defines a homeomorphism
$\overline{\mathcal{M}}_{\mathrm{Dol}}^{\mathrm{reg}}\simeq\overline{\mathcal{M}}_{\mathrm{Hit}}^{\mathrm{reg}}$.
On the singular spectral curve locus, however,
$\overline{\Xi}^{\mathrm{sing}}:\overline{\mathcal{M}}_{\mathrm{Dol}}^{\mathrm{sing}}\to\overline{\mathcal{M}}_{\mathrm{Hit}}^{\mathrm{sing}}$
is neither surjective nor injective.

It is convenient to analyze the behavior along rays in $\mathcal{B}$, where
the spectral curve is simply rescaled. Let $q\neq 0$ be a quadratic
differential and $\overline{\mathcal{M}}_{\mathrm{Dol},q}$ (resp.
$\overline{\mathcal{M}}_{\mathrm{Hit},q}$) be the points in
$\overline{\mathcal{M}}_{\mathrm{Dol}}$ (resp.
$\overline{\mathcal{M}}_{\mathrm{Hit}}$) with spectral curves $S_{tq}$,
$t\in\mathbb{C}^{\ast}$. The restriction of $\overline{\Xi}$ gives us the map
$\overline{\Xi}_{q}:\overline{\mathcal{M}}_{\mathrm{Dol},q}\to\overline{\mathcal{M}}_{\mathrm{Hit},q}$.
We shall study the continuous behavior of $\overline{\Xi}_{q}$ for points in
the fiber of $tq$ as $t\to\infty$. For convenience, we set
$\mathcal{M}_{q^{\ast}}:=\overline{\mathcal{M}}_{\mathrm{Dol},q}\cap\mathcal{M}_{\mathrm{Dol}}$.
When $q$ is irreducible, i.e. not a square, all elements in $\mathcal{M}_{q}$
are stable. Via the Hitchin [23] and Beauville-Narasimhan-Ramanan (BNR)
correspondence [1], this reduces the description of the fiber
$\mathcal{H}^{-1}(q)$ to the characterization of rank $1$ torsion free sheaves
on the integral curve $S_{q}$.

In [38], parameter spaces for rank $1$ torsion free sheaves on algebraic
curves with Gorenstein singularities were studied in the context of
compactified Jacobians, and the crucial notion of a parabolic module was
introduced. This was extensively investigated by Cook in [5, 4], partially
following ideas of Bhosle [2]. For simple plane curve singularities of the
type appearing in spectral curves, one makes use of the local classification
of torsion free modules of Greuel-Knörrer [17]. These methods were applied to
study the Hitchin fibration by Gothen-Oliveira in [16] (see also [28] for
recent study). In parallel, Horn [24] defines a stratification of
$\mathcal{M}_{q}=\bigcup_{D}\mathcal{M}_{q,D}$ by certain effective divisors
contained in the divisor of $q$ (see Section 5.5, and also [26] for the more
general situation). Using the results from these references, we have a
reinterpretation of Mochizuki’s construction [33]. This leads to the following
result.

######  Theorem 1.2.

Let $q\neq 0$ be an irreducible quadratic differential.

  1. (1)

If $q$ has only zeros of odd order, then $\overline{\Xi}_{q}$ is continuous.

  2. (2)

If $q$ has at least one zero of even order, then for each $D\neq 0$ there
exists an integer $n_{D}\geq 1$ so that for any Higgs bundle
$(\mathcal{F},\psi)\in\mathcal{M}_{q,D}$, there exist $n_{D}$ sequences of
Higgs bundles $(\mathcal{E}_{i}^{k},\varphi_{i}^{k})$ with $k=1,\ldots,n_{D}$
satisfying

     * •

$\lim_{i\to\infty}(\mathcal{E}_{i}^{k},\varphi_{i}^{k})=(\mathcal{F},\psi)$
for $k=1,\ldots,n_{D}$,

     * •

and if we write

$\eta^{k}:=\lim_{i\to\infty}\overline{\Xi}_{q}(\mathcal{E}_{i}^{k},\varphi_{i}^{k})\quad,\quad\xi:=\lim_{i\to\infty}\overline{\Xi}_{q}(\mathcal{F},t_{i}\psi)\
,$

for some sequence $t_{i}\in\mathbb{R}^{+}$, $t_{i}\to+\infty$, then
$\xi,\eta^{1},\ldots,\eta^{n_{D}}$ are $n_{D}+1$ different limiting
configurations.

When $q$ is reducible, the description of Higgs bundles in $\mathcal{M}_{q}$
becomes more complicated because, among other things, of the existence of
strictly semistable objects. To understand this, we use the local descriptions
of Gothen-Oliveira and Mochizuki (see [16, 33]). Our result, which focuses on
the stable locus, is the following.

######  Theorem 1.3.

Suppose $q\neq 0$ is reducible. Let
$\overline{\mathcal{M}}_{\mathrm{Dol},q}^{\mathrm{st}}$ denote the stable
locus of $\overline{\mathcal{M}}_{\mathrm{Dol},q}$. If $g\geq 3$, then the
restriction map
$\overline{\Xi}_{q}|_{\overline{\mathcal{M}}_{\mathrm{Dol},q}^{\mathrm{st}}}$
is discontinuous. However, if $g=2$, the map
$\overline{\Xi}_{q}|_{\overline{\mathcal{M}}_{\mathrm{Dol},q}^{\mathrm{st}}}$
is continuous.

We also note that there is recent work of Mochizuki and Szabó [34] on the
asymptotic behavior for families of Higgs bundles in the higher rank case.

This paper is organized as follows: in Section 2, we provide an overview of
Higgs bundles and BNR correspondence. In Section 3, we delve into the concepts
of filtered bundles and their compactness properties. Section 4 defines the
algebraic and analytic compactifications. Section 5 introduces parabolic
modules and examines their connection to spectral curves. The main results for
Hitchin fibers with irreducible singular spectral curves are established in
Section 6. In Section 7, the results for the reducible case are proven.
Finally, in Section 8, we construct the compactified Kobayashi-Hitchin map and
prove the main results. The Appendix, based on the work of Greuel-Knörrer,
calculates some invariants of rank $1$ torsion free sheaves on the spectral
curves we consider.

Acknowledgements. We extend our sincere gratitude to Takurō Mochizuki for his
valuable insights and stimulating discussions during the BIRS conference in
2021. The authors also wish to express their gratitude to a great many people
for their interest and helpful comments. Among them are Mark de Cataldo, Ron
Donagi, Simon Donaldson, Laura Fredrickson, Johannes Horn, Laura Schaposnik,
Shizhang Li, Jie Liu, Tony Pantev, Thomas Walpuski, Daxin Xu. S.H is supported
by NSFC grant No.12288201. R.W.’s research is supported by NSF grants
DMS-1906403 and DMS-2204346, and he thanks the Max Planck Institute for
Mathematics in Bonn for its hospitality and financial support. The authors
also gratefully acknowledge the support of the MSRI under NSF grant
DMS-1928930 during the program “Analytic and Geometric Aspects of Gauge
Theory”, Fall 2022.

##  2\. Background on Higgs bundles

This section gives a very brief overview of the Dolbeault and Hitchin moduli
spaces, spectral curve descriptions, and the nonabelian Hodge correspondence.
For more details on these topics, see [21, 23, 44, 48].

###  2.1. Higgs bundles

As in the Introduction, throughout this paper $\Sigma$ will denote a closed
Riemann surface of genus $g\geq 2$ with structure sheaf
$\mathcal{O}=\mathcal{O}_{\Sigma}$ and canonical bundle $K=K_{\Sigma}$. Let
$E\to\Sigma$ be a complex vector bundle. A Higgs bundle consists of a pair
$(\mathcal{E},\varphi)$, where $\mathcal{E}$ is a holomorphic bundle and
$\varphi\in H^{0}(\mathrm{End}(\mathcal{E})\otimes K)$. If
$\mathrm{rank}(\mathcal{E})=1$, then a Higgs field is just an abelian
differential $\omega$. The pair $(\mathcal{E},\varphi)$ is called an
$\mathrm{SL}(2,\mathbb{C})$ Higgs bundle if $\mathrm{rank}(E)=2$,
$\det(\mathcal{E})$ has a fixed isomorphism with the trivial bundle, and
$\mathrm{Tr}(\varphi)=0$. In this paper we will focus mainly on
$\mathrm{SL}(2,\mathbb{C})$ Higgs bundles, but the rank $1$ case will also be
important.

Let $(\mathcal{E},\varphi)$ be an $\mathrm{SL}(2,\mathbb{C})$ Higgs bundle. A
(proper) Higgs subbundle of $(\mathcal{E},\varphi)$ is a holomorphic line
bundle $\mathcal{L}\subset\mathcal{E}$ that is $\varphi$-invariant, i.e.
$\varphi:\mathcal{L}\to\mathcal{L}\otimes K$. In this case, the restriction
$\varphi_{\mathcal{L}}:=\varphi\bigr{|}_{\mathcal{L}}$, makes
$(\mathcal{L},\varphi_{\mathcal{L}})$ a rank $1$ Higgs bundle. Moreover,
$\varphi$ induces a Higgs bundle structure on the quotient
$\mathcal{E}/\mathcal{L}$. We say $(\mathcal{E},\varphi)$ is stable (resp.
semistable) if for all Higgs subbundles $\mathcal{L}$, $\deg\mathcal{L}<0$
(resp. $\deg\mathcal{L}\leq 0$). We say $(\mathcal{E},\varphi)$ is polystable
if
$(\mathcal{E},\varphi)\simeq(\mathcal{L},\omega)\oplus(\mathcal{L}^{-1},-\omega)$,
where $\mathcal{L}$ is a degree zero holomorphic line bundle and $\omega\in
H^{0}(K)$.

If $(\mathcal{E},\varphi)$ is strictly semistable, i.e. semistable but not
stable, the Seshadri filtration [40] gives a unique Higgs subbundle
$0\subset(\mathcal{L},\omega)\subset(\mathcal{E},\varphi)$ with
$\deg(\mathcal{L})=\frac{1}{2}\deg(\mathcal{E})=0$. Write
$(\mathcal{L}^{\prime},\omega^{\prime}):=(\mathcal{E},\varphi)/(\mathcal{L},\omega)$,
then we have $\omega^{\prime}=-\omega$ and
$\mathcal{L}^{\prime}=\mathcal{L}^{-1}$. The associated graded bundle
$\mathrm{Gr}(\mathcal{E},\varphi)=(\mathcal{L},\omega)\oplus(\mathcal{L}^{-1},-\omega)$
of this filtration is a polystable $\mathrm{SL}(2,\mathbb{C})$ Higgs bundle.
We say that $(\mathcal{E},\varphi)$ is S-equivalent to
$\mathrm{Gr}(\mathcal{E},\varphi)$.

Holomorphic bundles $\mathcal{E}$ with underlying $C^{\infty}$ bundle $E$ are
in 1-1 correspondence with $\bar{\partial}$-operators
$\bar{\partial}_{E}:\Omega^{0}(E)\to\Omega^{0,1}(E)$. We use the notation
$\mathcal{E}:=(E,\bar{\partial}_{E})$. Let $\mathcal{C}$ denote the space of
pairs $(\bar{\partial}_{E},\varphi)$, $\bar{\partial}_{E}\varphi=0$. Let
$\mathcal{C}^{s}$ and $\mathcal{C}^{ss}$ denote the subspaces of $\mathcal{C}$
where the Higgs bundles are stable (resp. semistable). The complex gauge
transformation group $\mathcal{G}_{\mathbb{C}}:=\mathrm{Aut}(E)$ has a right
action on $\mathcal{C}$ by defining for $g\in\mathcal{G}_{\mathbb{C}}$,
$(\bar{\partial}_{E},\varphi)g:=(g^{-1}\circ\bar{\partial}\circ
g,g^{-1}\circ\varphi\circ g)$.

There is a quasiprojective scheme $\mathcal{M}_{\mathrm{Dol}}$ whose closed
points are in 1-1 correspondence with polystable Higgs bundles constructed via
(finite dimensional) Geometric Invariant Theory (see [35, 45]). In [12] it was
shown that the infinite dimensional quotient
$\mathcal{C}^{\mathrm{ss}}\sslash\mathcal{G}_{\mathbb{C}}$, where the double
slash indicates that S-equivalent orbits are identified, admits the structure
of a complex analytic space that is biholomorphic to the analytification
$\mathcal{M}_{\mathrm{Dol}}^{\rm an}$ of $\mathcal{M}_{\mathrm{Dol}}$.
Henceforth, we shall work in the complex analytic category, identify the
algebro-geometric and gauge theoretic moduli spaces, and simply denote them
both by $\mathcal{M}_{\mathrm{Dol}}$. We note that the set of stable Higgs
bundles modulo gauge transformations,
$\mathcal{M}_{\mathrm{Dol}}^{s}:=\mathcal{C}^{\mathrm{s}}\sslash\mathcal{G}_{\mathbb{C}}$,
is a geometric quotient and an open subset of $\mathcal{M}_{\mathrm{Dol}}$.

Finally, notice that the a pair $(\mathcal{E},\varphi)$ is stable (resp.
semistable) if and only if the same is true for
$(\mathcal{E},\lambda\varphi)$, $\lambda\in\mathbb{C}^{\ast}$. Hence,
$\mathcal{M}_{\mathrm{Dol}}$ admits an action of $\mathbb{C}^{\ast}$ that
preserves $\mathcal{M}_{\mathrm{Dol}}^{s}$. Though
$\mathcal{M}_{\mathrm{Dol}}$ is only quasiprojective, the $\mathbb{C}^{\ast}$
action satisfies the Białynicki-Birula property:

######  Theorem 2.1 ([21, 44]).

For any $[(\mathcal{E},\varphi)]\in\mathcal{M}_{\mathrm{Dol}}$,

$\lim_{\lambda\to 0}\lambda\cdot[(\mathcal{E},\varphi)]:=\lim_{\lambda\to
0}[(\mathcal{E},\lambda\varphi)]$

exists in $\mathcal{M}_{\mathrm{Dol}}$.

###  2.2. Spectral curves and the Hitchin fibration

The Hitchin map is defined as

$\mathcal{H}:\mathcal{M}_{\mathrm{Dol}}\longrightarrow H^{0}(K^{2})\ ,\
[(\mathcal{E},\varphi)]\mapsto\det(\varphi)\ ,$

where $H^{0}(K^{2})=:\mathcal{B}$ is known as the Hitchin base. Hitchin [21,
23] showed that $\mathcal{H}$ is a proper map and a fibration by abelian
varieties over the open cone $\mathcal{B}^{\rm reg}\subset\mathcal{B}$
consisting of nonzero quadratic differentials with only simple zeros. The
discriminant locus $\mathcal{B}^{\rm
sing}:=\mathcal{B}\setminus\mathcal{B}^{\rm reg}$ consists of quadratic
differentials that are either identically zero or have at least one zero with
multiplicity. For $q\in\mathcal{B}$, let
$\mathcal{M}_{q}:=\mathcal{H}^{-1}(q)$. The “most singular fiber”
$\mathcal{M}_{0}$ is called the _nilpotent cone_.

Consider the total space $\mathrm{Tot}(K)$ of $K$, along with its projection
$\pi:\mathrm{Tot}(K)\to\Sigma$. The pullback bundle $\pi^{*}K$ has a
tautological section, which we denote by $\lambda\in
H^{0}(\mathrm{Tot}(K),\pi^{\ast}K)$. Given any $q\neq 0\in H^{0}(K^{2})$, the
_spectral curve_ $S_{q}$ associated with $q$ is the zero scheme of the section
$\lambda^{2}-\pi^{\ast}q\in H^{0}(\mathrm{Tot}(K),\pi^{*}K)$. This is a
reduced, but possibly reducible, projective algebraic curve. The restriction
of $\pi$ to $S_{q}$, also denoted by $\pi:S_{q}\to\Sigma$, is a double
covering branched along the zeros of $q$.

The spectral curve $S_{q}$ is smooth if and only if $q$ has only simple zeros.
It is reducible if and only if $q=-\omega\otimes\omega$ for some $\omega\in
H^{0}(K)$. We shall refer to such quadratic differentials as _reducible_ , and
_irreducible_ otherwise. There is a noteworthy observation regarding
irreducible spectral curves.

######  Proposition 2.2 (cf. [23]).

Let $(\mathcal{E},\varphi)$ be a Higgs bundle with $q=\det(\varphi)$, and
suppose $q$ is irreducible. Then $(\mathcal{E},\varphi)$ has no proper
invariant subbundles. In particular, $(\mathcal{E},\varphi)$ is stable.

###### Proof.

Suppose $\mathcal{L}\subset\mathcal{E}$ is $\varphi$-invariant, and let
$\varphi_{\mathcal{L}}$ be the restriction. Then

$\det\varphi=-\frac{1}{2}\mathrm{Tr}(\varphi^{2})=-(\varphi_{\mathcal{L}})^{2}\
,$

contradicting the assumption. ∎

Let us emphasize that being reducible is not the same as having only even
zeros. To see this, suppose that $\mathrm{Div}(q)=2\Lambda^{\prime}$. Then
$K\simeq\mathcal{O}(\Lambda^{\prime})\otimes\mathcal{I}$, where $\mathcal{I}$
is a 2-torsion point in the Jacobian. The spectral curve $S_{q}$ is reducible
if and only if $\mathcal{I}$ is trivial.

###  2.3. Rank 1 torsion free sheaves and the BNR correspondence

In this subsection, we provide some background on rank 1 torsion free sheaf
theory over spectral curves in the context of the Hitchin and BNR
correspondence, as developed in [23, 1].

Let $S$ be a reduced and irreducible complex projective curve and
$\mathcal{O}_{S}$ its structure sheaf. The moduli space of invertible sheaves
on $S$ is denoted by $\mathrm{Pic}(S)$, and
$\mathrm{Pic}^{d}(S)\subset\mathrm{Pic}(S)$ is the degree $d$ component. If
$\mathcal{F}$ is a coherent analytic sheaf on $S$, we can define its
cohomology groups $H^{i}(X,\mathcal{F})$. Since $\dim S=1$,
$H^{i}(X,\mathcal{F})=0$ for $i\geq 2$. The Euler characteristic is defined as
$\chi(\mathcal{F})=\dim H^{0}(X,\mathcal{F})-\dim H^{1}(X,\mathcal{F})$. The
degree of a torsion free sheaf $\mathcal{F}$ is given by
$\deg(\mathcal{F})=\chi(\mathcal{F})-\mathrm{rank}(\mathcal{F})\chi(\mathcal{O}_{S})$.
If $\mathcal{F}$ is locally free, then $\deg(\mathcal{F})$ coincides with the
degree of the invertible sheaf $\det(\mathcal{F})$.

Let $\overline{\mathrm{Pic}}^{d}(S)$ be the moduli space of degree $d$ rank 1
torsion free sheaves on $S$, and
$\overline{\mathrm{Pic}}(S)=\prod_{d\in\mathbb{Z}}\overline{\mathrm{Pic}}^{d}(S)$
[9]. Then $\overline{\mathrm{Pic}}^{d}(S)$ is an irreducible projective scheme
containing $\mathrm{Pic}^{d}(S)$ as an open subscheme. When $S$ is smooth, we
have $\overline{\mathrm{Pic}}^{d}(S)=\mathrm{Pic}^{d}(S)$. The relationship to
Higgs bundles is given by the following.

######  Theorem 2.3 ([23, 1]).

Let $q\in H^{0}(K^{2})$ be an irreducible quadratic differential with spectral
curve $S_{q}$. There is a bijective correspondence between points in
$\overline{\mathrm{Pic}}(S_{q})$ and isomorphism classes of rank 2 Higgs pairs
$(\mathcal{E},\varphi)$ with $\mathrm{Tr}(\varphi)=0$ and $\det(\varphi)=q$.
Explicitly: if $\mathcal{L}\in\overline{\mathrm{Pic}}(S_{q})$, then
$\mathcal{E}:=\pi_{\ast}(\mathcal{L})$ is a rank 2 vector bundle, and the
homomorphism $\pi_{\ast}\mathcal{L}\to\pi_{\ast}\mathcal{L}\otimes
K\cong\pi_{\ast}(\mathcal{L}\otimes\pi^{\ast}K)$ given by multiplication by
the canonical section $\lambda$ defines the Higgs field $\varphi$.

This correspondence gives the very useful exact sequence

(1)
$\begin{split}0\to\mathcal{L}(-\Delta)\to\pi^{\ast}\mathcal{E}\xrightarrow{\pi^{\ast}\varphi-\lambda}\pi^{\ast}\mathcal{E}\otimes\pi^{\ast}K\to\mathcal{L}\otimes\pi^{\ast}K\to
0\ ,\end{split}$

where $\mathcal{O}_{S}(\Delta):=K_{S}\otimes\pi^{\ast}K_{\Sigma}^{-1}$ is the
ramification divisor. This will be used below in Section 6.

Let $q$ be a quadratic differential with only simple zeros, and define a
divisor on $S$ by $\Lambda=\mathrm{Div}(\lambda)$. Then $\Lambda$ is the
ramification divisor of the map $\pi:S\to\Sigma$. By the Riemann-Hurwitz
formula, the genus of $S$ is $g(S)=4g-3$. Furthermore, for any
$\mathcal{L}\in\mathrm{Pic}(S)$, Riemann-Roch gives
$\deg(\pi_{\ast}\mathcal{L})=\deg(\mathcal{L})-(2g-2).$ The
$\mathrm{SL}(2,\mathbb{C})$ Higgs bundles are characterized by

(2)
$\mathcal{T}:=\\{\mathcal{L}\in\mathrm{Pic}^{2g-2}(S)\mid\det(\pi_{\ast}\mathcal{L})=\mathcal{O}_{\Sigma}\\}.$

By the Hitchin-BNR correspondence (Theorem 2.3), the map
$\chi_{BNR}:\mathcal{T}\to\mathcal{M}_{q}$ is a bijection.

The branched double cover $\pi:S\to\Sigma$ is given by an involution
$\sigma:S\to S$. We have the norm map
$\mathrm{Nm}_{S/\Sigma}:\mathrm{Jac}(S)\to\mathrm{Jac}(\Sigma)$, where
$\mathrm{Nm}_{S/\Sigma}(\mathcal{O}_{S}(D)):=\mathcal{O}_{\Sigma}(\pi(D))$.
The Prym variety is defined as

$\mathrm{Prym}(S/\Sigma):=\ker(\mathrm{Nm}_{S/\Sigma})=\\{\mathcal{L}\in\mathrm{Pic}(S)\mid\mathcal{L}\otimes\sigma^{\ast}\mathcal{L}=\mathcal{O}_{S}\\}\
.$

Also, we have
$\det(\pi_{\ast}\mathcal{L})\cong\mathrm{Nm}_{S/\Sigma}(\mathcal{L})\otimes
K^{-1}$. Thus, $\mathcal{T}$ can be expressed as

$\mathcal{T}=\\{\mathcal{L}\in\mathrm{Pic}^{2g-2}(S)\mid\mathrm{Nm}_{S/\Sigma}(\mathcal{L})\cong
K\\}\ .$

Hence, $\mathcal{T}$ is a torsor over $\mathrm{Prym}(S,\Sigma)$. Explicitly,
by choosing $\mathcal{L}_{0}\in\mathcal{T}$, we obtain an isomorphism
$\mathcal{T}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\mathrm{Prym}(S,\Sigma)$
given by $\mathcal{L}\to\mathcal{L}\otimes\mathcal{L}_{0}^{-1}$.

To summarize, we have the following:

######  Proposition 2.4.

Let $q$ be a quadratic differential with simple zeros. Then
$\mathcal{M}_{q}\cong\mathcal{T}\cong\mathrm{Prym}(S,\Sigma)$.

If $q\neq 0$ is irreducible but nongeneric, the spectral curve $S$ is singular
and irreducible. We may still define the set
$\overline{\mathcal{T}}\subset\overline{\mathrm{Pic}}^{2g-2}(S)$ as follows:

$\overline{\mathcal{T}}:=\\{\mathcal{L}\in\overline{\mathrm{Pic}}^{2g-2}(S)\mid\det(\pi_{\ast}\mathcal{L})\cong\mathcal{O}_{\Sigma}\\}\
,$

We also set $\mathcal{T}:=\overline{\mathcal{T}}\cap\mathrm{Pic}^{2g-2}$.Then
$\overline{\mathcal{T}}$ is the natural compactification of $\mathcal{T}$
induced by the inclusion
$\mathrm{Pic}^{2g-2}(S)\subset\overline{\mathrm{Pic}}(S)$. The BNR
correspondence, as stated in Theorem 2.3, implies that
$\chi_{\mathrm{BNR}}:\overline{\mathcal{T}}\to\mathcal{M}_{q}$ is an
isomorphism.

###  2.4. The Hitchin moduli space and the nonabelian Hodge correspondence

We now recall the well-known nonabelian Hodge correspondence (NAH), which
relates the space of flat $\mathrm{SL}(2,\mathbb{C})$ connections, Higgs
bundles, and solutions to the Hitchin equations. This result was developed in
the work of Hitchin [21], Simpson [42], Corlette [6], and Donaldson [8].

As above, let $E$ be a trivial, smooth, rank 2 vector bundle over a Riemann
surface $\Sigma$, and let $H_{0}$ be a fixed Hermitian metric on $E$. We
denote by $\mathfrak{sl}(E)$ (resp. $\mathfrak{su}(E)$) the bundle of
traceless (resp. traceless skew-hermitian) endomorphisms of $E$. Let $A$ be a
unitary (with respect to $H_{0}$) connection on $E$ that induces the trivial
connection on $\det E$, and let $\phi\in\Omega^{1}(i\mathfrak{su}(E))$. We
will sometimes also refer to $\phi$ as a Higgs field. The Hitchin equations
for the pair $(A,\phi)$ are given by:

(3) $\begin{split}F_{A}+\phi\wedge\phi=0\ ,\ d_{A}\phi=d_{A}^{\ast}\phi=0\
.\end{split}$

If we split the Higgs field into type: $\phi=\varphi+\varphi^{\dagger}$, with
$\varphi\in\Omega^{1,0}(\mathfrak{sl}(E))$, then (3) is equivalent to:

(4) $\begin{split}F_{A}+[\varphi,\varphi^{\dagger}]=0\ ,\
\bar{\partial}_{A}\varphi=0\ .\end{split}$

Notice that $(\bar{\partial}_{E},\varphi)$ then defines an
$\mathrm{SL}(2,\mathbb{C})$ Higgs bundle. The Hitchin moduli space, denoted by
$\mathcal{M}_{\mathrm{Hit}}$, is the moduli space of solutions to the Hitchin
equation, given by

$\mathcal{M}_{\mathrm{Hit}}:=\\{(A,\phi)\mid(A,\phi)\;\mathrm{satisfies}\;\eqref{eq_Hitchin_real}\\}/\mathcal{G},$

where $\mathcal{G}$ is the gauge group of unitary automorphisms of $E$. Recall
that a flat connection $\mathcal{D}$ is called completely reducible if and
only if it is a direct sum of irreducible flat connections. The NAH can be
summarized as follows:

######  Theorem 2.5 ([21, 43, 6, 8]).

A Higgs bundle $(\mathcal{E},\varphi)$ is polystable if and only if there
exists a Hermitian metric $H$ such that the corresponding Chern connection $A$
and Higgs field $\phi=\varphi+\varphi^{\dagger}$ solve the Hitchin equations
(3). Moreover, the connection $\mathcal{D}$ defined by
$\mathcal{D}=\nabla_{A}+\phi$ is a completely reducible flat connection, and
it is irreducible if and only if $(\mathcal{E},\varphi)$ is stable.

Conversely, a flat connection $\mathcal{D}$ is completely reducible if and
only if there exists a Hermitian metric $H$ on $E$ such that when we express
$\mathcal{D}=\nabla_{A}+\varphi+\varphi^{\dagger}$, we have
$\bar{\partial}_{\mathcal{E}}\varphi=0$. Moreover, the corresponding Higgs
bundle $(\mathcal{E},\varphi)$ is polystable, and it is stable if and only if
$\mathcal{D}$ is irreducible.

The nonabelian Hodge correspondence defines the following Kobayashi-Hitchin
homeomorphism

$\begin{split}\Xi:\mathcal{M}_{\mathrm{Dol}}\to\mathcal{M}_{\mathrm{Hit}}\
,\end{split}$

which is a diffeomorphism to irreducible solutions of (3) when restricted to
the stable locus.

Finally, we note that there is an action of $S^{1}$ on
$\mathcal{M}_{\mathrm{Hit}}$ defined by $(A,\phi)\to(A,e^{i\theta}\cdot\phi)$,
where $e^{i\theta}\cdot\phi=e^{i\theta}\varphi+e^{-i\theta}\varphi^{\dagger}$.
With respect to this and the $S^{1}\subset\mathbb{C}^{\ast}$ action on
$\mathcal{M}_{\mathrm{Dol}}$, the map $\Xi$ is $S^{1}$-equivariant.

##  3\. Filtered bundles and compactness

Filtered (or parabolic) bundles are described, for example, in [43]. They play
a key role in the analytic compactification. This section provides a brief
overview of filtered line bundles and demonstrates a compactness result.

###  3.1. Filtered line bundles and nonabelian Hodge

Let $Z$ be a finite collection of distinct points on a closed Riemann surface
$\Sigma$, and let $\Sigma^{\prime}=\Sigma\setminus Z$. Viewing $\Sigma$ as a
projective algebraic curve, an algebraic line bundle $L$ over
$\Sigma^{\prime}$ is a line bundle defined by regular transition functions on
Zariski open sets over $\Sigma^{\prime}$. The sheaf of sections of $L$ can be
extended in infinitely many different ways over $Z$ to obtain coherent
(invertible) sheaves on $\Sigma$. The sections of $L$ are then realized as
meromorphic sections of an extension, regular on $\Sigma^{\prime}$.

A _filtered line bundle_ $\mathcal{F}_{\ast}(L)$ is an algebraic line bundle
$L$ with a collection of coherent extensions $L_{\alpha}$ across the punctures
$Z$ such that $L_{\alpha}\subset L_{\beta}$ for $\alpha\geq\beta$, for a fixed
sufficiently small $\epsilon$, $L_{\alpha-\epsilon}=L_{\alpha}$, and
$L_{\alpha}=L_{\alpha+1}\otimes\mathcal{O}_{\Sigma}(Z)$. Let
$\mathrm{Gr}_{\alpha}=L_{\alpha+\epsilon}/L_{\alpha}$ denote the quotient
(torsion) sheaf. A value $\alpha$ where $\mathrm{Gr}_{\alpha}\neq 0$ is called
a jump. Since we are considering line bundles, for each $p$ in the support of
$\mathrm{Gr}_{\alpha_{p}}$, there is exactly one jump $\alpha_{p}$ in the
interval $[0,1)$. The collection of jumps $\\{\alpha_{p}\\}$ fully determines
the filtered bundle structure. If we denote by $\mathcal{L}:=L_{0}$, the
degree of a filtered line bundle is defined as

$\deg(\mathcal{F}_{\ast}(L))=\deg(\mathcal{L})+\sum_{p\in Z}\alpha_{p}\ .$

Alternatively, a _weighted line bundle_ is a pair $(\mathcal{L},\chi)$ where
$\mathcal{L}\to\Sigma$ is a holomorphic line bundle and $\chi:Z\to\mathbb{R}$
is a weight function. The degree of a weighted bundle is defined as

$\deg(\mathcal{L},\chi)=\deg(\mathcal{L})+\sum_{p\in Z}\chi_{p}\ .$

The concepts of filtered line bundles and weighted line bundles are nearly
equivalent. Namely, given a filtered line bundle $\mathcal{F}_{\ast}(L)$, we
define $\mathcal{L}:=\mathcal{L}_{0}$ and $\chi_{p}=\alpha_{p}$. Conversely,
given a weighted line bundle $(\mathcal{L},\chi)$, let
$\alpha_{p}=\chi_{p}+n_{p}$, where $n_{p}\in\mathbb{Z}$ is the unique integer
so that $0\leq\chi_{p}+n_{p}<1$. A filtered bundle $\mathcal{F}_{\ast}(L)$ is
then determined by setting $L_{0}=\mathcal{L}(-\sum_{p\in Z}n_{p}p)$ with
jumps $\alpha_{p}$. Clearly, $\deg(\mathcal{F}(L))=\deg(\mathcal{L},\chi)$. We
shall use the notation $\mathcal{F}_{\ast}(\mathcal{L},\chi)$ for the filtered
bundle associated to a weighted bundle $(\mathcal{L},\chi)$ in this way.

Different weighted bundles can give rise to the same filtered bundle. The
following is a fact that will be frequently used in this paper. If
$D=\sum_{x\in Z}d_{x}x$ is a divisor supported on $Z$, let

$\chi_{D}(x):=\begin{cases}d_{x}\ ,&x\in Z\ ;\\\ 0\ ,&x\in\Sigma\setminus Z\
.\end{cases}$

Then for any weighted bundle $(\mathcal{L},\chi)$ we have
$\mathcal{F}_{\ast}(\mathcal{L}(D),\chi-\chi_{D})=\mathcal{F}_{\ast}(\mathcal{L},\chi)$.

Let $(\mathcal{L}_{1},\chi_{1})$ and $(\mathcal{L}_{2},\chi_{2})$ be two
weighted lines bundles. We define the tensor product

$(\mathcal{L}_{1},\chi_{1})\otimes(\mathcal{L}_{2},\chi_{2}):=(\mathcal{L}_{1}\otimes\mathcal{L}_{2},\chi_{1}+\chi_{2})\
.$

Then the degree is additive on tensor products. For filtered bundles, we
_define_

(5)
$\mathcal{F}_{\ast}(\mathcal{L}_{1},\chi_{1})\otimes\mathcal{F}_{\ast}(\mathcal{L}_{2},\chi_{2}):=\mathcal{F}_{\ast}(\mathcal{L}_{1}\otimes\mathcal{L}_{2},\chi_{1}+\chi_{2})\
.$

The degree is again additive for the tensor product of filtered bundles. This
agrees with the usual definition of tensor product for parabolic bundles.

###  3.2. Harmonic metrics for weighted line bundles

######  Proposition 3.1.

Let $(\mathcal{L},\chi)$ be a degree 0 weighted bundle. Then there exists a
Hermitian metric $h$ on $\mathcal{L}_{\Sigma^{\prime}}$ such that:

  * (i)

the Chern connection $A_{h}$ of $(\mathcal{L},h)$ is flat: $F_{A_{h}}=0$;

  * (ii)

for $p\in Z$, and $(U_{p},z)$ a holomorphic coordinate centered at $p$,
$|z|^{-2\chi_{p}}h$ extends to a $\mathcal{C}^{\infty}$ Hermitian metric on
$\mathcal{L}|_{U_{p}}$;

  * (iii)

$h$ is uniquely determined up to a multiplication by a nonzero constant.

###### Proof.

We first choose a background Hermitian metric $h_{0}$ such that
$|z|^{-2\chi_{p}}h_{0}$ defines a $\mathcal{C}^{\infty}$ Hermitian metric
defined on $U_{p}$. Let $A_{h_{0}}$ be the Chern connection, and $F_{A_{0}}$
the curvature. Note that $F_{A_{0}}$ is smooth on $\Sigma$. By the Poincaré-
Lelong formula, we have
$\frac{\sqrt{-1}}{2\pi}\int_{\Sigma}F_{A_{0}}=\deg(\mathcal{L},\chi)=0$.
Therefore, there exists a $\mathcal{C}^{\infty}$ function $\rho$ such that
$\Delta\rho+\frac{\sqrt{-1}}{2\pi}\Lambda F_{A_{0}}=0$. We define
$h=h_{0}e^{\rho}$. For the corresponding Chern connection $A_{h}$, we have
$F_{A_{h}}=0$, which implies (i). (ii) follows from the property for $h_{0}$,
since $\rho$ is a smooth function on $\Sigma$. As $\rho$ is well-defined up to
a constant, $h$ is well-defined up to a constant, which implies (iii). ∎

The metric obtained above is called the _harmonic metric_. For a weighted
bundle $(\mathcal{L},\chi)$, the holomorphic bundle $\mathcal{L}$ and the
harmonic metric $h$ define a filtration as follows. For $\epsilon>0$
sufficiently small, let

$L_{\alpha}:=\\{s\in\mathcal{L}(\ast Z)\mid|s|_{h}\leq
Cr^{\alpha-\epsilon}\text{ for some $C$}\\}\ .$

Here, $r$ denotes the distance to $Z$ in any smooth conformal metric on
$\Sigma$. It is straightforward to check that this defines a filtered bundle
that matches $\mathcal{F}_{\ast}(\mathcal{L},\chi)$ under the correspondence
in the previous section.

Even though the harmonic metric is only well-defined up to a constant, the
Chern connection $A=(\mathcal{L},h)$ is independent of this choice. The
$(1,0)$ part of $A$, denoted $\nabla_{h}$, then defines logarithmic
connections $\nabla_{h}:L_{\alpha}\to L_{\alpha}\otimes K(Z).$

###  3.3. Convergence of weighted line bundles

In this subsection, we will consider the convergence of weighted line bundles.
The main result we prove here is a consequence of [34, Theorem 1.8]. For the
reader’s convenience, we present a short proof here for our situation.

Let $(\Sigma_{0},g_{0})$ be a metrized Riemann surface (i.e. a Riemann surface
$\Sigma_{0}$ with conformal metric $g_{0}$). We view $\Sigma_{0}$ as given by
an underlying surface $C$ with almost complex structure $J_{0}$. Consider a
neighborhood $U_{1}$ of $J_{0}$ in the moduli space of holomorphic structures
and a neighborhood $U_{2}$ of $g_{0}$ in the space of smooth metrics. We
denote the product of these neighborhoods by $U=U_{1}\times U_{2}$. We can
define the fiber bundle $\mathrm{Pic}_{U}\to U$, where each fiber is the
Picard group defined by the holomorphic structure. Let
$(\Sigma_{t}=(C,J_{t}),g_{t})$ be a family of metrized Riemann surfaces that
converge smoothly to $(\Sigma_{0},g_{0})$ as $t\to 0$. Let
$Z_{t}\subset\Sigma_{t}$ be a collection of a finite number of points that
converge to $Z_{0}$ in suitable symmetric products of $C$. For each $p\in
Z_{0}$, we can write $Z_{t}=\cup_{p\in Z_{0}}Z_{t,p}$ such that all points in
$Z_{t,p}$ converge to $p$. We define the convergence of weighted line bundles
as follows:

######  Definition 3.2.

A family of weighted line bundles $(\mathcal{L}_{t},\chi_{t})$ over
$\Sigma_{t}$ with weights $\chi_{t}:Z_{t}\to\mathbb{R}$ converges to
$(\mathcal{L}_{0},\chi_{0})$ if

  * (i)

$\mathcal{L}_{t}$ converges to $\mathcal{L}_{0}$ in $\mathrm{Pic}_{U}$,

  * (ii)

for all $p\in Z_{0}$ and $t$ sufficiently small, $\sum_{q\in
Z_{t,p}}\chi_{t}(q)=\chi_{0}(p)$.

A sequence of filtered bundles $\mathcal{F}_{\ast}(\mathcal{L}_{t})$ converges
to $\mathcal{F}_{\ast}(\mathcal{L}_{0})$ if the corresponding weighted bundles
converge. The following theorem provides insight into the compactness of a
sequence of weighted line bundles:

######  Theorem 3.3.

Let $(\mathcal{L}_{t},\chi_{t})$ be a sequence of weighted line bundles over
$\Sigma_{t}\setminus Z_{t}$, where $\deg(\mathcal{L}_{t},\chi_{t})=0$, and let
$h_{t}$ be the corresponding harmonic metrics. If $Z_{t}$ converges to
$Z_{0}$, we write $Z_{t}=\cup_{p\in Z_{0}}Z_{t,p}$. Then there exists a
weighted line bundle $(\mathcal{L}_{0},\chi_{0})$ over $Z_{0}$ with a harmonic
metric $h_{0}$ such that:

  * (i)

After being rescaled by $c_{t}>0$, $c_{t}h_{t}$ converges to $h_{0}$ over
$\Sigma_{0}\setminus Z_{0}$ in the $\mathcal{C}^{\infty}_{\mathrm{loc}}$
sense.

  * (ii)

Let $\nabla_{t}$ be the unitary connection of $h_{t}$. Then on
$\Sigma_{0}\setminus Z_{0}$, $\lim_{t\to 0}\nabla_{t}=\nabla_{0}$ in
$\mathcal{C}^{\infty}_{\mathrm{loc}}$.

###### Proof.

By the assumptions on weights, $\deg(\mathcal{L}_{t})$ is a fixed,
$t$-independent constant. Let $\gamma_{t}=(J_{t},g_{t})$ be a path in $U$.
Then $\mathrm{Pic}_{U}|_{\gamma_{t}}$ is compact, and there exists an
$\mathcal{L}_{0}\in\mathrm{Pic}(\Sigma_{0})$ such that $\mathcal{L}_{t}$
converges to $\mathcal{L}_{0}$. For $p\in Z_{0}$, define
$\chi_{0}(p)=\sum_{q\in Z_{t,p}}\chi_{t}(q)$, and thus obtain a weighted line
bundle $(\mathcal{L}_{0},\chi_{0})$. We can choose a family of approximate
harmonic metrics $h_{t}^{\mathrm{app}}$, such that
$|z|^{-2\chi_{p}}h_{t}^{\mathrm{app}}$ extends to a smooth metric in a
neighborhood of $p$ and $h_{t}^{\mathrm{app}}$ converges to
$h_{0}^{\mathrm{app}}$ in
$\mathcal{C}^{\infty}_{\mathrm{loc}}(\Sigma_{0}\setminus Z_{0})$. Moreover, we
write $h_{t}=h_{t}^{\mathrm{app}}e^{s_{t}}$. After a suitable rescale of
$h_{t}$, we can assume $\|s_{t}\|_{L^{2}}=1$. Let
$\rho_{t}:=\Delta_{t}h_{t}^{\mathrm{app}}$ be the curvature defined by the
metric $h^{\mathrm{app}}_{t}$. Then $s_{t}$ satisfies the equation
$\Delta_{t}s_{t}=\rho_{t}$ over $\Sigma$. As $\rho_{t}$ converges to
$\rho_{0}\in\mathcal{C}^{\infty}_{\mathrm{loc}}(\Sigma\setminus Z_{0})$, and
$g_{t}$ is a family with bounded geometry, we obtain the estimate

$\|s_{t}\|_{\mathcal{C}^{k+2,\alpha}(\Sigma)}\leq
C_{k,\alpha}(\|\rho_{t}\|_{\mathcal{C}^{k,\alpha}(\Sigma)}+1)\ ,$

where $C_{k,\alpha}$ is a $t$-independent constant. Therefore, passing to a
subsequence, $s_{t}$ converges to $s_{0}$ in $\mathcal{C}^{\infty}(\Sigma)$,
which implies (i). The assertion (ii) follows from (i). ∎

##  4\. The algebraic and analytic compactifications

In this section, we introduce compactifications of the Dolbeault and Hitchin
moduli spaces.

###  4.1. The algebraic compactification of the Dolbeault moduli space

In this subsection, we present the algebraic method for compactifying the
Dolbeault moduli space. This technique is based on the $\mathbb{C}^{\ast}$
action on $\mathcal{M}_{\mathrm{Dol}}$, and was introduced in [41, 39, 19, 7,
27]. The gauge theoretic approach can be found in [11].

######  Theorem 4.1 ([41, Thm. 11.2],[7]).

Let $V$ be an algebraic variety with $\mathbb{C}^{\ast}$ action. Suppose

  * (i)

the fixed point set of the $\mathbb{C}^{\ast}$ action is proper,

  * (ii)

for every $t\in\mathbb{C}^{\ast},\;v\in V$, the limit $\displaystyle\lim_{t\to
0}t\cdot v$ exists.

Then the space $\displaystyle U:=\\{v\in V\mid\lim_{t\to\infty}t\cdot
v\;\mathrm{does\;not\;exist}\\}$ is open in $V$, and the quotient
$U/\mathbb{C}^{\ast}$ is separated and proper.

We apply this to the Dolbeault moduli space. The first step is to note that
the possible isotropy subgroups are limited.

######  Lemma 4.2.

[19, Thm. 6.2] Let $\xi=[(\mathcal{E},\varphi)]$ be a Higgs bundle equivalence
class with $\mathcal{H}(\xi)\neq 0$. Then the stabilizer $\Gamma_{\xi}$ of
$\xi$ for the $\mathbb{C}^{\ast}$ action is either trivial or $\mathbb{Z}/2$.
The latter case holds if and only if $(\mathcal{E},\varphi)$ and
$(\mathcal{E},-\varphi)$ are complex gauge equivalent.

###### Proof.

For $t\in\Gamma_{\xi}$, $\mathcal{H}(t\cdot\xi)=t^{2}\mathcal{H}(\xi)$, hence
$t^{2}=1$ if $\mathcal{H}(\xi)\neq 0$. ∎

By this Lemma, the space
$(\mathcal{M}_{\mathrm{Dol}}\setminus\mathcal{H}^{-1}(0))/\mathbb{C}^{\ast}$
has an orbifold structure. In passing, we note that the fixed points of the
$\mathbb{Z}/2$ action correspond to real representations under the nonabelian
Hodge correspondence [21, Sec. 10].

By the properness of the Hitchin map $\mathcal{H}$ (see Theorem 2.1), it
follows that $\displaystyle\lim_{t\to\infty}t\cdot\xi$ exists if and only if
$\mathcal{H}(\xi)=0$. Now define

(6)
$\overline{\mathcal{M}}_{\mathrm{Dol}}=\left\\{(\mathcal{M}_{\mathrm{Dol}}\times\mathbb{C}^{\ast})\coprod(\mathcal{M}_{\mathrm{Dol}}\setminus\mathcal{H}^{-1}(0))\right\\}/\mathbb{C}^{\ast}\
.$

The analytic topology on the disjoint union is generated by open sets $U\times
W_{1}$ and $V\times(W_{2}\cap\mathbb{C}^{\ast})\amalg
V\cap(\mathcal{M}_{\mathrm{Dol}}\setminus\mathcal{H}^{-1}(0))$, where
$U,V\subset\mathcal{M}_{\mathrm{Dol}}$, $W_{1},W_{2}\subset\mathbb{C}$ are
open, and $0\not\in W_{1}$, $0\in W_{2}$. The topology on
$\overline{\mathcal{M}}_{\mathrm{Dol}}$ is then the quotient topology, and it
is straightforward to see that with this topology, it is compact.

Since
$(\mathcal{M}_{\mathrm{Dol}}\times\mathbb{C}^{\ast})/\mathbb{C}^{\ast}=\mathcal{M}_{\mathrm{Dol}}$,
there is a natural inclusion

$\iota:\mathcal{M}_{\mathrm{Dol}}\to\overline{\mathcal{M}}_{\mathrm{Dol}},\;\iota(\xi)=[(\xi,1)]\
,$

where brackets denote the equivalence class under the $\mathbb{C}^{\ast}$
action. The boundary of $\overline{\mathcal{M}}_{\mathrm{Dol}}$ is

$\partial\overline{\mathcal{M}}_{\mathrm{Dol}}=\overline{\mathcal{M}}_{\mathrm{Dol}}\setminus\iota(\mathcal{M}_{\mathrm{Dol}})=(\mathcal{M}_{\mathrm{Dol}}\setminus\mathcal{H}^{-1}(0))/\mathbb{C}^{\ast}\
.$

There is a _boundary map_

$\iota_{\partial}:\mathcal{M}_{\mathrm{Dol}}\setminus\mathcal{H}^{-1}(0)\longrightarrow\partial\overline{\mathcal{M}}_{\mathrm{Dol}},\;\xi\mapsto[(\xi,0)]\
,$

which is invariant under the $\mathbb{C}^{\ast}$ action, i.e.,
$\iota_{\partial}(\lambda\xi)=\iota_{\partial}(\xi)$ for
$\lambda\in\mathbb{C}^{\ast}$.

The $\mathbb{C}^{\ast}$ action on $\mathcal{M}_{\mathrm{Dol}}$ covers the
square of the action on $\mathcal{B}$. Hence, it is natural to compactify
$\mathcal{B}$ by projectivizing:

$\overline{\mathcal{B}}:=\mathbb{P}(H^{0}(K^{2})\oplus\mathbb{C})\ .$

The inclusion is given, as usual, by

$\iota_{0}:\mathcal{B}\to\overline{\mathcal{B}},\;\iota(q)=[q\times\\{1\\}]\
,$

where $q\times\\{1\\}\in H^{0}(K^{2})\oplus\mathbb{C}$. We also define
$\partial\overline{\mathcal{B}}=\overline{\mathcal{B}}\setminus\iota_{0}(\mathcal{B})\simeq\mathbb{P}(H^{0}(K^{2}))$,
with boundary projection map

$\iota_{0,\partial}:\mathcal{B}\setminus\\{0\\}\to\partial\overline{\mathcal{B}}\
,\ \iota_{0,\partial}(q)=[q\times\\{0\\}]\ .$

The Hitchin map $\mathcal{H}:\mathcal{M}_{\mathrm{Dol}}\to\mathcal{B}$ extends
to
$\overline{\mathcal{H}}:\overline{\mathcal{M}}_{\mathrm{Dol}}\to\overline{\mathcal{B}}$,
where
$\overline{\mathcal{H}}|_{\mathcal{M}_{\mathrm{Dol}}}:=\iota_{0}\circ\mathcal{H}$,
and for every
$[(\mathcal{E},\varphi)]/\mathbb{C}^{\ast}\in\partial\overline{\mathcal{M}}_{\mathrm{Dol}}$,

$\overline{\mathcal{H}}([(\mathcal{E},\varphi)]/\mathbb{C}^{\ast}):=[(\mathcal{H}(\varphi),0)]\subset\overline{\mathcal{B}}\
.$

This is well defined, since $\det(\varphi)\neq 0$ if
$[(\mathcal{E},\varphi)]/\mathbb{C}^{\ast}\in\partial\overline{\mathcal{M}}_{\mathrm{Dol}}$.
Moreover,

${\mathcal{M}_{\mathrm{Dol}}}$${\overline{\mathcal{M}}_{\mathrm{Dol}}}$${\mathcal{B}}$${\overline{\mathcal{B}}}$$\scriptstyle{\iota}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{\overline{\mathcal{H}}}$$\scriptstyle{\iota_{0}}$

commutes.

There is a good algebraic structure on this algebraic compactification:

######  Theorem 4.3 ([41, 39, 19, 7, 11]).

The compactified space $\overline{\mathcal{M}}_{\mathrm{Dol}}$ is a normal
projective variety, and $\partial\overline{\mathcal{M}}_{\mathrm{Dol}}$ is a
Cartier divisor of $\overline{\mathcal{M}}_{\mathrm{Dol}}$.

The following characterization of sequential convergence is useful.

######  Proposition 4.4.

Let $[(\mathcal{E}_{i},\varphi_{i})]\in\mathcal{M}_{\mathrm{Dol}}$ be a
sequence of Higgs bundles, and write $q_{i}=\det(\varphi_{i})$ and
$r_{i}=\|q_{i}\|_{L^{2}}^{\frac{1}{2}}$. Suppose $\limsup r_{i}\to\infty$.
Then up to subsequence:

  * (i)

there exists a Higgs bundle
$[(\widehat{\mathcal{E}}_{\infty},\hat{\varphi}_{\infty})]$ with
$\tilde{q}_{\infty}=\det(\tilde{\varphi}_{\infty})$ and
$\|\hat{q}_{\infty}\|_{L^{2}}=1$ such that
$\lim_{i\to\infty}[(\mathcal{E}_{i},r_{i}^{-1}\varphi_{i})]=[(\widehat{\mathcal{E}}_{\infty},\hat{\varphi}_{\infty})]$
in $\mathcal{M}_{\mathrm{Dol}}$ and
$\lim_{i\to\infty}r_{i}^{-1}q_{i}=\hat{q}_{\infty}$ in $H^{0}(K^{2})$;

  * (ii)

$\displaystyle\lim_{i\to\infty}\iota[(\mathcal{E}_{i},\varphi_{i})]$
$\displaystyle=\iota_{\partial}[(\widehat{\mathcal{E}}_{\infty},\hat{\varphi}_{\infty})]\
,\ \mbox{on}\ \overline{\mathcal{M}}_{\mathrm{Dol}}\ ,$
$\displaystyle\lim_{i\to\infty}\iota_{0}(q_{i})$
$\displaystyle=\iota_{0,\partial}(\hat{q}_{\infty})\ ,\ \mbox{on}\
\overline{\mathcal{B}}\ .$

###### Proof.

The first point follows since the Hitchin map $\mathcal{H}$ is proper and
$\mathcal{H}(r_{i}^{-1}\varphi_{i})$ is bounded. The second follows directly
from the definition. ∎

###  4.2. The analytic compactification of the Hitchin moduli space

We next describe the compactification of the Hitchin moduli space, as
developed in [29, 33, 46].

####  4.2.1. Decoupled Hitchin equations

We begin by defining the decoupled Hitchin equations. Recall the notation from
Section 2.4, let $E$ be a trivial, smooth, rank 2 vector bundle over a Riemann
surface $\Sigma$, and let $H_{0}$ be a background Hermitian metric on $E$. Let
$Z$ be a finite set of distinct points in $\Sigma$. For a smooth unitary
connection $A$ on $E|_{\Sigma\setminus Z}$ and smooth
$\phi=\varphi+\varphi^{\dagger}\in\Omega^{1}(i\mathfrak{su}(E))|_{\Sigma\setminus
Z}$, the _decoupled Hitchin equations_ on $\Sigma\setminus Z$ are:

(7) $\begin{split}F_{A}=0\ ,\ [\varphi,\varphi^{\dagger}]=0\ ,\
\bar{\partial}_{A}\varphi=0\ .\end{split}$

Solutions to (7) may be quite singular near $Z$, so we make the following
restriction:

######  Definition 4.5.

A solution $(A,\phi)$ to the decoupled Hitchin equations over $\Sigma\setminus
Z$ is called _admissible_ if $\phi\neq 0$, and $|\phi|$ extends to a
continuous function on $\Sigma$ with $|\phi|^{-1}(0)=Z$.

By a _limiting configuration_ we always mean an admissible solution to the
decoupled Hitchin equations. Clearly, $Z$ is determined by $(A,\phi)$.
Admissibility guarantees that $\det(\varphi)$ extends to a holomorphic
quadratic differential $q=\det(\varphi)$ on $\Sigma$, with $Z=q^{-1}(0)$ the
zero locus. Hence, the spectral curve $S_{q}$ is well-defined. We emphasize
that $Z$ may vary for different admissible solutions, but one always has that
$\\#Z\leq 4g-4$.

The equivalence relation on limiting configurations is that
$(A_{1},\phi_{1})\sim(A_{2},\phi_{2})$ if $Z_{1}=Z_{2}$ and
$(A_{1},\phi_{1})g=(A_{2},\phi_{2})$ for a smooth unitary gauge transformation
$g$ on $\Sigma\setminus Z_{1}$. The moduli space of decoupled Hitchin
equations is then

$\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}=\\{\text{admissible solutions to
}\eqref{eq_decoupled_Hitchin_equation}\\}/\sim\ .$

We denote by $\mathcal{M}_{\mathrm{Hit},q}^{\mathrm{Lim}}$ the elements in
$\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}$ with a fixed quadratic
differential $q$. In this case, the equivalence relation is induced by the
action of the unitary gauge group over $\Sigma\setminus Z$, $Z=q^{-1}(0)$.

There is a natural $\mathbb{C}^{\ast}$ action on the moduli space
$\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}$: given
$(A,\phi=\varphi+\varphi^{\dagger})\in\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}$
and $t\in\mathbb{C}^{\ast}$, we set
$t\cdot[(A,\phi)]=[(A,t\varphi+\bar{t}\varphi^{\dagger})]$, which is also a
solution to (7).

####  4.2.2. Compactification of the Hitchin moduli space

The following compactness result is due to Taubes [47] and Mochizuki [33] (see
also [20]).

######  Proposition 4.6.

Let $(A_{i},\varphi_{i})$ be a sequence of solutions to (3), with
$q_{i}=\det(\varphi_{i})\in H^{0}(K^{2})$. Then

  * (i)

if $\limsup\|q_{i}\|_{L^{2}(\Sigma)}<\infty$, then there is a subsequence
(also denoted $\\{i\\}$), a smooth solution $(A_{\infty},\phi_{\infty})$ to
(3), and a sequence $g_{i}$ of smooth unitary gauge transformations on
$\Sigma$, such that $(A_{i},\phi_{i})g_{i}$ converges smoothly to
$(A_{\infty},\phi_{\infty})$ on $\Sigma$;

  * (ii)

if $\lim\|q_{i}\|_{L^{2}(\Sigma)}=\infty$, then there is a subsequence (also
denoted $\\{i\\}$), and $q_{\infty}\in H^{0}(K^{2})$ so that

$\frac{q_{i}}{\|q_{i}\|_{L^{2}}}\longrightarrow q_{\infty}$

over $\Sigma$, and an admissible solution
$(A_{\infty},\phi_{\infty}=\varphi_{\infty}+\varphi^{\dagger}_{\infty})$ to
(7), with $Z_{\infty}:=q_{\infty}^{-1}(0)$, and smooth unitary gauge
transformations $g_{i}$ on $\Sigma\setminus Z_{\infty}$, such that over any
open set $\Omega\Subset\Sigma\setminus Z_{\infty}$, $(A_{i})g_{i}\to
A_{\infty}$, and

$\frac{g_{i}^{-1}\phi_{i}g_{i}}{\|\phi\|_{L^{2}}}\longrightarrow\phi_{\infty}$

smoothly on $\Omega$.

The norm on $H^{0}(\Sigma,K^{2})$ can be chosen arbitrarily, since it is a
finite dimensional space.

There is also a compactness result for sequences of solutions in
$\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}$.

######  Proposition 4.7.

Let
$[(A_{i},\phi_{i}=\varphi_{i}+\varphi_{i}^{\dagger})]\in\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}$
be a sequence of admissible solutions to (7), and let
$q_{i}=\det(\varphi_{i})$ be the corresponding quadratic differentials. Then
after passing to a subsequence, there are $t_{i}\in\mathbb{C}^{\ast}$, a
limiting configuration
$(A_{\infty},\phi_{\infty}=\varphi_{\infty}+\varphi_{\infty}^{\dagger})$ with
quadratic differential $q_{\infty}=\det(\varphi_{\infty})\neq 0$, and a
sequence $g_{i}$ of smooth gauge transformations on $\Sigma\setminus
Z_{\infty}$, such that:

  * (i)

$t_{i}^{2}q_{i}$ converges smoothly to $q_{\infty}$,

  * (ii)

over any open set $\Omega\Subset X\setminus Z_{\infty}$,
$(A_{i},t_{i}\cdot\phi_{i})g_{i}$ converges smoothly to
$(A_{\infty},\phi_{\infty})$.

###### Proof.

Write $q_{i}=\det(\varphi_{i})\in H^{0}(K^{2})$. Adjusting by $t_{i}$ if
necessary, we may assume $q_{i}$ converges to $q_{\infty}$ over $\Sigma$.
Also, since $F_{A_{i}}=0$ over $\Sigma\setminus Z_{i}$ and $Z_{i}$ converges
to $Z_{\infty}$, we can apply both Uhlenbeck compactness and the classical
bootstrapping method to obtain $A_{\infty}$ such that up to gauge $A_{i}$
converges smoothly to $A_{\infty}$ over $\Sigma\setminus Z_{\infty}$. Finally,
the convergence of $\varphi_{i}$ follows by the bound on $q_{i}$’s. ∎

####  4.2.3. The topology on the compactified space

We now carefully define the topology on the space
$\mathcal{M}_{\mathrm{Hit}}\coprod\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}/\mathbb{C}^{\ast}$.
Choose a metric in the conformal class on $\Sigma$. Let $W^{k,2}$ denote the
Sobolev spaces on $\Sigma$ of distributional sections with at least $k$
derivatives in $L^{2}$. For a finite set of points $Z\subset\Sigma$ (or indeed
any closed subset),

$\begin{split}W^{k,2}_{\mathrm{loc}}(\Sigma\setminus Z):=\\{f\mid f\in
W^{k,2}(K),\;K\subset\Sigma\setminus Z,\;K\;\mathrm{compact}\\}.\end{split}$

These definitions extend easily to the space of connections and
$\Omega^{1}(i\mathfrak{su}(E))$ for a Hermitian vector bundle $(E,H_{0})$ over
$\Sigma$ with a fixed smooth background connection.

Let ${\omega_{n}}$ be a nested collection of open sets with
$\omega_{n}\subset\overline{\omega_{n}}\subset\omega_{n+1}$, with
$\bigcup_{n}\omega_{n}=\Sigma\setminus Z$. We then define the seminorms
$\|f\|_{n}:=\|f\|_{W^{k,2}(\omega_{n})}$; in terms of these,
$W^{k,2}_{\mathrm{loc}}(\Sigma\setminus Z)$ a Fréchet space.

For any $q\in H^{0}(K^{2})\setminus\\{0\\}$, set $Z_{q}:=q^{-1}(0)$, and
consider the moduli space

$\mathbb{M}_{q}=\\{(A,\phi)\in\mathcal{M}_{q^{\ast}}\cap
W^{k,2}(\Sigma)\\}\cup\\{(A,\phi)\in\mathcal{M}_{\mathrm{Hit},q}^{\mathrm{Lim}}\cap
W^{k,2}_{\mathrm{loc}}(\Sigma\setminus Z_{q})\\}/\mathbb{C}^{\ast}\ .$

By classical bootstrapping of the gauge-theoretic elliptic equations,
$\mathbb{M}_{q}$ is independent of $k\geq 2$.

Next define $\mathbb{M}:=\mathcal{M}_{0}\cup\bigcup_{q\in
H^{0}(K^{2})\setminus\\{0\\}}\mathbb{M}_{q}$. Its topology is generated by two
types of open sets. For interior points
$\xi=[(A,\phi)]\in\mathcal{M}_{\mathrm{Hit}}\subset\mathbb{M}$ we use the open
sets

$\begin{split}V_{\xi,\epsilon}:=\\{[(A^{\prime},\phi^{\prime})]\in\mathcal{M}_{\mathrm{Hit}}\mid\|A^{\prime}-A\|_{W^{k,2}(\Sigma)}+\|\phi^{\prime}-\phi\|_{W^{k,2}(\Sigma)}<\epsilon\\}\end{split}$

from the topology of $\mathcal{M}_{\mathrm{Hit}}$. For any boundary point
$\xi_{0}\in\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}/\mathbb{C}^{\ast}$,
choose a representative $(A_{0},e^{i\theta}\phi_{0})$ for some constant
$\theta$. Let $q=\det(\phi_{0})$, and fix any open set
$\omega\Subset\Sigma\setminus Z_{q}$. Then, setting
$\mathcal{M}_{\mathrm{Hit}}^{\ast}=\mathcal{M}_{\mathrm{Hit}}\setminus\mathcal{H}^{-1}(0)$,

$\begin{split}U_{\xi_{0},\omega,\epsilon}:=&\\{(A,\phi)\in\mathcal{M}_{\mathrm{Hit}}^{\ast}\mid\|A-A_{0}\|_{W^{k,2}(\omega)}+\inf_{\theta\in
S^{1}}\|\|\phi\|_{L^{2}}^{-\frac{1}{2}}\phi-e^{i\theta}\phi_{0}\|_{W^{k,2}(\omega)}<\epsilon,\;\|\phi\|_{L^{2}}>\epsilon\\}\\\
&\bigcup\,\\{(A,\phi)\in\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}|\|A-A_{0}\|_{W^{k,2}(\omega)}+\|\phi-\phi_{0}\|_{W^{k,2}(\omega)}<\epsilon\\}\end{split}$

defines an open set around $\xi_{0}$. The sets $U_{\xi_{0},\omega,\epsilon}$
and $V_{\xi,\epsilon}$ generate the topology on $\mathbb{M}$.

######  Theorem 4.8.

The space $\mathbb{M}$ is a Hausdorff and compact.

###### Proof.

The Hausdorff property follows from the definition of the topology. By
Propositions 4.6 and 4.7, $\mathbb{M}$ is sequentially compact. Moreover,
using this explicit base for the topology $\mathbb{M}$ is first countable, and
hence compact. ∎

We may now define the compactification of the Hitchin moduli space as the
closure $\overline{\mathcal{M}}_{\mathrm{Hit}}\subset\mathbb{M}$; we write
$\partial\overline{\mathcal{M}}_{\mathrm{Hit}}$ for the boundary of the
closure, and
$\overline{\mathcal{M}}_{\mathrm{Hit},q}:=\overline{\mathcal{M}}_{\mathrm{Hit}}\cap\mathbb{M}_{q}$
for the subset of elements with a fixed quadratic differential.

The following result is described in [30, 36, 31].

######  Theorem 4.9 ([31, Prop. 3.3]).

If $q$ has only simple zeros, then
$\overline{\mathcal{M}}_{\mathrm{Hit},q}=\mathbb{M}_{q}$.

In other words, the compactification of any slice where $q$ does not lie in
the discriminant locus is “the obvious one”.

##  5\. Parabolic modules and stratification of BNR data

In this section, we review the notion of a parabolic module, as described in
[38, 5, 4, 16]. This concept leads to a partial normalization of the
generalized Jacobian and Prym varieties of the spectral curve.

###  5.1. Normalization of the spectral curve

Let $q\neq 0$ be a quadratic differential with an irreducible, singular
spectral curve $S=S_{q}$. The zeros of $q$ define a divisor
$\mathrm{Div}(q)=\sum_{i=1}^{r_{1}}m_{i}p_{i}+\sum_{j=1}^{r_{2}}n_{j}p_{j}^{\prime}$,
where the $m_{i}$ and $n_{j}$ are even and odd integers, respectively, and
hence $r_{1}$ and $r_{2}$ are the numbers of even and odd zeros, respectively,
counted without multiplicity. Write
$Z_{\mathrm{even}}=\\{p_{1},\dots,p_{r_{1}}\\}$,
$Z_{\mathrm{odd}}=\\{p_{1}^{\prime},\dots,p_{r_{2}}^{\prime}\\}$, and
$Z=Z_{\mathrm{even}}\cup Z_{\mathrm{odd}}$, so $\\#Z=r=r_{1}+r_{2}$.

The map $\pi:S\to\Sigma$ is a double covering branched along $Z$; hence, we
may view $p_{i}$ and $p_{i}^{\prime}$ as points in $S$. For $x\in S$, let
$\mathcal{O}_{x}$ be the local ring, $\mathcal{O}_{x}^{\ast}$ its group of
units, and $R_{x}$ the completion. We say that $S$ has an $A_{n}$ singularity
at $x$ if $R_{x}\cong\mathbb{C}[[r,s]]/(r^{2}-s^{n+1})$, where $n\geq 1$. If
$S$ has an $A_{1}$ singularity at $x$, we call it a _nodal_ singularity, and
if $S$ has an $A_{2}$ singularity at $x$, we call it a _cusp_ singularity.

Let $p:\widetilde{S}\to S$ be the normalization of $S$, and let
$\tilde{\pi}:=\pi\circ p$:

${\widetilde{S}}$${S}$${\Sigma}$$\scriptstyle{p}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{\pi}$

For even zeros $p_{i}$ we write
$p^{-1}(p_{i})=\\{\tilde{p}_{i}^{+},\tilde{p}_{i}^{-}\\}$, and for odd zeros
$p_{i}^{\prime}$ we write $p^{-1}(p_{i}^{\prime})=\tilde{p}_{i}^{\prime}$.
Since $\pi:S\to\Sigma$ is a branched double cover, the involution $\sigma$ on
$S$ extends to an involution of $\widetilde{S}$ which we also denote by
$\sigma$. Note that $\sigma(\tilde{p}_{i}^{\prime})=\tilde{p}_{i}^{\prime}$
while $\sigma(\tilde{p}_{i}^{\pm})=\tilde{p}_{i}^{\mp}$.

The ramification divisor
$\Lambda^{\prime}=\frac{1}{2}\sum_{i=1}^{r_{1}}m_{i}p_{i}+\sum_{j=1}^{r_{2}}(n_{j}-1)p_{j}^{\prime\prime}$,
is a divisor on $S$, and there is an exact sequence:

(8) $0\longrightarrow\mathcal{O}_{S}\longrightarrow
p_{\ast}\mathcal{O}_{\widetilde{S}}\longrightarrow\mathcal{O}_{\Lambda^{\prime}}\longrightarrow
0\ .$

The genus of $\widetilde{S}$ is
$g(\widetilde{S})=4g-3-\deg(\Lambda^{\prime})=2g-1+r_{2}/2$.

###  5.2. Jacobian under the pull-back of the normalization

We now recall some facts about the Jacobian under the pull-back of the
normalization, cf. [16]. Let $x\in Z\subset S$ be a singular point, i.e.
either $x\in Z_{\mathrm{even}}$ or $x=p^{\prime}_{j}$ with $n_{j}\geq 3$. Let
$\widetilde{\mathcal{O}}_{x}$ be the integral closure of $\mathcal{O}_{x}$. We
take $V:=\prod_{x\in
Z}\widetilde{\mathcal{O}}_{x}^{\ast}/\mathcal{O}_{x}^{\ast}$. Then we have the
following well-known short exact sequence induced by the pull-back of the
normalization.

(9) $\begin{split}0\longrightarrow
V\longrightarrow\mathrm{Jac}(S)\xrightarrow{p^{\ast}}\mathrm{Jac}(\widetilde{S})\longrightarrow
0\ .\end{split}$

This will play an important role later on.

####  5.2.1. Hitchin fiber

Now we examine the locally free part $\mathcal{T}$ of the Hitchin fiber under
the pull-back. Here, $\mathcal{T}$ is defined to be the set of
$L\in\mathrm{Pic}^{2g-2}(S)$ such that
$\det(\pi_{\ast}L)=\mathcal{O}_{\Sigma}$. For any $L\in\mathrm{Pic}(S)$, from
(8) we see that
$\det(\tilde{\pi}_{\ast}p^{\ast}L)\cong\det(\pi_{\ast}L)\otimes\mathcal{O}_{\Sigma}(\Lambda^{\prime})$.
We define a new set, $\widetilde{\mathcal{T}}$, as follows:

$\widetilde{\mathcal{T}}:=\\{\widetilde{L}\in\mathrm{Pic}^{2g-2}(\widetilde{S})\mid\det(\tilde{\pi}_{\ast}L)\cong\mathcal{O}(\Lambda^{\prime})\\}.$

It follows that $p^{\ast}$ maps $\mathcal{T}$ to $\widetilde{\mathcal{T}}$.
Furthermore, if $L_{1},L_{2}\in\mathrm{Pic}(S)$ satisfy $p^{\ast}L_{1}\cong
p^{\ast}L_{2}$, then we have $\pi_{\ast}L_{1}\cong\pi_{\ast}L_{2}$. This means
that the fiber of $p^{\ast}:\mathrm{Jac}(S)\to\mathrm{Jac}(\widetilde{S})$ is
the same as $p^{\ast}:\mathcal{T}\to\widetilde{\mathcal{T}}$, resulting in the
following exact sequence:

(10) $0\longrightarrow V\longrightarrow\mathcal{T}\xrightarrow{\hskip
11.38092ptp^{\ast}\ }\widetilde{\mathcal{T}}\longrightarrow 0\ .$

###  5.3. Torsion free sheaves

Now we present Cook’s parametrization of rank 1 torsion free sheaves on curves
with Gorenstein singularities (see [4, p. 40] and also [5, 38]). An explicit
computation of the invariants used in this paper is provided in Appendix A.
Let $x\in Z$ be a singular point of $S$, and let $L\to S$ be a rank 1 torsion
free sheaf. We again let $\mathcal{O}_{x}$ denote the local ring at $x$,
$\widetilde{\mathcal{O}}_{x}$ its integral closure, $\mathcal{K}_{x}$ its
fraction field, and
$\delta_{x}=\dim_{\mathbb{C}}(\widetilde{\mathcal{O}}_{x}/\mathcal{O}_{x})$.
According to [18, Lemma 1.1], there exists a fractional ideal $I_{x}$ that is
isomorphic to $L_{x}$, uniquely defined up to multiplication by a unit of
$\widetilde{\mathcal{O}}_{x}$, such that $\mathcal{O}_{x}\subset
I_{x}\subset\widetilde{\mathcal{O}}_{x}$. We define
$\ell_{x}:=\dim_{\mathbb{C}}(I_{x}/\mathcal{O}_{x})$ and
$b_{x}:=\dim_{\mathbb{C}}(\mathrm{Tor}(I_{x}\otimes_{\mathcal{O}_{x}}\widetilde{\mathcal{O}}_{x}))$.
Then, $\ell_{x}$ and $b_{x}$ are invariants of $L$.

Let $\mathcal{K}_{x}$ be the field of fractions of $\mathcal{O}_{x}$. The
_conductor_ of $I_{x}\subset\widetilde{\mathcal{O}}_{x}$ is defined to be

$C(I_{x})=\\{u\in\mathcal{K}_{x}\mid u\cdot\widetilde{\mathcal{O}}_{x}\subset
I_{x}\\}\ .$

The singularity is characterized by the following dimensions:

$\underbrace{C(\mathcal{O}_{x})\subset\overbrace{C(I_{x})\subset\overbrace{\mathcal{O}_{x}\subset\underbrace{I_{x}\subset\widetilde{\mathcal{O}}_{x}}_{\delta_{x}-\ell_{x}}}^{\delta_{x}}}^{2\delta_{x}-2\ell_{x}}}_{2\delta_{x}}\
.$

For $x=p_{i}\in Z_{\mathrm{even}}$, we have $\delta_{p_{i}}=m_{i}/2$, and
there are two maximal ideals $\mathfrak{m}_{\pm}$ in
$\widetilde{\mathcal{O}}_{x}$ corresponding to the points
$\tilde{p}_{i}^{\pm}$. We let
$(\widetilde{\mathcal{O}}_{p_{i}}/C(I_{p_{i}}))_{\mathfrak{m}_{\pm}}$ be the
localization by the ideals $\mathfrak{m}_{\pm}$, and define
$a_{\tilde{p}_{i}^{\pm}}:=\dim_{\mathbb{C}}(\widetilde{\mathcal{O}}_{p_{i}}/C(I_{p_{i}}))_{\mathfrak{m}_{\pm}}$.
Moreover, we have
$\dim_{\mathbb{C}}(\widetilde{\mathcal{O}}_{p_{i}}/C(\mathcal{O}_{p_{i}}))_{\mathfrak{m}_{\pm}}=m_{i}/2=\delta_{p_{i}}$.
By Appendix A, $a_{\tilde{p}_{i}^{\pm}}=(m_{i}/2)-\ell_{p_{i}}$, and therefore
$a_{\tilde{p}_{i}^{+}}+a_{\tilde{p}_{i}^{-}}=2\delta_{p_{i}}-2\ell_{p_{i}}$,
and also $b_{p_{i}}=\ell_{p_{i}}$. Define

$V(L_{p_{i}})=\\{(c_{i}^{+},c_{i}^{-})\mid c_{i}^{\pm}\in\mathbb{Z}_{\geq 0}\
,\ c_{i}^{+}+c_{i}^{-}=\ell_{p_{i}}\\}\ .$

For $x=p_{i}^{\prime}\in Z_{\mathrm{odd}}$, we have
$\delta_{p_{i}^{\prime}}=(n_{i}-1)/2$, and the maximal ideal $\mathfrak{m}$ of
$\widetilde{\mathcal{O}}_{x}$ is unique. Define
$a_{\tilde{p}_{i}^{\prime}}:=\dim_{\mathbb{C}}(\widetilde{\mathcal{O}}_{p_{i}^{\prime}}/C(I_{p_{i}^{\prime}}))_{\mathfrak{m}}$.
By Appendix A, we have
$a_{\tilde{p}_{i}^{\prime}}=2\delta_{p_{i}^{\prime}}-2\ell_{p_{i}^{\prime}}$
and $b_{p_{i}^{\prime}}=\ell_{p_{i}^{\prime}}$. Moreover,
$\dim_{\mathbb{C}}(\widetilde{\mathcal{O}}_{p_{i}^{\prime}}/C(\mathcal{O}_{p_{i}^{\prime}}))_{\mathfrak{m}}=n_{i}-1=2\delta_{p_{i}^{\prime}}$.
In this case we set $V(L_{p_{i}^{\prime}})=\\{\ell_{p_{i}^{\prime}}\\}$.

Now consider modules compatible with $L_{x}$. Let
$\eta:\widetilde{\mathcal{O}}_{x}\to\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x})$
be the quotient map. Define

$S(L_{x}):=\\{\mathcal{O}_{x}\text{-submodules}\
F_{x}\subset\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x})\mid\dim_{\mathbb{C}}(F_{x})=\delta\
,\ \eta^{-1}(F_{x})\cong L_{x}\\}\ .$

Hence, if $J_{x}=\eta^{-1}(F_{x})$ with $F_{x}\in S(L_{x})$, there exists an
ideal $\mathfrak{t}_{x}$ such that $J_{x}=\mathfrak{t}_{x}\cdot L_{x}$. For
$x=p_{i}\in Z_{\mathrm{even}}$, we obtain two integers
$c_{i}^{\pm}=\dim_{\mathbb{C}}(\widetilde{\mathcal{O}}_{x}/(\mathfrak{t}_{x}\cdot\widetilde{\mathcal{O}}_{x}))_{\mathfrak{m}_{\pm}}$.
By [4, Lemma 6], $(c_{i}^{+},c_{i}^{-})\in V(L_{p_{i}})$, for
$x=p_{i}^{\prime}\in Z_{\mathrm{odd}}$,
$\dim_{\mathbb{C}}(\widetilde{\mathcal{O}}_{x}/(\mathfrak{t}_{x}\cdot\widetilde{\mathcal{O}}_{x}))=\ell_{p_{i}^{\prime}}\in
V(L_{p_{i}^{\prime}})$, and these only depend on $F_{x}$. Hence, there is a
well-defined map:

$\kappa_{x}:S(L_{x})\longrightarrow V(L_{x})\ :\
\begin{cases}F_{x}\to(c_{i}^{+},c_{i}^{-})&\mathrm{when}\;x=p_{i},\\\
F_{x}\to\ell_{p_{i}^{\prime}}&\mathrm{when}\;x=p_{i}^{\prime}\ .\end{cases}$

######  Lemma 5.1 ([4, Lemma 6]).

For $x\in Z$, the components of $S(L_{x})$ are parameterized by elements in
$V(L_{x})$.

Set $V(L):=\prod_{x\in Z}V(L_{x})$ and $S(L):=\prod_{x\in Z}S(L_{x})$. Write
$N(L):=|V(L)|$ for the number of points in $V(L)$. there is a map

$\kappa:=\prod_{x\in Z}\kappa_{x}:S(L)\longrightarrow V(L)\ .$

For any $\mathbf{c}\in V(L)$, write
$\mathbf{c}=(c_{1}^{\pm},\ldots,c_{r_{1}}^{\pm},\ell_{p_{1}^{\prime}},\ldots,\ell_{p_{r_{2}}^{\prime}})$.
Associate to this the divisor

$D_{\mathbf{c}}=\sum_{i=1}^{r_{1}}(c_{i}^{+}\tilde{p}_{i}^{+}+c_{i}^{-}\tilde{p}_{i}^{-})+\sum_{i=1}^{r_{2}}\ell_{p_{i}^{\prime}}\tilde{p}_{i}^{\prime}$

on $\widetilde{S}$. Composing $\kappa$ with the map above, we define

(11) $\varkappa:S(L)\longrightarrow\mathrm{Div}(\widetilde{S})\ :\ \prod_{x\in
Z}F_{x}\mapsto\mathbf{c}\mapsto D_{\mathbf{c}}\ .$

The following result is straightforward but important:

######  Proposition 5.2.

$L$ is locally free if and only if $\varkappa=0$ on $S(L)$.

###### Proof.

$L$ is locally free if and only if $\ell_{x}=0$ for $x\in Z$. The claim then
follows directly from the definition of $D_{\mathbf{c}}$. ∎

###  5.4. Parabolic modules

In this subsection, we define the notion of a parabolic module, following [38,
5, 4].

First note that
$\dim_{\mathbb{C}}(\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x}))=2\delta_{x}$.
Let $\mathrm{Gr}(\delta_{x},\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x}))$
be the Grassmannian of $\delta_{x}$ dimensional subspaces of the vector space
$\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x})$. Then
$\widetilde{\mathcal{O}}^{\ast}_{x}$ acts on
$\mathrm{Gr}(\delta_{x},\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x}))$ by
multiplication, with fixed points corresponding to $\delta_{x}$-dimensional
$\mathcal{O}_{x}$ submodules of
$\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x})$. We write $\mathcal{P}(x)$
for the (reduced) variety of fixed points. This is a closed subvariety of
$\mathrm{Gr}(\delta_{x},\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x}))$.

Suppose $x$ is an $A_{n}$ singularity. For notational convenience, we write
$\mathcal{P}(A_{n}):=\mathcal{P}(x)$. We have the following:

######  Proposition 5.3 ([4, Prop. 2]).

The following holds:

  * (i)

$\mathcal{P}(A_{n})$ is connected and depends only on $\delta_{x}$. Also,
$\dim_{\mathbb{C}}\mathcal{P}(A_{n})=n$, and we have isomorphisms
$\mathcal{P}(A_{2n-1})\cong\mathcal{P}(A_{2n})$.

  * (ii)

If $P(A_{0})$ is defined to be a point, then the inclusions
$\mathcal{P}(A_{0})\subset\mathcal{P}(A_{2})\subset\cdots\subset\mathcal{P}(A_{2n})$
give a cell decomposition of $\mathcal{P}(A_{2n})$.

  * (iii)

The singular locus
$\mathrm{Sing}(\mathcal{P}(A_{2n}))\cong\mathcal{P}(A_{2n-4})$. In particular,
it has codimension $\geq 2$. Moreover,
$\mathcal{P}(A_{1})=\mathcal{P}(A_{2})\cong\mathbb{C}P^{1}$, and
$\mathcal{P}(A_{4})$ is a quadric cone.

Define $\mathscr{P}(S)=\prod_{x\in Z}\mathcal{P}(x)$. This only depends on the
curve singularity of $S$. Let $J\in\mathrm{Pic}(\widetilde{S})$. We have an
isomorphism
$J_{x}\otimes\mathcal{O}_{\Lambda^{\prime},x}\cong\widetilde{\mathcal{O}}_{x}/C(\mathcal{O}_{x})$
as $\mathcal{O}_{x}$-modules. More explicitly, as vector spaces,

$J_{\tilde{p}_{i}^{+}}^{\oplus\frac{m_{i}}{2}}\oplus
J_{\tilde{p}_{i}^{-}}^{\oplus\frac{m_{i}}{2}}\cong\widetilde{\mathcal{O}}_{p_{i}}/C(\mathcal{O}_{p_{i}^{\prime}})\
,\
J_{\tilde{p}_{i}^{\prime}}^{\oplus(n_{i}-1)}\cong\widetilde{\mathcal{O}}_{p_{i}^{\prime}}/C(\mathcal{O}_{p_{i}^{\prime}})\
.$

######  Definition 5.4.

A parabolic module $\mathrm{PMod}(\widetilde{S})$ consists of pairs $(J,v)$,
where $J\in\mathrm{Jac}(\widetilde{S})$ and $v=\prod_{x\in Z}v_{x}$, with
$v_{x}\in J_{x}\otimes\mathcal{O}_{\Lambda^{\prime},x}$.

By [4, p. 41], $\mathrm{PMod}(\widetilde{S})$ has a natural algebraic
structure. Let
$\mathrm{pr}:\mathrm{PMod}(\widetilde{S})\to\mathrm{Jac}(\widetilde{S})$ be
the projection to the first component. Then $\mathrm{pr}$ defines a fibration
of $\mathrm{PMod}(\widetilde{S})$ with fiber $\mathscr{P}(S)$. Moreover, there
is a finite morphism
$\tau:\mathrm{PMod}(\widetilde{S})\to\overline{\mathrm{Jac}}(S)$ defined by
sending $(J,v)\to L$, where $L$ is the kernel of the restriction map
$p_{\ast}J\to(J\otimes\mathcal{O}_{\Lambda})/v$:

$0\longrightarrow L\longrightarrow
p_{\ast}J\longrightarrow(J\otimes\mathcal{O}_{\Lambda})/v\longrightarrow 0\ .$

There is a diagram:

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The map $\tau$ may be regarded as the compactification of the pull-back
normalization map $p^{\ast}$ in (9).

######  Theorem 5.5 ([4, Thm. 1]).

For the map $\tau:\mathrm{PMod}(\widetilde{S})\to\overline{\mathrm{Jac}}(S)$
defined above,

  * (i)

$\tau$ is a finite morphism, where the fiber over $L$ consists of $N(L)$
points,

  * (ii)

The restriction $\tau:\tau^{-1}\mathrm{Jac}(S)\to\mathrm{Jac}(S)$ is an
isomorphism. Moreover, for $L\in\mathrm{Jac}(S)$, we have
$\mathrm{pr}\circ\tau^{-1}(L)=p^{\ast}(L)$.

  * (iii)

Suppose $\tau(J,v)=L$. For $x\in Z$, we have $v\in S(L)$. Let
$D_{v}=\varkappa(v)$ be the divisor defined in (11). Then

$0\longrightarrow p^{\ast}L/\mathrm{Tor}(p^{\ast}L)\longrightarrow
J\longrightarrow\mathcal{O}_{D_{v}}\longrightarrow 0\ .$

In particular, $p^{\ast}L/\mathrm{Tor}(p^{\ast}L)=J(-D_{v})$.

Suppose all of the zeros of the quadratic differential $q$ are odd. Then for
$L\in\overline{\mathrm{Jac}}(S)$, $N(L)=1$, and we can deduce the following.

######  Corollary 5.6.

If $q^{-1}(0)=\\{p_{1}^{\prime},\ldots,p_{r}^{\prime}\\}$ and all zeroes have
odd multiplicity, then
$\tau:\mathrm{PMod}(\widetilde{S})\to\overline{\mathrm{Jac}}(S)$ is a
bijection. Moreover, for $L\in\overline{\mathrm{Jac}}(S)$ with $\tau(J,v)=L$,
we have

$p^{\ast}L/\mathrm{Tor}(p^{\ast}L)=J(-\sum\ell_{p_{i}^{\prime}}\tilde{p}_{i}^{\prime})\
.$

We will now present an example of a parabolic module.

######  Example 5.7 ([4, Ex. 2]).

Suppose $q$ contains $4g-2$ simple zeros and one zero $x$ of order $2$. Then
the spectral curve $S$ has one nodal singularity at $x$. Denote
$p:\widetilde{S}\to S$ the normalization, with
$p^{-1}(x)=\\{\tilde{x}_{+},\tilde{x}_{-}\\}$. Then
$\mathscr{P}(S)=\mathbb{C}P^{1}$, and we obtain a fibration
$\mathbb{C}P^{1}\to\mathrm{PMod}(\widetilde{S})\to\mathrm{Jac}(\widetilde{S})$.
Let $L\in\overline{\mathrm{Jac}}(S)$. If we write
$\widetilde{L}:=p^{\ast}L/\mathrm{Tor}(p^{\ast}L)$, then
$\tau^{-1}(L)={(\widetilde{L}\otimes\mathcal{O}(\tilde{x}_{+}),v_{+}),(\widetilde{L}\otimes\mathcal{O}(\tilde{x}_{-}),v_{-})}$.
We can define two sections:

$s_{\pm}:\overline{\mathrm{Jac}}(S)\longrightarrow\mathrm{PMod}(\widetilde{S}),\;s_{\pm}:L\longrightarrow(\widetilde{L}\otimes\mathcal{O}(\tilde{x}_{\pm}),v_{\pm})\
.$

Then $\overline{\mathrm{Jac}}(S)$ is the quotient of
$\mathrm{PMod}(\widetilde{S})$ given by the identification

$\overline{\mathrm{Jac}}(S)\cong\mathrm{PMod}(\widetilde{S})/(s_{+}\sim\mathcal{O}(\tilde{x}_{-}-\tilde{x}_{+})s_{-})\
.$

In particular, $\mathrm{PMod}(\widetilde{S})$ is not a fibration over
$\overline{\mathrm{Jac}}(S)$.

######  Proposition 5.8.

The singular set of $\mathrm{PMod}(\widetilde{S})$ has codimension at least
$2$. Moreover, if the spectral curve $S$ contains only cusp or nodal
singularities, then $\mathrm{PMod}(\widetilde{S})$ is smooth.

###### Proof.

As the singularities of $\mathrm{PMod}(\widetilde{S})$ come from the space
$\mathscr{P}(S)$, the claim follows from Proposition 5.3. ∎

Since we focus on $\mathrm{SL}(2,\mathbb{C})$ Higgs bundles, we must consider
the parabolic module compactification of the fibration

$0\longrightarrow V\longrightarrow\mathcal{P}\xrightarrow{\hskip
8.5359ptp^{\ast}\ }\mathrm{Prym}(\widetilde{S}/\Sigma)\longrightarrow 0\ .$

Setting,
$\widehat{\mathrm{PMod}}(\widetilde{S}):=\tau^{-1}(\overline{\mathcal{P}})$,
then there is a diagram from [16, p. 17]

(12)
${\mathscr{P}(S)}$${\widehat{\mathrm{PMod}}(\widetilde{S})}$${\mathrm{Prym}(\widetilde{S}/\Sigma)}$${\overline{\mathcal{P}}}$$\scriptstyle{\mathrm{pr}}$$\scriptstyle{\tau}$

Theorem 5.5 proves that $\mathrm{pr}\circ\tau^{-1}|_{\mathcal{P}}=p^{\ast}$.

###  5.5. Stratifications of the BNR data

Parabolic modules define a stratification of $\overline{\mathcal{P}}$ and
$\overline{\mathcal{T}}$. In the following, $\pi:S\to\Sigma$ is a branched
double cover, $\sigma$ the associated involution on $S$, and by $\sigma$ we
also denote its extension to an involution on the normalization
$\widetilde{S}$ of $S$.

For a rank 1 torsion free sheaf $L\in\overline{\mathrm{Pic}}(S)$, consider the
map

$p^{\star}_{\mathrm{tf}}:\overline{\mathrm{Pic}}(S)\longrightarrow\mathrm{Pic}(\widetilde{S})\
,\ p^{\star}_{\mathrm{tf}}(L):=p^{\ast}L/\mathrm{Tor}(p^{\ast}L)\ ,$

i.e. the torsion free part of the pull-back to the normalization. By [37],
$p^{\star}_{\mathrm{tf}}(L)=p^{\ast}L$ at $x\in\widetilde{S}$ if and only if
$L$ is locally free at $p(x)\in S$.

######  Definition 5.9 ([24]).

An effective divisor $D\in\mathrm{Div}(\widetilde{S})$ is called a
$\sigma$-divisor if

  * (i)

$D\leq\Lambda$ and $\sigma^{\ast}D=D$;

  * (ii)

and for any $x\in\mathrm{Fix}(\sigma)$, $D|_{x}=d_{x}x$, where $d_{x}\equiv
0\mod 2$.

The $\sigma$-divisors play an important role in describing the singular
Hitchin fibers.

######  Proposition 5.10 ([24, 33]).

Let $L\in\overline{\mathcal{P}}$ and write
$\widetilde{L}:=p^{\star}_{\mathrm{tf}}L$. Then we have
$\widetilde{L}\otimes\sigma^{\ast}\widetilde{L}=\mathcal{O}(\Lambda-D)$ for
$D$ a $\sigma$-divisor.

For a $\sigma$-divisor $D$, define

(13)
$\displaystyle\begin{split}\widetilde{\mathcal{T}}_{D}&=\\{J\in\mathrm{Pic}(\widetilde{S})\mid
J\otimes\sigma^{\ast}J=\mathcal{O}(\Lambda-D)\\}\ ;\\\
\widetilde{\mathcal{P}}_{D}&=\\{J\in\mathrm{Pic}(\widetilde{S})\mid
J\otimes\sigma^{\ast}J=\mathcal{O}(-D)\\}\ .\end{split}$

Then by [24, Prop. 5.6], $\widetilde{\mathcal{T}}_{D}$ and
$\widetilde{\mathcal{P}}_{D}$ are abelian torsors over
$\mathrm{Prym}(\widetilde{S}/\Sigma)$ with dimension
$g(\widetilde{S})-g(S)=g-1+\frac{1}{2}r_{2}$. In addition, we define

(14)
$\displaystyle\begin{split}\overline{\mathcal{T}}_{D}&=\\{L\in\overline{\mathcal{T}}\mid
p^{\star}_{\mathrm{tf}}L\in\widetilde{\mathcal{T}}_{D}\\}\ ;\\\
\overline{\mathcal{P}}_{D}&=\\{L\in\overline{\mathcal{P}}\mid
p^{\star}_{\mathrm{tf}}L\in\widetilde{\mathcal{P}}_{D}\\}\ .\end{split}$

Then the partial order on divisors defines a stratification of
$\overline{\mathcal{T}}$ (resp. $\overline{\mathcal{P}}$) by:
$\cup_{D^{\prime}\leq D}\overline{\mathcal{T}}_{D^{\prime}}$ (resp.
$\cup_{D^{\prime}\leq D}\overline{\mathcal{P}}_{D^{\prime}}$). The top strata
are $\overline{\mathcal{T}}_{D=0}$ (resp. $\overline{\mathcal{P}}_{D=0}$), and
these consist of the locally free sheaves. From the definition,
$\mathcal{T}=\overline{\mathcal{T}}_{D=0}$ and
$\mathcal{P}=\overline{\mathcal{P}}_{D=0}$.

######  Theorem 5.11 ([24, Thm. 6.2]).

(i) Suppose $q$ contains at least one zero of odd order. Then for each stratum
indexed by $\sigma$-divisor $D$, if we let $n_{ss}$ be the number of $p$ such
that $D|_{p}=\Lambda|_{p}$, then there are holomorphic fiber bundles

(15)
$\displaystyle\begin{split}(\mathbb{C}^{\ast})^{k_{1}}\times\mathbb{C}^{k_{2}}\longrightarrow\overline{\mathcal{T}}_{D}\xrightarrow{\hskip
5.69046ptp^{\star}_{\mathrm{tf}}\ }\widetilde{\mathcal{T}}_{D}\ ;\\\
(\mathbb{C}^{\ast})^{k_{1}}\times\mathbb{C}^{k_{2}}\longrightarrow\overline{\mathcal{P}}_{D}\xrightarrow{\hskip
5.69046ptp^{\star}_{\mathrm{tf}}\ }\widetilde{\mathcal{P}}_{D}\ ,\end{split}$

where $k_{1}=r_{1}-n_{ss}$,
$k_{2}=2g-2-\frac{1}{2}\deg(D)-r_{1}+n_{ss}-\frac{r_{2}}{2}$, and
$r_{1},r_{2}$ are the number of even and odd zeros. (ii) Suppose $q$ is
irreducible but all zeros are of even order. Then there exists an analytic
space $\overline{\mathcal{T}}_{D}^{\prime}$ and a double branched covering
$p:\overline{\mathcal{T}}_{D}\to\overline{\mathcal{T}}_{D}^{\prime}$, with
$\overline{\mathcal{T}}_{D}^{\prime}$ a holomorphic fibration

$(\mathbb{C}^{\ast})^{k_{1}}\times\mathbb{C}^{k_{2}}\longrightarrow\overline{\mathcal{T}}_{D}^{\prime}\xrightarrow{\hskip
7.11317ptp^{\star}_{\mathrm{tf}}\ }\widetilde{\mathcal{T}}_{D}\ .$

In particular,
$\dim(\overline{\mathcal{P}}_{D})=\dim(\overline{\mathcal{T}}_{D})=3g-3-\frac{1}{2}\deg(D).$

As explained in [24], via the BNR correspondence the stratification above
translates into a stratification of the Hitchin fiber. Let
$\chi_{\mathrm{BNR}}:\overline{\mathcal{T}}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\mathcal{M}_{q}$
be the bijection in Theorem 2.3. Let $D$ be a $\sigma$-divisor. Define
$\mathcal{M}_{q,D}:=\chi_{\mathrm{BNR}}(\overline{\mathcal{T}}_{D})$. Then the
stratification of $\overline{\mathcal{T}}$ induces a stratification on
$\mathcal{M}_{q}=\bigcup_{D}\mathcal{M}_{q,D}$.

For each $\sigma$-divisor $D$, since $\sigma^{\ast}D=D$, we can write
$D^{\prime}:=\frac{1}{2}\tilde{\pi}(D)$. Then $D^{\prime}$ is an effective
divisor with $\mathrm{supp\;}D^{\prime}\subset Z$. Moreover, for $x\in
q^{-1}(0)$, $D_{x}^{\prime}\leq\frac{1}{2}\lfloor\mathrm{ord}_{x}(q)\rfloor$.
Therefore, $\mathcal{M}_{q}$ may be regarded as also being stratified by
divisors $D^{\prime}$ defined over $\Sigma$.

###  5.6. The structure of the parabolic module projection

We now explain the relationship between the divisor $D_{v}$ in Theorem 5.5 and
the $\sigma$-divisor. Given $L\in\overline{\mathcal{P}}$, define

(16)
$\displaystyle\begin{split}\mathscr{N}_{L}&:=\\{(J,v)\in\widehat{\mathrm{PMod}}(\widetilde{S})\mid\tau(J,v)=L\\}\
;\\\ \mathscr{D}_{L}&:=\\{D_{v}\mid(J,v)\in\mathscr{N}_{L}\\}\ ,\end{split}$

where $\mathscr{N}_{L}$ is $\tau^{-1}(L)$, and $\mathscr{D}_{L}$ is the
collection of divisors $D_{v}$ such that
$J(-D_{v})=p^{\star}_{\mathrm{tf}}(L)$. If $L$ is locally free, then
$J=p^{\ast}L$, and $\mathscr{D}_{L}$ is empty. Moreover, if
$\tau(J,v)=\tau(J^{\prime},v)$, then $J^{\prime}=J(D_{v^{\prime}}-D_{v})$.

The divisor $D_{v}$ satisfies the following symmetry property:

######  Proposition 5.12.

Let $D$ be a $\sigma$-invariant divisor and $L\in\overline{\mathcal{P}}_{D}$.
For any $D_{v}\in\mathscr{D}_{L}$, we have $D_{v}+\sigma^{\ast}D_{v}=D$.

###### Proof.

Let $\tau(J,v)=L$. Then by Theorem 5.5, we have $\widetilde{L}=J(-D_{v})$,
where $\widetilde{L}=p^{\star}_{\mathrm{tf}}(L)$. As
$L\in\overline{\mathcal{P}}_{D}$ and
$J\in\mathrm{Prym}(\widetilde{S}/\Sigma)$, we have
$\widetilde{L}\otimes\sigma^{\ast}\widetilde{L}=\mathcal{O}(-D)$ and
$J\otimes\sigma^{\ast}J=\mathcal{O}_{\widetilde{S}}$, which implies
$D_{v}+\sigma^{\ast}D_{v}=D$. ∎

As a consequence, we have the following:

######  Corollary 5.13.

Suppose $q$ has only zeroes of odd order. Then for
$L\in\overline{\mathcal{P}}_{D}$ and $D_{v}\in\mathscr{D}_{L}$, we have
$\sigma^{\ast}D_{v}=D_{v}$ and $D_{v}=\frac{1}{2}D$. In addition,
$\tau:\widehat{\mathrm{PMod}}(\widetilde{S})\to\overline{\mathcal{P}}$ is a
bijection.

###### Proof.

Since each zero has odd order
$\mathrm{supp\;}(D_{v})\subset\mathrm{Fix}(\sigma)$, which implies
$D_{v}=\sigma^{\ast}D_{v}$. By Proposition 5.12, we must have
$D_{v}=\frac{1}{2}D$. ∎

There are relationships between the integers appearing in the construction of
the parabolic module:

######  Lemma 5.14 ([17]).

Let
$D=\sum_{i=1}^{r_{1}}d_{i}(\tilde{p}_{i}^{+}+\tilde{p}_{i}^{-})+\sum_{i=1}^{r_{2}}d_{i}^{\prime}\tilde{p}_{i}^{\prime}$
be a $\sigma$-divisor, and let $L\in\overline{\mathcal{P}}_{D}$. Then we have

  * (i)

$\ell_{p_{i}}=d_{i}$ and $\ell_{p_{i}^{\prime}}=d_{i}^{\prime}/2$;

  * (ii)

$a_{\tilde{p}_{i}^{+}}=a_{\tilde{p}_{i}^{-}}=(m_{i}/2)-d_{i}$ and
$a_{\tilde{p}_{i}^{\prime}}=n_{i}-1-d_{i}^{\prime}$.

###### Proof.

Since $L\in\overline{\mathcal{P}}_{D}$, we have
$\dim\mathrm{Tor}(p^{\ast}L_{p_{i}})=d_{i}$ and
$\dim\mathrm{Tor}(p^{\ast}L_{p_{i}^{\prime}})=d_{i}^{\prime}/2$. The claim
then follows from Proposition A.1. ∎

The elements in $\mathscr{D}_{L}$ can be explicitly computed.

######  Proposition 5.15.

Let
$D=\sum_{i=1}^{r_{1}}d_{i}(\tilde{p}_{i}^{+}+\tilde{p}_{i}^{-})+\sum_{i=1}^{r_{2}}d_{i}^{\prime}\tilde{p}_{i}^{\prime}$
be a $\sigma$-divisor, and let $L\in\overline{\mathcal{P}}_{D}$. Then
$N_{L}=\prod_{i=1}^{r_{1}}(d_{i}+1)$. The number $N_{L}$ depends only on the
$\sigma$-divisor $D$.

###### Proof.

By Lemma 5.14, $V(L)$ can be rewritten as

$\begin{split}V(L)=\\{(c_{1}^{\pm},\ldots,c_{r_{1}}^{\pm},c^{\prime}_{1}=l_{p_{1}^{\prime}},\ldots,c^{\prime}_{r_{2}}=l_{p^{\prime}_{r_{2}}})\mid
c_{i}^{+}+c_{i}^{-}=d_{i},c_{i}^{\pm}\in\mathbb{Z}_{\geq 0}\\}\ ,\end{split}$

which implies the claim. The condition $D_{v}+\sigma^{\ast}D_{v}=D$ is
automatically satisfied. ∎

If we define $n_{L}$ to be the number of $D_{v}\in\mathscr{D}_{L}$ such that
$\sigma^{\ast}D_{v}\neq D_{v}$, then we have the following:

######  Proposition 5.16.

  * (i)

$n_{L}$ is even;

  * (ii)

if $L\in\overline{\mathcal{P}}_{D}$ with

$D=\sum_{i=1}^{r_{1}}d_{i}(\tilde{p}_{i}^{+}+\tilde{p}_{i}^{-})+\sum_{i=1}^{r_{2}}d_{i}^{\prime}\tilde{p}_{i}^{\prime}\
,$

and if there exists $i_{0}\in\\{1,\dots,r_{1}\\}$ such that $d_{i_{0}}$ is not
even, then $n_{L}=N_{L}$; otherwise, $n_{L}=N_{L}-1$.

###### Proof.

To prove (i), note that if $\sigma^{\ast}D_{v}\neq D_{v}$, then
$\sigma^{\ast}(\sigma^{\ast}D_{v})\neq\sigma^{\ast}D_{v}$, which means that
$n_{L}$ is even. For (ii), by Proposition 5.15, $D_{v}=\sigma^{\ast}D_{v}$ for
$D_{v}\in\mathscr{D}_{L}$ if and only if $c_{i}^{+}=c_{i}^{-}=d_{i}/2$.
Therefore, $n_{L}\neq N_{L}$ if and only if all $d_{i}$ are even, which
implies (ii). ∎

##  6\. Irreducible singular fibers and the Mochizuki map

In this section, we provide a reinterpretation of the limiting configuration
construction of a Higgs bundle on an irreducible fiber, as introduced by
Mochizuki in [33] (see also [24]). We also investigate the relationship
between limiting configurations and the stratification.

###  6.1. Abelianization of a Higgs bundle

Let $q$ be a fixed irreducible quadratic differential with spectral curve $S$,
with normalization $p:\widetilde{S}\to S$. We define
$\widetilde{K}:=\widetilde{\pi}^{\ast}K$ (but note that $\widetilde{K}\neq
K_{\widetilde{S}}$) and $\widetilde{q}:=\widetilde{\pi}^{\ast}q\in
H^{0}(\widetilde{K}^{2})$, where $\widetilde{\pi}$ is the double branched
covering of $\widetilde{S}\to\Sigma$ associated to the branching set $Z$ of
$q$. Choose a square root $\omega\in H^{0}(\widetilde{K})$ such that
$\widetilde{q}=-\omega\otimes\omega$. Let $\Lambda:=\mathrm{Div}(\omega)$ and
$\widetilde{Z}:=\mathrm{supp}(\Lambda)$. We can then write

$\Lambda=\sum_{i=1}^{r_{1}}\frac{m_{i}}{2}(\tilde{p}_{i}^{+}+\tilde{p}_{i}^{-})+\sum_{j=1}^{r_{2}}n_{j}\tilde{p}_{j}^{\prime}\
.$

If $\sigma:\widetilde{S}\to\widetilde{S}$ is the involution, then
$\sigma^{\ast}\omega=-\omega$.

Let $(\mathcal{E},\varphi)$ be a Higgs bundle with $\det\varphi=q$. Consider
the pullback
$(\widetilde{\mathcal{E}},\tilde{\varphi}):=(\widetilde{\pi}^{\ast}\mathcal{E},\widetilde{\pi}^{\ast}\varphi)$.
We have $\tilde{\varphi}\in
H^{0}(\mathrm{End}(\widetilde{\mathcal{E}})\otimes\widetilde{K})$ and
$\widetilde{q}=\det(\tilde{\varphi})$. Since
$\widetilde{q}=-\omega\otimes\omega$, $\pm\omega$ are well-defined eigenvalues
of $\tilde{\varphi}$ over $\widetilde{S}$. Let $\tilde{\lambda}$ be the
canonical section of $\mathrm{Tot}(\widetilde{K})$. The spectral curve for
$(\widetilde{\mathcal{E}},\tilde{\varphi})$ is defined by the equation

$\widetilde{S}^{\prime}:=\\{\tilde{\lambda}^{2}-\tilde{q}=0\\}.$

The set
$\widetilde{S}^{\prime}=\mathrm{Im}(\omega)\cup\mathrm{Im}(-\omega)\subset\mathrm{Tot}(\widetilde{K})$
decomposes into two irreducible pieces.

Having fixed a choice of $\omega$, the eigenvalues of $\tilde{\varphi}$ are
globally well-defined, and we can define the line bundle
$\widetilde{L}_{+}\subset\mathcal{E}$ as
$\widetilde{L}_{+}:=\ker(\tilde{\varphi}-\omega)$. Since
$\sigma^{\ast}\omega=-\omega$,
$\widetilde{L}_{-}=\sigma^{\ast}\widetilde{L}_{+}=\ker(\tilde{\varphi}+\omega)$,
and there is an isomorphism
$\widetilde{\mathcal{E}}|_{\widetilde{S}\setminus\widetilde{Z}}\cong\widetilde{L}_{+}\oplus\widetilde{L}_{-}|_{\widetilde{S}\setminus\widetilde{Z}}$.

There is a local description of $(\widetilde{\mathcal{E}},\tilde{\varphi})$:

######  Lemma 6.1 ([24, Lemma 5.1, Thm. 5.3],[33, Lemma 4.2]).

Let $x\in\widetilde{Z}$ and write $\Lambda|_{x}=m_{x}x$. Let $U$ be a
holomorphic coordinate neighborhood of $x$. Then there exists a frame
$\mathfrak{e}\in H^{0}(U,\widetilde{K})$ such that, under a suitable
trivialization of $\mathcal{E}|_{U}\cong U\times\mathbb{C}^{2}$, we can write

(17) $\textrm{ }\tilde{\varphi}=z^{d_{x}}\begin{pmatrix}0&1\\\
z^{2m_{x}-2d_{x}}&0\end{pmatrix}\otimes\mathfrak{e}.$

Moreover, if we define $D:=\sum_{x\in\widetilde{Z}}d_{x}x$, then $D$ is a
$\sigma$-divisor.

######  Lemma 6.2 ([33, Sec. 4.1]).

For the $\widetilde{L}_{\pm}$ defined above, we have
$\widetilde{L}_{+}\otimes\widetilde{L}_{-}=\mathcal{O}(D-\Lambda)$. Moreover,
if we denote $\widetilde{L}_{0}:=\widetilde{L}_{+}(\Lambda-D)$ and
$\widetilde{L}_{1}:=\sigma^{\ast}\widetilde{L}_{0}$, then
$\widetilde{L}_{+}=\widetilde{\mathcal{E}}\cap\widetilde{L}_{0}$,
$\widetilde{L}_{-}=\widetilde{\mathcal{E}}\cap\widetilde{L}_{1}$, and we have
the exact sequences

$\displaystyle
0\longrightarrow\widetilde{L}_{+}\longrightarrow\widetilde{\mathcal{E}}\longrightarrow\widetilde{L}_{1}\longrightarrow
0\ ;$ $\displaystyle
0\longrightarrow\widetilde{L}_{-}\longrightarrow\widetilde{\mathcal{E}}\longrightarrow\widetilde{L}_{0}\longrightarrow
0\ .$

###### Proof.

The inclusion of $\widetilde{L}_{\pm}\to\mathcal{E}$ defines an exact sequence
of sheaves

$0\longrightarrow\mathcal{O}(\widetilde{L}_{+})\oplus\mathcal{O}(\widetilde{L}_{-})\longrightarrow\mathcal{O}(\mathcal{E})\longrightarrow\mathcal{T}\longrightarrow
0\ ,$

where $\mathcal{T}$ is a torsion sheaf with
$\mathrm{supp\;}\mathcal{T}\subset\widetilde{Z}$. From the local description
in (17), in the same trivialization, $\widetilde{L}_{\pm}$ are spanned by the
bases $s_{\pm}=\begin{pmatrix}1\\\ \pm z^{m_{x}-d_{x}}\end{pmatrix}.$
Therefore, as $\det(\mathcal{E})=\mathcal{O}_{\Sigma}$, we obtain
$\widetilde{L}_{+}\otimes\widetilde{L}_{-}=\mathcal{O}(D-\Lambda)$. Since
$s_{+},s_{-}$ are linear independent away from $z$,
$\widetilde{\mathcal{E}}/\widetilde{L}_{+}$ is locally generated by the
section $z^{d_{x}-m_{x}}s_{-}$. Therefore,
$\widetilde{\mathcal{E}}/\widetilde{L}_{+}\cong\widetilde{L}_{-}(\Lambda-D)=\widetilde{L}_{1}$.
Using the involution, we obtain the other exact sequence. ∎

Therefore, if
$\widetilde{L}\otimes\sigma^{\ast}\widetilde{L}=\mathcal{O}(D-\Lambda)$, we
have
$\widetilde{L}_{0}=\widetilde{L}(\Lambda-D)\in\widetilde{\mathcal{T}}_{D}$. In
summary, the construction above leads us to consider the composition of the
following maps:

$\begin{split}\delta&:\mathcal{M}_{q}\longrightarrow\mathrm{Pic}(\widetilde{S})\xrightarrow{\otimes\mathcal{O}(\Lambda-D)}\widetilde{\mathcal{T}}_{D}\
;\\\
&(\mathcal{E},\varphi)\mapsto\widetilde{L}_{+}\mapsto\widetilde{L}_{+}(\Lambda-D)\
,\end{split}$

where the first map is given by taking the kernel of
$(\tilde{\pi}^{\ast}\varphi-\omega)|_{\tilde{\pi}^{\ast}\mathcal{E}}$.

This construction is directly related to the torsion free pull-back. Recall
that $\chi_{\mathrm{BNR}}:\overline{\mathcal{T}}\to\mathcal{M}_{q}$ is the BNR
correspondence map, and
$p^{\star}_{\mathrm{tf}}:\overline{\mathrm{Pic}}(S)\to\mathrm{Pic}(\widetilde{S})$
is the torsion free pull-back. Then we have

######  Proposition 6.3.

$\delta\circ\chi_{\mathrm{BNR}}=p^{\star}_{\mathrm{tf}}$. In particular, if
$J\in\overline{\mathcal{T}}_{D}$, then
$\delta\circ\chi_{\mathrm{BNR}}(J)\in\widetilde{\mathcal{T}}_{D}$.

###### Proof.

Let $J\in\overline{\mathcal{T}}$, and write
$(\mathcal{E},\varphi)=\chi_{\mathrm{BNR}}(J)$,
$(\widetilde{\mathcal{E}},\tilde{\varphi}):=\tilde{\pi}^{\ast}(\mathcal{E},\varphi)$.
Recall the BNR exact sequence on $S$ (see (1)):

$\begin{split}0\longrightarrow
J(-\Delta)\longrightarrow\pi^{\ast}\mathcal{E}\xrightarrow{\pi^{\ast}\varphi-\lambda}\pi^{\ast}\mathcal{E}\otimes\pi^{\ast}K\longrightarrow
J\otimes\pi^{\ast}K\longrightarrow 0\ .\end{split}$

As $p^{\ast}$ is right exact, we obtain

$\widetilde{\mathcal{E}}\xrightarrow{\hskip
8.5359pt\tilde{\varphi}-\tilde{\lambda}\
}\widetilde{\mathcal{E}}\otimes\tilde{K}\longrightarrow
p^{\ast}J\otimes\tilde{K}\longrightarrow 0\ .$

Since the spectral curve is
$\widetilde{S}^{\prime}=\mathrm{Im}(\omega)\cup\mathrm{Im}(-\omega)$, we can
consider the restriction to the component $\mathrm{Im}(\omega)$ and write
$\tilde{\lambda}=\omega,\;\widetilde{L}_{\pm}:=\ker(\tilde{\varphi}\mp\omega)$.
We obtain an exact sequence

$0\longrightarrow\widetilde{L}_{+}\longrightarrow\widetilde{\mathcal{E}}\xrightarrow{\hskip
7.11317pt\tilde{\varphi}-\omega\
}\widetilde{\mathcal{E}}\otimes\tilde{K}\longrightarrow
p^{\ast}J\otimes\tilde{K}\longrightarrow 0\ ,$

which breaks into short exact sequences

$\displaystyle
0\longrightarrow\widetilde{L}_{+}\longrightarrow\widetilde{\mathcal{E}}\longrightarrow\mathrm{Im}(\tilde{\varphi}-\omega)\longrightarrow
0\ ;$ $\displaystyle
0\longrightarrow\mathrm{Im}(\tilde{\varphi}-\omega)\longrightarrow\widetilde{\mathcal{E}}\otimes\tilde{K}\longrightarrow
p^{\ast}J\otimes\tilde{K}\longrightarrow 0\ .$

Using the local trivialization in Lemma 6.1,
$\mathrm{Im}(\tilde{\varphi}-\omega)$ is locally spanned by
$\begin{pmatrix}z^{d_{x}}\\\ -z^{m_{x}}\end{pmatrix}\mathfrak{e}$. From Lemma
6.2, if we write $\widetilde{L}_{0}:=\widetilde{L}_{+}(\Lambda-D)$ and
$\widetilde{L}_{1}:=\sigma^{\ast}\widetilde{L}_{0}$, then

$\delta\circ\chi_{BNR}(J)=\widetilde{L}_{+}(\Lambda-D)\ .$

Moreover, there is an isomorphism
$\mathrm{Im}(\tilde{\varphi}-\omega)\cong\widetilde{L}_{1}$. Letting
$\widetilde{L}_{1}^{\prime}$ be the saturation of $\widetilde{L}_{1}$, then we
obtain the commutative diagram:

${0}$${\widetilde{L}_{1}}$${\widetilde{\mathcal{E}}\otimes\tilde{K}}$${p^{\ast}J\otimes\tilde{K}}$${0}$${0}$${\widetilde{L}_{1}^{\prime}}$${\widetilde{\mathcal{E}}\otimes\tilde{K}}$${p^{\star}_{\mathrm{tf}}J\otimes\tilde{K}}$${0}$$\scriptstyle{i}$$\scriptstyle{\cong}$

where $i:\widetilde{L}_{1}\to\widetilde{L}_{1}^{\prime}$ is the natural
inclusion. Moreover, in the same trivialization, $\widetilde{L}_{1}^{\prime}$
is spanned by the section $\begin{pmatrix}1\\\
-z^{m_{x}-d_{x}}\end{pmatrix}\mathfrak{e}$. Therefore,
$\widetilde{L}_{1}^{\prime}\cong\widetilde{L}_{-}\otimes\tilde{K}$ and from
Lemma 6.2, $p^{\star}_{\mathrm{tf}}J=\delta\circ\chi_{\mathrm{BNR}}(J)$. ∎

If $(\mathcal{E},\varphi)$ is a Higgs bundle with
$(\mathcal{E},\varphi)=\chi_{\mathrm{BNR}}(L)$, and
$\widetilde{L}_{0}=\delta\circ\chi_{BNR}(L)$, then by Proposition 6.3,
$\widetilde{L}_{0}=p^{\star}_{\mathrm{tf}}(L)$. We define a Higgs bundle
$(\widetilde{\mathcal{E}}_{0},\tilde{\varphi}_{0})$ as follows

$\begin{split}\widetilde{\mathcal{E}}_{0}=\widetilde{L}_{0}\oplus\sigma^{\ast}\widetilde{L}_{0},\;\tilde{\varphi}_{0}=\begin{pmatrix}\omega&0\\\
0&-\omega\end{pmatrix}.\end{split}$

Moreover, $\widetilde{\mathcal{E}}$ is an $\mathcal{O}_{\widetilde{S}}$
submodule of $\widetilde{\mathcal{E}}_{0}$ with a natural inclusion
$\iota:\widetilde{\mathcal{E}}\to\widetilde{\mathcal{E}}_{0}$ satisfying the
following:

  * (i)

the induced morphism
$\widetilde{\mathcal{E}}\to\widetilde{L}_{0},\;\widetilde{\mathcal{E}}\to\sigma^{\ast}\widetilde{L}_{0}$
is surjective,

  * (ii)

the restriction of $\iota|_{\widetilde{S}\setminus\widetilde{Z}}$ is an
isomorphism,

  * (iii)

$\tilde{\varphi}_{0}\circ\iota=\iota\circ\tilde{\varphi}$.

Following [33, Sec. 4.1], we call
$(\widetilde{\mathcal{E}}_{0},\tilde{\varphi}_{0})$ the _abelianization of the
Higgs bundle_ $(\mathcal{E},\varphi)$.

###  6.2. The construction of the algebraic Mochizuki map

In this subsection, we define the algebraic Mochizuki map, as introduced in
[33]. Recall that for any divisor $D=\sum_{x\in Z}d_{x}x$, there is a
canonical weight function

$\chi_{D}(x):=\begin{cases}d_{x}&x\in\mathrm{supp\;}D\ ;\\\
0&x\notin\mathrm{supp\;}D\ .\end{cases}$

We also have the stratification
$\overline{\mathcal{T}}=\cup_{D}\overline{\mathcal{T}}_{D}$, for $\sigma$
divisors $D$. Let $\mathscr{F}(\widetilde{S})$ be the space of all degree zero
filtered line bundles over $\widetilde{S}$. The _algebraic Mochizuki map_
$\Theta^{\mathrm{Moc}}$ is defined as

$\Theta^{\mathrm{Moc}}:\overline{\mathcal{T}}\longrightarrow\mathscr{F}(\widetilde{S})\
,\
L\mapsto\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}(L),\tfrac{1}{2}\chi_{D-\Lambda})\
.$

######  Example 6.4.

When $q$ has only simple zeroes, this construction generalizes that of [30]
(see also [13]). In the case of a quadratic differential with simple zeros,
the spectral curve $S$ is smooth, and every torsion free sheaf is locally
free, so that $\mathcal{T}=\overline{\mathcal{T}}$. If
$Z=\\{p_{1},\dots,p_{4g-4}\\}$ are the branch points of $S$, and
$\Lambda=\sum_{i=1}^{4g-4}p_{i}$, then the weight function
$\frac{1}{2}\chi_{-\Lambda}$ assigns a weight of $-\frac{1}{2}$ at each
$p_{i}$. For $L\in\mathcal{T}$,
$\Theta^{\mathrm{Moc}}(L)=\mathcal{F}_{\ast}(L,\frac{1}{2}\chi_{-\Lambda}).$

Below are some additional properties of $\Theta^{\mathrm{Moc}}$.

######  Proposition 6.5.

$\Theta^{\mathrm{Moc}}|_{\overline{\mathcal{T}}_{D}}$ is a continuous map.

###### Proof.

This follows directly from the definition of $\Theta^{\mathrm{Moc}}$ and
Theorem 3.3. ∎

From Theorem 5.11, we know that for a $\sigma$-divisor $D$, the preimage of
the map
$p^{\star}_{\mathrm{tf}}:\overline{\mathcal{T}}_{D}\to\widetilde{\mathcal{T}}_{D}$
has dimension $2g-2-\frac{1}{2}\deg(D)-r_{2}/2$, where $r_{2}$ is the number
of odd zeros of $q$. Even for the top stratum $D=0$, $p^{\star}_{\mathrm{tf}}$
is not injective if the spectral curve is not smooth. Indeed, if
$L_{1},L_{2}\in\overline{\mathcal{T}}_{D}$ with
$p^{\star}_{\mathrm{tf}}(L_{1})=p^{\star}_{\mathrm{tf}}(L_{2})$, then based on
the construction we have
$\Theta^{\mathrm{Moc}}(L_{1})=\Theta^{\mathrm{Moc}}(L_{2})$. In summary, we
have the following result:

######  Proposition 6.6.

If $q\in H^{0}(K^{2})$ is irreducible, then $\Theta^{\mathrm{Moc}}$ is
injective if and only if $q$ has simple zeros.

###  6.3. Convergence of subsequences

Fix a locally free $L_{0}\in\mathcal{T}$. Using the isomorphism
$\psi_{L_{0}}:\overline{\mathcal{T}}\to\overline{\mathcal{P}}$ defined by
$\psi_{L_{0}}(L)=LL_{0}^{-1}$, we can extend the Mochizuki map
$\Theta^{\mathrm{Moc}}$ to $\overline{\mathcal{P}}$. For
$J\in\overline{\mathcal{P}}_{D}$, we write
$\widetilde{J}:=p^{\star}_{\mathrm{tf}}(J)$ and choose the weight function
$\frac{1}{2}\chi_{D}$. We then define:

$\Theta^{\mathrm{Moc}}_{0}:\overline{\mathcal{P}}\longrightarrow\mathscr{F}(\widetilde{S})\
,\ J\mapsto\mathcal{F}_{\ast}(\widetilde{J},\tfrac{1}{2}\chi_{D})\ .$

######  Proposition 6.7.

The map $\Theta^{\mathrm{Moc}}_{0}$ satisfies the following properties:

  * (i)

Let $J\in\overline{\mathcal{P}}$ and $L:=L_{0}J$, then

$\Theta^{\mathrm{Moc}}_{0}(J)=\Theta^{\mathrm{Moc}}(L)\otimes\Theta^{\mathrm{Moc}}(L_{0})^{-1}\
,$

where $\otimes$ is the tensor product for filtered line bundles (5).

  * (ii)

Suppose $L=\tau(I,v)$ with
$(\mathcal{I},v)\in\widehat{\mathrm{PMod}}(\widetilde{S})$ and
$L\in\overline{\mathcal{P}}_{D}$, then

$\Theta^{\mathrm{Moc}}_{0}\circ\tau(I,v)=\mathcal{F}_{\ast}(I(-D_{v}),\tfrac{1}{2}\chi_{D_{v}+\sigma^{\ast}D_{v}})\
,$

where $D_{v}$ is the corresponding divisor defined in Theorem 5.5.

  * (iii)

If $\sigma^{\ast}D_{v}=D_{v}$, then
$\Theta^{\mathrm{Moc}}_{0}\circ\tau(\mathcal{I},v)=\mathcal{F}_{\ast}(\mathcal{I},0)$,
where $0$ means all parabolic weights are zero.

###### Proof.

As $L_{0}$ is locally free, we have
$p^{\star}_{\mathrm{tf}}J=(p^{\ast}L_{0})^{-1}\otimes
p^{\star}_{\mathrm{tf}}L$. By definition,

$\Theta^{\mathrm{Moc}}_{0}(J)=\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}J,\tfrac{1}{2}\chi_{D})\
,\
\Theta^{\mathrm{Moc}}(L)=\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}L,\tfrac{1}{2}\chi_{D-\Lambda})\
,\
\Theta^{\mathrm{Moc}}(L_{0})=\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}L_{0},\tfrac{1}{2}\chi_{-\Lambda})\
,$

which implies (i). For (ii), by Theorem 5.5,
$p^{\star}_{\mathrm{tf}}L=I(-D_{v})$, and from Proposition 5.12, we have
$D=D_{v}+\sigma^{\ast}D_{v}$, which implies (ii). When
$\sigma^{\ast}D_{v}=D_{v}$, we compute

$\mathcal{F}_{\ast}(I(-D_{v})\ ,\
\tfrac{1}{2}\chi_{D_{v}+\sigma^{\ast}D_{v}})=\mathcal{F}_{\ast}(I(-D_{v})\ ,\
\chi_{D_{v}})=\mathcal{F}_{\ast}(I,0)\ ,$

which implies (iii). ∎

We now give a criterion for the continuity of the map $\Theta^{\mathrm{Moc}}$.
By Proposition 6.7, it is sufficient to study the map
$\Theta^{\mathrm{Moc}}_{0}$. Recall that for $L\in\overline{\mathcal{P}}$, we
have

$\mathscr{N}_{L}:=\\{(J,v)\in\widehat{\mathrm{PMod}}(\widetilde{S})\mid\tau(J,v)=L\\},\quad\mathscr{D}_{L}:=\\{D_{v}\mid(J,v)\in\mathscr{N}_{L}\\},$

and the number $n_{L}$ is defined to be the number of divisors
$D_{v}\in\mathscr{D}_{L}$ such that $\sigma^{\ast}D_{v}\neq D_{v}$.

######  Proposition 6.8.

Let $D$ be a $\sigma$-divisor, $L\in\overline{\mathcal{P}}_{D}$, and assume
that $\Theta^{\mathrm{Moc}}_{0}$ is continuous at $L$. Then, for
$(J,v)\in\mathscr{N}_{L}$ and $D_{v}\in\mathscr{D}_{L}$, we have
$\sigma^{\ast}D_{v}=D_{v}$, i.e., $n_{L}=0$.

###### Proof.

As the top stratum $\mathcal{P}$ is dense in $\overline{\mathcal{P}}$, there
exists a family of $L_{i}\in\mathcal{P}$ such that $\lim_{i\to\infty}L_{i}=L$.
Let $(J_{i},v_{i})\in\widehat{\mathrm{PMod}}(\widetilde{S})$ be such that
$\tau(J_{i},v_{i})=L_{i}$. Then,
$\lim_{i\to\infty}(J_{i},v_{i})=(J_{\infty},v_{\infty})$, and
$\tau(J_{\infty},v_{\infty})=L$. As $L_{i}$ is locally free, we have
$D_{v_{i}}=0$. Moreover, by Theorem 5.5, we have
$p^{\star}_{\mathrm{tf}}L=J_{\infty}(-D_{v_{\infty}})$, and from Proposition
5.12, we have $D=D_{v_{\infty}}+\sigma^{\ast}D_{v_{\infty}}$. By Proposition
6.7, we have

$\Theta^{\mathrm{Moc}}_{0}(L_{i})=\Theta^{\mathrm{Moc}}_{0}\circ\tau(J_{i},v_{i})=\mathcal{F}_{\ast}(J_{i},0)\
,$

and we compute

$\lim_{i\to\infty}\Theta^{\mathrm{Moc}}_{0}(L_{i})=\mathcal{F}_{\ast}(J_{\infty},0)=\mathcal{F}_{\ast}(J_{\infty}(-D_{v_{\infty}}),\chi_{D_{v_{\infty}}})\
.$

Moreover, by Proposition 6.7, we have

$\Theta^{\mathrm{Moc}}_{0}(L)=\mathcal{F}_{\ast}(J_{\infty}(-D_{v_{\infty}}),\frac{1}{2}(\chi_{D_{v_{\infty}}}+\chi_{{\sigma^{\ast}D_{v_{\infty}}}})).$

Since $\Theta^{\mathrm{Moc}}_{0}$ is continuous on $L$, we have
$\lim_{i\to\infty}\Theta^{\mathrm{Moc}}_{0}(L_{i})=\Theta^{\mathrm{Moc}}(L)$,
which implies that $\chi_{D_{v_{\infty}}}=\chi_{\sigma^{\ast}D_{v_{\infty}}}$.
∎

By Proposition 5.16, $n_{L}>0$ if and only if $q$ has at least one zero of
even order. Hence, the following is immediate.

######  Corollary 6.9.

Suppose $q$ is irreducible and has a zero of even order, then
$\Theta^{\mathrm{Moc}}_{0}$ is not continuous.

By contrast, we have the following.

######  Proposition 6.10.

If $q$ is irreducible with all zeroes of odd order, then
$\Theta^{\mathrm{Moc}}_{0}$ is continuous.

###### Proof.

Since all zeroes of $q$ are odd, for any $L\in\overline{\mathcal{P}}$, we have
$n_{L}=0$. Let $L_{\infty}\in\overline{\mathcal{P}}$ be fixed and let
$L_{i}\in\overline{\mathcal{P}}$ be any sequence such that
$\lim_{i\to\infty}L_{i}=L_{\infty}$. Since
$\tau:\widehat{\mathrm{PMod}}(\widetilde{S})\to\overline{\mathcal{P}}$ is
bijective, we take $(J_{i},v_{i})\in\widehat{\mathrm{PMod}}(\widetilde{S})$
with $\tau(J_{i},v_{i})=L_{i}$. Moreover, we assume
$\lim_{i\to\infty}(J_{i},v_{i})=(J_{\infty},v_{\infty})$ with
$\tau(J_{\infty},v_{\infty})=L_{\infty}$. Since $q$ only contains odd zeros,
it follows that $\mathrm{supp\;}D_{v}\subset\mathrm{Fix}(\sigma)$. By
Proposition 6.7, we have
$\Theta^{\mathrm{Moc}}_{0}(L_{i})=\mathcal{F}_{\ast}(J_{i},0)$. Therefore, we
have:

$\lim_{i\to\infty}\Theta^{\mathrm{Moc}}_{0}(L_{i})=\lim_{i\to\infty}\mathcal{F}_{\ast}(J_{i},0)=\mathcal{F}_{\ast}(J_{\infty},0)=\Theta^{\mathrm{Moc}}_{0}(L_{\infty}).$

This concludes the proof. ∎

######  Theorem 6.11.

Suppose $q$ is irreducible. For the map
$\Theta^{\mathrm{Moc}}:\mathcal{M}_{q}\rightarrow\mathscr{F}(\widetilde{S})$,
we have:

  * (i)

$\Theta^{\mathrm{Moc}}$ is injective if and only if $q$ only has only simple
zeros;

  * (ii)

if $q$ has only zeroes of odd order, $\Theta^{\mathrm{Moc}}$ is continuous;

  * (iii)

if $q$ contains a zero of even order, $\Theta^{\mathrm{Moc}}$ is not
continuous.

###### Proof.

(i) follows from Proposition 6.6. (ii) follows from Proposition 6.10. (iii)
follows from Corollary 6.9. ∎

######  Proposition 6.12.

Suppose $n_{L}>0$. Then for $k=1,\ldots,n_{L}$, there exist sequences
${L_{i}^{k}}$ with $\lim_{i\to\infty}L_{i}^{k}=L$ such that if we denote
$\mathcal{F}_{\ast}^{k}:=\lim_{i\to\infty}\Theta^{\mathrm{Moc}}_{0}(L_{i}^{k})$,
$\mathcal{F}_{\ast}^{0}:=\Theta^{\mathrm{Moc}}_{0}(L)$, then
$\mathcal{F}_{\ast}^{k_{1}}\neq\mathcal{F}_{\ast}^{k_{2}}$ for $k_{1}\neq
k_{2}$. Moreover, there exist
$\\{D_{1},\ldots,D_{n_{L}}\\}\subset\mathscr{D}_{L}$ such that
$\mathcal{F}_{\ast}^{k}=\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}L,\chi_{D_{k}})$.

###### Proof.

By the definition of $n_{L}$, we can find $(J^{k},v^{k})$ with
$\tau(J^{k},v^{k})=L$. If we define $D_{k}:=D_{v^{k}}$, then
$\sigma^{\ast}D_{k}\neq D_{k}$. Moreover, by Theorem 5.5, we have
$p^{\star}_{\mathrm{tf}}L=J^{k}(-D_{k})$. As $\tau^{-1}(\mathcal{P})$ is dense
in $\widehat{\mathrm{PMod}}(\widetilde{S})$, for each $(J^{k},v^{k})$, we can
find a sequence $(J_{i}^{k},v_{i}^{k})\in\tau^{-1}(\mathcal{P})$ such that
$\lim_{i\to\infty}(J_{i}^{k},v_{i}^{k})=(J^{k},v^{k})$ and we define
$L_{i}^{k}:=\tau(J_{i}^{k},v_{i}^{k})$. Since $L_{i}^{k}$ is locally free,
$D_{v_{i}^{k}}=0$, and thus
$\Theta^{\mathrm{Moc}}_{0}(L_{i}^{k})=\mathcal{F}_{\ast}(J_{i}^{k},0)$. We
compute

$\lim_{i\to\infty}\Theta^{\mathrm{Moc}}_{0}(L_{i}^{k})=\mathcal{F}_{\ast}(J^{k},0)=\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}L,\chi_{D_{k}})$

and
$\Theta^{\mathrm{Moc}}_{0}(L)=\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}L,\frac{1}{2}\chi_{D})$.
Based on our assumptions, we have $D_{k_{1}}\neq D_{k_{2}}$ for $k_{1}\neq
k_{2}$ and $\sigma^{\ast}D_{k}\neq D_{k}$, which implies that
$\chi_{D_{k_{1}}}\neq\chi_{D_{k_{2}}}$ for $k_{1}\neq k_{2}$ and
$\chi_{D_{k}}\neq\frac{1}{2}\chi_{D}$. ∎

We now present a computation for the case of a simple nodal curve.

######  Example 6.13.

Let $q$ be a quadratic differential with $2g-4$ simple zeros, and let $x$ be
an even zero of $q$ of order two. Then $S$ has a singular point, which we also
denote by $x$. Let $p:\widetilde{S}\to S$ be the normalization map and
$p^{-1}(x)=\\{x_{1},x_{2}\\}$. Consider the $\sigma$-divisor $D=x_{1}+x_{2}$,
and let $L\in\overline{\mathcal{P}}_{D}$. Then $n_{L}=2$, and we can write
$\mathscr{N}_{L}={(J_{1},v_{1}),(J_{2},v_{2})}$, where $D_{v_{1}}=x_{1}$ and
$D_{v_{2}}=x_{2}$. Moreover, we have
$p^{\star}_{\mathrm{tf}}L=J_{1}\otimes\mathcal{O}(-x_{1})=J_{2}\otimes\mathcal{O}(-x_{2})$.
Let $(\alpha,\beta)$ denote the parabolic weight that is equal to $\alpha$ at
$x_{1}$, $\beta$ at $x_{2}$, and $\frac{1}{2}$ at all other zeros. Then the
filtered bundles obtained in Proposition 6.12 are

$\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}L,(1,0))\ ,\
\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}L,(0,1))\ ,\
\mathcal{F}_{\ast}(p^{\star}_{\mathrm{tf}}L,(\tfrac{1}{2},\tfrac{1}{2}))\ .$

###  6.4. Mochizuki’s convergence theorem for irreducible fibers

In this subsection, we recall Mochizuki’s construction of the limiting
configuration metric [33, Section 4.2.1, 4.3.2] and the convergence theorem.

####  6.4.1. Limiting configuration metric

Let $q$ be an irreducible quadratic differential and
$(\mathcal{E},\varphi)\in\mathcal{M}_{q}$ a Higgs bundle with
$(\mathcal{E},\varphi)=\chi_{\mathrm{BNR}}(L)$. We write
$\widetilde{L}_{0}=p^{\star}_{\mathrm{tf}}L$ and
$(\widetilde{\mathcal{E}},\tilde{\varphi}):=p^{\ast}(\mathcal{E},\varphi)$.
Then the abelianization of $(\mathcal{E},\varphi)$, which is a Higgs bundle
over $\widetilde{S}$, can be written as
$\widetilde{\mathcal{E}}_{0}=\widetilde{L}_{0}\oplus\sigma^{\ast}\widetilde{L}_{0},\;\tilde{\varphi}_{0}=\mathrm{diag}(\omega,\;-\omega)$.
The natural inclusion
$\iota:(\widetilde{\mathcal{E}},\tilde{\varphi})\to(\widetilde{\mathcal{E}}_{0},\tilde{\varphi}_{0})$
is an isomorphism over $\widetilde{S}\setminus\widetilde{Z}$. Moreover, we
write $D$ be the $\sigma$-divisor of $(\mathcal{E},\varphi)$.

From the construction of $\Theta^{\mathrm{Moc}}(L)$ and Proposition 6.12, we
have $n_{L}$ different divisors $D_{k}$ for $k=1,\cdots,n_{L}$ with
$\sigma^{*}D_{k}\neq D_{k}$ and $D_{k}+\sigma^{*}D_{k}=D$. Moreover, we can
find $n_{L}+1$ different filtered bundles with deg $0$. Define

$\mathcal{F}_{*,0}:=\Theta^{\mathrm{Moc}}(L)=\mathcal{F}_{*}(\widetilde{L}_{0},\chi_{\frac{1}{2}(D-\Lambda)}),\;\mathcal{F}_{*,k}:=\mathcal{F}_{*}(\widetilde{L}_{0},\chi_{D_{k}-\frac{1}{2}\Lambda)}),$

which are all degree zero filtered bundles with different level of
filtrations.

Now, we will introduce the construction in [33, Section 4.2.1, 4.3.2]. For
$k=0,\cdots,n_{L}$, we define $\tilde{h}_{k}$ to be the harmonic metric for
the filtered bundle $\mathcal{F}_{*,k}$; this is well-defined up to a
multiplicative constant. To fix this constant, assume that
$\sigma^{\ast}\tilde{h}_{k}\otimes\tilde{h}_{k}=1$. This gives a unique choice
of $\tilde{h}_{0}$. We then define the metric
$\widetilde{H}_{k}=\mathrm{diag}(\tilde{h}_{k},\;\sigma^{\ast}\tilde{h}_{k})$
on $\widetilde{\mathcal{E}}_{0}$, with $\det(\widetilde{H}_{k})=1$. For the
resulting harmonic bundle
$(\widetilde{\mathcal{E}}_{0},\varphi_{0},\widetilde{H}_{k})$, we define
$\tilde{\nabla}_{k}$ to be the unitary connection determined by
$\widetilde{H}_{k}$. Since $\widetilde{H}_{k}$ is diagonal, over
$\widetilde{S}\setminus\widetilde{Z}$, it follows that
$F_{\tilde{\nabla}_{k}}=0$, and we have
$[\varphi_{0},\varphi_{0}^{\dagger_{\widetilde{H}_{k}}}]=0$. Furthermore, as
$\iota$ is an isomorphism on $\widetilde{S}\setminus\widetilde{Z}$, the metric
$\widetilde{H}_{k}$ also defines a metric on
$(\widetilde{\mathcal{E}},\tilde{\varphi})$ over
$\widetilde{S}\setminus\widetilde{Z}$.

For any $\tilde{x}\in\widetilde{S}\setminus\widetilde{Z}$ with
$x:=p(\tilde{x})$, we have isomorphisms

$(\widetilde{\mathcal{E}}_{0},\tilde{\varphi}_{0})|_{\sigma(\tilde{x})}\cong(\widetilde{\mathcal{E}}_{0},\tilde{\varphi}_{0})|_{\tilde{x}}\cong(\widetilde{\mathcal{E}},\tilde{\varphi})|_{\tilde{x}}\cong(\mathcal{E},\varphi)|_{x}.$

Therefore, $\widetilde{H}_{k}$ induces a metric $H^{\mathrm{Lim}}_{k}$ on
$\Sigma\setminus Z$, and we may consider $H^{\mathrm{Lim}}_{k}$ as the push-
forward of $\tilde{h}_{k}$. In [25, Theorem 5.2], the push-forward metric of
$\Theta^{\mathrm{Moc}}(L)$ is explicitly written in local coordinates.

Recall the notation from Section 2.4, let $E$ be a trivial, smooth, rank 2
vector bundle over a Riemann surface $\Sigma$, and let $K$ be a background
Hermitian metric on $E$. Over $\Sigma\setminus Z$, we write
$\nabla_{k}^{\mathrm{Lim}}$ for the Chern connection defined by
$H_{k}^{\mathrm{Lim}}$, which is unitary w.r.t. $H_{0}$ and
$\phi^{\mathrm{Lim}}_{k}=\varphi_{k}^{\mathrm{Lim}}+\varphi_{k}^{\dagger_{\mathrm{Lim}}}$
be the corresponding Higgs field in the unitary gauge. They satisfy the
decoupled Hitchin equations over $\Sigma\setminus Z$. Thus from any Higgs
bundle $(\mathcal{E},\varphi)$, we obtain $n_{L}+1$ limiting configurations

$(\nabla_{k}^{\mathrm{Lim}},\phi_{k}^{\mathrm{Lim}}=\varphi+\varphi_{k}^{\dagger_{\mathrm{Lim}}})\in\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}.$

The flat connection, which is defined over $\Sigma\setminus Z$, may be
understood by using the nonabelian Hodge correspondence for filtered vector
bundles [43]. Given the filtered line bundles $\mathcal{F}_{*,k}$, define
filtered vector bundles
$\widetilde{\mathcal{E}}_{*,k}:=\mathcal{F}_{*,k}\oplus\sigma^{*}\mathcal{F}_{*,k}$,
which can be explicitly written as

(18)
$\begin{split}\widetilde{\mathcal{E}}_{*,0}:&=\mathcal{F}_{*}(\widetilde{L}_{0},\chi_{\frac{1}{2}(D-\Lambda)})\oplus\mathcal{F}_{*}(\sigma^{*}\widetilde{L}_{0},\chi_{\frac{1}{2}(D-\Lambda)})\
;\\\
\widetilde{\mathcal{E}}_{*,k}:&=\mathcal{F}_{*}(\widetilde{L}_{0},\chi_{D_{k}-\frac{1}{2}\Lambda})\oplus\mathcal{F}_{*}(\sigma^{*}\widetilde{L}_{0},\chi_{\sigma^{*}D_{k}-\frac{1}{2}\Lambda})\
,\ k\neq 0\ .\end{split}$

These are polystable filtered vector bundles over
$\widetilde{S}\setminus\widetilde{Z}$. As for each $k=0,\cdots,n_{L}$,
$\sigma^{*}\widetilde{\mathcal{E}}_{*,k}=\widetilde{\mathcal{E}}_{*,k}$, the
filtered bundles $\widetilde{\mathcal{E}}_{*,k}$ induce a filtered vector
bundles $\mathcal{E}_{*,k}$ over $\Sigma\setminus Z$. The flat connection
$\nabla_{k}^{\mathrm{Lim}}$ will be the unique harmonic unitary connection
corresponding to the $\mathcal{E}_{*,k}$. Moreover, for $0\leq k_{1}\neq
k_{2}\leq n_{L}$, based on the definition of $D_{k_{1}}$ and $D_{k_{2}}$, we
can always find $\tilde{x}\in\widetilde{Z}_{\mathrm{even}}$, a preimage of an
even zero $x$ of $q$, such that $\widetilde{\mathcal{E}}_{*,k_{1}}$ and
$\widetilde{\mathcal{E}}_{*,k_{2}}$ have different filtered structures near
$\tilde{x}$. Since over even zeros, $\widetilde{S}\to\Sigma$ is not a branched
covering, we conclude that near $x$, $\mathcal{E}_{*,k_{1}}$ and
$\mathcal{E}_{*,k_{2}}$ are different filtered bundles. By [43, Main theorem],
the harmonic connections $\nabla_{k_{1}}$ and $\nabla_{k_{2}}$ are not gauge
equivalent.

We therefore conclude the following:

######  Proposition 6.14.

For $0\leq k_{1}\neq k_{2}\leq n_{L}$,
$(\nabla_{k_{1}}^{\mathrm{Lim}},\phi_{k_{1}}^{\mathrm{Lim}})$ and
$(\nabla_{k_{2}}^{\mathrm{Lim}},\phi_{k_{2}}^{\mathrm{Lim}})$ are not gauge
equivalent in $\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}$.

This leads us to define the _analytic Mochizuki map_ $\Upsilon^{\mathrm{Moc}}$
as

(19)
$\Upsilon^{\mathrm{Moc}}:\mathcal{M}_{q}\longrightarrow\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}\
:\
[(\mathcal{E},\varphi)]\mapsto[(\nabla_{0}^{\mathrm{Lim}},\phi_{0}^{\mathrm{Lim}})],$

which we recall is the limiting configuration defined by
$\Theta^{\mathrm{Moc}}(L)$.

####  6.4.2. The continuity of the limiting configurations

We now introduce the main result of Mochizuki [33]. Fix
$(\mathcal{E},\varphi)=\chi_{\mathrm{BNR}}(L)\in\mathcal{M}_{q}$. For any real
parameter $t$, $(\mathcal{E},t\varphi)$ is a stable Higgs bundle. By the
Kobayashi-Hitchin correspondence, there exists a unique metric $H_{t}$ solving
the Hitchin equation. Denote by $\nabla_{t}$ the unitary connection defined by
$H_{t}$ and write $\mathcal{D}_{t}=\nabla_{t}+t\phi_{t}$ for the full
$\mathrm{SL}(2,\mathbb{C})$ flat connection. We then have:

######  Theorem 6.15 ([33]).

The family $(\mathcal{E},t\varphi)$ has a unique limiting configuration as
limit for $t\to\infty$. Moreover, for any compact set $K\subset\Sigma\setminus
Z$, let $d:=\min_{x\in K}|q|(x)$. Then there exist $t$-independent constants
$C_{k}$ and $C^{\prime}_{k}$ such that

$|(\nabla_{t},\phi_{t})-\Upsilon^{\mathrm{Moc}}(\mathcal{E},\varphi)|_{\mathcal{C}^{k}}\leq
C_{k}e^{-C^{\prime}_{k}td}\ .$

As the map $\Upsilon^{\mathrm{Moc}}$ is the composition of
$\Theta^{\mathrm{Moc}}\circ\chi_{\mathrm{BNR}}^{-1}$ with the nonabelian Hodge
correspondence, the behavior of $\Upsilon^{\mathrm{Moc}}$ is the same as
$\Theta^{\mathrm{Moc}}$. Recall the decomposition
$\mathcal{M}_{q}=\bigcup\mathcal{M}_{q,D}$ from the end of Section 5. By
Theorem 6.11, Proposition 6.12 and Proposition 6.14, we obtain:

######  Theorem 6.16.

Let $q$ be an irreducible quadratic differential. The map
$\Upsilon^{\mathrm{Moc}}:\mathcal{M}_{q}\to\mathcal{M}_{\mathrm{Hit}}^{\mathrm{Lim}}$
satisfies the following properties:

  * (i)

if all the zeros of $q$ are odd, then $\Upsilon^{\mathrm{Moc}}$ is continuous;

  * (ii)

if at least one zero of $q$ is even, then for each
$(\mathcal{E},\varphi)\in\mathcal{M}_{q,D}$, there exists an integer $n_{D}$
that only depends on $D$, and $n_{D}$ sequences
$\\{(\mathcal{E}_{i}^{k},\varphi_{i}^{k})\\}$ for $k=1,\dots,n_{D}$, such that
$\lim_{i\to\infty}(\mathcal{E}_{i}^{k},\varphi_{i}^{k})=(\mathcal{E}_{\infty},\varphi_{\infty})$,
and

$\lim_{i\to\infty}\Upsilon^{\mathrm{Moc}}(\mathcal{E}_{i}^{k_{1}},\varphi_{i}^{k_{1}})\neq\lim_{i\to\infty}\Upsilon^{\mathrm{Moc}}(\mathcal{E}_{i}^{k_{2}},\varphi_{i}^{k_{2}})\neq\Upsilon^{\mathrm{Moc}}(\mathcal{E}_{\infty},\varphi_{\infty})\
,\ \text{for $k_{1}\neq k_{2}$}\ .$

##  7\. Reducible singular fiber and the Mochizuki map

We now investigate the properties of the Hitchin fiber associated with a
reducible quadratic differential, as discussed in [16]. Additionally, we will
provide an overview of Mochizuki’s technique for constructing limiting
configurations of Hitchin fibers for reducible quadratic differentials, as
detailed in [33]. We also analyze the continuity of the Mochizuki map.

###  7.1. Local description of a Higgs bundle

Write $q=-\omega\otimes\omega$ with $\omega\in H^{0}(K)$,
$\Lambda=\mathrm{Div}(\omega)$, $Z=\mathrm{supp\;}(\Lambda)$, and
$\mathcal{M}_{q}=\mathcal{H}^{-1}(q)$. Compared to the irreducible case,
$\mathcal{M}_{q}$ contains strictly semistable Higgs bundles, so we let
$\mathcal{M}_{q}^{\rm st}$ denote the stable locus. We point out that there is
a sign ambiguity in the choice of $\omega$, which actually plays an important
role in the following.

####  7.1.1. Local description

Given a Higgs bundle $(\mathcal{E},\varphi)$ with $\det(\varphi)=q$, define
line bundles

(20) $L_{\pm}:=\ker(\varphi\pm\omega)\ .$

Then the inclusion maps $L_{\pm}\to\mathcal{E}$ are injective. Similarly, we
may define an abelianization of $(\mathcal{E},\varphi)$ by
$(\mathcal{E}_{0}=L_{+}\oplus
L_{-},\varphi_{0}=\mathrm{diag}(\omega,-\omega))$. We then have a natural
inclusion $\iota:\mathcal{E}_{0}\to\mathcal{E}$, which is is an isomorphism on
$\Sigma\setminus Z$, and $\varphi\circ\iota=\iota\circ\varphi_{0}$.

It follows from [16, Prop. 7.10] that $L_{\pm}$ are the only
$\varphi$-invariant subbundles of $\mathcal{E}$. If we write
$d_{\pm}:=\deg(L_{\pm})$, then $(\mathcal{E},\varphi)$ is stable (resp.
semistable) if and only if $d_{\pm}<0\ (\text{resp.}\ \leq 0)$. As
$\det(\mathcal{E})=\mathcal{O}$, the map $\det(\iota):L_{+}\otimes
L_{-}\to\mathcal{O}$ defines a divisor $D=\mathrm{Div}(\det(\iota))$ such that
$L_{+}\otimes L_{-}=\mathcal{O}(-D).$ Therefore, we obtain

$d_{+}+d_{-}+\deg D=0\ ,$

and $0\leq D\leq\Lambda$. The Higgs bundle $(\mathcal{E},\varphi)$ is
semistable if and only if $-\deg D\leq d_{+}\leq 0$ and stable if the
equalities are strict. For the rest of this section, we always write
$D=\sum_{p\in Z}\ell_{p}p$.

####  7.1.2. Semistable settings

As the fiber $\mathcal{M}_{q}$ might contain strictly semistable Higgs
bundles, we now explicitly enumerate all of the possible $S$-equivalence
classes. When $D=0$, then $L_{-}=L_{+}^{-1}$ and $\deg(L_{+})=0$. The
corresponding Higgs bundle is polystable and can be explicitly written as

$\Bigl{(}L\oplus L^{-1},\begin{pmatrix}\omega&0\\\
0&-\omega\end{pmatrix}\Bigr{)}\ ,$

where $L\in\mathrm{Jac}(\Sigma)$. When $D\neq 0$, suppose
$\deg(L_{+})=-\deg(D)$. Then $L_{-}=L_{+}^{-1}(-D)$ and $\deg(L_{-})=0$. Under
$S$-equivalence, the polystable Higgs bundle is

$\Bigl{(}L_{+}(D)\oplus L_{+}^{-1}(-D),\begin{pmatrix}\omega&0\\\
0&-\omega\end{pmatrix}\Bigr{)}\ ,$

where $L_{+}\in\mathrm{Pic}^{-\deg(D)}(\Sigma)$.

###  7.2. Reducible spectral curves

In this subsection, we introduce the algebraic data in [16] which describes
the singular fiber with a reducible spectral curve. This plays a similar role
to the parabolic modules. See [16, Sec. 7.1] for more details.

For any divisor $D$, and line bundle $L$, define the space

$H^{0}(D,L)=\bigoplus_{p\in\mathrm{supp\;}D}\mathcal{O}(L)_{p}/\sim\ ,$

where $s_{1}\sim s_{2}$ if and only if $\mathrm{ord}_{p}([s_{1}]-[s_{2}])\geq
D_{p}$.

Let $L\in\mathrm{Pic}(\Sigma)$, define the following subspaces of
$H^{0}(\Lambda,L^{2}K)$:

$\begin{split}&\mathcal{V}(D,L):=\\{s\in
H^{0}(\Lambda,L^{2}K)\mid\mathrm{ord}_{p}(s)=\Lambda_{p}-D_{p},\;\mathrm{if}\;D_{p}>0;\;s(p)=0,\;\mathrm{if}\;D_{p}=0\\},\\\
&\mathcal{W}(D,L)=\\{s\in H^{0}(\Lambda,L^{2}K)\mid
s|_{\mathrm{supp\;}(\Lambda-D)}=0\\}\ .\end{split}$

One checks that $\mathcal{W}(D,L)=\cup_{D^{\prime}\leq
D}\mathcal{V}(D^{\prime},L)$. Moreover, the space $\mathcal{V}(D,L)$ is a
linear subspace of $H^{0}(\Lambda,L^{2}K)$ with a hyperplane removed. In
addition, $\mathbb{C}^{\ast}$ acts on $\mathcal{V}(D,L)$ by multiplication,
and $\dim(\mathcal{V}(D,L)/\mathbb{C}^{\ast})=\deg(D)-1$.

We define the fibrations

$\begin{split}p_{m}:\mathscr{V}(D,m)\longrightarrow\mathrm{Pic}^{m}(\Sigma),\;p_{m}:\mathscr{W}(D,m)\longrightarrow\mathrm{Pic}^{m}(\Sigma)\end{split}$

such that for $L\in\mathrm{Pic}^{m}(\Sigma)$, the fibers are
$\mathcal{V}(D,L)$ and $\mathcal{W}(D,L)$.

####  7.2.1. Algebraic data from the extension

The Higgs bundle $(\mathcal{E},\varphi)$ can be understood in terms of an
extension of exact sequence. As $\det(\mathcal{E})=\mathcal{O}$, we have the
exact sequence

$0\longrightarrow L_{+}\longrightarrow\mathcal{E}\longrightarrow
L_{+}^{-1}\longrightarrow 0\ .$

For each $p\in Z$, with $U\subset\Sigma$ a neighborhood of $p$,
$(\mathcal{E},\varphi)$ can be written as the following splitting of
$\mathcal{C}^{\infty}$ bundles

$\begin{split}\mathcal{E}=L_{+}\oplus_{\mathcal{C}^{\infty}}L_{+}^{-1}\ ,\
\bar{\partial}_{\mathcal{E}}=\begin{pmatrix}\bar{\partial}_{L_{+}}&b\\\
0&\bar{\partial}_{L_{+}^{-1}}\end{pmatrix}\ ,\
\varphi=\begin{pmatrix}\omega&c\\\ 0&-\omega\end{pmatrix}\ .\end{split}$

We would like to consider the restriction of $\varphi+\omega\cdot\mathrm{id}$
to $\Lambda$. As $\omega|_{\Lambda}=0$,
$\varphi+\omega\cdot\mathrm{id}|_{L_{+}}=0$ and the image of
$\varphi+\omega\mathrm{id}\subset(L_{+}\oplus 0)\otimes
K\subset\mathcal{E}\otimes K$. Therefore, the restriction of
$\varphi+\omega\cdot\mathrm{id}$ to $\Lambda$ defines a holomorphic map
$s:L_{+}^{-1}|_{\Lambda}\to L_{+}K|_{\Lambda},$ or equivalently a section
$s\in H^{0}(\Lambda,L_{+}^{2}K).$ Moreover, by [16, Lemma 7.12],
$\mathrm{Div}(s)=\Lambda-D$. Therefore, given any
$(\mathcal{E},\varphi)\in\mathcal{M}_{q}$, we obtain an
$L\in\mathrm{Pic}^{m}(\Sigma)$ and an element in $\mathcal{V}(D,L)$. Moreover,
the stability condition implies that $0\leq D\leq\Lambda$, we have $-\deg
D\leq\deg L\leq 0.$

####  7.2.2. Inverse construction

The inverse of the construction above also holds; for further details, see
[16, Sec. 7] and [24, Sec. 5]. Given $L\in\mathrm{Pic}^{m}(\Sigma)$ and
$q\in\mathcal{V}(D,L)$, we define a Higgs bundle via extensions as follows.
From $q,L$, we have a short exact sequence of complexes of sheaves:

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