# Endoscopic decompositions and the Hausel–Thaddeus conjecture

Davesh Maulik  Massachusetts Institute of Technology<EMAIL_ADDRESS>and
Junliang Shen  Massachusetts Institute of Technology<EMAIL_ADDRESS>Yale
University<EMAIL_ADDRESS>
###### Abstract.

We construct natural operators connecting the cohomology of the moduli spaces
of stable Higgs bundles with different ranks and genera which, after numerical
specialization, recover the topological mirror symmetry conjecture of
Hausel–Thaddeus concerning $\mathrm{SL}_{n}$\- and $\mathrm{PGL}_{n}$-Higgs
bundles. This provides a complete description of the cohomology of the moduli
space of stable $\mathrm{SL}_{n}$-Higgs bundles in terms of the tautological
classes, and gives a new proof of the Hausel–Thaddeus conjecture, proven
recently by Gröchenig–Wyss–Ziegler via p-adic integration.

Our method is to relate the decomposition theorem for the Hitchin fibration,
using vanishing cycle functors, to the decomposition theorem for the twisted
Hitchin fibration whose supports are simpler.

###### Contents

  1. 0 Introduction
  2. 1 Hitchin-type moduli spaces
  3. 2 Support theorems for Hitchin moduli spaces
  4. 3 Endoscopic decompositions
  5. 4 Vanishing cycles and Hitchin moduli spaces
  6. 5 The P=W conjecture and the Hausel-Thaddeus conjecture

##  0\. Introduction

###  0.1. Overview

Throughout, we work over the complex numbers ${\mathbb{C}}$. Let $C$ be a
nonsingular projective curve of genus $g\geq 2$. Let $n,d$ be integers with
$n>0$ and $\mathrm{gcd}(n,d)=1$.

The cohomology of the moduli space $\widetilde{{\mathcal{N}}}_{n,d}$ of rank
$n$ degree $d$ stable vector bundles on $C$ has been studied intensively for
decades. By [1, 2], the cohomology
$H^{*}(\widetilde{{\mathcal{N}}}_{n,d},{\mathbb{C}})$ is generated by the
_tautological classes_ — the Künneth factors of the Chern characters of a
universal family. Relations between the tautological classes were explored in
[27, 14].

A natural moduli space closely related to $\widetilde{{\mathcal{N}}}_{n,d}$ is
the moduli of stable $\mathrm{SL}_{n}$-bundles

${\mathcal{N}}_{n,L}\subset\widetilde{{\mathcal{N}}}_{n,d}$

which parameterizes rank $n$ stable vector bundles with fixed determinant
$L\in\mathrm{Pic}^{d}(C)$. The finite abelian group

$\Gamma=\mathrm{Pic}^{0}(C)[n]$

acts on ${\mathcal{N}}_{n,L}$ via tensor product, which induces a
$\Gamma$-action on the cohomology $H^{*}({\mathcal{N}}_{n,L},{\mathbb{C}})$.
The $\Gamma$-invariant part $H^{*}({\mathcal{N}}_{n,L},{\mathbb{C}})^{\Gamma}$
recovers the cohomology of the quotient ${\mathcal{N}}_{n,L}/\Gamma$, which
can be viewed as the moduli space of stable $\mathrm{PGL}_{n}$-bundles. The
tautological classes associated with a universal family generate the
$\Gamma$-invariant cohomology
$H^{*}({\mathcal{N}}_{n,L},{\mathbb{C}})^{\Gamma}$.

The following theorem by Harder–Narasimhan [19] shows that every class in
$H^{*}({\mathcal{N}}_{n,L},{\mathbb{C}})$ is $\Gamma$-invariant.

######  Theorem 0.1 (Harder–Narasimhan [19]).

The $\Gamma$-action on $H^{*}({\mathcal{N}}_{n,L},{\mathbb{C}})$ is trivial.

As a consequence of Theorem 0.1, we obtain immediately that the tautological
classes generate the total cohomology
$H^{*}({\mathcal{N}}_{n,L},{\mathbb{C}})$.

The purpose of this paper is to study the $\Gamma$-action on the cohomology of
the moduli space of stable $\mathrm{SL}_{n}$-Higgs bundles from the viewpoint
of the Hausel-Thaddeus conjecture [21]. We denote by ${\mathcal{M}}_{n,L}$ the
(coarse) moduli space parameterizing stable Higgs bundles

$({\mathcal{E}},\theta:{\mathcal{E}}\rightarrow{\mathcal{E}}\otimes\Omega_{C}):\quad\mathrm{det}({\mathcal{E}})\simeq
L,\quad\mathrm{trace}(\theta)=0$

on the curve $C$. It is a nonsingular quasi-projective variety admitting a
natural hyper-Kähler structure [25, 32]. As in the case of vector bundles, the
group $\Gamma$ acts on ${\mathcal{M}}_{n,L}$ via tensor product

${\mathcal{L}}\cdot({\mathcal{E}},\theta)=({\mathcal{L}}\otimes{\mathcal{E}},\theta),\quad{\mathcal{L}}\in\Gamma.$

The induced $\Gamma$-action on $H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})$
yields the following canonical decomposition

(1)
$H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})=H^{*}({\mathcal{M}}_{n,L}/\Gamma,{\mathbb{C}})\oplus\bigoplus_{\kappa\neq
0}H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$

where $\kappa\in\hat{\Gamma}=\mathrm{Hom}(\Gamma,{\mathbb{C}}^{*})$ runs
through all nontrivial characters of $\Gamma$, and
$H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$ denotes the
$\kappa$-isotypic component. By [31] (and the isomorphism [8, (70)]), the
tautological classes associated with a universal family of
${\mathcal{M}}_{n,L}$ generate the $\Gamma$-invariant cohomology

$H^{*}({\mathcal{M}}_{n,L}/\Gamma,{\mathbb{C}})=H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})^{\Gamma}.$

However, contrary to Theorem 0.1, the $\Gamma$-variant part of (1) is
nontrivial, and carries a rich structure, predicted by topological mirror
symmetry [21].

In this paper, we focus on the structure of
$H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$ for $\kappa\neq 0$. We
introduce natural operators which determine
$H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$ in terms of the cohomology
of the moduli space of stable $\mathrm{GL}_{r}$-Higgs bundles on a certain
curve for some $r\leq n$. These operators respect the perverse and the Hodge
filtrations, and, upon specialization to Hodge polynomials, they recover the
Hausel–Thaddeus conjecture [21]. In particular, this gives a new proof using
perverse sheaves of the topological mirror symmetry conjecture of Hausel-
Thaddeus, which was recently proven by [17] using p-adic integration.

###  0.2. Hitchin fibrations

The moduli space ${\mathcal{M}}_{n,L}$ carries a Lagrangian fibration

(2) $h:{\mathcal{M}}_{n,L}\rightarrow{\mathcal{A}}=\bigoplus_{i\geq
2}H^{0}(C,{\Omega_{C}^{\otimes i}})$

given by Hitchin’s integrable system, which is now referred to as the _Hitchin
fibration_. The $\Gamma$-action on ${\mathcal{M}}_{n,L}$ is fiberwise with
respect to (2). There are two types of moduli spaces closely related to the
cohomological study of ${\mathcal{M}}_{n,L}$ from the perspective of mirror
symmetry [21] and representation theory [33, 34].

The moduli spaces of the first type are the fixed loci of an element
$\gamma\in\Gamma$. For any $\gamma\in\Gamma$, we denote by
${\mathcal{M}}_{\gamma}\subset{\mathcal{M}}_{n,L}$ the $\gamma$-fixed
subvariety, which maps to the Hitchin base via

${\mathcal{M}}_{\gamma}\xrightarrow{h_{\gamma}}{\mathcal{A}}_{\gamma}=\mathrm{Im}(h|_{{\mathcal{M}}_{\gamma}})\xhookrightarrow{i_{\gamma}}{\mathcal{A}}.$

The $\Gamma$-action on ${\mathcal{M}}_{n,L}$ induces a $\Gamma$-action on
${\mathcal{M}}_{\gamma}$.

The second type of moduli spaces are associated with a cyclic Galois cover
$\pi:C^{\prime}\to C$ of the original curve given by $\gamma\in\Gamma$. We
assume $\mathrm{deg}(\pi)=\mathrm{ord}(\gamma)=m$ and $n=mr$. Let
${\mathcal{M}}_{r,L}(\pi)$ be the moduli space parameterizing rank $r$ stable
Higgs bundles $({\mathcal{E}},\theta)$ on $C^{\prime}$ such that

$\mathrm{det}(\pi_{*}{\mathcal{E}})\simeq
L,\quad\mathrm{trace}(\pi_{*}\theta)=0.$

It admits a Hitchin fibration

$h_{\pi}:{\mathcal{M}}_{r,L}(\pi)\rightarrow{\mathcal{A}}(\pi)$

with a fiberwise $\Gamma$-action; see Section 1 for more details on these
moduli spaces. From the viewpoint of representation theory, the moduli spaces
${\mathcal{M}}_{r,L}(\pi)$ are related to the study of the corresponding
endoscopic groups for $\mathrm{SL}_{n}$ over $C$ [34]. They are nonsingular
but disconnected. The Galois group

$G_{\pi}=\mathrm{Aut}(\pi)\simeq{\mathbb{Z}}/m{\mathbb{Z}}$

acts on both the source ${\mathcal{M}}_{r,L}(\pi)$ and the target
${\mathcal{A}}(\pi)$, whose quotients recover ${\mathcal{M}}_{\gamma}$ and
${\mathcal{A}}_{\gamma}$ respectively. We denote by

$q_{\mathcal{A}}:{\mathcal{A}}(\pi)\rightarrow{\mathcal{A}}_{\gamma}$

the quotient map of the base ${\mathcal{A}}(\pi)$. We also consider the
largest open subset ${\mathcal{A}}(\pi)^{*}\subset{\mathcal{A}}(\pi)$ upon
which the $G_{\pi}$-action is free, and set
$A_{\gamma}^{*}:={\mathcal{A}}(\pi)^{*}/G_{\pi}\subset A_{\gamma}$.

###  0.3. Endoscopic decompositions

In order to understand the decomposition (1) sheaf-theoretically, we consider
the canonical decompositions of the direct image complexes

$\mathrm{Rh}_{*}{\underline{{\mathbb{C}}}}\in
D_{c}^{b}({\mathcal{A}}),\quad\mathrm{Rh_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\in
D_{c}^{b}({\mathcal{A}}_{\gamma}),\quad\mathrm{Rh_{\pi}}_{*}{\underline{{\mathbb{C}}}}\in
D_{c}^{b}({\mathcal{A}}(\pi))$

into eigen-subcomplexes with respect to the $\Gamma$-actions. We first clarify
some notation before stating the main theorems. Throughout, we use
$D^{b}_{c}(-)$ to denote the bounded derived category of constructible
sheaves. We say that ${\mathcal{K}}\xrightarrow{\simeq}{\mathcal{K}}^{\prime}$
is an _isomorphism_ for two objects in a derived category if it is a quasi-
isomorphism between the complexes ${\mathcal{K}}$ and
${\mathcal{K}}^{\prime}$. Given a complex with a $\Gamma$-action and a
character $\kappa\in\hat{\Gamma}$, we denote by $(-)_{\kappa}$ the
$\kappa$-isotypic component. We call
$(-)_{\mathrm{st}}=(-)_{0\in\hat{\Gamma}}$ its _stable part_ , which is the
subcomplex fixed by the $\Gamma$-action. The Weil pairing identifies
canonically the group $\Gamma$ and its dual,

(3) $\hat{\Gamma}=\Gamma;$

see Section 1.3.

Our first result is the following theorem, which relates the stable part of
the endoscopic cohomology with the pullback of the $\kappa$-isotypic
contribution for $\mathrm{SL}_{n}$. This extends the endoscopic decomposition
of [39] in the case of $\mathrm{SL}_{n}$ from the elliptic locus to a much
larger open subset on the Hitchin base.

######  Theorem 0.2 (Theorem 3.1).

Let $\kappa\in\hat{\Gamma}$ and $\gamma\in\Gamma$ be identified by (3), let
$\pi:C^{\prime}\to C$ be the cyclic Galois cover associated with $\gamma$, and
let $d_{\gamma}=\mathrm{codim}_{{\mathcal{A}}}({\mathcal{A}}_{\gamma})$. There
are isomorphisms in $D^{b}_{c}({\mathcal{A}}(\pi)^{*})$ which are canonical up
to scaling (see Definition 2.9):

(4)
$q_{\mathcal{A}}^{*}\left(\mathrm{Rh_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}(\pi)^{*}}\xrightarrow{\simeq}\left(\mathrm{Rh_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}(\pi)^{*}}[-2d_{\gamma}]\xrightarrow{\simeq}\left(\mathrm{Rh_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}\Big{|}_{{\mathcal{A}}(\pi)^{*}}[-2d_{\gamma}]$

with the first isomorphism $G_{\pi}$-equivariant.

In the equation (4), the $G_{\pi}$-equivariant structure for the first term is
given by the pullback map along the $G_{\pi}$-quotient
$q_{\mathcal{A}}:{\mathcal{A}}(\pi)^{*}\to{\mathcal{A}}_{\gamma}^{*}$. The
$G_{\pi}$-equivariant structure for the second term is induced by the
$G_{\pi}$-action on ${\mathcal{M}}_{r,L}(\pi)$.

The following theorem is a further extension of Theorem 0.2, which provides a
complete description of the $\kappa$-isotypic component of
$\mathrm{Rh}_{*}{\underline{{\mathbb{C}}}}$ in terms of the $\gamma$-fixed
subvariety ${\mathcal{M}}_{\gamma}\subset{\mathcal{M}}_{n,L}$.

######  Theorem 0.3 (Theorem 3.2).

Let $\kappa\in\hat{\Gamma}$ and $\gamma\in\Gamma$ be identified by (3), and
let $d_{\gamma}=\mathrm{codim}_{{\mathcal{A}}}({\mathcal{A}}_{\gamma})$. We
have an isomorphism

(5)
$c_{\kappa}:\left(\mathrm{Rh}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i_{\gamma}}_{*}\left(\mathrm{Rh_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d_{\gamma}]\in
D_{c}^{b}({\mathcal{A}})$

which is canonical up to scaling.

The construction of the operator

$c_{\kappa}:\left(\mathrm{Rh}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i_{\gamma}}_{*}\left(\mathrm{Rh_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d_{\gamma}]\in
D_{c}^{b}({\mathcal{A}})$

realizing the isomorphism of Theorem 0.3 is the main theme of this paper. It
is of a geometric nature, given by a combination of algebraic correspondences
and vanishing cycle functors. Since it induces a correspondence between the
$\kappa$-part of the cohomology of an $\mathrm{SL}_{n}$-Hitchin fiber and the
$\kappa$-part of the cohomology of the corresponding endoscopic Hitchin fiber,
we call Theorems 0.2 and 0.3 the _endoscopic decomposition_ associated with
$\mathrm{SL}_{n}$ and the character $\kappa$. A major difference between
Theorem 0.3 and the work of Ngô [34] and Yun [39] is that they mainly work
with $D$-twisted Hitchin fibrations with $\mathrm{deg}(D)>2g-2$ or with just
the elliptic locus of the $K_{C}$-twisted Hitchin fibration, while we are
interested in entire space in the latter setting. The structure of the
supports of the direct image complexes is much more complicated in the
$K_{C}$-case over the total Hitchin base; see [7].

In the following, we give some applications of Theorems 0.3.

###  0.4. Structure of the cohomology of ${\mathcal{M}}_{n,L}$

Let $\kappa\in\hat{\Gamma}$ and $\gamma\in\Gamma$ be identified by (3). Let
$\pi:C^{\prime}\to C$ be the degree $m$ cyclic Galois cover associated with
$\gamma$. Assume $n=mr$. We denote by
$\widetilde{{\mathcal{M}}}^{\prime}_{r,d}$ the moduli space of stable
($\mathrm{GL}_{r}$-)Higgs bundles

$({\mathcal{E}},\theta):\quad\mathrm{rank}({\mathcal{E}})=r,\quad\mathrm{deg}({\mathcal{E}})=d$

on the curve $C^{\prime}$.

Recall the decomposition (1). The following theorem is a structural result for
$H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$.

######  Theorem 0.4 (Theorem 5.4).

The operator (5) induces a surjective morphism

$\mathfrak{p}_{\kappa}:H^{i}(\widetilde{{\mathcal{M}}}^{\prime}_{r,d},{\mathbb{C}})\twoheadrightarrow
H^{i+2d_{\gamma}}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}.$

Moreover, let
$P_{k}H^{*}(\widetilde{{\mathcal{M}}}^{\prime}_{r,d},{\mathbb{C}})$ and
$P_{k}H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$ be the perverse
filtrations defined via the Hitchin fibrations, then

$\mathfrak{p}_{\kappa}\left(P_{k}H^{i}(\widetilde{{\mathcal{M}}}^{\prime}_{r,d},{\mathbb{C}})\right)=P_{k+d_{\gamma}}H^{i+2d_{\gamma}}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}.$

We refer to [10, 5] for perverse filtrations; see also Section 5.1 for a brief
review.

If $\kappa=0$, we have $\pi=\mathrm{id}:C\xrightarrow{\simeq}C$. The operator
$\mathfrak{p}_{\kappa=0}$ in this special case recovers the restriction map

$j^{*}:H^{i}(\widetilde{{\mathcal{M}}}_{n,d},{\mathbb{C}})\twoheadrightarrow
H^{i}({\mathcal{M}}_{n,L},{\mathbb{C}})^{\Gamma}$

associated with the embedding
$j:{\mathcal{M}}_{n,L}\hookrightarrow\widetilde{{\mathcal{M}}}_{n,d}=\widetilde{{\mathcal{M}}}^{\prime}_{r,d}$.

By Markman’s theorem [31], the cohomology
$H^{*}(\widetilde{{\mathcal{M}}}^{\prime}_{r,d},{\mathbb{C}})$ is generated by
the tautological classes associated with a universal family on
$\widetilde{{\mathcal{M}}}^{\prime}_{r,d}$. Hence Theorem 0.4 shows that each
isotypic component $H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$ for
$\kappa\neq 0$ is governed by the tautological classes of a _different_ moduli
space of Higgs bundles through the operator $\mathfrak{p}_{\kappa}$. More
discussions concerning Theorem 0.4 and the P=W conjecture [5] are given in
Section 5.

###  0.5. The Hausel–Thaddeus conjecture

In [21], Hausel and Thaddeus showed that the moduli spaces of stable
$\mathrm{SL}_{n}$\- and $\mathrm{PGL}_{n}$-Higgs bundles are mirror partners
in the sense of the Strominger–Yau–Zaslow mirror symmetry. As a consequence,
these two moduli spaces should have identical Hodge numbers.

As explained in [21], the moduli space of degree $d$ stable
$\mathrm{PGL}_{n}$-Higgs bundles can be realized as the quotient
${\mathcal{M}}_{n,L}/\Gamma$ which is naturally a Deligne–Mumford stack.
Therefore, Hausel–Thaddeus conjectured that, for any two line bundles
$L,L^{\prime}$ with

$\mathrm{deg}(L)=d,~{}~{}~{}\mathrm{deg}(L^{\prime})=d^{\prime},\quad\mathrm{gcd}(d,n)=\mathrm{gcd}(d^{\prime},n)=1,$

the Hodge numbers of ${\mathcal{M}}_{n,L}$ are the same as the stringy Hodge
numbers of the stack $[{\mathcal{M}}_{n,L^{\prime}}/\Gamma]$ (twisted by a
certain gerbe $\alpha$):

(6)
$h^{i,j}({\mathcal{M}}_{n,L})=h_{\mathrm{st}}^{i,j}([{\mathcal{M}}_{n,L^{\prime}}/\Gamma],\alpha);$

see [21, Section 4] for precise definitions of the gerbe $\alpha$ and the
stringy Hodge numbers. Later, Hausel further conjectured a refinement of (6):
the Hodge numbers of $H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$
coincide with the Hodge numbers of the gerby sector
$[({\mathcal{M}}_{n,L^{\prime}})_{\gamma}/\Gamma]$. Here
$({\mathcal{M}}_{n,L^{\prime}})_{\gamma}\subset{\mathcal{M}}_{n,L^{\prime}}$
is the $\gamma$-fixed subvariety, and $\kappa$ and $\gamma$ are matched via
(3). We refer to [23, Conjecture 4.5] concerning Hodge numbers and [23,
Conjecture 5.9] for a further refinement involving perverse filtrations.

The following theorem is a direct consequence of Theorem 0.3 which proves the
Hausel–Thaddeus conjecture and a refinement of it; see the formulation (1.2.2)
and the appendix in [29] for an explanation of how the right-hand side is
equivalent to the gerby description above.

######  Theorem 0.5.

Assume that $e$ is a multiplicative inverse of $d^{\prime}$ modulo $n$. Let
$\gamma$ and $\kappa$ be matched via (3). The following identity holds in the
Grothendieck group of complex Hodge structures $K_{0}(\mathrm{HS})$:

(7)
$[P_{k}H^{i}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}]=[P_{k-d_{\gamma}}H^{i-2d_{\gamma}}(({\mathcal{M}}_{n,L^{\prime}})_{\gamma},{\mathbb{C}})_{de\kappa}(-d_{\gamma})].$

Here $(\bullet)$ stands for the Tate twist111Recall that for $V\in
K_{0}(\mathrm{HS})$, we have
$h^{i,j}(V(\bullet))=h^{i+\bullet,j+\bullet}(V)$., and each piece of the
perverse filtrations admits a natural Hodge structure by the theory of mixed
Hodge modules. In particular, for any $q\in{\mathbb{Z}}$ coprime to $n$, we
have

(8)
$[P_{k}H^{i}({\mathcal{M}}_{n,L},{\mathbb{C}})]=\sum_{\gamma\in\Gamma}[P_{k-d_{\gamma}}H^{i-2d_{\gamma}}(({\mathcal{M}}_{n,L^{\prime}})_{\gamma},{\mathbb{C}})_{q\kappa}(-d_{\gamma})]\in
K_{0}(\mathrm{HS}).$

######  Remark 0.6.

By the last paragraph of [21, Proof of Proposition 8.2], the shift

$d_{\gamma}=\mathrm{codim}_{{\mathcal{A}}}({\mathcal{A}}_{\gamma})=\frac{1}{2}\mathrm{codim}_{{\mathcal{M}}_{n,L}}({\mathcal{M}}_{\gamma})$

in (7) coincides with the “fermionic shift” $F(\gamma)$ in the formulation of
Hausel and Thaddeus.

A refined version of the Hausel–Thaddeus conjecture was previously proven by
Gröchenig–Wyss–Ziegler [17, Theorem 7.24] by p-adic integration, and was
generalized by Loeser–Wyss [29, Remark 5.3.4] by motivic integration. Note
that our refined version (7) is slightly different from the versions of [17,
29], since the right-hand side of (7) depends on the degree of $L$ while the
corresponding term in [17, 29] is independent of this degree. Instead, our
refined version is closer to the conjecture formulated by Hausel in [23,
Conjectures 4.5 and 5.9]. Motivated by the Hausel–Thaddeus conjecture,
connections between the moduli of Higgs bundles and the $\gamma$-fixed locus
with $\gamma\in\Gamma$ were discussed in [15] via Fourier–Mukai transform.

###  0.6. Idea of the proof

Our approach proceeds in two steps. We first show analogs of Theorems 0.2 and
0.3 for the moduli space ${\mathcal{M}}^{D}_{n,L}$ of $D$-twisted
$\mathrm{SL}_{n}$-Higgs bundles with $\mathrm{deg}D>2g-2$ (see Section 1 for
precise definitions). As mentioned earlier, one expects this case to be
simpler than the original setting, due to work of Chaudouard–Laumon [12] and
de Cataldo [6], which determines the supports appearing in the decomposition
theorem for the twisted Hitchin map. After proving the corresponding support
theorem for endoscopic moduli, we study the endoscopic decomposition of Ngô
[34] and Yun [39] over the elliptic locus, and extend it over the full twisted
Hitchin base.

Unfortunately, this approach is not sufficient when $D=K_{C}$, since the
supports of the Hitchin map remain mysterious [7]. Moreover, although
${\mathcal{M}}_{n,L}$ embeds inside ${\mathcal{M}}^{D}_{n,L}$ for certain
effective divisor $D$ with $\mathrm{deg}(D)>2g-2$, we cannot simply pull back
equation (5).

Instead, we realize ${\mathcal{M}}_{n,L}$ as the critical locus of a regular
function

(9) $\mu_{\pi,{\mathcal{M}}}:{\mathcal{M}}^{D}_{n,L}\to{\mathbb{A}}^{1};$

see Theorem 4.5. This allows us to express the cohomology of
${\mathcal{M}}_{n,L}$ as the vanishing cohomology of (9). In addition, since
the function $\mu_{\pi,{\mathcal{M}}}$ factors through the Hitchin base, we
can use the vanishing cycles functor to relate the decomposition theorem for
${\mathcal{M}}_{n,L}$ in terms of that for ${\mathcal{M}}^{D}_{n,L}$. By
applying this technique to the twisted version of equation (5), we obtain the
full result.

###  0.7. Relation to other work

As discussed in Section 0.5, the Hausel–Thaddeus conjecture as well as its
refinements has been proven by Gröchenig–Wyss–Ziegler [17] via p-adic
integration. Using similar approach, they also gave a new proof of Ngô’s
geometric stabilization theorem [18] which plays a crucial role in Ngô’s proof
of the fundamental lemma of the Langlands program [34]. Our approach goes
along the inverse direction — we prove the Hausel–Thaddeus conjecture by
extending Ngô’s method [34, 39] in the proof the geometric stabilization
theorem via perverse sheaves and the support theorems. This carries out the
proposal of Hausel in [23, Section 5.4]. A benefit of the sheaf-theoretic
approach is that it allows us to construct concrete geometric operators which
realize the Hausel–Thaddeus conjecture (Theorems 0.2 and 0.3) and which
provide better understanding of
$H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$ for each non-trivial
$\kappa$.

In the case of $D$-twisted Higgs bundles, our result (Theorem 3.2) removes a
technical condition of [17, Theorems 7.21] on the parity of $\mathrm{deg}(D)$.

###  0.8. Acknowledgement

We are grateful to Mark Andrea de Cataldo, Michael Gröchenig, and Zhiwei Yun
for helpful discussions. We also thank the anonymous referee for careful
reading and numerous useful suggestions. D.M. would like to thank Chris and
Martin Kratt for providing logistical support. J.S. was supported by NSF
DMS-2000726 and DMS-2134315.

##  1\. Hitchin-type moduli spaces

Throughout, we work over the complex numbers ${\mathbb{C}}$. In this section,
we fix the curve $C$ of genus $g\geq 2$, the rank $n$, and the line bundle
$L\in\mathrm{Pic}^{d}(C)$ which serves as the determinant of the Higgs bundles
as in Section 0.1. We study several Hitchin-type moduli spaces relevant to
Theorems 0.2 and 0.3.

###  1.1. $D$-Higgs bundles

For our purpose, it is important to consider generalized Higgs bundles
$({\mathcal{E}},\theta)$ with the Higgs field $\theta$ twisted by a divisor
$D$ not necessarily the canonical divisor $K_{C}$. Such flexibility also plays
a crucial role in the proof of the fundamental lemma [33, 34].

Let $D$ either be an effective divisor of degree $\mathrm{deg}(D)>2g-2$ or
$D=K_{C}$. A $D$-Higgs bundle is a pair $({\mathcal{E}},\theta)$ where
${\mathcal{E}}$ is a vector bundle and $\theta$ is a $D$-twisted Higgs field

$\theta:{\mathcal{E}}\rightarrow{\mathcal{E}}\otimes{\mathcal{O}}_{C}(D).$

We denote by $\mathrm{char}(\theta)$ the tuple of the coefficients for the
characteristic polynomial associated with $({\mathcal{E}},\theta)$,

$\mathrm{char}(\theta)=(a_{1},\dots,a_{n}),\quad
a_{i}=\mathrm{trace}(\wedge^{i}\theta)\in H^{0}(C,{\mathcal{O}}_{C}(iD)).$

Parallel to the case of $K_{C}$-Higgs bundles, the stability condition for
$D$-Higgs bundles is with respect to the slope
$\mu({\mathcal{E}})=\mathrm{deg}({\mathcal{E}})/\mathrm{rank}({\mathcal{E}})$.
By [32], there is a nonsingular quasi-projective moduli space
$\widetilde{{\mathcal{M}}}_{n,d}^{D}$ parameterizing stable $D$-Higgs bundles
of rank $n$, degree $d$, with the Hitchin map

(10)
$\widetilde{h}^{D}:\widetilde{{\mathcal{M}}}_{n,d}^{D}\rightarrow\widetilde{{\mathcal{A}}}^{D}=\oplus_{i=1}^{n}H^{0}(C,{\mathcal{O}}_{C}(iD)),\quad({\mathcal{E}},\theta)\mapsto\mathrm{char}(\theta),$

which is proper and surjective.

The moduli space of stable $\mathrm{SL}_{n}$ $D$-Higgs bundles is defined to
be the subvariety

(11)
${\mathcal{M}}^{D}_{n,L}=\\{({\mathcal{E}},\theta)\in\widetilde{{\mathcal{M}}}_{n,d}^{D};\quad\mathrm{det}({\mathcal{E}})\simeq
L,\quad\mathrm{trace}(\theta)=0\\}\subset\widetilde{{\mathcal{M}}}_{n,d}^{D}.$

It is nonsingular and irreducible by [6, Section 2.1], which has a Hitchin map
induced from the Hitchin map of the ambient space (10),

(12)
$h^{D}:{\mathcal{M}}^{D}_{n,L}\rightarrow{\mathcal{A}}^{D}=\oplus_{i=2}^{n}H^{0}(C,{\mathcal{O}}_{C}(iD)).$

It is clear that the variety ${\mathcal{M}}^{D}_{n,L}$ is the fiber over the
closed point $(L,0)$ of the smooth map

(13)
$q:\widetilde{{\mathcal{M}}}_{r,d}^{D}\rightarrow\widetilde{{\mathcal{M}}}_{1,d}^{D}=\mathrm{Pic}^{d}(C)\times
H^{0}(C,{\mathcal{O}}_{C}(D)),\quad({\mathcal{E}},\theta)\mapsto(\mathrm{det}({\mathcal{E}}),\mathrm{trace}(\theta)).$

A major difference between the $D=K_{C}$ case and the case of
$\mathrm{deg}(D)>2g-2$ is that, the Hitchin fibration for either
$\mathrm{GL}_{n}$ or $\mathrm{SL}_{n}$ is _Lagrangian_ with respect to the
canonical hyper-Kähler structure for $D=K_{C}$, while the dimension of the
Hitchin base is always larger than the dimension of a fiber in the case of
$\mathrm{deg(D)}>2g-2$. In view of the support theorems (Section 2.2), such a
difference will influence substantially the study of the topology of Hitchin
fibrations.

From now on, all $D$-Higgs bundles will be uniformly called _Higgs bundles_
for convenience.

###  1.2. Relative Hitchin moduli spaces

In this section, we study the relative Hitchin moduli space associated with a
Galois cover $\pi:C^{\prime}\to C$. This parameterizes _stable_ Higgs bundles
with respect to the endoscopic group of $\mathrm{SL}_{n}$ over $C$ attached to
a character $\kappa\in\hat{\Gamma}$; see [33, 34].

Let $C^{\prime}$ be a nonsingular curve with a cyclic Galois cover

$\pi:C^{\prime}\to C$

whose Galois group is $G_{\pi}\cong\mathbb{Z}/m{\mathbb{Z}}$. We denote the
divisor $\pi^{*}D$ by $D^{\prime}$. For an element $\sigma\in
H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(D^{\prime}))$, the pushforward
along $\pi$ gives an element

$\pi_{*}\sigma\in H^{0}(C,\pi_{*}{\mathcal{O}}_{C^{\prime}}(D^{\prime})).$

The trace of $\pi_{*}\sigma$ recovers its projection to the direct summand
component

$\mathrm{trace}(\pi_{*}\sigma)\in H^{0}(C,{\mathcal{O}}_{C}(D))\subset
H^{0}(C,\pi_{*}{\mathcal{O}}_{C^{\prime}}(D^{\prime})).$

The moduli space $\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})$ of
rank $r$ degree $d$ stable Higgs bundles on $C^{\prime}$ admits a map

(14)
$q_{\pi}:\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})\rightarrow\widetilde{{\mathcal{M}}}_{1,d}^{D}(C)$

which is the composition of (13) for the curve $C^{\prime}$ and the
pushforward

(15)
$\pi_{*}:\widetilde{{\mathcal{M}}}_{1,d}^{D^{\prime}}(C^{\prime})\rightarrow\widetilde{{\mathcal{M}}}_{1,d}^{D}(C),\quad({\mathcal{L}},\sigma)\mapsto(\mathrm{det}(\pi_{*}{\mathcal{L}}),\mathrm{trace}(\pi_{\ast}\sigma)).$

Since both (13) and (15) are smooth, the composition $q_{\pi}$ is also smooth.

We define the relative Hitchin moduli space of rank $r$ and degree $d$
associated with $\pi$ as the subvariety of
$\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})$ given as a fiber of
(14):

${\mathcal{M}}_{r,L}^{D}(\pi)=q_{\pi}^{-1}(L,0)\subset\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}).$

The variety ${\mathcal{M}}_{r,L}^{D}(\pi)$, which recovers the
$\mathrm{SL}_{n}$-Hitchin moduli space (11) when $\pi=\mathrm{id}$, is
nonsingular due to the smoothness of $q_{\pi}$.

Next we describe the Hitchin fibration associated with
${\mathcal{M}}_{r,L}^{D}(\pi)$ which generalizes (12). Recall from (10) the
$\mathrm{GL}_{r}$-Hitchin fibration
$h^{D^{\prime}}:\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})\to\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})$
associated with the curve $C^{\prime}$. The restriction of $h^{D^{\prime}}$ to
${\mathcal{M}}_{r,L}^{D}(\pi)$ induces the Hitchin map

(16) $h_{\pi}^{D}:{\mathcal{M}}_{r,L}^{D}(\pi)\to{\mathcal{A}}^{D}(\pi)$

which fits into the commutative diagram

(17)
${{\mathcal{M}}_{r,L}^{D}(\pi)}$${\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})}$${{\mathcal{A}}^{D}(\pi)}$${\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime}).}$$\scriptstyle{h_{\pi}^{D}}$$\scriptstyle{h^{D^{\prime}}}$

The Hitchin base ${\mathcal{A}}^{D}(\pi)$ can be concretely described as

(18)
${\mathcal{A}}^{D}(\pi)=H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(D^{\prime}))_{\mathrm{var}}\oplus\left(\oplus_{i=2}^{r}H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(iD^{\prime}))\right)$

where
$H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(D^{\prime}))_{\mathrm{var}}$ is
the variant part with respect to the natural Galois group $G_{\pi}$-action
induced by the $G_{\pi}$-action on $C^{\prime}$; see [20, Section 5]. Since
the line bundles ${\mathcal{O}}_{C^{\prime}}(iD^{\prime})$ have canonical
$G_{\pi}$-linearizations, there is a natural $G_{\pi}$-action on the Hitchin
base (18).

######  Proposition 1.1.

We have the following properties.

  1. (a)

The moduli space ${\mathcal{M}}_{r,L}^{D}(\pi)$ is a disjoint union of $m$
nonsingular isomorphic components

(19) ${\mathcal{M}}_{r,L}^{D}(\pi)=\bigsqcup_{i=1}^{m}M_{i}.$

  2. (b)

The restrictions of $h_{\pi}^{D}$ to all components
$h_{i}:M_{i}\to{\mathcal{A}}^{D}(\pi)$ are
${\mathcal{A}}^{D}(\pi)$-isomorphic. More precisely, for each pair $1\leq
i,j\leq m$, there exists an isomorphism
$\phi_{ij}:M_{i}\xrightarrow{\simeq}M_{j}$ induced by tensoring with a line
bundle ${\mathcal{L}}_{ij}\in\Gamma$ satisfying the commutative diagram

(20)
${M_{i}}$${M_{j}}$${{\mathcal{A}}^{D}(\pi).}$$\scriptstyle{h_{i}}$$\scriptstyle{\phi_{ij}}$$\scriptstyle{h_{j}}$

###### Proof.

Recall that ${\mathcal{M}}_{r,L}^{D}(\pi)$ is the fiber of

$q_{\pi}=(\pi_{*})\circ q$

(see (14)) over the point $(L,0)\in\widetilde{{\mathcal{M}}}_{1,d}^{D}(C)$.
The map $q$ is surjective and smooth, whose fibers are isomorphic to the
moduli of stable $\mathrm{SL}_{r}$-Higgs bundles of degree $d$ on the curve
$C^{\prime}$. In particular, each fiber of $q$ is nonsingular and irreducible.

The morphism $\pi_{*}$ given in (15) respects the product structures

(21) $\pi_{*}:\mathrm{Pic}^{d}(C^{\prime})\times
H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(D^{\prime}))\rightarrow\mathrm{Pic}^{d}(C)\times
H^{0}(C,{\mathcal{O}}_{C}(D)),$

where the morphism between the second factors form a trivial affine bundle.
For the first factors of (21), a fiber of
$\pi_{*}:\mathrm{Pic}^{d}(C^{\prime})\rightarrow\mathrm{Pic}^{d}(C)$ is
isomorphic to the degree $d$ Prym variety associated with the Galois cover
$\pi:C^{\prime}\to C$, which is the disjoint union of $m$ isomorphic abelian
varieties; see [21, Section 7]. Hence the moduli space
${\mathcal{M}}_{r,L}^{D}(\pi)$ has $m$ nonsingular connected components.

Assume $n=mr$. Tensoring with a line bundle
${\mathcal{L}}\in\Gamma=\mathrm{Pic}^{0}(C)[n]$ induces an
${\mathcal{A}}^{D}(\pi)$-automorphism

(22)
$\phi_{\mathcal{L}}:{\mathcal{M}}_{r,L}^{D}(\pi)\xrightarrow{\simeq}{\mathcal{M}}_{r,L}^{D}(\pi),\quad\phi_{\mathcal{L}}({\mathcal{E}},\theta)=({\mathcal{L}}\otimes{\mathcal{E}},\theta).$

Moreover, for a general point $a\in{\mathcal{A}}^{D}(\pi)$ corresponding to a
degree $n=rm$ spectral cover

$g_{a}:C^{\prime}_{a}\to C^{\prime}\xrightarrow{\pi}C,$

the fiber ${(h_{\pi}^{D})}^{-1}(a)$ is identical to a fiber of the morphism

${g_{a}}_{*}:\mathrm{Pic}^{d}(C^{\prime}_{a})\rightarrow\mathrm{Pic}^{d}(C),\quad{\mathcal{L}}\mapsto\mathrm{det}({g_{a}}_{*}{\mathcal{L}}),$

where $\Gamma$ acts transitively on the set of its connected components
(_c.f._ [20, Lemmas 2.1 and 2,2]). This ensures that $\Gamma$ acts
transitively on $\\{M_{i}\\}_{1\leq i\leq m}$. In particular, for any pair
$1\leq i,j\leq m$, there exits a line bundle ${\mathcal{L}}_{ij}\in\Gamma$
such that the isomorphism $\phi_{ij}=\phi_{{\mathcal{L}}_{ij}}$ given in (22)
satisfies the commutative diagram (20). (a) and (b) are proved. ∎

###  1.3. Weil pairing and cyclic covers

Recall from Section 0.1 that the group $\Gamma=\mathrm{Pic}^{0}(C)[n]$ acts on
the $\mathrm{SL}_{n}$-moduli space ${\mathcal{M}}^{D}_{n,L}$ via tensor
product. For $\gamma\in\Gamma$, the $\gamma$-fixed subvariety
${\mathcal{M}}^{D}_{\gamma}\subset{\mathcal{M}}^{D}_{n,L}$ carries an induced
Hitchin map

${\mathcal{M}}^{D}_{\gamma}\xrightarrow{h^{D}_{\gamma}}{\mathcal{A}}^{D}_{\gamma}=\mathrm{Im}(h^{D}|_{{\mathcal{M}}_{\gamma}})\xhookrightarrow{i^{D}_{\gamma}}{\mathcal{A}}^{D}.$

As indicated by Theorem 0.3, the cohomology of ${\mathcal{M}}^{D}_{\gamma}$ is
related to a $\kappa$-isotypic component of the cohomology of
${\mathcal{M}}^{D}_{n,L}$ with respect to the $\Gamma$-action.

In order to describe this connection, we need a correspondence (3) between an
element $\gamma\in\Gamma$ and a character $\kappa\in\hat{\Gamma}$, which we
review in the following.

Let $\mu_{n}\subset{\mathbb{C}}^{*}$ denote the finite group of the $n$-th
roots of unity. We have the _Weil pairing_ on the group of $n$-torsion points
of $\mathrm{Pic}^{0}(C)$,

$\langle~{}~{},~{}~{}\rangle_{\Gamma}:\Gamma\times\Gamma\rightarrow\mu_{n}.$

Under the identification

$\mathrm{Pic}^{0}(C)[n]=H_{1}(C,{\mathbb{Z}}/n{\mathbb{Z}}),$

the Weil pairing coincide with the intersection pairing on
$H_{1}(C,{\mathbb{Z}}/n{\mathbb{Z}})$. In particular, it is non-degenerate,
which induces a character

$\kappa=\langle\gamma,~{}~{}\rangle_{\Gamma}:\Gamma\rightarrow\mu_{n}\subset{\mathbb{C}}^{*}$

for each $\gamma\in\Gamma$. This gives the identification (3).

We also note that an element $\gamma\in\Gamma$ naturally corresponds to a
cyclic Galois cover of $C$ whose degree is the order of $\gamma\in\Gamma$. In
fact, for fixed $\gamma\in\Gamma$, we denote by $L_{\gamma}$ the $n$-torsion
line bundle associated with $\gamma$, and let $m$ be its order which divides
$n$. Taking the $m$-th roots of unity fiberwise inside the total space of
$L_{\gamma}$ with respect to the zero section $C$, we obtain a cyclic Galois
cover

$\pi:C^{\prime}\rightarrow C$

with the Galois group $G_{\pi}\simeq\mathbb{Z}/m{\mathbb{Z}}$. Conversely,
every degree $m$ étale cyclic Galois cover arises this way.

###  1.4. Characters

In this section we give a concrete description for the character $\kappa$ in
terms of the Prym variety associated with the corresponding Galois cover
$\pi:C^{\prime}\to C$.

As before, we assume that $\pi:C^{\prime}\to C$ is a degree $m$ cyclic Galois
cover associated with $\kappa\in\hat{\Gamma}$ as in Section 1.3, and we assume
that $n=mr$. The character $\kappa\in\hat{\Gamma}$ matches with
$\gamma\in\Gamma$ via (3). We consider the Prym variety

(23)
$\mathrm{Prym}(C^{\prime}/C)=\mathrm{Ker}\left(\mathrm{det}(\pi_{*}-):\mathrm{Pic}^{0}(C^{\prime})\to\mathrm{Pic}^{0}(C)\right)$

with the component group

$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))\simeq{\mathbb{Z}}/m{\mathbb{Z}}.$

For an $n$-torsion line bundle ${\mathcal{L}}\in\Gamma$, the projection
formula yields

$\mathrm{det}(\pi_{*}\pi^{*}{\mathcal{L}})={\mathcal{L}}^{\otimes
m}\in\mathrm{Pic}^{0}(C)[r].$

In particular, the line bundle $\pi^{*}{\mathcal{L}}^{\otimes
r}\in\mathrm{Pic}^{0}(C^{\prime})$ represents a point in
$\mathrm{Prym}(C^{\prime}/C))$, which yields a natural group homomorphism

(24)
$\Gamma\rightarrow\pi_{0}(\mathrm{Prym}(C^{\prime}/C)),\quad{\mathcal{L}}\mapsto[\pi^{*}{\mathcal{L}}^{\otimes
r}]\in\pi_{0}(\mathrm{Prym}(C^{\prime}/C)).$

The morphism (24) admits a factorization

$\Gamma=\mathrm{Pic}^{0}(C)[n]\xrightarrow{[r]}\mathrm{Pic}^{0}(C)[m]\rightarrow\pi_{0}(\mathrm{Prym}(C^{\prime}/C)).$

The first map above is the multiplication by $r$ which is clearly surjective.
The second map sends ${\mathcal{L}}\in\mathrm{Pic}^{0}(C)[m]$ to the
equivalent class of the line bundle
$\pi^{*}{\mathcal{L}}\in\mathrm{Prym}(C^{\prime}/C)$, and its surjectivity is
given by the proof of [20, Theorem 1.1 (2)]. Hence (24) is surjective.

Recall that $\gamma\in\Gamma$ is of order $m$, so

$\gamma\in\mathrm{Pic}^{0}(C)[m]\subset\Gamma.$

The following lemma is obtained by viewing the Weil pairing on
$\mathrm{Pic}^{0}(C)[i]$ for any $i\in{\mathbb{N}}$, via Poincaré duality, as
the intersection pairing

$H_{1}(C,{\mathbb{Z}}/i{\mathbb{Z}})\times
H_{1}(C,{\mathbb{Z}}/i{\mathbb{Z}})\rightarrow{\mathbb{Z}}/i{\mathbb{Z}}\simeq\mu_{i}.$

######  Lemma 1.2.

Assume $\gamma^{\prime}\in\Gamma$. Let
$\langle~{}~{},~{}~{}\rangle_{\mathrm{Pic}^{0}(C)[m]}$ denotes the Weil
pairing on $\mathrm{Pic}^{0}(C)[m]$, and we view $r\gamma^{\prime}$ naturally
as an element in $\mathrm{Pic}^{0}(C)[m]$. We have

$\langle\gamma,\gamma^{\prime}\rangle_{\Gamma}=\langle\gamma,r\gamma^{\prime}\rangle_{\mathrm{Pic}^{0}(C)[m]}.$

######  Proposition 1.3.

The character $\kappa\in\hat{\Gamma}$ (corresponding to $\pi:C^{\prime}\to C$)
factors through a character of $\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$ of order
$m$ via the morphism (24),

$\kappa:\Gamma\rightarrow\pi_{0}(\mathrm{Prym}(C^{\prime}/C))\left(\simeq{\mathbb{Z}}/m{\mathbb{Z}}\right)\to{\mathbb{C}}^{*}.$

###### Proof.

Recall from Section 1.3 that the character $\kappa$ is given by the Weil
pairing $\langle\gamma,~{}~{}~{}\rangle_{\Gamma}$ where $\gamma\in\Gamma$
corresponds to $\pi$. Since $\gamma$ is of order $m$, _i.e.,_

$\gamma\in\mathrm{Pic}^{0}(C)[m]\subset\Gamma,$

Lemma 1.2 implies for any element $\gamma^{\prime}\in\Gamma$ that

(25)
$\kappa(\gamma^{\prime})=\langle\gamma,\gamma^{\prime}\rangle_{\Gamma}=\langle\gamma,r\gamma^{\prime}\rangle_{\mathrm{Pic}^{0}(C)[m]}.$

We consider the subgroup

$K=\mathrm{Ker}\left\\{\pi^{*}:\mathrm{Pic}^{0}(C)\to\mathrm{Pic}^{0}(C^{\prime})\right\\}\subset\mathrm{Pic}^{0}(C).$

It is the cyclic finite subgroup $\langle\gamma\rangle$ of
$\mathrm{Pic}^{0}(C)$ generated by $\gamma$. Since $\gamma$ is of order $m$,
we have $K\subset\mathrm{Pic}^{0}(C)[m]$. By the proof of [20, Theorem 1.1
(1)], there is a canonical isomorphism

(26) $\pi_{0}(\mathrm{Prym}(C^{\prime}/C))=\hat{K},$

which, for any ${\mathcal{L}}\in\mathrm{Pic}^{0}(C)[m]$, identifies
$[\pi^{*}{\mathcal{L}}]\in\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$ with the
character of $K$ sending the generator $\gamma\in K$ to

$\langle\gamma,{\mathcal{L}}\rangle_{\mathrm{Pic}^{0}(C)[m]}\in\mu_{m}\subset{\mathbb{C}}^{*}.$

Comparing with (25), this implies that $\kappa:\Gamma\to{\mathbb{C}}^{*}$ is
the composition of (24) and the order $m$ character of
$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$ given by

$\gamma\in
K=\mathrm{Hom}({\pi_{0}(\mathrm{Prym}(C^{\prime}/C))},{\mathbb{C}}^{*}).$

Here we dualize (26) in the last identity. This completes the proof. ∎

Since ${\pi_{0}(\mathrm{Prym}(C^{\prime}/C))}\cong{\mathbb{Z}}/m{\mathbb{Z}}$,
its character group is also cyclic. We obtain the following corollary of
Proposition 1.3.

######  Corollary 1.4.

A character $\rho:\Gamma\to{\mathbb{C}}^{*}$ lies in the cyclic subgroup
$\langle\kappa\rangle\subset\hat{\Gamma}$ if and only if $\rho$ is induced by
a character of $\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$ factoring through (24).

Now we consider the kernel of the morphism (24), which we denote by

(27) $\Omega\subset\Gamma.$

The subgroup $\Omega$ acts on the moduli space ${\mathcal{M}}^{D}_{r,L}(\pi)$
via tensor product.

######  Lemma 1.5.

The $\Omega$-action on ${\mathcal{M}}^{D}_{r,L}(\pi)$ preserves each connected
component $M_{i}$ of (19).

###### Proof.

Recall that ${\mathcal{M}}_{r,L}^{D}(\pi)$ is a fiber of

$q_{\pi}=(\pi_{*})\circ
q:\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})\xrightarrow{q}\widetilde{{\mathcal{M}}}_{1,d}^{D^{\prime}}(C^{\prime})\xrightarrow{\pi_{*}}\widetilde{{\mathcal{M}}}_{1,d}^{D}(C).$

The fibers of the first map are connected. The second map respects the product
structure (21). Both the maps $q$ and $\pi_{*}$ are $\Gamma$-equivariant.
Hence the $m$ connected components of (19) matches the connected components of
the degree $d$ Prym variety

$\mathrm{Prym}^{d}(C^{\prime}/C)=(\pi_{*})^{-1}(L),\quad\pi_{*}:\mathrm{Pic}^{d}(C^{\prime})\to\mathrm{Pic}^{d}(C).$

Since $\mathrm{Prym}^{d}(C^{\prime}/C)$ is a torsor of
$\mathrm{Prym}(C^{\prime}/C)$ (defined in (23)), and by definition the group
$\Omega$ acts trivially on the component group
$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$, we obtain that the action of $\Omega$
preserves each connected component of $\mathrm{Prym}^{d}(C^{\prime}/C)$. The
proposition then follows from the fact that the restriction of $q$ to
${\mathcal{M}}^{D}_{r,d}(\pi)$

$q|_{(\pi_{*})^{-1}(L,0)}:{\mathcal{M}}^{D}_{r,L}(\pi)\to(\pi_{*})^{-1}(L,0)=\mathrm{Prym}^{d}(C^{\prime}/C)\times
H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(D^{\prime}))_{\mathrm{var}}$

is $\Omega$-equivariant. ∎

###  1.5. Endoscopic moduli spaces and $\gamma$-fixed loci

In this section, we connect the $\gamma$-fixed subvariety
${\mathcal{M}}^{D}_{\gamma}$ to the relative Hitchin moduli spaces introduced
in Section 1.2.

We fix $\gamma\in\Gamma$ of order $m$. Let $\pi:C^{\prime}\rightarrow C$ be
the cyclic Galois cover with the Galois group
$G_{\pi}\simeq\mathbb{Z}/m{\mathbb{Z}}$ corresponding to $\gamma$ as in
Section 1.3. Assume $n=mr$. We consider the relative Hitchin moduli space
${\mathcal{M}}^{D}_{r,L}(\pi)$ with the Hitchin fibration

(28)
$h_{\pi}^{D}:{\mathcal{M}}^{D}_{r,L}(\pi)\rightarrow{\mathcal{A}}^{D}(\pi).$

By [21, Proposition 7.1], the Galois group $G_{\pi}$ acts freely on the
relative moduli space ${\mathcal{M}}^{D}_{r,L}(\pi)$ whose quotient recovers
the $\gamma$-fixed subvariety ${\mathcal{M}}^{D}_{\gamma}$. The group
$G_{\pi}$ also acts on the base ${\mathcal{A}}^{D}(\pi)$ with the Hitchin map
(28) $G_{\pi}$-equivariant. In summary, we have the following commutative
diagrams

(29)
${{\mathcal{M}}^{D}_{r,L}(\pi)}$${{\mathcal{M}}_{\gamma}^{D}}$${{\mathcal{M}}^{D}_{n,L}}$${{\mathcal{A}}^{D}{(\pi)}}$${{\mathcal{A}}^{D}_{\gamma}}$${{\mathcal{A}}^{D}}$$\scriptstyle{q_{\mathcal{M}}}$$\scriptstyle{h^{D}_{\pi}}$$\scriptstyle{h^{D}_{\gamma}}$$\scriptstyle{h^{D}}$$\scriptstyle{q_{\mathcal{A}}}$$\scriptstyle{i_{\gamma}}$

where $q_{\mathcal{M}}$ and $q_{\mathcal{A}}$ are the quotient maps with
respect to the natural actions of the Galois group $G_{\pi}$. Let
${\mathcal{A}}^{D}(\pi)^{*}$ be the largest open subset of
${\mathcal{A}}^{D}(\pi)$ upon which the $G_{\pi}$-action is free. Then the
left diagram is Cartesian after restricting to

${\mathcal{A}}^{D}(\pi)^{*}\xrightarrow{~{}~{}~{}q_{\mathcal{A}}~{}~{}~{}}{{\mathcal{A}}_{\gamma}^{D}}^{*}:={\mathcal{A}}^{D}(\pi)^{*}/G_{\pi}.$

######  Remark 1.6.

We see from the description (18) that ${\mathcal{A}}^{D}(\pi)^{*}$ is non-
empty. In fact, it suffices to find vectors in the affine space
${\mathcal{A}}^{D}(\pi)$ that are not fixed by any nontrivial element of the
cyclic group $G_{\pi}$. We consider the pushforward of the structure sheaf
${\mathcal{O}}_{C^{\prime}}$ along the Galois cover $\pi:C^{\prime}\to C$,
which admits a splitting
$\pi_{*}{\mathcal{O}}_{C^{\prime}}=\oplus_{\chi}L_{\chi}$ where $\chi$ runs
through all characters of $G_{\pi}$ and $L_{\chi}$ is a degree 0 line bundle
corresponding to $\chi$. The projection formula yields

$\pi_{*}{\mathcal{O}}_{C^{\prime}}(iD^{\prime})=\pi_{*}\pi^{*}{\mathcal{O}}_{C}(iD)=\bigoplus_{\chi}{\mathcal{O}}_{C}(iD)\otimes
L_{\chi},\quad\forall i\geq 1.$

In particular, for _any_ character $\chi$ of $G_{\pi}$, we have

(30)
$H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(iD^{\prime}))_{\chi}=H^{0}(C,{\mathcal{O}}_{C}(iD)\otimes
L_{\chi})\neq 0$

where we used the Riemann–Roch formula. The non-emptiness of
${\mathcal{A}}^{D}(\pi)^{*}$ follows from (30) and (18).

######  Lemma 1.7.

The direct image complex

$\mathrm{Rh^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\in
D^{b}_{c}({\mathcal{A}}^{D}(\pi))$

is $G_{\pi}$-equivariant, and we have a canonical isomorphism

$\left({q_{\mathcal{A}}}_{*}{\mathrm{Rh}^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)^{G_{\pi}}\xrightarrow{\simeq}{\mathrm{Rh}^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\in
D^{b}_{c}({\mathcal{A}}^{D}_{\gamma}).$

###### Proof.

Since the trivial local system on ${\mathcal{M}}^{D}_{r,L}(\pi)$ is
$G_{\pi}$-equivariant, the pushforward
${q_{\mathcal{M}}}_{*}{\underline{{\mathbb{C}}}}$ along the quotient map
$q_{\mathcal{M}}$ admits a natural $G_{\pi}$-action with a canonical
isomorphism

(31)
$\left({q_{\mathcal{M}}}_{*}{\underline{{\mathbb{C}}}}\right)^{G_{\pi}}\xrightarrow{\simeq}{\underline{{\mathbb{C}}}}\in
D^{b}_{c}({\mathcal{M}}^{D}_{\gamma}).$

The map $h^{D}_{\pi}$ is $G_{\pi}$-equivalent, therefore we obtain in
$D^{b}_{c}({\mathcal{A}}^{D}_{\gamma})$ that

$\left({q_{\mathcal{A}}}_{*}\mathrm{Rh^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)^{G_{\pi}}=\left(\mathrm{Rh^{D}_{\gamma}}_{*}{q_{\mathcal{M}}}_{*}{\underline{{\mathbb{C}}}}\right)^{G_{\pi}}=\mathrm{Rh^{D}_{\gamma}}_{*}\left({q_{\mathcal{M}}}_{*}{\underline{{\mathbb{C}}}}\right)^{G_{\pi}}\xrightarrow{\simeq}\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}$

where the last isomorphism is induced by (31). ∎

##  2\. Support theorems for Hitchin moduli spaces

###  2.1. Supports

Let $f:X\to Y$ be a proper morphism between nonsingular quasi-projective
varieties. The decomposition theorem by Bernstein, Beilinson, Deligne, and
Gabber [3] implies that the direct image complex
$\mathrm{Rf}_{\ast}{\underline{{\mathbb{C}}}}$ is (non-canonically) isomorphic
to a direct sum of shifted simple perverse sheaves,

(32)
$\mathrm{Rf}_{*}{\underline{{\mathbb{C}}}}\simeq\bigoplus_{Z_{i}}\mathrm{IC}_{Z_{i}}(L_{i})[d_{i}]\in
D^{b}_{c}(Y),$

where $d_{i}\in{\mathbb{Z}}$, $Z_{i}\subset Y$ is an irreducible subvariety,
and $L_{i}$ is a local system on an open subset $U_{i}$ of $Z_{i}$. Every
$Z_{i}$ above is called a _support_ of $f:X\to Y$. We say that a direct
summand ${\mathcal{F}}$ of the object

$\mathrm{Rf}_{*}{\underline{{\mathbb{C}}}}={\mathcal{F}}\oplus{\mathcal{F}}^{\prime}$

has _full support_ if each perverse constituent
$\mathrm{IC}_{Z_{i}}(L_{i})[d_{i}]$ of ${\mathcal{F}}$ has support $Z_{i}=Y$.

In this section, we analyze the supports of various Hitchin fibrations
introduced in Section 1 when $\mathrm{deg}(D)>2g-2$.

###  2.2. Support theorems

For the Hitchin fibration $h^{D}$ (resp. $h^{D}_{\pi}$), we define the
elliptic locus of the Hitchin bases ${\mathcal{A}}^{D}$ (resp.
${\mathcal{A}}^{D}(\pi)$), denoted by ${\mathcal{A}}^{D,\mathrm{ell}}$ (resp.
${\mathcal{A}}^{D,\mathrm{ell}}(\pi))$, to be the open subset consisting of
integral spectral curves.

Following the methods of Ngô [34] and Chaudouard–Laumon [12], de Cataldo
showed in [6] that all the supports for the $\mathrm{SL}_{n}$-Hitchin
fibration (12) are governed by the elliptic locus
${\mathcal{A}}^{D,\mathrm{ell}}\subset{\mathcal{A}}^{D}$ when
$\mathrm{deg}(D)>2g-2$.

######  Theorem 2.1 ([6, Theorem 1.0.2]).

Let $D$ be an effective divisor on $C$ of degree $\mathrm{deg}(D)>2g-2$. Then
the generic points of the supports of

$h^{D}:{\mathcal{M}}^{D}_{n,L}\to{\mathcal{A}}^{D}$

are contained in ${\mathcal{A}}^{D,\mathrm{ell}}\subset{\mathcal{A}}^{D}$.

Now we consider the $\Gamma$-action on the moduli space
${\mathcal{M}}^{D}_{n,L}$. This action is fiberwise with respect to the
Hitchin map $h^{D}$, which induces a canonical decomposition of the direct
image complex

(33)
$\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}=\bigoplus_{\kappa\in\hat{\Gamma}}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\in
D^{b}_{c}({\mathcal{A}}^{D});$

see [28, Lemma 3.2.5]. We define the _stable part_
$\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}$ as
the component in (33) corresponding to the trivial character
$\kappa=0\in\hat{\Gamma}$.

Combining Theorem 2.1 with Ngô’s support theorems over the elliptic loci [34,
Theorems 7.8.3 and 7.8.5], we obtain the following complete description of the
supports for every $\kappa$-part of (33).

######  Corollary 2.2.

The only support of
$\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}$ is
${\mathcal{A}}^{D}_{\gamma}$ where $\gamma$ corresponds to $\kappa$ via (3).

The trivial character $0\in\hat{\Gamma}$ corresponds to $0\in\Gamma$, and
${\mathcal{A}}^{D}_{0}={\mathcal{A}}^{D}$. Hence as a special case of
Corollary 2.2, the stable part
$\left(\mathrm{Rh^{D}_{*}}{\mathbb{C}}\right)_{\mathrm{st}}$ has full support
${\mathcal{A}}^{D}$.

We fix a character $\kappa\in\hat{\Gamma}$ of order $m$. Let
$\pi:C^{\prime}\to C$ be the cyclic Galois cover associated with $\kappa$ as
in Section 1.3. Assume $n=mr$. By the projection formula, we have

$\mathrm{det}\left(\pi_{*}(\pi^{*}{\mathcal{L}}\otimes{\mathcal{E}})\right)=\mathrm{det}\left({\mathcal{L}}\otimes\pi_{*}{\mathcal{E}}\right)={\mathcal{L}}^{\otimes
mr}\otimes\mathrm{det}(\pi_{*}{\mathcal{E}})=\mathrm{det}(\pi_{*}{\mathcal{E}})$

for ${\mathcal{L}}\in\Gamma=\mathrm{Pic}^{0}(C)[n]$ and ${\mathcal{E}}$ a rank
$r$ vector bundle on $C^{\prime}$. Therefore, the group $\Gamma$ acts on the
moduli space ${\mathcal{M}}^{D}_{r,L}(\pi)$ via tensor product

${\mathcal{L}}\cdot({\mathcal{E}},\theta)=(\pi^{*}{\mathcal{L}}\otimes{\mathcal{E}},\theta),\quad{\mathcal{L}}\in\Gamma.$

We have similarly a $\kappa$-decomposition as (33) for the Hitchin fibration
(16) associated with ${\mathcal{M}}^{D}_{r,L}(\pi)$,

$\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}=\bigoplus_{\kappa\in\hat{\Gamma}}\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\in
D^{b}_{c}({\mathcal{A}}^{D}(\pi)).$

The main result of this section is to prove a support theorem for the Hitchin
map (16) associated with $\pi:C^{\prime}\to C$.

######  Theorem 2.3.

Let $D$ be an effective divisor on $C$ of degree $\mathrm{deg}(D)>2g-2$.
Assume that the degree $m$ Galois cover $\pi:C^{\prime}\to C$ is associated
with $\kappa\in\hat{\Gamma}$ via (3). Assume $n=mr$. Then we have the
following concerning the supports of the Hitchin map
$h_{\pi}^{D}:{\mathcal{M}}_{r,L}^{D}(\pi)\to{\mathcal{A}}^{D}(\pi)$.

  1. (a)

The generic points of the supports of
${\mathrm{Rh^{D}_{\pi}}}_{*}{\underline{{\mathbb{C}}}}$ are contained in the
elliptic locus ${\mathcal{A}}^{D,\mathrm{ell}}(\pi)$.

  2. (b)

The stable part
$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\mathbb{C}}\right)_{\mathrm{st}}$ has full
support ${\mathcal{A}}^{D}(\pi)$.

The validity of support theorems (Theorems 2.1 and 2.3) are the main
ingredients in the proof of the endoscopic decomposition in the twisted case
of $\mathrm{deg}(D)>2g-2$ when $\mathrm{deg}(D)$ is even; see the proof of
Theorem 3.3.

###  2.3. Weak abelian fibrations

We recall the notion of weak abelian fibration introduced in [34], which is
modelled on the properties of Hitchin fibrations.

We follow closely the exposition of [6, Section 2.6]. Let $A$ be an
irreducible nonsingular variety. Assume that

$h:M\to A,\quad g:P\to A$

are morphisms of the same relative dimensions
$\mathrm{dim}(h)=\mathrm{dim}(g)$ satisfying the following properties:

  1. (a)

The map $g:P\to A$ is a smooth commutative group scheme with connected fibers;

  2. (b)

The map $h$ is proper, and $M$ is nonsingular;

  3. (c)

The group scheme $P$ acts on $M$ fiberwise with affine stabilizers for every
geometric point of $M$.

We call a triple $(M,A,P)$ as above a _weak abelian fibration_ , if the Tate
module

$T_{\overline{{\mathbb{Q}}}_{l}}(P)=\mathrm{R^{2d_{g}-1}g}_{!}\overline{{\mathbb{Q}}}_{l}(d_{g})$

as an $l$-adic sheaf is polarizable; see [6, Section 2.6] the paragraph
following Lemma 2.6.2.

Over a closed point $a\in A$, we consider the Chevalley decomposition for the
restricted group scheme $P_{a}$,

$0\to P_{a}^{\mathrm{aff}}\to P_{a}\to P_{a}^{\mathrm{ab}}\to 0,$

where $P_{a}^{\mathrm{aff}}$ is the maximal connected affine linear subgroup
of $P_{a}$ and $P_{a}^{\mathrm{ab}}$ is an abelian variety.

We recall in the following the support inequality of Ngô [34].

######  Theorem 2.4 ([34, Theorem 7.2.2]).

Let (M,A,P) be a weak abelian fibration. Assume the irreducible subvariety
$Z\subset A$ is a support of $h:M\to A$, then

(34) $\mathrm{dim}(h)-\mathrm{dim}(A)+\mathrm{dim}(Z)\geq
d_{Z}^{\mathrm{ab}}(P).$

Here $d_{Z}^{\mathrm{ab}}(P)=\mathrm{dim}(P_{a}^{\mathrm{ab}})$ with $a\in Z$
a general point.

###  2.4. Hitchin fibrations

Following [12, 6], we show that the Hitchin fibration

(35) $h_{\pi}^{D}:{\mathcal{M}}_{r,L}^{D}(\pi)\to{\mathcal{A}}^{D}(\pi)$

associated with $\pi:C^{\prime}\to C$ admits the structure of a weak abelian
fibration.

Recall the commutative diagram (17). The $\mathrm{GL}_{r}$-Hitchin base
$\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})$ parameterizes spectral
curves in the total space $V(D^{\prime})$ of the line bundle
${\mathcal{O}}_{C^{\prime}}(D^{\prime})$. We assume that

(36) ${\mathcal{C}}\to\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})$

is the universal spectral curve. Let
$g_{\mathcal{C}}:\mathrm{Pic}^{0}_{\mathcal{C}}\to\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})$
be the relative degree 0 Picard scheme associated with (36), which acts on
$\widetilde{{\mathcal{M}}}^{D^{\prime}}_{r,d}(C^{\prime})$ via tensor product.
The following result is obtained in [12]; see also [6, Section 3] for a
detailed review.

######  Proposition 2.5 ([12]).

The triple

(37)
$\left(\widetilde{{\mathcal{M}}}^{D^{\prime}}_{r,d}(C^{\prime}),\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime}),\mathrm{Pic}^{0}_{\mathcal{C}}\right),\quad\widetilde{h}^{D^{\prime}}:\widetilde{{\mathcal{M}}}^{D^{\prime}}_{r,d}(C^{\prime})\rightarrow\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime}),\quad
g_{\mathcal{C}}:\mathrm{Pic}^{0}_{\mathcal{C}}\to\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})$

is a weak abelian fibration.

In order to study (35), we “fix the determinant” after pushing forward along
$\pi:C^{\prime}\to C$. Since the relative Hitchin moduli space
${\mathcal{M}}_{r,L}^{D}(\pi)$ is a closed fiber of (14), we consider the
morphism of ${\mathcal{A}}^{D}(\pi)$-group schemes

$N_{\pi}:\mathrm{Pic}^{0}_{\mathcal{C}}\times_{\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})}{\mathcal{A}}^{D}(\pi)\to\mathrm{Pic}^{0}(C)\times{\mathcal{A}}^{D}(\pi)$

given by composition of the ${\mathcal{A}}^{D}(\pi)$-morphisms

$\mathrm{Pic}^{0}_{\mathcal{C}}\times_{\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})}A^{D}(\pi)\to\mathrm{Pic}^{0}(C^{\prime})\times{\mathcal{A}}^{D}(\pi)\rightarrow\mathrm{Pic}^{0}(C)\times{\mathcal{A}}^{D}(\pi).$

Here the first map is the restriction of the norm map $N_{p}$ ([6, (42)]) to
$A^{D}(\pi)$, and the second map is

$({\mathcal{L}},a)\mapsto(\mathrm{det}(\pi_{*}{\mathcal{L}}),a).$

By the discussion of [6, Section 4.1], the map $N_{\pi}$ is smooth. We let

(38) $g_{\pi}:P\rightarrow{\mathcal{A}}^{D}(\pi)$

be the kernel of $N_{\pi}$, and we let the group scheme
$g^{0}_{\pi}:P^{0}\to{\mathcal{A}}^{D}(\pi)$ be the identity component of $P$.
The nonsingular group schemes $P$ and $P^{0}$ act on
${\mathcal{M}}^{D}_{r,L}(\pi)$ induced by the
$\mathrm{Pic}^{0}_{\mathcal{C}}$-action on
$\widetilde{{\mathcal{M}}}^{D^{\prime}}_{r,d}(C^{\prime})$.

######  Proposition 2.6.

The triple

(39) $\left({\mathcal{M}}^{D}_{r,L}(\pi),A^{D}(\pi),P^{0}\right),\quad
h_{\pi}^{D}:{\mathcal{M}}^{D}_{r,L}(\pi)\to A^{D}(\pi),\quad
g^{0}_{\pi}:P^{0}\to A^{D}(\pi)$

is a weak abelian fibration.

###### Proof.

The weak abelian fibration structure for (39) is essentially inherited from
that for $(\ref{GL_weak})$. The proof is parallel to [6, Section 4]. Here we
summarize some necessary minor modifications.

It is clear that (a) and (c) in Section 2.3 follow from the construction. We
need to verify (b), and show that the Tate module associated with
$g_{\pi}:P\to{\mathcal{A}}^{D}(\pi)$ is polarizable.

#### (i) Affine stabilizers.

For a closed point in
${\mathcal{M}}^{D}_{r,L}(\pi)\subset\widetilde{{\mathcal{M}}}^{D^{\prime}}_{r,d}(C^{\prime})$,
its stabilizer with respect to the $P^{0}$-action on
${\mathcal{M}}^{D}_{r,L}(\pi)$ is a subgroup of the corresponding stabilizer
with respect the $\mathrm{Pic}^{0}_{\mathcal{C}}$-action on
$\widetilde{{\mathcal{M}}}^{D^{\prime}}_{r,d}(C^{\prime})$, whose affineness
follows from the fact that (37) is a weak abelian fibration.

#### (ii) Polarizability of the Tate module.

This follows from the proof of [6, Theorem 4.7.2]. In fact, for a closed point
$a\in{\mathcal{A}}^{D}(\pi)$, the Tate module of the abelian part
$P^{\mathrm{ab}}_{a}$ is an orthogonal direct summand component of
$T_{\overline{{\mathbb{Q}}}_{l}}(\mathrm{Pic}^{0,\mathrm{ab}}_{{\mathcal{C}},a})$
with respect to the non-degenerate Tate-Weil pairing on
$T_{\overline{{\mathbb{Q}}}_{l}}(\mathrm{Pic}^{0,\mathrm{ab}}_{{\mathcal{C}},a})$.
Hence the restriction of the Tate-Weil pairing on
$T_{\overline{{\mathbb{Q}}}_{l}}(\mathrm{Pic}^{0,\mathrm{ab}}_{{\mathcal{C}},a})$
to $T_{\overline{{\mathbb{Q}}}_{l}}(P^{\mathrm{ab}}_{a})$ is non-degenerate. ∎

By [12, Section 9] (see also [6, Section 5.2]), the $\mathrm{GL}_{r}$-Hitchin
base admits a stratification

$\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})=\bigsqcup_{\underline{m},\underline{n}}\widetilde{{\mathcal{A}}}_{\underline{m},\underline{n}}$

with $\underline{m}=(m_{1},m_{2},\dots,m_{s})$,
$\underline{n}=(n_{1},n_{2},\dots,n_{s})$ satisfying

  1. (a)

$n_{i}\geq n_{i+1}$ for any $i$;

  2. (b)

$m_{i}\geq m_{i+1}$ whenever $n_{i}=n_{i+1}$;

  3. (c)

$\sum_{i=1}^{s}m_{i}n_{i}=r$.

Each $\widetilde{{\mathcal{A}}}_{\underline{m},\underline{n}}$ is a locally
closed subset formed by spectral curves of the topological type
$(\underline{m},\underline{n})$:

(40) $\widetilde{{\mathcal{A}}}_{\underline{m},\underline{n}}=\left\\{E\subset
V(D^{\prime}):~{}~{}~{}E=\sum_{i}m_{i}E_{i},~{}~{}~{}E_{i}\subset
V(D^{\prime})\right\\}$

where $V(D^{\prime})$ is the total space of
${\mathcal{O}}_{C^{\prime}}(D^{\prime})$ and $E_{i}$ is an integral spectral
curve of degree $n_{i}$ over $C^{\prime}$. The stratification (40) induces a
stratification on
${\mathcal{A}}^{D}(\pi)\subset{\mathcal{A}}^{D^{\prime}}(C^{\prime})$,

${\mathcal{A}}^{D}(\pi)=\bigsqcup_{\underline{m},\underline{n}}{{\mathcal{A}}}(\pi)_{\underline{m},\underline{n}},\quad{{\mathcal{A}}}(\pi)_{\underline{m},\underline{n}}={\mathcal{A}}^{D}(\pi)\cap\widetilde{{\mathcal{A}}}_{\underline{m},\underline{n}}.$

We have the following multi-variable inequality.

######  Proposition 2.7 (_c.f._ [6, Corollary 5.4.4]).

Let $Z\subset{\mathcal{A}}^{D}(\pi)$ be an irreducible subvariety whose
general points are of the type $(\underline{m},\underline{n})$. Then we have

(41)
$d^{\mathrm{ab}}_{Z}\left(\mathrm{Pic}^{0}_{\mathcal{C}}\right)\geq\sum_{i}\left({d_{\widetilde{h}^{D^{\prime}}_{n_{i}}(C^{\prime})}}-d_{\widetilde{{\mathcal{A}}}^{D^{\prime}}_{n_{i}}(C^{\prime})}\right)+\mathrm{dim}(Z)+\left(\mathrm{deg}(D)-g+1\right).$

Here ${d_{\widetilde{h}^{D^{\prime}}_{n_{i}}(C^{\prime})}}$ and
$d_{\widetilde{{\mathcal{A}}}^{D^{\prime}}_{n_{i}}(C^{\prime})}$ are the
dimensions of a fiber and the base respectively of the
$\mathrm{GL}_{n_{i}}$-Hitchin fibration (10) associated with the curve
$C^{\prime}$ and the divisor $D^{\prime}=\pi^{*}D$, and

$d^{\mathrm{ab}}_{Z}\left(\mathrm{Pic}^{0}_{\mathcal{C}}\right)=\mathrm{dim}\left((\mathrm{Pic}^{0}_{{\mathcal{C}},a})^{\mathrm{ab}}\right)$

with $a\in Z$ a general point.

###### Proof.

When $\pi=\mathrm{id}$, the subspace

(42)
${\mathcal{A}}^{D}(\mathrm{id})=\\{\mathrm{char}(\theta)\in\widetilde{{\mathcal{A}}}^{D}:~{}~{}~{}\mathrm{trace}(\theta)=0\\}\subset\widetilde{{\mathcal{A}}}^{D}$

coincides with the $\mathrm{SL}_{n}$-Hitchin base, and the inequality (41) is
the $\delta$-inequality for $\mathrm{SL}_{n}$ proven in [6, Corollary 5.4.4
(76)]. As explained in the last paragraph of [6, Proof of Corollary 5.4.4],
the main ingredient in the proof of enhancing the $\delta$-inequality for
$\mathrm{GL}_{n}$ to that for $\mathrm{SL}_{n}$ is [6, Theorem 5.4.2], which
asserts that the restriction of the $\delta$-regular weak abelian fibrations
to their elliptic loci in the sub-Hitchin base (42) remain $\delta$-regular.
This follows from the product structure [6, (74)] for the group schemes
associated with the spectral curves, which says that the variations of the
group schemes associated with the spectral curves are trivial along the
$H^{0}(C,{\mathcal{O}}_{C}(D))$-direction.

Now for a general $\pi:C^{\prime}\to C$ with $D^{\prime}=\pi^{*}D^{\prime}$ as
we consider here, we have the canonical decomposition for the Hitchin base

(43) ${\mathcal{A}}^{D}(\pi)\oplus H=\widetilde{{\mathcal{A}}}^{D^{\prime}}$

with $H=H^{0}(C,{\mathcal{O}}_{C}(D))$ a direct summand component of
$H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(D^{\prime}))$. Applying the
product structure [6, (74)] to the curve $C^{\prime}$ and the divisor
$D^{\prime}$, we obtain an analogous product structure for (43), that the
variations of the group schemes associated with the spectral curves are
trivial along the $H$-direction. Hence (41) holds in the relative case
$\pi:C^{\prime}\to C$ by the same reason as for $\mathrm{SL}_{n}$. ∎

###  2.5. Proof of Theorem 2.3 (a)

The proof of Theorem 2.3 (a) is parallel to the proofs of the main theorems in
[12, 6], which we provide in the following for the reader’s convenience. The
crucial ingredient is to combine Ngô’s support inequality (34) and the multi-
variable inequality (41).

We assume $Z\subset{\mathcal{A}}^{D}(\pi)$ is an irreducible support of (35)
whose general points have types $(\underline{m},\underline{n})$. By Theorem
2.4, Proposition 2.6, and Proposition 2.7, we have

$\begin{split}\mathrm{dim}({h^{D}_{\pi}})-&\mathrm{dim}({A^{D}(\pi)})+\mathrm{dim}(Z)\geq
d^{\mathrm{ab}}_{Z}\left(\mathrm{Pic}^{0}_{\mathcal{C}}\right)-g\\\
&\geq\sum_{i}\left({d_{\widetilde{h}^{D^{\prime}}_{n_{i}}(C^{\prime})}}-d_{\widetilde{{\mathcal{A}}}^{D^{\prime}}_{n_{i}}(C^{\prime})}\right)+\mathrm{dim}(Z)+(\mathrm{deg}(D)-g+1)-g\end{split}$

where we use
$d^{\mathrm{ab}}(P^{0}_{a})=d^{\mathrm{ab}}(\mathrm{Pic}^{0}_{{\mathcal{C}},a})-g$.
Hence

(44)
$\mathrm{dim}({h^{D}_{\pi}})-\mathrm{dim}({{\mathcal{A}}^{D}(\pi)})\geq\sum_{i=1}^{s}\left({d_{\widetilde{h}^{D^{\prime}}_{n_{i}}(C^{\prime})}}-d_{\widetilde{{\mathcal{A}}}^{D^{\prime}}_{n_{i}}(C^{\prime})}\right)+(\mathrm{deg}(D)-2g+1).$

Here we recall that $s$ is the number of irreducible components of the
spectral curve. We now apply the dimension formulas for
$\mathrm{GL}_{*}$-Hitchin fibrations [6, Section 6.1] to compute both sides of
(44). The left-hand side is equal to:

$\displaystyle\mathrm{LHS}$
$\displaystyle=\left({d_{\widetilde{h}^{D^{\prime}}_{r}(C^{\prime})}}-d_{\widetilde{{\mathcal{A}}}^{D^{\prime}}_{r}(C^{\prime})}\right)+\left(\mathrm{dim}H^{0}(C,{\mathcal{O}}_{C}(D))-g\right)$
$\displaystyle=\left(-r\mathrm{deg}(D^{\prime})+2r(g^{\prime}-1)+1\right)+\left(\mathrm{deg}(D)-2g+1\right)$

where $g^{\prime}$ is the genus of $C^{\prime}$ and we used the last formula
of [6, (77)]. Similarly for the right-hand side we have:

$\displaystyle\mathrm{RHS}$
$\displaystyle=\sum_{i=1}^{s}\left(-n_{i}\mathrm{deg}(D^{\prime})+2n_{i}(g^{\prime}-1)+1\right)+\left(\mathrm{deg}(D)-2g+1\right)$
$\displaystyle=\left((-\sum_{i=1}^{s}n_{i})\left((\mathrm{deg}(D^{\prime})-2(g^{\prime}-1)\right)+s\right)+\left(\mathrm{deg}(D)-2g+1\right).$

In particular, (44) implies that

(45)
$1-s\geq\left(r-\sum_{i}n_{i}\right)\left(\mathrm{deg}(D^{\prime})-(2g^{\prime}-2)\right).$

Since

$\mathrm{deg}(D^{\prime})-(2g^{\prime}-2)=\mathrm{deg}(\pi)(\mathrm{deg}(D)-(2g-2))>0$

by the assumption on $D$, the inequality (45) forces that $s=1$ and $m_{1}=1$.
This implies that the generic point of $Z$ lies in
${\mathcal{A}}^{D,\mathrm{ell}}(\pi)$, which completes the proof of Theorem
2.3 (a). ∎

###  2.6. Proof of Theorem 2.3 (b)

Due to Theorem 2.3 (a), it suffices to prove (b) over the elliptic locus with
respect to the restricted Hitchin map

(46)
$h^{\mathrm{ell}}_{\pi}=h_{\pi}^{D}|_{{\mathcal{A}}^{D,\mathrm{ell}}(\pi)}:{\mathcal{M}}^{D,\mathrm{ell}}_{r,L}(\pi)\to{\mathcal{A}}^{D,\mathrm{ell}}(\pi).$

Recall the group scheme (38) of the relative Prym variety. By the support
theorem [34, Propositions 7.2.2 and 7.2.3], we only need to show that the
constructible sheaf of the top degree cohomology

(47)
$\left(\mathrm{R^{2d_{h_{\pi}^{D}}}h^{\mathrm{ell}}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}},\quad
d_{h_{\pi}^{D}}=\mathrm{dim}(h_{\pi}^{D})$

is the trivial local system on the elliptic locus
${\mathcal{A}}^{D,\mathrm{ell}}(\pi)$.

It is clear that (47) contains the trivial local system given by the sum of
point classes for all irreducible components of the fibers of (46). Hence it
suffices to prove that the stable part of the degree $2d_{h_{\pi}^{D}}$
cohomology group is 1-dimensional for each fiber of (46).

Assume $a\in{\mathcal{A}}^{D,\mathrm{ell}}(\pi)$. Let $C^{\prime}_{a}$ be the
corresponding integral spectral curve with the spectral cover

(48) $C^{\prime}_{a}\xrightarrow{p_{a}}C^{\prime}\xrightarrow{\pi}C.$

The Hitchin fiber ${\mathcal{M}}^{D}_{r,d}(\pi)_{a}$ contains a Zariski dense
open subset

${\mathcal{M}}^{\mathrm{reg}}_{a}\subset{\mathcal{M}}^{D}_{r,L}(\pi)_{a}$

parameterizing line bundles on the spectral curve $C^{\prime}_{a}$, which is a
torsor of the group scheme $P_{a}$. Hence ${\mathcal{M}}^{D}_{r,L}(\pi)_{a}$
has $|\pi_{0}(P_{a})|$ irreducible components.

We fix a base point in $x\in{\mathcal{M}}_{a}^{\mathrm{reg}}$. Since
${\mathcal{M}}_{a}^{\mathrm{reg}}$ is a torsor of the group scheme $P_{a}$,
the base point $x$ trivializes the torsor and therefore the
$\pi_{0}(P_{a})$-action on $x$ yields an isomorphism

$[x]:\pi_{0}({\mathcal{M}}^{\mathrm{reg}}_{a})\xrightarrow{\simeq}\pi_{0}(P_{a}),$

which further induces

(49)
$H^{2d_{h_{\pi}^{D}}}\left({\mathcal{M}}^{D}_{r,L}(\pi)_{a},{\mathbb{C}}\right)=\bigoplus_{v\in\pi_{0}(P_{a})}{\mathbb{C}}v.$

The action of $P_{a}$ on the cohomology
$H^{*}\left({\mathcal{M}}^{D}_{r,L}(\pi)_{a},{\mathbb{C}}\right)$ factors
through the group $\pi_{0}(P_{a})$ of connected components, which acts
naturally on the right-hand side of (49). In particular, the action of the
discrete subgroup $\Gamma\subset P_{a}$ on (49) factors through the natural
action of $\pi_{0}(P_{a})$. By the proof of [20, Theorem 1.1 (2)], the
morphism

$\Gamma=\mathrm{Pic}^{0}(C)[n]\twoheadrightarrow\pi_{0}(P_{a})$

induced by the pullback $p_{a}^{*}\circ\pi^{*}$ along (48) is a surjection.
Therefore we obtain that

$H^{2d_{h_{\pi}^{D}}}\left({\mathcal{M}}^{D}_{r,L}(\pi)_{a},{\mathbb{C}}\right)_{\mathrm{st}}=H^{2d_{h_{\pi}^{D}}}\left({\mathcal{M}}^{D}_{r,L}(\pi)_{a},{\mathbb{C}}\right)^{\Gamma}\subseteq
H^{2d_{h_{\pi}^{D}}}\left({\mathcal{M}}^{D}_{r,L}(\pi)_{a},{\mathbb{C}}\right)^{\pi_{0}(P_{a})}={\mathbb{C}},$

where the last equality is given by the $\pi_{0}(P_{a})$-equivariant
isomorphism (49). This implies

(50)
$\left(\mathrm{R^{2d_{h_{\pi}^{D}}}h^{\mathrm{ell}}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}={\underline{{\mathbb{C}}}}$

and completes the proof of (b). ∎

######  Remark 2.8.

The vector space

$H^{2d_{h_{\pi}^{D}}}\left({\mathcal{M}}^{D}_{r,L}(\pi)_{a},{\mathbb{C}}\right)_{\mathrm{st}}=H^{2d_{h_{\pi}^{D}}}\left({\mathcal{M}}^{D}_{r,L}(\pi)_{a},{\mathbb{C}}\right)^{\Gamma}$

may fail to be 1-dimensional when
$a\in{\mathcal{A}}^{D}(\pi)\setminus{\mathcal{A}}^{D,\mathrm{ell}}(\pi)$. In
particular, the constructible sheaf
$\left(\mathrm{R^{2d_{h_{\pi}^{D}}}h^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}$
is not a rank 1 local system over the total Hitchin base
${\mathcal{A}}^{D}(\pi)$. Hence the proof of Theorem 2.3 (b) relies heavily on
the support theorem — Theorem 2.3 (a).

###  2.7. Transfer from the $\kappa$-part to the stable part

In Section 2.7, we assume that $D$ is an effective divisor of degree
$\mathrm{deg}(D)>2g-2$ or $D=K_{C}$. Our main purpose is to prove Proposition
2.10 which allows us to transfer naturally from the $\kappa$-part to the
stable part of the complex
$\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}$. This extends [39,
Proposition 2.3.2] to the total Hitchin base for certain endoscopic Hitchin
moduli spaces associated with $\mathrm{SL}_{n}$. We note that this transfer
does not rely on the support theorem.

Recall the decomposition (19) of Proposition 1.1 (a). By Lemma 1.5, the group
$\Omega$ (introduced in (27)) acts on each direct image complex
${\mathrm{Rh}_{i}}_{*}{\underline{{\mathbb{C}}}}$, and we consider its
$\Omega$-invariant part

$({\mathrm{Rh}_{i}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}\in
D_{c}^{b}({\mathcal{A}}^{D}(\pi)).$

For any pair $1\leq i,j\leq m$, the isomorphism of Proposition 1.1 (b):

$\phi_{ij}=\phi_{{\mathcal{L}}_{ij}}:M_{i}\xrightarrow{\simeq}M_{j},\quad({\mathcal{E}},\phi)\mapsto({\mathcal{E}}\otimes{\mathcal{L}}_{ij},\theta)$

induced by a line bundle ${\mathcal{L}}_{ij}\in\Gamma$ yields an isomorphism

$\phi_{ij}^{*}:{\mathrm{Rh}_{j}}_{*}{\underline{{\mathbb{C}}}}\xrightarrow{\simeq}{\mathrm{Rh}_{i}}_{*}{\underline{{\mathbb{C}}}}.$

It preserves the $\Omega$-invariant parts,

(51)
$\phi_{ij}^{*}:({\mathrm{Rh}_{j}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}\xrightarrow{\simeq}({\mathrm{Rh}_{i}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}.$

We note that the isomorphism (51) only depends on the class of the line bundle
${\mathcal{L}}_{ij}\in\Gamma$ in the quotient group

$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))=\Gamma/\Omega.$

Hence the $\Gamma$-action on
$\bigoplus_{j=1}^{m}({\mathrm{Rh}_{j}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}$
passes through $\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$. Since the group
$\Omega$ preserves each component $M_{i}$, it follows from Proposition 1.1 (b)
that the elements of the cyclic group
$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))=\Gamma/\Omega$ acts transitively on the
set $\\{M_{i}\\}_{i=1}^{m}$. We may view
$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$ as the group of connected components of
${\mathcal{M}}^{D}_{r,L}(\pi)$. For any fixed $1\leq i_{0}\leq m$, the
isomorphisms (51) yields a canonical
$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$-equivariant isomorphism

(52)
$\bigoplus_{j=1}^{m}({\mathrm{Rh}_{j}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}=(\mathrm{Rh_{i_{0}}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}\otimes\left(\bigoplus_{v\in\pi_{0}(\mathrm{Prym}(C^{\prime}/C))}{\mathbb{C}}v\right)$

where the action on the right-hand side is the natural one.

Before stating Proposition 2.10, we introduce the following definition for
convenience.

######  Definition 2.9.

Let $X$ be an algebraic variety, and let
${\mathcal{F}}_{1},{\mathcal{F}}_{2}\in D^{b}_{c}(X)$ be two objects. We say
that two morphisms

$f_{1}:{\mathcal{F}}_{1}\to{\mathcal{F}}_{2},\quad
f_{2}:{\mathcal{F}}_{1}\to{\mathcal{F}}_{2}$

are _equivalent up to scaling_ , if there exists $\lambda\in{\mathbb{C}}^{*}$
such that $f_{1}=\lambda f_{2}$. We say that there is an isomorphism between
two objects ${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{2}$:

$f:{\mathcal{F}}_{1}\to{\mathcal{F}}_{2}$

which is _canonical up to scaling_ , if our construction induces a set of
isomorphisms $f_{i}:{\mathcal{F}}_{1}\xrightarrow{\simeq}{\mathcal{F}}_{2}$
which are all equivalent up to scaling.

######  Proposition 2.10.

Let $D$ be an effective divisor on $C$ of degree $\mathrm{deg}(D)>2g-2$ or
$D=K_{C}$. Assume that $\pi:C^{\prime}\to C$ is the degree $m$ Galois cover
associated with $\gamma\in\Gamma$, which corresponds to the character
$\kappa\in\hat{\Gamma}$ via (3). Assume $n=mr$. Then for any two elements
$\kappa_{1},\kappa_{2}$ in the cyclic group
$\langle\kappa\rangle\subset\hat{\Gamma}$ generated by $\kappa$, there is an
isomorphism for the corresponding isotypic components:

(53)
$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa_{1}}=\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa_{2}}$

which is canonical up to scaling. In particular, (53) induces an isomorphism
which is canonical up to scaling:

$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}=\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}.$

###### Proof.

We consider the $\Omega$-invariant part

(54)
$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)^{\Omega}\in
D_{c}^{b}({\mathcal{A}}^{D}(\pi))$

of the direct image complex $\mathrm{Rh_{\pi}^{D}}_{*}{\mathbb{C}}$. On one
hand, the group $\Omega$ acts on each complex
${\mathrm{Rh}_{i}}_{*}{\mathbb{C}}$ and we have

(55)
$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)^{\Omega}=\bigoplus_{j=1}^{m}({\mathrm{Rh}_{j}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}.$

On the other hand, by Corollary 1.4, an isotypic component
$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa^{\prime}}$
contributes to (54) if and only if $\kappa^{\prime}$ lies in
$\langle\kappa\rangle$. Hence

(56)
$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)^{\Omega}=\bigoplus_{\kappa^{\prime}\in\langle\kappa\rangle}\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa^{\prime}}.$

Combining (52), (55), and (56), we obtain a natural
$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$-equivariant isomorphism

$\bigoplus_{\kappa^{\prime}\in\langle\kappa\rangle}\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa^{\prime}}=(\mathrm{Rh_{i_{0}}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}\otimes\left(\bigoplus_{v\in\pi_{0}(\mathrm{Prym}(C^{\prime}/C))}{\mathbb{C}}v\right).$

In particular, if we take the $\kappa^{\prime}$-parts on both sides, since the
$\kappa^{\prime}$-part of the regular representation is one-dimensional, this
yields a natural isomorphism up to scaling

(57)
$f_{i_{0},\kappa^{\prime}}:\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa^{\prime}}\xrightarrow{\simeq}(\mathrm{Rh_{i_{0}}}_{*}{\underline{{\mathbb{C}}}})^{\Omega}.$

This gives an isomorphism (53) up to scaling, which, _a priori_ , still
depends on the choice of $1\leq i_{0}\leq m$.

Different choices of $i_{0}$ influence the isomorphism (53) via the action of
an element

$g\in\Gamma/\Omega=\pi_{0}(\mathrm{Prym}(C^{\prime}/C))$

on both objects of (57). After isolating the $\kappa^{\prime}$-isotypic
component, we conclude this only changes the isomorphism (53) by a possible
scalar ambiguity. ∎

###  2.8. Changing the degree

Assume $\mathrm{deg}(D)>2g-2$. As another application of Ngô’s support
theorem, we analyze the $G_{\pi}$-equivariant complex

(58)
$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\in
D_{c}^{b}({\mathcal{A}}^{D}(\pi))$

when the degree of the line bundle $L\in\mathrm{Pic}^{d}(C)$ is changed.

Due to Theorem 2.3 (b) and Proposition 2.10, the object (58) has full support
${\mathcal{A}}^{D}(\pi)$. Hence (58) is completely determined by its
restriction to the open subset $U^{\mathrm{sm}}\subset{\mathcal{A}}^{D}(\pi)$
where the spectral curves are nonsingular.

Ngô’s analysis of supports for direct image complexes [34, Section 7] works
for each $\kappa$-part (see [34, Proposition 7.2.3]). In particular, as a
corollary of the “freeness” [34, Proposition 7.4.10], the isomorphism class of
the restriction of (58) to $U^{\mathrm{sm}}$ is determined by the group scheme
$P^{0}|_{U^{\mathrm{sm}}}$ of (39) and the constructible sheaf

(59)
$\left(\mathrm{R^{2d_{h_{\pi}^{D}}}h^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{U^{\mathrm{sm}}}\in\mathrm{Sh}_{c}(U^{\mathrm{sm}})$

which are both equipped with the $G_{\pi}$-actions. See [11, Appendix] for a
precise form expressing (58) in terms of the direct image complex associated
with

$g^{0}_{\pi}\Big{|}_{U^{\mathrm{sm}}}:P^{0}|_{U^{\mathrm{sm}}}\rightarrow
U^{\mathrm{sm}}$

and (59).

The following proposition will only be used in Section 5.5.

######  Proposition 2.11.

Assume $\mathrm{deg}(D)>2g-2$. Let $q$ be an integer coprime to $n$. We have
an isomorphism of the $G_{\pi}$-equivariant objects

(60)
$\left(\mathrm{Rh_{\pi,L}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{q\kappa}\simeq\left(\mathrm{Rh_{\pi,L^{\otimes
q}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\in
D_{c}^{b}({\mathcal{A}}^{D}(\pi)).$

Here $h_{\pi,L^{\otimes q}}^{D}:{\mathcal{M}}^{D}_{r,L^{\otimes
q}}(\pi)\to{\mathcal{A}}^{D}(\pi)$ stands for the Hitchin fibration associated
with the line bundle $L^{\otimes q}$.

###### Proof.

For notational convenience, we use $e$ to denote $2d_{h_{\pi}^{D}}$. After
restricting to $U^{\mathrm{sm}}$ we have

(61) $\left(\mathrm{R^{e}h_{\pi,L^{\otimes
q}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa^{\prime}}\simeq{\underline{{\mathbb{C}}}},\quad\quad\forall\kappa^{\prime}\in\langle\kappa\rangle$

by Proposition 2.10 and (50). We need to analyze the $G_{\pi}$-equivariant
structure on the rank 1 trivial local systems (61).

Now we consider the constructible sheaf

(62) $\mathrm{R^{e}h_{\pi,L^{\otimes
q}}^{D}}_{*}{\underline{{\mathbb{C}}}}\Big{|}_{U^{\mathrm{sm}}}\in\mathrm{Sh}_{c}(U^{\mathrm{sm}})$

with the $G_{\pi}$-equivariant structure.

By Proposition 1.1 (b), the sheaf (62) is a trivial local system of rank $m$
(corresponding to the $m$ connected components of
${\mathcal{M}}^{D}_{r,L^{\otimes q}}(\pi)$). We may view (62) as an
$m$-dimensional vector space $V_{q}\simeq{\mathbb{C}}^{m}$ on which the cyclic
groups $G_{\pi}$ and $\Gamma$ act. Therefore, to prove (60) we only need to
show that the isotypic component $(V_{q})_{\kappa}$ is $G_{\pi}$-equivariantly
isomorphic to $(V_{1})_{q\kappa}$.

Recall the degree $dq$ Prym variety $\mathrm{Prym}^{dq}(C^{\prime}/C)$
associated with the line bundle $L^{\otimes q}$, whose $m$ connected
components are identified with the $m$ connected components of
${\mathcal{M}}^{D}_{r,L^{\otimes q}}(\pi)$. To connect $V_{q}$ and $V_{1}$, we
consider the the “multiplication by $q$” map:

(63)
$\mathrm{mult}_{q}:\mathrm{Prym}^{d}(C^{\prime}/C)\to\mathrm{Prym}^{dq}(C^{\prime}/C),\quad{\mathcal{L}}\mapsto{\mathcal{L}}^{\otimes
q}$

which is clearly $G_{\pi}$-equivariant.

We note that $\mathrm{mult}_{q}$ induces an identification of the $m$
connected components for the Prym varieties on both sides of (63). In fact,
choosing base points $x\in\mathrm{Prym}^{q}(C^{\prime}/C)$ and
$qx\in\mathrm{Prym}^{dq}(C^{\prime}/C)$ trivializes both
$\mathrm{Prym}(C^{\prime}/C)$-torsors, and the map $\mathrm{mult}_{q}$ induces
a “multiplication by $q$” map on the cyclic group
$\pi_{0}(\mathrm{Prym}(C^{\prime}/C))\simeq{\mathbb{Z}}/m{\mathbb{Z}}$. The
claim follows from the fact that $\mathrm{gcd}(m,q)=1$.

In conclusion, (63) induces a $G_{\pi}$-equivariant isomorphism

$[\mathrm{mult}_{q}]:V_{1}\xrightarrow{\simeq}V_{q}$

whose $\Gamma$-action on the right-hand side is given by the $\Gamma$-action
on the left-hand side composed with the “multiplication by $q$”
$\Gamma\xrightarrow{\cdot q}\Gamma$. In particular we have a
$G_{\pi}$-equivariant isomorphism between $(V_{q})_{\kappa}$ and
$(V_{1})_{q\kappa}$. This completes the proof of the proposition. ∎

The constraint $\mathrm{deg}(D)>2g-2$ will be removed by Remark 4.9, despite
the fact that we no longer have full supports in that case.

##  3\. Endoscopic decompositions

###  3.1. Overview: main results

In Sections 3 and 4, we establish a generalized version of Theorem 0.3 for any
effective divisor $D$ with $\mathrm{deg}(D)>2g-2$ or $D=K_{C}$.

Let $D$ be as above, and let $\pi:C^{\prime}\to C$ be a degree $m$ cyclic
Galois cover with $n=mr$. Recall the Hitchin fibrations

$h^{D}:{\mathcal{M}}^{D}_{n,L}\to{\mathcal{A}}^{D},\quad
h_{\pi}^{D}:{\mathcal{M}}^{D}_{r,L}(\pi)\rightarrow{\mathcal{A}}^{D}(\pi),$

the fiberwise $\Gamma$-actions, and the corresponding $\kappa$-decompositions.
The Galois group $G_{\pi}$ acts naturally on ${\mathcal{M}}^{D}_{r,d}(\pi)$
and ${\mathcal{A}}^{D}(\pi)$ such that the Hitchin map $h_{\pi}^{D}$ is
$G_{\pi}$-equivariant; see Section 1.5. By Lemma 1.7, the direct image complex
$\mathrm{Rh_{\pi}^{D}}_{*}{\mathbb{C}}$ is $G_{\pi}$-equivariant, so is each
$\kappa$-isotypic part

$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\in
D^{b}_{c}({\mathcal{A}}^{D}(\pi))$

due to the commutativity of the $\Gamma$\- and the $G_{\pi}$-actions. We also
note that

$q_{\mathcal{A}}^{*}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\in
D^{b}_{c}({\mathcal{A}}^{D}(\pi))$

is naturally $G_{\pi}$-equivariant induced by the pullback map from the
$G_{\pi}$-quotient

$q_{\mathcal{A}}:{\mathcal{A}}^{D}(\pi)\rightarrow{{\mathcal{A}}^{D}_{\gamma}}.$

Recall the open subsets ${\mathcal{A}}^{D}(\pi)^{*}$ and
${{\mathcal{A}}^{D}_{\gamma}}^{*}$ for the Hitchin bases and the free
$G_{\pi}$-quotient map between them from Section 1.5.

The following theorem is a generalization of Theorem 0.2.

######  Theorem 3.1.

Let $\kappa\in\hat{\Gamma}$ and $\gamma\in\Gamma$ be identified by (3), let
$\pi:C^{\prime}\to C$ be the Galois cover associated with $\gamma$, and let
$d^{D}_{\gamma}=\mathrm{codim}_{{\mathcal{A}}^{D}}({\mathcal{A}}^{D}_{\gamma})$.
There are isomorphisms in $D^{b}_{c}({\mathcal{A}}^{D}(\pi)^{*})$ which are
canonical up to scaling:

(64)
$q_{\mathcal{A}}^{*}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}\xrightarrow{\simeq}\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}[-2d^{D}_{\gamma}]\xrightarrow{\simeq}\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}[-2d^{D}_{\gamma}]$

with the first isomorphism $G_{\pi}$-equivariant.

The second isomorphism of (64) is obtained immediately from Proposition 2.10,
which actually holds over the total space ${\mathcal{A}}^{D}(\pi)$:

$\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]\xrightarrow{\simeq}\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}[-2d^{D}_{\gamma}].$

The following theorem is a sheaf-theoretic enhancement of the Hausel–Thaddeus
conjecture.

######  Theorem 3.2.

Let
$i^{D}_{\gamma}:{\mathcal{A}}^{D}_{\gamma}\hookrightarrow{\mathcal{A}}^{D}$ be
the closed embedding. With the same notation as in Theorem 3.1, there is an
isomorphism which is canonical up to scaling:

(65)
$c^{D}_{\kappa}:\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]\in
D_{c}^{b}({\mathcal{A}}^{D}).$

We first observe that (65) induces the first isomorphism of (64). So Theorem
3.1 is recovered by Theorem 3.2. In fact, we restrict $c^{D}_{\kappa}$ to the
open subset
${{\mathcal{A}}_{\gamma}^{D}}^{*}\subset{\mathcal{A}}_{\gamma}^{D}$ and pull
it back along the free $G_{\pi}$-quotient map

(66)
$q_{\mathcal{A}}:{\mathcal{A}}^{D}(\pi)^{*}\to{{\mathcal{A}}^{D}_{\gamma}}^{*}.$

Since the left diagram of (29) is Cartesian restricting to (66), we recover
the first map of (64) via proper base change.

Theorems 3.1 and 3.2 recovers Theorems 0.2 and 0.3 when $D=K_{C}$. When
$\mathrm{deg}(D)>2g-2$, Theorem 3.2 provide a concrete description of the
contribution of each support ${\mathcal{A}}^{D}_{\gamma}$ to

$\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\in
D_{c}^{b}({\mathcal{A}}^{D}).$

This enhances the main theorem of de Cataldo [6].

As discussed above, for proving Theorems 3.1 and 3.2, we only need to
construct $G_{\pi}$-equivariant isomorphisms

$c^{D}_{\kappa}:\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]\in
D_{c}^{b}({\mathcal{A}}^{D}),$

which we treat in this section for the following special cases.

######  Theorem 3.3.

Theorems 3.1 and 3.2 hold when $\mathrm{deg}(D)$ is even and greater than
$2g-2$.

In Section 4, we reduce the other cases of Theorem 3.2, which includes the
possibly most interesting case $D=K_{C}$, to Theorem 3.3.

###  3.2. Spectral curves and line bundles

Recall the universal spectral curve (36) for $\mathrm{GL}_{r}$-Higgs bundles
over the curve $C^{\prime}$. We denote its restriction to the subspace
${\mathcal{A}}^{D}(\pi)\subset\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})$
by

(67) ${\mathcal{C}}_{\pi}\rightarrow{\mathcal{A}}^{D}(\pi).$

The morphism (67) is $G_{\pi}$-equivariant with respect to the natural Galois
group $G_{\pi}$ actions on both the base ${\mathcal{A}}^{D}(\pi)$ and the
universal curve ${\mathcal{C}}_{\pi}$.

We consider the largest Zariski open subset

${\mathcal{A}}^{\heartsuit}(\pi)\subset A^{D}(\pi)$

such that: 333We note that our notation ${\mathcal{A}}^{\heartsuit}(\pi)$ has
different meaning as the similar notation used in [34].

  1. (a)

The action of $G_{\pi}$ is free on ${\mathcal{A}}^{\heartsuit}(\pi)$;

  2. (b)

The restriction of the spectral curves (67)

(68)
${\mathcal{C}}^{\heartsuit}_{\pi}\rightarrow{\mathcal{A}}^{\heartsuit}(\pi)$

is smooth.

By taking the $G_{\pi}$-quotients, the family (68) descends to a family of
nonsingular curves

(69)
${\mathcal{C}}^{\heartsuit}_{\gamma}\rightarrow{\mathcal{A}}^{\heartsuit}_{\gamma}$

where
${\mathcal{A}}^{\heartsuit}_{\gamma}={\mathcal{A}}^{\heartsuit}(\pi)/G_{\pi}$
is an open dense subset of
${\mathcal{A}}^{D}_{\gamma}={\mathcal{A}}^{D}(\pi)/G_{\pi}$. We denote by

(70)
${\mathcal{C}}^{\circ}_{\gamma}\rightarrow{\mathcal{A}}^{\heartsuit}_{\gamma}$

the restriction of the universal $\mathrm{SL}_{n}$-spectral curves over
${\mathcal{A}}^{D}$ to ${\mathcal{A}}^{\heartsuit}_{\gamma}$. The families
(69) and (70) are connected by the following lemma.

######  Lemma 3.4.

There is a natural ${\mathcal{A}}^{\heartsuit}_{\gamma}$-morphism

(71)
${{\mathcal{C}}^{\heartsuit}_{\gamma}}$${{\mathcal{C}}^{\circ}_{\gamma}}$${{\mathcal{A}}_{\gamma}^{\heartsuit}}$$\scriptstyle{u_{\mathcal{C}}}$

whose restriction to each closed fiber

$u_{a}:{\mathcal{C}}^{\heartsuit}_{\gamma,a}\rightarrow{\mathcal{C}}^{\circ}_{\gamma,a},\quad
a\in{\mathcal{A}}^{\heartsuit}_{\gamma}$

is a normalization of curves.

###### Proof.

We first recall the construction of [20, Section 5.1] that, for a given degree
$r$ spectral curve $C^{\prime}_{\alpha}\rightarrow C^{\prime}$ lying in
$V(D^{\prime})$, there is a natural birational morphism

$C^{\prime}_{\alpha}\to C_{\alpha}$

with $C_{\alpha}$ a degree $n$ spectral curve over $C$ lying in the total
space $V(D)$. In fact, given $C^{\prime}_{\alpha}\to C^{\prime}$, let

$g^{*}C^{\prime}_{\alpha}\rightarrow C^{\prime}$

be another degree $r$ spectral cover over $C^{\prime}$ obtained as the
pullback of $C^{\prime}_{\alpha}\rightarrow C^{\prime}$ along the Galois
automorphism

$g:C^{\prime}\xrightarrow{\simeq}C^{\prime},\quad g\in G_{\pi}.$

The $G_{\pi}$-invariant curve

$\widetilde{C}^{\prime}_{\alpha}=\bigcup_{g\in
G_{\pi}}g^{*}C^{\prime}_{\alpha}$

is a degree $n(=mr)$ spectral cover over $C^{\prime}$, which descends to a
degree $n$ spectral cover $C_{\alpha}\to C$ via taking the $G_{\pi}$-quotient.
Moreover, we see from the construction of $\Phi_{\Gamma}$ in [20, Section 5.1]
that the point $[C^{\prime}_{\alpha}]\in{\mathcal{A}}^{\heartsuit}(\pi)$ maps
to $[C_{\alpha}]\in{\mathcal{A}}^{\heartsuit}_{\gamma}$ via the natural
quotient map
$q^{D}_{\mathcal{A}}:{\mathcal{A}}^{\heartsuit}(\pi)\to{\mathcal{A}}^{\heartsuit}_{\gamma}$.
The composition

$C^{\prime}_{\alpha}\hookrightarrow\widetilde{C}^{\prime}_{\alpha}\rightarrow
C_{\alpha}$

is birational, hence it is a normalization by the smoothness of
$C^{\prime}_{\alpha}$.

The construction above works for families of spectral curves over the Hitchin
bases. Hence we obtain a commutative diagram

${{\mathcal{C}}^{\heartsuit}_{\pi}}$${{\mathcal{C}}^{\circ}_{\gamma}}$${{\mathcal{A}}^{\heartsuit}(\pi)}$${{\mathcal{A}}^{\heartsuit}_{\gamma}}$$\scriptstyle{q_{\mathcal{A}}}$

where the left vertical morphism is $G_{\pi}$-equivariant, and the horizontal
morphisms are $G_{\pi}$-quotient maps. The lemma follows from descending the
left vertical arrow. ∎

For a closed point $a\in{\mathcal{A}}^{\heartsuit}_{\gamma}$, we denote the
corresponding spectral curves over $C^{\prime}$ and $C$ by $C^{\prime}_{a}$
and $C_{a}$ respectively with the morphism

$u_{a}:C^{\prime}_{a}\rightarrow C_{a}$

given by Lemma 3.4. We consider the commutative diagram

(72)
${C^{\prime}_{a}}$${C_{a}}$${C.}$$\scriptstyle{s^{\prime}_{a}}$$\scriptstyle{u_{a}}$$\scriptstyle{s_{a}}$

Here $s_{a}:C_{a}\to C$ is the spectral cover over $C$, and
$s^{\prime}_{a}:C^{\prime}_{a}\to C$ is the composition of the spectral cover
$C^{\prime}_{a}\to C^{\prime}$ and the cyclic Galois cover $\pi:C^{\prime}\to
C$. Both $s^{\prime}_{a}$ and $s_{a}$ are finite of degree $n$.

We also consider the following line bundles

${\omega_{\pi,a}}=\mathrm{det}({s^{\prime}_{a}}_{*}{\mathcal{O}}_{C^{\prime}_{a}})\in\mathrm{Pic}(C),\quad\omega_{a}=\mathrm{det}({s_{a}}_{*}{\mathcal{O}}_{C_{a}})\in\mathrm{Pic}(C).$

The line bundle $\omega_{\pi,a}$ is defined for every spectral curve over
${\mathcal{A}}^{D}(\pi)$, which gives a family of line bundles over the affine
space ${\mathcal{A}}^{D}(\pi)$. Hence it is constant over
${\mathcal{A}}^{D}(\pi)$ and does not depend on the choice of the spectral
curve. Similarly, the line bundle $\omega_{a}$ is also independent of the
spectral curve over $C$. So we may write

$\omega_{\pi}=\omega_{\pi,a},\quad\omega=\omega_{a}.$

The following lemma is obtained via a direct calculation.

######  Lemma 3.5.

We have

$\mathrm{deg}(\omega_{\pi})=n(1-r)\frac{\mathrm{deg}(D)}{2},\quad\mathrm{deg}(\omega)=n(1-n)\frac{\mathrm{deg}(D)}{2}.$

In particular, if $\mathrm{deg}(D)$ is even, both line bundles $\omega_{\pi}$
and $\omega$ have degrees divisible by $n$

###### Proof.

Since the second equality is a special case of the first one, we only prove
the degree formula for $\mathrm{deg}(\omega_{\pi})$.

By the discussion before Lemma 3.5, the line bundle $\omega_{\pi}$ does not
depend on the choice of the spectral curve. Let

$s^{\prime}_{a}:C^{\prime}_{a}\to C^{\prime}\xrightarrow{\pi}C$

be the spectral cover where $C^{\prime}_{a}$ is a nonsingular curve lying in
$V(D^{\prime})$ of genus

$g(C^{\prime}_{a})=r(r-1)\frac{\mathrm{deg}(D^{\prime})}{2}+r\left(g(C^{\prime})-1\right)+1;$

see the formula of $d_{h_{n}}$ in [6, Section 6.1]. Then applying the Riemann-
Roch formula to
$\chi(C^{\prime}_{a},{\mathcal{O}}_{C^{\prime}_{a}})=\chi(C,\omega_{\pi})$, we
obtain that

$1-g(C^{\prime}_{a})=\mathrm{deg}(\omega_{\pi})+(1-g)$

which implies the lemma. ∎

The following lemma concerns pushing forward a line bundle
${\mathcal{N}}\in\mathrm{Pic}(C_{a})$ and its pullback

$u_{a}^{*}{\mathcal{N}}\in\mathrm{Pic}(C_{a}^{\prime})$

to the curve $C_{a}^{\prime}$.

######  Lemma 3.6.

With the same notation as in (72), we have

$\mathrm{det}({s_{a}}_{*}{\mathcal{N}})=\mathrm{det}({s^{\prime}_{a}}_{*}u_{a}^{*}{\mathcal{N}})\otimes\omega\otimes\omega_{\pi}^{-1}.$

###### Proof.

Recall the norm maps

$\mathrm{Nm}:\mathrm{Pic}(C^{\prime}_{a})\rightarrow\mathrm{Pic}(C),\quad\mathrm{Nm}:\mathrm{Pic}(C_{a})\rightarrow\mathrm{Pic}(C)$

from [20, Section 3]. By [20, Lemma 3.4], we have

$\mathrm{Nm}({\mathcal{N}})=\mathrm{Nm}(u_{a}^{*}{\mathcal{N}}).$

Then [20, Corollary 3.12] implies that

$\mathrm{det}({s_{a}}_{*}{\mathcal{N}})\otimes\omega^{-1}=\mathrm{Nm}({\mathcal{N}})=\mathrm{Nm}(u_{a}^{*}{\mathcal{N}})=\mathrm{det}({s^{\prime}_{a}}_{*}u_{a}^{*}{\mathcal{N}})\otimes\omega_{\pi}^{-1}.\qed$

For $n\geq 1$ and $L\in\mathrm{Pic}(C)$, we consider the regular parts

${\mathcal{M}}^{D,\mathrm{reg}}_{n,L}\subset{\mathcal{M}}^{D}_{n,L},\quad{\mathcal{M}}^{D,\mathrm{reg}}_{r,L}(\pi)\subset{\mathcal{M}}^{D}_{r,L}(\pi)$

which are open subvarieties parameterizing Higgs bundles obtained as the
pushforward of _line bundles_ supported on the spectral curves. We define the
line bundle

(73) $L^{\prime}=L\otimes\omega\otimes\omega_{\pi}^{-1}\in\mathrm{Pic}(C).$

The following is a corollary of Lemma 3.6.

######  Corollary 3.7.

The pullback $u^{*}_{\mathcal{C}}$ of (71) induces a $G_{\pi}$-equivariant
morphism of the regular parts

$g_{u}:{\mathcal{M}}^{D,\mathrm{reg}}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{\heartsuit}(\pi)\rightarrow{\mathcal{M}}^{D,\mathrm{reg}}_{r,L^{\prime}}(\pi)\times_{{\mathcal{A}}^{D}(\pi)}{\mathcal{A}}^{\heartsuit}(\pi)$

where $L^{\prime}$ is given by (73). The morphism
${\mathcal{A}}^{\heartsuit}(\pi)\rightarrow{\mathcal{A}}^{D}$ used in the base
change of the left-hand side is the composition

${\mathcal{A}}^{\heartsuit}(\pi)\xrightarrow{q_{\mathcal{A}}}{\mathcal{A}}^{\heartsuit}_{\gamma}\hookrightarrow{\mathcal{A}}^{D}.$

######  Remark 3.8.

Since $\mathrm{gcd}(n,\mathrm{deg}(L))=1$ and

$\mathrm{deg}(L^{\prime})=\mathrm{deg}(L)+\mathrm{deg}(\omega)-\mathrm{deg}(\omega_{\pi}),$

Lemma 3.5 implies that

$\mathrm{gcd}(r,\mathrm{deg}(L^{\prime}))=\mathrm{gcd}(n,\mathrm{deg}(L^{\prime}))=1$

when $\mathrm{deg}(D)$ is even.

Finally, we note that both varieties ${\mathcal{M}}^{D,\mathrm{reg}}_{n,L}$
and ${\mathcal{M}}^{D,\mathrm{reg}}_{r,L^{\prime}}(\pi)$ admit
$\Gamma$-actions induced by the tensor product with $n$-torsion line bundles.
The $\Gamma$-actions are clearly fiberwise with respect to the Hitchin maps

(74)
${\mathcal{M}}^{D,\mathrm{reg}}_{n,L}\to{\mathcal{A}}^{D},\quad{\mathcal{M}}^{D,\mathrm{reg}}_{r,L^{\prime}}(\pi)\to{\mathcal{A}}^{D}(\pi).$

Since the pullback
$u_{a}^{*}:\mathrm{Pic}(C_{a})\to\mathrm{Pic}(C_{a}^{\prime})$ induced by (72)
is compatible with the $\Gamma$-action on both sides, we conclude the
following lemma.

######  Lemma 3.9.

The morphism $g_{u}$ given in Corollary 3.7 is $\Gamma$-equivariant.

###  3.3. Proof of Theorem 3.3 (step 1): Cohomological correspondences

Throughout Section 3.3 to Section 3.5, we assume that $D$ is an effective
divisor with $\mathrm{deg}(D)$ even and $\mathrm{deg}(D)>2g-2$. Our first step
in the proof of Theorem 3.3 is to construct a correspondence between the
direct image complexes associated with the two Hitchin maps attached to (74).
Then we show that this correspondence induces an isomorphism for the
$\kappa$-parts following Ngô and Yun ([39, Appendix A]).

We consider the graph of $g_{u}$ in Corollary 3.7 which gives a subvariety

$\mathrm{Graph}(g_{u})\subset\left({\mathcal{M}}^{D,\mathrm{reg}}_{r,L^{\prime}}(\pi)\times_{{\mathcal{A}}^{D}(\pi)}{\mathcal{A}}^{\heartsuit}(\pi)\right)\times_{{\mathcal{A}}^{\heartsuit}(\pi)}\left({\mathcal{M}}^{D,\mathrm{reg}}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{\heartsuit}(\pi)\right).$

Taking its Zariski closure, we obtain a closed subvariety

$\Sigma=\overline{\mathrm{Graph}(g_{u})}\subset{\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)\times_{{\mathcal{A}}^{D}(\pi)}\left({\mathcal{M}}^{D}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{D}(\pi)\right)$

which fits into the commutative diagram

(75)
${\Sigma}$${{\mathcal{M}}^{D}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{D}(\pi)}$${{\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)}$${{\mathcal{A}}^{D}(\pi).}$$\scriptstyle{h^{D}}$$\scriptstyle{h^{D}_{\pi,L^{\prime}}}$

Here we use $h^{D}_{\pi,L^{\prime}}$ to denote the Hitchin fibration

$h^{D}_{\pi,L^{\prime}}:=h^{D}_{\pi}:{\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)\to{\mathcal{A}}^{D}(\pi)$

indicating its dependence on the line bundle $L^{\prime}$. All the morphisms
in (75) are proper.

By Corollary 3.7 and Lemma 3.9, the morphism $g_{u}$ is equivariant under the
actions of $G_{\pi}$ and $\Gamma$. Hence, as the Zariski closure of the graph
of $g_{u}$, the subvariety $\Sigma$ is invariant under the natural actions of
$G_{\pi}$ and $\Gamma$ on the ambient space
${\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)\times_{{\mathcal{A}}^{D}(\pi)}\left({\mathcal{M}}^{D}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{D}(\pi)\right)$.
Since the projections

$\begin{split}{\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)\times_{{\mathcal{A}}^{D}(\pi)}\left({\mathcal{M}}^{D}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{D}(\pi)\right)&\to{\mathcal{M}}^{D}_{r,L^{\prime}},\\\
{\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)\times_{{\mathcal{A}}^{D}(\pi)}\left({\mathcal{M}}^{D}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{D}(\pi)\right)&\to{\mathcal{M}}^{D}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{D}(\pi)\end{split}$

to both factors are $G_{\pi}$\- and $\Gamma$-equivariant, we obtain that the
projections from the invariant subvariety $\Sigma$ to both factors are also
$G_{\pi}$\- and $\Gamma$-equivariant.

######  Theorem 3.10.

The correspondence (75) induces a morphism

(76)
$[\Sigma]_{\\#}:q_{\mathcal{A}}^{*}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)\rightarrow\mathrm{Rh_{\pi,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}[-2d^{D}_{\gamma}]$

which is equivariant under the natural actions of $G_{\pi}$ and $\Gamma$.
Assume that the element $\gamma\in\Gamma$ inducing the Galois cover
$\pi:C^{\prime}\to C$ matches with $\kappa$ via (3). Then the
$G_{\pi}$-equivariant morphism for the $\kappa$-parts

(77)
$[\Sigma]_{\\#,\kappa}:q_{\mathcal{A}}^{*}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\to\left(\mathrm{Rh_{\pi,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]$

induced by (76) is an isomorphism after restricting to
${\mathcal{A}}^{D}(\pi)^{*}$:

(78)
$[\Sigma]_{\\#,\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}:q_{\mathcal{A}}^{*}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}\xrightarrow{\simeq}\left(\mathrm{Rh_{\pi,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}[-2d^{D}_{\gamma}].$

###### Proof.

The first part follows from the general theory of cohomological
correspondences. We refer to [38, Appendix A] as a reference; see also the
paragraph before [39, Proposition 3.3.1]. In particular, since the variety
$\Sigma$ is invariant under the $G_{\pi}$\- and the $\Gamma$-actions, and both
projections from $\Sigma$ to
${\mathcal{M}}^{D}_{n,L}\times_{{\mathcal{A}}^{D}}{\mathcal{A}}^{D}(\pi)$ and
${\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)$ are $G_{\pi}$\- and
$\Gamma$-equivariant, we conclude that the cohomological correspondence (76)
is also $G_{\pi}$\- and $\Gamma$-equivariant.

Now we restrict $[\Sigma]_{\\#,\kappa}$ to the open subset
${\mathcal{A}}^{D}(\pi)^{*}$ and show that it is an isomorphism. It suffices
to prove that the restriction of $[\Sigma]_{\\#,\kappa}$ induces an
isomorphism on every perverse cohomology, _i.e._ ,

(79)
$^{p}{\mathcal{H}}^{i}([\Sigma]_{\\#,\kappa})\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}:q_{\mathcal{A}}^{*}{{}^{p}{\mathcal{H}}^{i}}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}\xrightarrow{\simeq}{{}^{p}{\mathcal{H}}^{i-2d^{D}_{\gamma}}}\left(\mathrm{Rh_{\pi,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}.$

Here we used that $q_{\mathcal{A}}$ is étale restricting to
${\mathcal{A}}^{D}(\pi)^{*}$. By Corollary 2.2, the left-hand side of (79) has
${\mathcal{A}}^{D}(\pi)^{*}$ as the only support. Theorem 2.3 (b) and
Proposition 2.10 yield that the right-hand side of (79) also has
${\mathcal{A}}^{D}(\pi)^{*}$ as the only support. Therefore, both sides of
(79) are intermediate extensions of certain local systems defined on an open
subset of ${\mathcal{A}}^{D}(\pi)$. As a consequence, in order to prove (79),
we only need to show that

${}^{p}{\mathcal{H}}^{i}([\Sigma]_{\\#,\kappa})\Big{|}_{U}:q_{\mathcal{A}}^{*}{{}^{p}{\mathcal{H}}^{i}}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{U}\xrightarrow{\simeq}{{}^{p}{\mathcal{H}}^{i-2d^{D}_{\gamma}}}\left(\mathrm{Rh_{\pi,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{U}$

with $U\subset{\mathcal{A}}^{D}(\pi)^{*}$ a Zariski open subset. This reduces
the proof of (79) to showing that, for a general point
$a\in{\mathcal{A}}^{D}(\pi)^{*}$, the correspondence between the Hitchin
fibers

(80)
$[\Sigma_{a}]:H^{i}\left(({\mathcal{M}}^{D}_{n,L})_{a},{\mathbb{C}}\right)\rightarrow
H^{i-2d^{D}_{\gamma}}\left({\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)_{a},{\mathbb{C}}\right)$

induced by the fundamental class of $\Sigma_{a}$ is an isomorphism between the
$\kappa$-parts. Here $\Sigma_{a}$ is the restriction of $\Sigma$ over $a$. Let
$a$ be a general point lying in the open subset
${\mathcal{A}}^{\heartsuit}(\pi)\subset{\mathcal{A}}^{D}(\pi)$. The pullback
of the diagram (71) along
$\\{a\\}\rightarrow{\mathcal{A}}^{\heartsuit}_{\gamma}$ is a normalization

$u_{a}:C^{\prime}_{a}\to C_{a}$

of curves where $C_{a}$ has at worst nodal singularities. In this case, the
description of the correspondence (80) is concretely given in [39, Appendix A]
and the isomorphism of the $\kappa$-parts follows from a direct calculation
[39, Lemma 3.4.1].444For the correspondence (80), we only concern Hitchin
fibers over a general closed point $a\in{\mathcal{A}}^{D}(\pi)$. Since
tensoring with degree 1 line bundles on the spectral curves over
$a\in{\mathcal{A}}^{D}(\pi)$ identifies the Hitchin fibers in the case of
degree 0 and the Hitchin fibers in the case of coprime rank and degree, the
result of [39, Lemma 3.4.1] concerning degree 0 Higgs bundles applies here.
This completes the proof. ∎

######  Corollary 3.11.

With the same assumption as in Theorem 3.10, the $G_{\pi}$-equivariant
isomorphism (78) induces an isomorphism:

$[\Sigma]^{G_{\pi}}_{\\#,\kappa}:\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma,L^{\prime}}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]\in
D^{b}_{c}({\mathcal{A}}^{D}).$

###### Proof.

We pushforward the morphism (77) along the $G_{\pi}$-quotient map
$q_{\mathcal{A}}:{\mathcal{A}}^{D}(\pi)\to{\mathcal{A}}^{D}_{\gamma}$ and take
the $G_{\pi}$-invariant parts on both sides. This gives a morphism

$[\Sigma]^{G_{\pi}}_{\\#,\kappa}:\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\to{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma,L^{\prime}}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]$

where both sides are semi-simple with ${\mathcal{A}}^{D}_{\gamma}$ the only
support by Corollary 2.2, Theorem 2.3 (b), and Proposition 2.10. Hence similar
to the proof of Theorem 3.10, it suffices to check that it is an isomorphism
restricting to ${{\mathcal{A}}^{D}_{\gamma}}^{*}$ which is equivalent to (78).
∎

######  Remark 3.12.

We cannot conclude that (77) is an isomorphism over the total space
${\mathcal{A}}^{D}(\pi)$ from the isomorphism (78) due to the following
reason. For a non-smooth finite morphism $f:X\to Y$ and a semi-simple perverse
sheaf ${\mathcal{K}}$ on $Y$, the pullback $f^{*}{\mathcal{K}}$ may fail to be
semi-simple. A typical example is the case where

$f:{\mathbb{A}}^{1}\to{\mathbb{A}}^{1},\quad z\mapsto z^{2}$

and ${\mathcal{K}}$ is the intermediate extension of a 2-torsion rank 1 local
system on ${\mathbb{C}}^{*}\subset{\mathbb{A}}^{1}$. The object
$f^{*}{\mathcal{K}}$ is not determined by its restriction to the open subset
${\mathbb{C}}^{*}$. In particular, although the restriction of the natural
morphism $f^{*}{\mathcal{K}}\to\underline{{\mathbb{C}}}$ to ${\mathbb{C}}^{*}$
is an isomorphism, the morphism
$f^{*}{\mathcal{K}}\to\underline{{\mathbb{C}}}$ itself fails to be an
isomorphism on ${\mathbb{A}}^{1}$. The issue here is caused by the fact that
$f$ is not étale at $0\in{\mathbb{A}}^{1}$.

###  3.4. Proof of Theorem 3.3 (Step 2): changing from $L^{\prime}$ to $L$

By Lemma 3.5, we have

(81)
$\mathrm{deg}(L^{\prime})-\mathrm{deg}(L)=0~{}~{}~{}\mathrm{mod}~{}~{}~{}n.$

Hence there exists a line bundle

(82) ${\mathcal{N}}_{0}\in\mathrm{Pic}^{k}(C),\quad
k=\frac{\mathrm{deg}(L^{\prime})-\mathrm{deg}(L)}{n}\in{\mathbb{Z}}$

such that $L^{\prime}=L\otimes{\mathcal{N}}_{0}^{\otimes n}$. The line bundle
${\mathcal{N}}_{0}$ induces an ${\mathcal{A}}^{D}(\pi)$-isomorphism between
the relative Hitchin moduli spaces

(83)
${{\mathcal{M}}^{D}_{r,L}(\pi)}$${{\mathcal{M}}^{D}_{r,L^{\prime}}(\pi)}$${{\mathcal{A}}^{D}(\pi)}$$\scriptstyle{h^{D}_{\pi,L}}$$\scriptstyle{\phi_{{\mathcal{N}}_{0}}}$$\scriptstyle{h^{D}_{\pi,L^{\prime}}}$

via tensor product
$\phi_{{\mathcal{N}}_{0}}({\mathcal{E}},\theta)=({\mathcal{E}}\otimes{\mathcal{N}}_{0},\theta)$.
It is clear that $\phi_{{\mathcal{N}}_{0}}$ is $G_{\pi}$\- and
$\Gamma$-equivariant.

######  Proposition 3.13.

There is a $G_{\pi}$-equivariant isomorphism

(84)
$\left(\mathrm{Rh_{\pi,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}=\left(\mathrm{Rh_{\pi,L}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\in
D^{b}_{c}({\mathcal{A}}^{D}(\pi))$

induced by (83). Up to scaling, it is independent of the choice of line bundle
(82). In particular, (84) induces

$\left(\mathrm{Rh_{\gamma,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}=\left(\mathrm{Rh_{\gamma,L}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\in
D^{b}_{c}({\mathcal{A}}_{\gamma}^{D}).$

###### Proof.

The pullback along $\phi_{{\mathcal{N}}_{0}}$ in the diagram (83) induces a
$G_{\pi}$\- and $\Gamma$\- equivariant isomorphism

(85)
$\mathrm{Rh_{\pi,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}=\mathrm{Rh_{\pi,L}^{D}}_{*}{\underline{{\mathbb{C}}}}$

which gives our desired isomorphism. For another choice
${\mathcal{N}}^{\prime}_{0}$ of the line bundle (82), we have

${\mathcal{N}}_{0}^{-1}\otimes{\mathcal{N}}^{\prime}_{0}\in\Gamma.$

Hence the difference of the isomorphisms $\phi^{*}_{{\mathcal{N}}_{0}}$ and
$\phi^{*}_{{\mathcal{N}}^{\prime}_{0}}$ is induced by an automorphism of
${\mathcal{M}}^{D}_{r,L}$ given by an element $g\in\Gamma$. In particular, our
choice of isomorphism is affected only by scaling.

The last claim follows from Lemma 1.7. ∎

######  Remark 3.14.

In view of Lemma 3.5, the equation (81) and the existence of the line bundle
(82) rely heavily on the assumption that $\mathrm{deg}(D)$ is even.

###  3.5. Completing the proof of Theorem 3.3

Theorem 3.3 follows from Corollary 3.11 and Proposition 3.13.

More precisely, we constructed an isomorphism

$c^{D}_{\kappa}:\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]\in
D_{c}^{b}({\mathcal{A}}^{D})$

which is canonical up to scaling as the composition

$\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow[\simeq]{~{}~{}[\Sigma]^{G_{\pi}}_{\\#,\kappa}~{}~{}}{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh_{\gamma,L^{\prime}}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]\xrightarrow[\simeq]{\mathrm{Prop}.~{}~{}3.13.}{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh_{\gamma,L}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}].$

Here the first isomorphism $[\Sigma]^{G_{\pi}}_{\\#,\kappa}$ is given by
Corollary 3.11, and the second isomorphism is given by Proposition 3.13. This
completes the proof of Theorem 3.3. ∎

##  4\. Vanishing cycles and Hitchin moduli spaces

###  4.1. Overview

In this section, we follow the same notation as in Section 3. We complete the
proof of Theorem 3.2 (and therefore Theorem 3.1 as explained after Theorem
3.2) by constructing an operator

$c^{D}_{\kappa}:\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}]\in
D_{c}^{b}({\mathcal{A}}^{D})$

which is constructed up to scaling for any effective divisor $D$ with
$\mathrm{deg}(D)>2g-2$ or $D=K_{C}$. Our main tool is Theorem 4.5, where we
apply a vanishing cycle functor to connect the moduli of $D$-Higgs bundles to
the moduli of $(D+p)$-Higgs bundles with $p\in C$ a closed point. This reduces
the general cases of Theorem 3.2 to the special cases already treated in
Theorem 3.3.

In this section, it is convenient to work with the moduli stacks of stable
$\mathrm{SL}_{n}$-Higgs bundles and relative stable Higgs bundles associated
with $\pi:C^{\prime}\to C$. As these are nonsingular Deligne-Mumford stacks,
gerbes over the coarse moduli spaces, this has no effect on the direct image
complexes (64). Therefore, throughout this section, we still use the notation
${\mathcal{M}}^{D}_{n,L}$ and ${\mathcal{M}}^{D}_{r,L}(\pi)$ to denote the
corresponding moduli stacks for stable Higgs bundles and relative stable Higgs
bundles.

###  4.2. Restrictions of Higgs bundles to a point

Let $p$ be an abstract reduced point $\mathrm{Spec}({\mathbb{C}})$. Any rank
$n$ vector bundle on $p$ is an $n$-dimensional vector space. Hence the
category of $\mathrm{SL}_{n}$-Higgs bundles on $p$ can be thought of as the
category of matrices in $\mathfrak{sl}_{n}$ up to
$\mathrm{SL}_{n}$-conjugation, whose moduli stack is given by the quotient

${\mathcal{M}}_{n,p}=[\mathfrak{sl}_{n}/\mathrm{SL}_{n}].$

Here $\mathrm{SL}_{n}$ acts on $\mathfrak{sl}_{n}$ via conjugation. The
Hitchin fibration associated with ${\mathcal{M}}_{n,p}$ is

$h_{p}:{\mathcal{M}}_{n,p}\to{\mathcal{A}}_{p}$

where
${\mathcal{A}}_{p}=\mathfrak{sl}_{n}\mathbin{/\mkern-6.0mu/}\mathrm{SL}_{n}$
is the affine $\mathrm{GIT}$-quotient parameterizing all characteristic
polynomials

$(a_{2},a_{3},\dots,a_{n})\in{\mathcal{A}}_{p},\quad
a_{i}=\mathrm{trace}(\wedge^{i}\theta_{p})$

of the traceless endomorphism
$\theta_{p}\in\mathrm{End}({\mathbb{A}}^{n})_{0}$ associated with a matrix in
${\mathcal{M}}_{n,p}$. We refer to [34, Section 2.2] for more details
concerning the stack ${\mathcal{M}}_{n,p}$ and the morphism $h_{p}$.

Now we consider $p$ as a closed point on the curve $C$. We fix a
trivialization

(86) ${\mathcal{O}}_{C}(D)_{p}\xrightarrow{\simeq}{\mathbb{C}}.$

Then the restriction map with respect to the closed embedding

$i_{p}:\\{p\\}\hookrightarrow C$

induces the following commutative diagram

(87)
${{\mathcal{M}}_{n,L}^{D}}$${{\mathcal{M}}_{n,p}}$${{\mathcal{A}}^{D}}$${{\mathcal{A}}_{p}}$$\scriptstyle{\mathrm{ev}_{p}}$$\scriptstyle{h^{D}}$$\scriptstyle{h_{p}}$

where the trivialization (86) induces an evaluation (at $p$) map
$\mathrm{ev}_{p}$:

$\mathrm{ev}_{p}({\mathcal{E}},\theta)=i_{p}^{*}({\mathcal{E}},\theta)\in{\mathcal{M}}_{n,p}.$

To generalize the diagram (87) for the relative moduli space
${\mathcal{M}}^{D}_{r,L}(\pi)$ with $n=mr$, we consider the Lie group

$H_{\pi}=\\{(g_{1},g_{2},\dots,g_{m})\in\mathrm{GL}_{r}^{\times
m}:~{}~{}\prod_{i}\mathrm{det}(g_{i})=1\\}\subset\mathrm{GL}_{r}^{\times m},$

which is naturally a subgroup of $\mathrm{SL}_{n}$ with Lie algebra

$\mathfrak{h}_{\pi}=\\{(g_{1},g_{2},\dots,g_{m})\in\mathfrak{gl}_{r}^{\times
m}:~{}~{}\sum_{i}\mathrm{trace}(g_{i})=0\\}.$

The quotient stack

${\mathcal{M}}_{r,p}(\pi)=[\mathfrak{h}_{\pi}/H_{\pi}]$

is the moduli of $\mathrm{SL}_{n}$-Higgs bundles on the point $p$ obtained as
the pushforward of rank $r$-Higgs bundles on $m$ distinct reduced points
$\sqcup_{i=1}^{m}p_{i}$ along the projection

$\pi_{p}:\sqcup_{i=1}^{m}p_{i}\to p.$

Similar to (87), we have the following diagram given by the restriction to
$p\in C$:

(88)
${{\mathcal{M}}_{r,L}^{D}(\pi)}$${{\mathcal{M}}_{r,p}(\pi)}$${{\mathcal{A}}^{D}}$${{\mathcal{A}}_{p}(\pi).}$$\scriptstyle{\mathrm{ev}_{p}}$$\scriptstyle{h^{D}_{\pi}}$$\scriptstyle{h_{\pi,p}}$

Here for a Higgs bundle $({\mathcal{E}},\theta)\in{\mathcal{M}}^{D}_{r,L}$ on
$C^{\prime}$, again (86) induces an evaluation map:

$\mathrm{ev}_{p}({\mathcal{E}},\theta)=i_{p}^{*}\left(\pi_{*}{\mathcal{E}},\pi_{*}\theta\right)\in{\mathcal{M}}_{r,p}(\pi),$

and the “Hitchin map over a point” $h_{p}$ is the natural projection

$h_{\pi,p}:[\mathfrak{h}_{\pi}/H_{\pi}]\rightarrow{\mathcal{A}}_{p}(\pi):=\mathfrak{h}_{\pi}\mathbin{/\mkern-6.0mu/}H_{\pi}.$

The diagram (88) recovers (87) when $C^{\prime}=C$ and $\pi:C^{\prime}\to C$
is the identity.

######  Proposition 4.1.

Assume that $D$ is a divisor on $C$ satisfying

  1. (a)

$D-p=K_{C}$, or

  2. (b)

$D-p$ is effective and $\mathrm{deg}(D-p)>2g-2$.

Then the evaluation map
$\mathrm{ev}_{p}:{\mathcal{M}}^{D}_{r,L}(\pi)\to{\mathcal{M}}_{r,p}(\pi)$ in
(88) is smooth.

###### Proof.

We first review the deformation theory of ${\mathcal{M}}^{D}_{r,L}(\pi)$
following [4] and [34, Section 4.14]. The deformation theory of a rank $r$
Higgs bundle $({\mathcal{E}},\theta)$ on $C^{\prime}$ is governed by the
tangent complex

(89)
$\left[{\mathcal{E}}{nd}({\mathcal{E}})\xrightarrow{\mathrm{ad}(\theta)}{\mathcal{E}}{nd}({\mathcal{E}})\otimes{\mathcal{O}}_{C^{\prime}}(D^{\prime})\right]$

lying in degrees -1 and 0. Since now we only consider rank $r$ Higgs bundles
$({\mathcal{E}},\theta)$ on $C^{\prime}$ with the fixed determinant and trace
after pushing forward to $C$, to govern the deformation theory of
${\mathcal{M}}^{D}_{r,L}(\pi)$ we need to remove the traces of
${\mathcal{E}}{nd}({\mathcal{E}})$ after pushing forward the complex (89) to
$C$. Hence the deformation theory of ${\mathcal{M}}^{D}_{r,L}(\pi)$ is
governed by

(90)
$C^{\bullet}({\mathcal{E}},\theta,D)=\left[(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}}))_{0}\xrightarrow{\pi_{*}\mathrm{ad}(\theta)}(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}}))_{0}\otimes{\mathcal{O}}_{C}(D)\right],\quad({\mathcal{E}},\theta)\in{\mathcal{M}}^{D}_{r,L}(\pi),$

where $(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}}))_{0}$ denotes the kernel with
respect to the trace

$\mathrm{tr}_{C}:\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}})\xrightarrow{\pi_{*}\mathrm{tr}_{C^{\prime}}}\pi_{*}{\mathcal{O}}_{C^{\prime}}\to{\mathcal{O}}_{C}$

on $C$. The complex (90) is the tangent complex of
${\mathcal{M}}^{D}_{r,L}(\pi)$. The automorphism space, the tangent space, and
the obstruction space are thus given by the cohomology groups

$H^{0}(C,C^{\bullet}({\mathcal{E}},\theta,D)),\quad
H^{1}(C,C^{\bullet}({\mathcal{E}},\theta,D)),\quad\textup{and}~{}~{}~{}~{}H^{2}(C,C^{\bullet}({\mathcal{E}},\theta,D))$

respectively. Since the evaluation map $\mathrm{ev}_{p}$ is induced by the
restriction to the point $p$ via $i_{p}:\\{p\\}\hookrightarrow C$, the tangent
map of $\mathrm{ev}_{p}$ is

(91)
$\mathrm{Tan}_{\mathrm{ev}_{p}}:H^{1}(C,C^{\bullet}({\mathcal{E}},\theta,D))\rightarrow
H^{1}(p,i_{p}^{*}C^{\bullet}({\mathcal{E}},\theta,D))$

induced by the restriction morphism between the tangent complexes. Here
$i_{p}^{*}C^{\bullet}({\mathcal{E}},\theta,D)=[\mathfrak{h}_{\pi}\xrightarrow{\mathrm{ad}}\mathfrak{h}_{\pi}]$
recovers the tangent complex of ${\mathcal{M}}_{r,p}(\pi)$; see [7, Appendix
8.2]. To prove the smoothness of $\mathrm{ev}_{p}$, we show in the following
that (91) is surjective.

The restriction map between the tangent complexes

$C^{\bullet}({\mathcal{E}},\theta,D))\to{i_{p}}_{*}i_{p}^{*}C^{\bullet}({\mathcal{E}},\theta,D))$

fits into the exact triangle

(92) $E^{\bullet}\to
C^{\bullet}({\mathcal{E}},\theta,D)\to{i_{p}}_{*}i_{p}^{*}C^{\bullet}({\mathcal{E}},\theta,D)\xrightarrow{+1}$

where $E^{\bullet}$ is given by

$E^{\bullet}=C^{\bullet}({\mathcal{E}},\theta,D)\otimes{\mathcal{O}}_{C}(-p)=\left[(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}}))_{0}\otimes{\mathcal{O}}_{C}(-p)\xrightarrow{\pi_{*}\mathrm{ad}(\theta)}(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}}))_{0}\otimes{\mathcal{O}}_{C}(D-p)\right].$

The long exact sequence associated with (92) contains

$H^{1}(C,C^{\bullet}({\mathcal{E}},\theta,D))\xrightarrow{\mathrm{Tan}_{\mathrm{ev}_{p}}}H^{1}(C,{i_{p}}_{*}i_{p}^{*}C^{\bullet}({\mathcal{E}},\theta,D))\rightarrow
H^{2}(C,E^{\bullet}).$

Hence, in view of the Serre duality, it suffices to show the vanishing of

(93)
$H^{2}(C,E^{\bullet})^{\vee}=H^{0}(C,(E^{\bullet})^{\vee}\otimes\Omega_{C})$

where

$(E^{\bullet})^{\vee}\otimes\Omega_{C}=\left[(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}}))_{0}\otimes{\mathcal{O}}_{C}(p-D+K_{C})\xrightarrow{-\mathrm{ad}(\pi_{*}\theta)}(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}})_{0}\otimes{\mathcal{O}}_{C}(p+K_{C})\right].$

To calculate (93), we consider the hyper-cohomology group
$H^{0}(C,(\widetilde{E}^{\bullet})^{\vee}\otimes\Omega_{C})$ with

$\widetilde{E}^{\bullet}=\left[(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}}))\otimes{\mathcal{O}}_{C}(-p)\xrightarrow{\pi_{*}\mathrm{ad}(\theta)}(\pi_{*}{\mathcal{E}}{nd}({\mathcal{E}}))\otimes{\mathcal{O}}_{C}(D-p)\right].$

It is clear that $H^{0}(C,(\widetilde{E}^{\bullet})^{\vee}\otimes\Omega_{C})$
contains (93) as a direct sum component, and the complement is contributed by
the trace parts on $C$. The hyper-cohomology group
$H^{0}(C,(\widetilde{E}^{\bullet})^{\vee}\otimes\Omega_{C})$ can be written as

(94)
$H^{0}\left(C^{\prime},\left[{\mathcal{E}}{nd}({\mathcal{E}})\otimes\pi^{*}{\mathcal{O}}_{C}(p-D+K_{C})\xrightarrow{-\mathrm{ad}(\theta)}{\mathcal{E}}{nd}({\mathcal{E}})\otimes\pi^{*}{\mathcal{O}}_{C}(p+K_{C})\right]\right)$

via the projection formula associated with $\pi:C^{\prime}\to C$. By [16,
Theorem 5.1] (see also the calculation of [17, Lemma 7.3]), (94) can be
interpreted as the group of homomorphisms of stable Higgs bundles

(95)
$\mathrm{Hom}_{C^{\prime}}\left(({\mathcal{E}},\theta),({\mathcal{E}}\otimes\pi^{*}{\mathcal{O}}_{C}(K_{C}-D+p),\theta)\right).$

In the case (a), these two stable Higgs bundles in (95) coincide, so the
$\mathrm{Hom}$ space (95) is 1-dimensional given by the identity map. Hence we
have the vanishing of (93) by removing the $1$-dimensional trace parts on $C$
from (95). In the case (b), we have

$\mathrm{deg}({\mathcal{E}})>\mathrm{deg}\left({\mathcal{E}}\otimes\pi^{*}{\mathcal{O}}_{C}(K_{C}-D+p)\right).$

Hence (95) vanishes due to the stability condition, which further implies the
vanishing of (93). ∎

Assume that the divisor $D$ satisfies (a) and (b) of Proposition 4.1. The
moduli of stable $(D-p)$-Higgs bundles admits a natural closed embedding into
the moduli of $D$-Higgs bundles. More precisely, let $({\mathcal{E}},\theta)$
be a stable $(D-p)$-Higgs bundle on $C$, then we may view it naturally as a
$D$-Higgs bundle $({\mathcal{E}},\theta^{\prime})$ by setting the new Higgs
field as the composition

$\theta^{\prime}:{\mathcal{E}}\xrightarrow{\theta}{\mathcal{E}}\otimes{\mathcal{O}}_{C}(D-p)\rightarrow{\mathcal{E}}\otimes{\mathcal{O}}_{C}(D)$

where the second map is induced by
${\mathcal{O}}_{C}(-p)\hookrightarrow{\mathcal{O}}_{C}$. By definition, the
(slope-)stability conditions of $({\mathcal{E}},\theta)$ and
$({\mathcal{E}},\theta^{\prime})$ coincide. Hence we obtain a closed embedding

(96)
${\mathcal{M}}^{D-p}_{n,L}\hookrightarrow{\mathcal{M}}^{D}_{n,L},\quad({\mathcal{E}},\theta)\mapsto({\mathcal{E}},\theta^{\prime}).$

Similarly, we also have the relative version with respect to
$\pi:C^{\prime}\to C$:

(97)
${\mathcal{M}}^{D-p}_{r,L}(\pi)\hookrightarrow{\mathcal{M}}^{D}_{r,L}(\pi),\quad({\mathcal{E}},\theta)\mapsto({\mathcal{E}},\theta^{\prime})$

where
$\theta^{\prime}:{\mathcal{E}}\to{\mathcal{E}}\otimes\pi^{*}{\mathcal{O}}_{C}(D)$
is given by the composition

$\theta^{\prime}:{\mathcal{E}}\xrightarrow{\theta}{\mathcal{E}}\otimes\pi^{*}{\mathcal{O}}_{C}(D-p)\to{\mathcal{E}}\otimes\pi^{*}{\mathcal{O}}_{C}(D).$

As before, the relative case (97) recovers (96) by setting $\pi=\mathrm{id}$.

A Higgs bundle in ${\mathcal{M}}^{D}_{r,L}(\pi)$ which sits inside
${\mathcal{M}}^{D-p}_{r,L}(\pi)$ can be characterized by the vanishing of the
restricted Higgs field over the point $p$. Therefore it is natural to use the
evaluation map $\mathrm{ev}_{p}$ to describe (96) and (97) as in the following
lemma.

######  Lemma 4.2.

Assume the divisor $D$ satisfies (a) and (b) of Proposition 4.1. We denote by

$0_{H}=[0/H_{\pi}]\hookrightarrow{\mathcal{M}}_{r,p}(\pi)=[\mathfrak{h}_{\pi}/H_{\pi}]$

the closed substack corresponding to the $0$ matrix. Then the closed embedding
(97) is realized as a closed fiber of $\mathrm{ev}_{p}$ over $0_{H}$.

###  4.3. Functions and critical loci

We consider the quotient map

$\mathfrak{sl}_{n}\rightarrow\mathfrak{sl}_{n}\mathbin{/\mkern-6.0mu/}\mathrm{SL}_{n}$

sending a matrix to the coefficients

$(a_{2},a_{3},\dots,a_{n})\in\mathfrak{sl}_{n}\mathbin{/\mkern-6.0mu/}\mathrm{SL}_{n}=\mathrm{Spec}\left({\mathbb{C}}[\mathfrak{sl}_{n}]^{\mathrm{SL}_{n}}\right),\quad\mathrm{deg}(a_{i})=i$

of its characteristic polynomial. The term $a_{i}$ defines a degree $i$
polynomial function on the Lie algebra $\mathfrak{sl}_{n}$. We define the
quadratic function on the Lie algebra $\mathfrak{sl}_{n}$ given by $a_{2}$ as

(98) $\mu=a_{2}:\mathfrak{sl}_{n}\to{\mathbb{A}}^{1},$

which induces a function $\mu_{\pi}:\mathfrak{h}_{\pi}\to{\mathbb{A}}^{1}$ via
the composition

(99)
$\mu_{\pi}:\mathfrak{h}_{\pi}\hookrightarrow\mathfrak{sl}_{n}\xrightarrow{\mu}{\mathbb{A}}^{1}.$

By definition, the functions $\mu$ and $\mu_{\pi}$ are invariant under the
conjugation actions by the Lie groups $\mathrm{SL}_{n}$ and $H_{\pi}$
respectively.

Since any matrix $g\in\mathfrak{h}_{\pi}\subset\mathfrak{sl}_{n}$ satisfies
$\mathrm{trace}(g)=0$, the quadratic function $\mu_{\pi}$ (up to scaling) can
be written as

(100) $g\mapsto\mathrm{trace}(g^{2}).$

For the semi-simple Lie algebra $\mathfrak{sl}_{n}$, (100) is the Killing form
which is clearly non-degenerate. In the following we prove the non-degeneracy
for general $\mathfrak{h}_{\pi}$.

######  Lemma 4.3.

The critical locus of the quadratic function $\mu_{\pi}$ is the isolated
reduced point $0\in\mathfrak{h}_{\pi}$, _i.e._ ,

$\mathrm{Crit}(\mu_{\pi})\left(=\\{d\mu_{\pi}=0\\}\right)=\\{0\\}\subset\mathfrak{h}_{\pi}.$

Consequently, the perverse sheaf of vanishing cycles
$\varphi_{\mu_{\pi}}(\underline{{\mathbb{C}}}[\mathrm{dim}({\mathfrak{h}_{\pi}}])$
(see [13, Theorem 5.2.21]) is the skyscraper sheaf supported on the closed
point $0\in\mathfrak{h}_{\pi}$.

###### Proof.

To prove the first part of Lemma 4.3, it suffices to show that the quadratic
form (100) on $\mathfrak{h}_{\pi}$ is non-degenerate.

We consider the decomposition

(101) $\mathfrak{h}_{\pi}=\mathfrak{h}^{\prime}_{\pi}\oplus\mathfrak{c}$

where $\mathfrak{c}\subset\mathfrak{h}_{\pi}$ is the Cartan subalgebra of
trace-free diagonal matrices, and $\mathfrak{h}^{\prime}_{\pi}$ consists of
the matrices in $\mathfrak{h}_{\pi}$ such that the entries of the diagonals
vanish. For a matrix $g\in\mathfrak{h}_{\pi}$ with the decomposition

$g=g^{\prime}+c,\quad
g^{\prime}\in\mathfrak{h}^{\prime}_{\pi},~{}~{}~{}c\in\mathfrak{c},$

a direct calculation yields $\mathrm{trace}(g^{\prime}c)=0$. Therefore we have

$\mathrm{trace}(g^{2})=\mathrm{trace}({g^{\prime}}^{2})+\mathrm{trace}(c^{2}).$

So it suffices to show that the quadratic forms (100) are non-degenerate for
both $\mathfrak{h}^{\prime}_{\pi}$ and $\mathfrak{c}$.

We notice that the Cartan subalgebra $\mathfrak{c}$ of $\mathfrak{h}_{\pi}$ is
the same as that of $\mathfrak{sl}_{n}$. Also, equipped with the quadratic
forms (100), $\mathfrak{h}^{\prime}_{\pi}$ is a direct summand component of
the Lie algebra $\mathfrak{sl}_{r}^{\times m}$ via the decomposition (101) for
$\mathfrak{sl}_{r}^{\times m}$. Since both $\mathfrak{h}_{\pi}^{\prime}$ and
$\mathfrak{c}$ are direct summand components of semi-simple Lie algebras where
the killing forms (100) are non-degenerate, we conclude the non-degeneracy of
(100) for $\mathfrak{h}^{\prime}_{\pi}$ and $\mathfrak{c}$, which further
implies the non-degeneracy of $\mathfrak{h}_{\pi}$ through (101).

This completes the first part of the lemma, and reduce the second part to the
case

$\mu_{\pi}:{\mathbb{A}}^{N}\to{\mathbb{A}}^{1},\quad(z_{1},\dots,z_{N})\mapsto\sum_{i=1}^{N}z_{i}^{2}.$

In this case, the Milnor fiber is a sphere [30] whose reduced homology
computes the vanishing cycle [13, Proposition 4.2.2]. ∎

The $H_{\pi}$-invariant function (99) induces the functions

$\mu_{1}:[\mathfrak{h}_{\pi}/H_{\pi}]\to{\mathbb{A}}^{1},\quad\mu_{2}:\mathfrak{h}_{\pi}\mathbin{/\mkern-6.0mu/}H_{\pi}\to{\mathbb{A}}^{1}$

which form the commutative diagram

(102)
${{\mathcal{M}}_{r,p}(\pi)}$${{\mathcal{A}}_{p}(\pi)}$${{\mathbb{A}}^{1}.}$$\scriptstyle{\mu_{1}}$$\scriptstyle{h_{\pi,p}}$$\scriptstyle{\mu_{2}}$

The pullback of (102) along the diagram (88) yields the functions

(103)
$\mu_{\pi,{\mathcal{M}}}:{\mathcal{M}}^{D}_{r,L}(\pi)\to{\mathbb{A}}^{1},\quad\mu_{\pi,{\mathcal{A}}}:{\mathcal{A}}^{D}(\pi)\to{\mathbb{A}}^{1}$

fitting into the commuatative diagram

(104)
${{\mathcal{M}}_{r,L}^{D}(\pi)}$${{\mathcal{A}}^{D}(\pi)}$${{\mathbb{A}}^{1}.}$$\scriptstyle{\mu_{\pi,{\mathcal{M}}}}$$\scriptstyle{h^{D}_{\pi}}$$\scriptstyle{\mu_{\pi,{\mathcal{A}}}}$

Before stating and proving the main theorem (Theorem 4.5) of this section, we
first note the following standard facts.

######  Lemma 4.4.

Let $f:V\to{\mathbb{A}}^{1}$ be a regular function.

  1. (a)

Assume $V$ admits an action of a finite group $G$ which is fiberwise with
respect to $f$. Then the nearby and vanishing cycle functors
$\Phi_{f},\varphi_{f}$ are $G$-equivariant.

  2. (b)

Assume ${\mathcal{F}}\in D^{b}_{c}(V)$, and assume that
$g=\lambda\cdot\mathrm{id}\in\mathrm{End}({\mathcal{F}})$ is a scaling
automorphism of ${\mathcal{F}}$ with $\lambda\in{\mathbb{C}}^{*}$. Then
applying the nearby or vanishing cycle functor to $g$ yields also an scaling
endomorphism:

$\Phi_{f}(g)=\lambda\cdot\mathrm{id}:\Phi_{f}{\mathcal{F}}\xrightarrow{\simeq}\Phi_{f}{\mathcal{F}},\quad\varphi_{f}(g)=\lambda\cdot\mathrm{id}:\varphi_{f}{\mathcal{F}}\xrightarrow{\simeq}\varphi_{f}{\mathcal{F}}.$

  3. (c)

Assume that $g:W\to V$ is smooth with $f^{\prime}=f\circ
g:W\to{\mathbb{A}}^{1}$ the composition, then we have the smooth pullback
formula for vanishing cycles:

$g^{*}\circ\varphi_{f}=\varphi_{f^{\prime}}\circ g^{*}:D^{b}_{c}(V)\to
D_{c}^{b}(f^{\prime-1}(0_{{\mathbb{A}}^{1}})).$

Parts (a) and (c) follow directly from the definition of these functors [26,
(1.1.5)]; part (b) holds more generally for any ${\mathbb{C}}$-linear exact
functor between triangulated categories.

######  Theorem 4.5.

Assume the divisor $D$ satisfies (a) and (b) of Proposition 4.1.

  1. (a)

The closed embedding (97) can be realized as the critical locus of the
function
$\mu_{\pi,{\mathcal{M}}}:{\mathcal{M}}^{D}_{r,L}(\pi)\to{\mathbb{A}}^{1}$,
_i.e._ , we have

${\mathcal{M}}^{D-p}_{r,L}(\pi)=\mathrm{Crit}\left(\mu_{\pi,{\mathcal{M}}}\right)\hookrightarrow{\mathcal{M}}^{D}_{r,L}(\pi).$

  2. (b)

We have a natural isomorphism

(105)
$\varphi_{\mu_{\pi,{\mathcal{M}}}}\underline{{\mathbb{C}}}=\underline{{\mathbb{C}}}[-r_{0}].$

Here the first and the second $\underline{{\mathbb{C}}}$ stand for the trivial
local systems on ${\mathcal{M}}^{D}_{r,L}(\pi)$ and
${\mathcal{M}}^{D-p}_{r,L}(\pi)$ respectively, and $r_{0}$ is the codimension
of (97).

  3. (c)

For any character $\kappa\in\hat{\Gamma}$, the isomorphism (105) induces a
natural isomorphism

$\varphi_{\mu_{\pi,{\mathcal{A}}}}\left(\mathrm{Rh^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}=\left(\mathrm{Rh^{D-p}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-r_{0}]$

where $\mu_{\pi,{\mathcal{A}}}$ is given in (104) and $r_{0}$ is the same as
in (b).

###### Proof.

By definition, the function
$\mu_{\pi,{\mathcal{M}}}:{\mathcal{M}}^{D}_{r,L}(\pi)\to{\mathbb{A}}^{1}$ is
the composition

${\mathcal{M}}^{D}_{r,L}(\pi)\xrightarrow{\mathrm{ev}_{p}}{\mathcal{M}}_{r,p}(\pi)\xrightarrow{\mu_{1}}{\mathbb{A}}^{1}$

where the first morphism is smooth. Hence we have

$\mathrm{Crit}(\mu_{\pi,{\mathcal{M}}})=\mathrm{Crit}\left(\mu_{1}\circ\mathrm{ev}_{p}\right)=\mathrm{ev}_{p}^{-1}\mathrm{Crit}(\mu_{1})=\mathrm{ev}_{p}^{-1}(0_{H}).$

Here the last identity follows from Lemma 4.3. This implies (a) by Lemma 4.2.