# The P=W conjecture for $\mathrm{GL}_{n}$

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

We prove the $P=W$ conjecture for $\mathrm{GL}_{n}$ for all ranks $n$ and
curves of arbitrary genus $g\geq 2$. The proof combines a strong perversity
result on tautological classes with the curious Hard Lefschetz theorem of
Mellit. For the perversity statement, we apply the vanishing cycles
constructions in our earlier work to global Springer theory in the sense of
Yun, and prove a parabolic support theorem.

###### Contents

  1. 0 Introduction
  2. 1 Perverse filtrations and vanishing cycles
  3. 2 Strong perversity for Chern classes
  4. 3 Global Springer theory
  5. 4 Parabolic support theorem

##  0\. Introduction

Throughout, we work over the complex numbers ${\mathbb{C}}$.

###  0.1. The P=W conjecture

The purpose of this paper is to present a proof of the $P=W$ conjecture by de
Cataldo–Hausel–Migliorini [6] for arbitrary rank $n$ and genus $g\geq 2$.

Let $C$ be a nonsingular irreducible projective curve of genus $g\geq 2$. For
two coprime integers $n\in{\mathbb{Z}}_{\geq 1}$ and $d\in{\mathbb{Z}}$, there
are two moduli spaces $M_{\mathrm{Dol}}$ and $M_{B}$, called the Dolbeault and
the Betti moduli spaces, attached to $C,n,$ and $d$.

The Dolbeault moduli space parameterizes stable Higgs bundles
$({\mathcal{E}},\theta)$ on $C$, where ${\mathcal{E}}$ is a vector bundle on
$C$ of rank $n$ and degree $d$,
$\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\Omega_{C}^{1}$ is a Higgs field,
and stability is defined with respect to the slope
$\mu({\mathcal{E}},\theta)={\mathrm{deg}({\mathcal{E}})}/{\mathrm{rk({\mathcal{E}})}}$.
The moduli space $M_{\mathrm{Dol}}$ admits the structure of a completely
integrable system

$h:M_{\mathrm{Dol}}\to A:=\bigoplus_{i=1}^{n}H^{0}(C,{\Omega_{C}^{1}}^{\otimes
i}),\quad({\mathcal{E}},\theta)\mapsto\mathrm{char.polynomial}(\theta),$

which is referred to as the Hitchin system [19, 20]. The Hitchin map $h$ is
surjective and proper; it is also Lagrangian with respect to the canonical
holomorphic symplectic form on $M_{\mathrm{Dol}}$ induced by the hyper-Kähler
metric. The _perverse filtration_ is an increasing filtration

$P_{0}H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}})\subset
P_{1}H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}})\subset\cdots\subset
H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}})$

on the (singular) cohomology of $M_{\mathrm{Dol}}$ governed by the topology of
the Hitchin system $h$; see Section 1.1 for a brief review.

The Betti moduli space $M_{B}$ is the (twisted) character variety associated
with $\mathrm{GL}_{n}({\mathbb{C}})$ and degree $d$. It parametrizes
isomorphism classes of irreducible local systems

$\rho:\pi_{1}(C\backslash\\{p\\})\rightarrow\mathrm{GL}_{n}({\mathbb{C}})$

where $\rho$ sends a loop around a chosen point $p$ to
$e^{\frac{2\pi\sqrt{-1}d}{n}}\mathrm{Id}_{n}$. Concretely, we have

$M_{B}:=\Big{\\{}a_{k},b_{k}\in\mathrm{GL}_{n}({\mathbb{C}}),~{}k=1,2,\dots,g:~{}~{}\prod_{j=1}^{g}[a_{j},b_{j}]=e^{\frac{2\pi\sqrt{-1}d}{n}}\mathrm{Id}_{n}\Big{\\}}\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}\mathrm{GL}_{n}({\mathbb{C}}).$

It is an affine variety whose mixed Hodge structure admits a nontrivial weight
filtration

$W_{0}H^{*}(M_{B},{\mathbb{Q}})\subset
W_{1}H^{*}(M_{B},{\mathbb{Q}})\subset\cdots\subset H^{*}(M_{B},{\mathbb{Q}}).$

Non-abelian Hodge theory [34, 35] gives a canonical diffeomorphism between the
two very different algebraic varieties $M_{\mathrm{Dol}}$ and $M_{B}$, which
identifies their cohomology:

(1) $H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}})=H^{*}(M_{B},{\mathbb{Q}}).$

The $P=W$ conjecture by de Cataldo–Hausel–Migliorini [6] refines the
identification (1); it predicts that the perverse filtration associated with
the Hitchin system is matched with the (double-indexed) weight filtration
associated with the Betti moduli space. This establishes a surprising
connection between topology of Hitchin systems and Hodge theory of character
varieties.

######  Conjecture 0.1.

[The P=W conjecture for $\mathrm{GL}_{n}$, [6]] For any
$k,m\in{\mathbb{Z}}_{\geq 0}$, we have

$P_{k}H^{m}(M_{\mathrm{Dol}},{\mathbb{Q}})=W_{2k}H^{m}(M_{B},{\mathbb{Q}})=W_{2k+1}H^{m}(M_{B},{\mathbb{Q}}).$

Conjecture 0.1 has previously been proven for $n=2$ and arbitrary genus $g\geq
2$ by de Cataldo–Hausel–Migliorini [6], and recently for arbitrary rank $n$
and genus 2 by de Cataldo–Maulik–Shen [9]. The compatibility between the $P=W$
conjecture and Galois conjugation on the Betti side was proven in [11], which
implies that $P=W$ does not depend on the degree $d$ as long as it is coprime
to $n$. We refer to [6, Section 1] and the paragraphs following [9, Theorem
0.2] for discussions concerning connections between Conjecture 0.1 and other
directions. In particular, by [3] and [26, Section 9.3], Conjecture 0.1
implies the correspondence between Gopakumar–Vafa invariants and
Pandharipande–Thomas invariants [26, Conjecture 3.13] for the local Calabi–Yau
3-fold $T^{*}C\times{\mathbb{C}}$ in any curve class $n[C]$.

The main result of this paper is a full proof of Conjecture 0.1.

######  Theorem 0.2.

Conjecture 0.1 holds.

Conjecture 0.1 has several variants which, to the best of our knowledge, are
still open. These include the version formulated for possibly singular moduli
spaces, intersection cohomology, and general reductive groups [6, 7, 15], and
the version formulated for moduli stacks [12]. There also exist parabolic
versions of Conjecture 0.1 and we expect that our argument applies to these
settings using parabolic variants of the ingredients here [28, 31].

For the case of type $A_{n-1}$
($\mathrm{GL}_{n},\mathrm{PGL}_{n},\mathrm{SL}_{n}$) and a degree $d$ coprime
to $n$, Conjecture 0.1 (_i.e._ the $P=W$ conjecture for $\mathrm{GL_{n}}$) is
equivalent to the $P=W$ conjecture for $\mathrm{PGL}_{n}$; see the paragraph
after [9, Theorem 0.2]. The case of $\mathrm{SL}_{n}$ is more subtle — it is
closely related to the endoscopic decomposition of the $\mathrm{SL}_{n}$
Hitchin moduli spaces. A systematic discussion on this aspect can be found in
[23, Section 5]. In the special case when $n=p$ a prime number, [10, Theorem
0.2] implies that the $P=W$ conjecture for $\mathrm{GL}_{p}$,
$\mathrm{SL}_{p}$, and $\mathrm{PGL}_{p}$ are all equivalent. In particular,
we have the following immediate consequence of Theorem 0.2, which generalizes
the result of [6] for $\mathrm{SL}_{2}$.

######  Theorem 0.3.

The $P=W$ conjecture holds for a curve $C$ of any genus at least two and
$\mathrm{SL}_{p}$ with $p$ a prime number.

###  0.2. Idea of the proof

Our proof of Conjecture 0.1 has 4 major steps.

  1. Step 1.

_Strong perversity of Chern classes_ : Using work of Markman [22], Shende
[33], and Mellit [28], we may reduce the $P=W$ conjecture to a statement about
the interaction between Chern classes of the universal family and the perverse
filtration. This allows us to reduce the $P=W$ conjecture using a sheaf-
theoretic statement which concerns strong perversity of Chern classes.

  2. Step 2.

_Vanishing cycle techniques_ : We apply the formalism of vanishing cycles to
reduce the sheaf-theoretic formulation of Step 1 for the Hitchin system, to
the case of twisted Hitchin systems associated with meromorphic Higgs bundles.
Our motivation is from the key observation by Ngô [30] and Chaudouard-Laumon
[2], that the decomposition theorem for such twisted Hitchin systems is more
manageable.

Steps 1 and 2 are carried out in Sections 1 and 2.

  3. Step 3.

_Global Springer theory_ : The global Springer theory of Yun [37, 38, 39]
produces rich symmetries for certain Hitchin moduli spaces; this proves the
sheaf-theoretic formulation of Step 1 over the _elliptic locus_. For our
purpose, we need a version of global Springer theory for the stable locus over
the total Hitchin base. This is described in Section 3.

  4. Step 4.

_A support theorem_ : Lastly, in order to extend part of Yun’s symmetries
induced by Chern classes from the elliptic locus to the total Hitchin base, we
prove a support theorem parallel to [2] for certain parabolic Hitchin moduli
spaces. This is completed in Section 4.

The idea of combining vanishing cycle functors and support theorems was also
applied in [23] to give a proof of the topological mirror symmetry conjecture
of Hausel–Thaddeus [17, 16].

###  0.3. Acknowledgements

We would like to thank Mark Andrea de Cataldo, Bhargav Bhatt, and Max Lieblich
for various discussions. We are especially grateful to Zhiwei Yun for
explaining his thesis to us, both in his office and at the playground. J.S.
was supported by the NSF grant DMS-2134315.

##  1\. Perverse filtrations and vanishing cycles

In this section, we introduce the notion of _strong perversity_ and show its
compatibility with vanishing cycle functors (Proposition 1.4). This plays a
crucial role in Section 2 in lifting the $P=W$ conjecture sheaf-theoretically
and reduce it to the twisted case.

###  1.1. Perverse filtrations

Let $f:X\to Y$ be a proper morphism between irreducible nonsingular quasi-
projective varieties (or Deligne–Mumford stacks) with $\mathrm{dim}X=a$ and
$\mathrm{dim}Y=b$. Let $r$ be the defect of semismallness of $f$:

$r:=\mathrm{dim}X\times_{Y}X-\mathrm{dim}X.$

In particular, we have $r=a-b$ when $f$ has equi-dimensional fibers. The
perverse filtration

$P_{0}H^{m}(X,{\mathbb{Q}})\subset
P_{1}H^{m}(X,{\mathbb{Q}})\subset\dots\subset H^{m}(X,{\mathbb{Q}})$

is an increasing filtration on the cohomology of $X$ governed by the topology
of the morphism $f$; it is defined to be

$P_{i}H^{m}(X,{\mathbb{Q}}):=\mathrm{Im}\left\\{H^{m-a+r}(X,{{}^{\mathbf{p}}\tau_{\leq
i}}(Rf_{*}{\mathbb{Q}}_{X}[a-r]))\to H^{m}(X,{\mathbb{Q}})\right\\}$

where ${}^{\mathbf{p}}\tau_{\leq*}$ is the perverse truncation functor [1].

A cohomology class $\gamma\in H^{l}(X,{\mathbb{Q}})$ can be viewed as a
morphism $\gamma:{\mathbb{Q}}_{X}\to{\mathbb{Q}}_{X}[l]$, which naturally
induces

(2) $\gamma:Rf_{*}{\mathbb{Q}}_{X}\to Rf_{*}{\mathbb{Q}}_{X}[l]$

after pushing forward along $f$. For an integer $c\geq 0$, we say that
$\gamma\in H^{l}(X,{\mathbb{Q}})$ has _strong perversity_ $c$ with respect to
$f$ if its induced morphism (2) satisfies

(3) $\gamma\left({{}^{\mathbf{p}}\tau_{\leq
i}Rf_{*}{\mathbb{Q}}_{X}}\right)\subset{{}^{\mathbf{p}}\tau_{\leq
i+(c-l)}}\left(Rf_{*}{\mathbb{Q}}_{X}[l]\right),\quad\forall i;$

more precisely, the condition (3) says that the composition

${{}^{\mathbf{p}}\tau_{\leq i}Rf_{*}{\mathbb{Q}}_{X}}\hookrightarrow
Rf_{*}{\mathbb{Q}}_{X}\xrightarrow{\gamma}Rf_{*}{\mathbb{Q}}_{X}[l]$

factors through ${{}^{\mathbf{p}}\tau_{\leq
i+(c-l)}}\left(Rf_{*}{\mathbb{Q}}_{X}[l]\right)\hookrightarrow
Rf_{*}{\mathbb{Q}}_{X}[l]$ for any $i$. Notice that $\gamma$ automatically has
strong perversity $l$, so this condition is interesting only when $c<l$.

######  Lemma 1.1.

If $\gamma\in H^{l}(X,{\mathbb{Q}})$ has strong perversity $c$ with respect to
$f$, then taking cup-product with $\gamma$ satisfies

$\gamma\cup-:H^{m}(X,{\mathbb{Q}})\to H^{m+l}(X,{\mathbb{Q}}),\quad
P_{i}H^{m}(X,{\mathbb{Q}})\mapsto P_{i+c}H^{m+l}(X,{\mathbb{Q}}).$

###### Proof.

This follows from taking global cohomology for (3) and noticing that

$\displaystyle H^{m-a+r}\left(Y,{{}^{\mathbf{p}}\tau_{\leq
i+(c-l)}}\left(Rf_{*}{\mathbb{Q}}_{X}[l+a-r]\right)\right)$
$\displaystyle=H^{(m-a+r)+l}$ $\displaystyle\left(Y,{{}^{\mathbf{p}}\tau_{\leq
i+c}}\left(Rf_{*}{\mathbb{Q}}_{X}[a-r]\right)\right).\qed$

In general, the perverse filtration $P_{\bullet}H^{*}(X,{\mathbb{Q}})$ may not
be multiplicative, _i.e._ , for $\gamma_{j}\in P_{c_{j}}H^{*}(X,{\mathbb{Q}})$
(j=1,2,…,s), it may not be true that

$\gamma_{1}\cup\gamma_{2}\cup\cdots\cup\gamma_{s}\in
P_{c_{1}+\cdots+c_{s}}H^{*}(X,{\mathbb{Q}});$

see [5, Exercise 5.6.8]. In fact, it was proven in [9, Theorem 0.6] that
Conjecture 0.1 is equivalent to the multiplicativity of the perverse
filtration associated with the Hitchin system. The following easy observation
illustrates the advantage of considering strong perversity in view of the
multiplicativity issue.

######  Lemma 1.2.

If the class $\gamma_{j}\in H^{l_{j}}(X,{\mathbb{Q}})$ ($j=1,2,\dots,s$) has
strong perversity $c_{j}$ with respect to $f$, then the cup product

$\gamma_{1}\cup\gamma_{2}\cup\cdots\cup\gamma_{s}\in
H^{l_{1}+\cdots+l_{s}}(X,{\mathbb{Q}})$

has strong perversity $\sum_{j}c_{j}$.

###  1.2. Vanishing cycles

Throughout section 1.2, we let $g:X\to{\mathbb{A}}^{1}$ be a morphism such
that $X$ is nonsingular and irreducible with $X_{0}=g^{-1}(0)$ the closed
fiber over $0\in{\mathbb{A}}^{1}$. We consider the vanishing cycle functor

$\varphi_{g}:D^{b}_{c}(X)\rightarrow D^{b}_{c}(X_{0})$

which preserves the perverse $t$-structures,

$\varphi_{g}:\mathrm{Perv}(X)\rightarrow\mathrm{Perv}(X_{0}).$

Here $\mathrm{Perv}(-)$ stands for the abelian category of perverse sheaves.
We denote by

$\varphi_{g}:=\varphi_{g}(\mathrm{IC}_{X})=\varphi_{g}({\mathbb{Q}}_{X}[\mathrm{dim}X])\in\mathrm{Perv}(X_{0})$

the perverse sheaf of vanishing cycles. We use $X^{\prime}\subset X$ to denote
the support of the vanishing cycle complex $\varphi_{g}$ so that
$\varphi_{g}\in\mathrm{Perv}(X^{\prime})$.

Recall that for any bounded constructible object ${\mathcal{K}}\in
D^{b}_{c}(X)$, a class $\gamma\in H^{l}(X,{\mathbb{Q}})$ induces a morphism

$\gamma:{\mathcal{K}}\to{\mathcal{K}}[l]$

via taking the tensor product with
$\gamma:{\mathbb{Q}}_{X}\to{\mathbb{Q}}_{X}[l]$. The following lemma shows the
compatibility between the vanishing cycle functor and restriction of
cohomology classes.

######  Lemma 1.3.

With the same notation as above, let $i:X^{\prime}\hookrightarrow X$ be the
closed embedding. The morphism

$i^{*}\gamma:\varphi_{g}\rightarrow\varphi_{g}[l]\in D^{b}_{c}(X^{\prime})$

induced by the class $i^{*}\gamma$ (applied to the object
${\mathcal{K}}=\varphi_{g}$) coincides with

$\varphi_{g}(\gamma):\varphi_{g}\to\varphi_{g}[l]\in D_{c}^{b}(X^{\prime})$

obtained by applying the functor $\varphi_{g}$ to
$\gamma:{\mathbb{Q}}_{X}\to{\mathbb{Q}}_{X}[l]$.

###### Proof.

Let $\iota:X_{0}\hookrightarrow X$ be the closed embedding of the closed fiber
over $0$. By [21, Definition 8.6.2], the vanishing cycle functor can be
written as

(4) $\varphi_{g}(-)=\iota^{*}R{\mathcal{H}}{om}({\mathcal{C}},-)\in
D^{b}_{c}(X_{0})$

with ${\mathcal{C}}$ a fixed complex of sheaves. The morphism obtained by
applying $R{\mathcal{H}}{om}({\mathcal{C}},-)$ to
$\gamma:{\mathbb{Q}}_{X}\to{\mathbb{Q}}_{X}[l]$ is equivalent to the morphism
induced by applying $\gamma$ to
$R{\mathcal{H}}{om}({\mathcal{C}},{\mathbb{Q}}_{X})$. Similarly, the functor
$\iota^{*}$ sends the morphism induced by $\gamma$ to the morphism induced by
$\iota^{*}\gamma$.

Therefore, if we denote by $\iota^{\prime}:X^{\prime}\hookrightarrow X$ the
closed embedding and view $\varphi_{g}$ as a perverse sheaf on $X^{\prime}$,
we have an equivalent morphisms

(5)
$\varphi_{g}(\gamma)=\iota^{*}\gamma:\iota^{\prime}_{*}\varphi_{g}\to\iota^{\prime}_{*}\varphi_{g}[l]\in
D^{b}_{c}(X_{0}).$

Finally, after applying $\iota^{\prime*}:D^{b}_{c}(X_{0})\rightarrow
D^{b}_{c}(X^{\prime})$ to (5) and noticing
$\iota^{\prime*}\iota^{\prime}_{*}=\mathrm{id}$, we obtain that the class
$i^{*}\gamma=\iota^{\prime*}\iota^{*}\gamma$ induces

$\varphi_{g}(\gamma):\varphi_{g}\to\varphi_{g}[l]\in D_{c}^{b}(X^{\prime}).$

This completes the proof. ∎

######  Proposition 1.4.

Let $g:X\to{\mathbb{A}}^{1}$ and $X^{\prime}$ be as above. Assume that
$X^{\prime}$ is nonsingular and

(6)
$\varphi_{g}\simeq\mathrm{IC}_{X^{\prime}}={\mathbb{Q}}_{X^{\prime}}[\mathrm{dim}X^{\prime}]\in\mathrm{Perv}(X^{\prime}).$

Assume further that we have the commutative diagram

${X^{\prime}}$${X}$${Y^{\prime}}$${Y}$$\scriptstyle{i}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$

such that $f$ is proper and $g=\nu\circ f$ with $\nu:Y\to{\mathbb{A}}^{1}$.
Then if a class $\gamma\in H^{l}(X,{\mathbb{Q}})$ has strong perversity $c$
with respect to $f$, its restriction $i^{*}\gamma\in
H^{l}(X^{\prime},{\mathbb{Q}})$ has strong perversity $c$ with respect to
$f^{\prime}$.

###### Proof.

By definition, the morphism

(7) $\gamma:Rf_{*}{\mathbb{Q}}_{X}\to Rf_{*}{\mathbb{Q}}_{X}[l]\in
D^{b}_{c}(Y)$

induced by $\gamma$ satisfies

(8) $\gamma\left({{}^{\mathbf{p}}\tau_{\leq
i}Rf_{*}{\mathbb{Q}}_{X}}\right)\subset{{}^{\mathbf{p}}\tau_{\leq
i+(c-l)}}(Rf_{*}{\mathbb{Q}}_{X}[l]),\quad\forall i.$

Now we apply the vanishing cycle functor $\varphi_{\nu}$ to (7). On one hand,
we have the base change
$Rf^{\prime}_{*}\circ\varphi_{g}\simeq\varphi_{\nu}\circ Rf_{*}$ and the fact
that the vanishing cycle functor preserves the perverse $t$-structures.
Therefore (8) implies that the morphism

$\varphi_{g}(\gamma):Rf^{\prime}_{*}\varphi_{g}\to
Rf^{\prime}_{*}\varphi_{g}[l]\in D^{b}_{c}(X^{\prime})$

satisfies

$\varphi_{g}(\gamma)\left({{}^{\mathbf{p}}\tau_{\leq
i}}Rf^{\prime}_{*}\varphi_{g}\right)\subset{{}^{\mathbf{p}}\tau_{\leq
i+(c-l)}}(Rf^{\prime}_{*}\varphi_{g}[l]),\quad\forall i.$

On the other hand, using the isomorphism (6) and Lemma 1.3, the above equation
means precisely that the morphism
$i^{*}\gamma:{\mathbb{Q}}_{X^{\prime}}\to{\mathbb{Q}}_{X^{\prime}}[l]$
satisfies

$(i^{*}\gamma)\left({{}^{\mathbf{p}}\tau_{\leq
i}Rf^{\prime}_{*}{\mathbb{Q}}_{X^{\prime}}}\right)\subset{{}^{\mathbf{p}}\tau_{\leq
i+(c-l)}}(Rf^{\prime}_{*}{\mathbb{Q}}_{X^{\prime}}[l]),\quad\forall i;$

that is, the class $i^{*}\gamma$ has strong perversity $c$ with respect to
$f^{\prime}$. ∎

##  2\. Strong perversity for Chern classes

In this section, we fix the rank $n$ and the degree $d$ with $(n,d)=1$. In
Theorem 2.6, we rephrase and then enhance Conjecture 0.1 to a statement
involving ${\mathcal{L}}$-twisted Hitchin systems and strong perversity of
Chern classes. It will be proven in Sections 3 and 4.

###  2.1. Tautological classes

As discussed in [9, Section 0.3], the $P=W$ conjecture for $\mathrm{GL}_{n}$
can be reduced to a statement involving tautological classes on
$M_{\mathrm{Dol}}$ and the perverse filtration associated with
$h:M_{\mathrm{Dol}}\to A$, without reference to the Betti moduli $M_{B}$. In
this subsection, we recall this reduction step.

For convenience, we work with the $\mathrm{PGL}_{n}$ Dolbeault moduli space to
avoid normalization of a universal family as in [9]; we refer to [33] for a
detailed discussion concerning the formulation of the $P=W$ conjecture in
terms of tautological classes for the $\mathrm{PGL}_{n}$ Dolbeault moduli
space.

Fix ${\mathcal{N}}\in\mathrm{Pic}^{d}(C)$. Let $\widecheck{M}_{\mathrm{Dol}}$
be the moduli stack of stable Higgs bundles $({\mathcal{E}},\theta)$ with
$\mathrm{det}({\mathcal{E}})\simeq{\mathcal{N}}$ and
$\mathrm{trace}(\theta)=0$, rigidified with respect to the generic
$\mu_{n}$-stabilizer; this is the same as taking its coarse moduli space. We
refer to this (nonsingular) variety as the $\mathrm{SL}_{n}$ Dolbeault moduli
space of degree $d$. The finite group $\Gamma:=\mathrm{Pic}^{0}(C)[n]$ acts
naturally on $\widecheck{M}_{\mathrm{Dol}}$ via tensor product. The
$\mathrm{PGL}_{n}$ Dolbeault moduli space of degree $d$ is recovered by taking
the quotient stack

(9) $\widehat{M}_{\mathrm{Dol}}:=\widecheck{M}_{\mathrm{Dol}}/\Gamma$

which is a nonsingular Deligne–Mumford stack. The $\mathrm{PGL}_{n}$ Hitchin
system

$\widehat{h}:\widehat{M}_{\mathrm{Dol}}\to\widehat{A}:=\bigoplus_{i=2}^{n}H^{0}(C,{\Omega_{C}^{1}}^{\otimes
i})$

is induced by the Hitchin map associated with $\widecheck{M}_{\mathrm{Dol}}$
as the $\Gamma$-action is fiberwise with respect to $h$. Analogous to the
$\mathrm{GL}_{n}$ case, we have the perverse filtration
$P_{*}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$ associated with
$\widehat{h}$. The universal $\mathrm{PGL}_{n}$-bundle ${\mathcal{U}}$ on
$C\times\widehat{M}_{\mathrm{Dol}}$ induces Chern characters

$\mathrm{ch}_{k}({\mathcal{U}})\in
H^{2k}(C\times\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}}),\quad k\geq 2.$

The tautological classes $c_{k}(\gamma)$ are defined to be

$c_{k}(\gamma):=\int_{\gamma}\mathrm{ch}_{k}({\mathcal{U}})=q_{M*}(q_{C}^{*}\gamma\cup\mathrm{ch}_{k}({\mathcal{U}}))\in
H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}}),\quad\gamma\in
H^{*}(C,{\mathbb{Q}}),$

where $q_{(-)}$ are the projections from $C\times\widehat{M}_{\mathrm{Dol}}$.

Now we consider the $\mathrm{PGL}_{n}$ Betti moduli space of degree $d$ with
the isomorphism on cohomology provided by non-abelian Hodge theory (_c.f._[6,
Theorem 1.2.4]):

(10)
$H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})=H^{*}(\widehat{M}_{B},{\mathbb{Q}}).$

Define the Hodge sub-vector space

${}^{k}\mathrm{Hdg}^{m}(\widehat{M}_{B}):=W_{2k}H^{m}(M_{B},{\mathbb{Q}})\cap
F^{k}H^{m}(M_{B},{\mathbb{C}})\subset H^{m}(\widehat{M}_{B},{\mathbb{Q}}).$

The following theorem, collecting results of Markman and Shende, provides a
complete description of $H^{*}(\widehat{M}_{B},{\mathbb{Q}})$ in terms of the
Chern classes $\mathrm{ch}_{k}({\mathcal{U}})$ and the weight filtration.

######  Theorem 2.1 ([22, 33]).

We use the same notation as above.

  1. (i)

The tautological classes $c_{k}(\gamma)\in
H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$ generate
$H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$ as a ${\mathbb{Q}}$-algebra.

  2. (ii)

The class $c_{k}(\gamma)$, passing through the non-abelian Hodge
correspondence (10), lies in ${}^{k}\mathrm{Hdg}^{*}(\widehat{M}_{B})$. In
particular, we have a canonical decomposition

$H^{*}(\widehat{M}_{B},{\mathbb{Q}})=\bigoplus_{m,k}{{}^{k}\mathrm{Hdg}^{m}(\widehat{M}_{B})}.$

###### Proof.

The first part was proven in [22], and the second part was proven in [33]. ∎

Theorem 2.1 (ii) yields immediately that

$W_{2k}H^{m}(\widehat{M}_{B},{\mathbb{Q}})=W_{2k+1}H^{m}(\widehat{M}_{B},{\mathbb{Q}}).$

Moreover, by Theorem 2.1, the $P=W$ conjecture implies that each class
$\prod_{i=1}^{s}c_{k_{i}}(\gamma_{i})$ lies in the perverse piece
$P_{\Sigma_{i}{k_{i}}}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$; the
latter is in fact equivalent to the $P=W$ conjecture. Indeed, suppose we have

(11) $\prod_{i=1}^{s}c_{k_{i}}(\gamma_{i})\in
P_{\Sigma_{i}{k_{i}}}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$

for any product of tautological classes. Then we know that
$W_{2k}H^{*}(\widehat{M}_{B},{\mathbb{Q}})\subset
P_{k}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$. The curious hard
Lefschetz theorem proven by Mellit [28] forces the two filtrations
$P_{\bullet}$ and $W_{2\bullet}$ to coincide as long as one contains the
other. This was mentioned in the last paragraph of [28, Section 1]; we include
its proof here for the reader’s convenience.

######  Lemma 2.2.

We denote by $V$ the ${\mathbb{Q}}$-vector space (10) with the perverse and
the weight filtrations $P_{\bullet}$ and $W_{2\bullet}$. If $W_{2k}\subset
P_{k}$ for all $k$, then $W_{2k}=P_{k}$ for all $k$.

###### Proof.

Assume $r=\mathrm{dim}\widehat{M}$; both filtrations terminate at the $r$-th
pieces, _i.e._ , $W_{2r}=P_{r}=V$. We first show that

$W_{0}=P_{0},\quad W_{2(r-1)}=P_{r-1}.$

In fact, since $W_{\bullet}\subset P_{\bullet}$, we have

$\mathrm{dim}W_{0}\leq\mathrm{dim}P_{0}=\mathrm{dim}V/P_{r-1}\leq\mathrm{dim}V/W_{2(r-1)};$

moreover, by the curious hard Lefschetz theorem, each inequality has to be an
equality. So our claim follows.

Then we proceed by applying the same argument to $W_{1},P_{1}$ and
$W_{2(r-2)},P_{r-1}$. The lemma follows by a simple induction. ∎

In conclusion, we have reduced the $P=W$ conjecture to the following:

######  Conjecture 2.3 (Equivalent version of P=W).

Condition (11) holds for all products of tautological classes.

###  2.2. Strong perversity for Chern classes for ${\mathcal{L}}$-twisted
Hitchin systems

For our purposes, it is important to consider Dolbeault moduli spaces of Higgs
bundles, twisted by an effective line bundle ${\mathcal{L}}$ (_i.e._ ,
$H^{0}(C,{\mathcal{L}})\neq 0$). These moduli spaces have already appeared in
[2, 18, 17, 37, 38]; we review the construction here briefly.

Set $\Omega_{\mathcal{L}}$ to be the line bundle
$\Omega^{1}_{C}\otimes{\mathcal{L}}$ on the curve $C$. We denote by
$M^{\mathcal{L}}_{\mathrm{Dol}}$ the moduli space of stable twisted Higgs
bundles

$({\mathcal{E}},\theta),\quad\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\Omega_{\mathcal{L}},\quad\mathrm{rk}({\mathcal{E}})=n,~{}~{}\mathrm{deg}({\mathcal{E}})=d,$

with respect to the slope stability condition. The corresponding Hitchin map

$h^{\mathcal{L}}:M^{\mathcal{L}}_{\mathrm{Dol}}\to
A^{\mathcal{L}}:=\bigoplus_{i=1}^{n}H^{0}\left(C,{\Omega_{\mathcal{L}}}^{\otimes
i}\right),\quad({\mathcal{E}},\theta)\mapsto\mathrm{char.polynomial}(\theta),$

is still proper as in the untwisted case; but it fails to be a Lagrangian
fibration when $\mathrm{deg}({\mathcal{L}})>0$. The ${\mathcal{L}}$-twisted
$\mathrm{SL}_{n}$ and $\mathrm{PGL}_{n}$ Dolbeault moduli spaces
$\widecheck{M}^{\mathcal{L}}_{\mathrm{Dol}}$ and
$\widehat{M}^{\mathcal{L}}_{\mathrm{Dol}}$ can be constructed similarly. The
moduli space

$\widecheck{M}^{\mathcal{L}}_{\mathrm{Dol}}:=\\{({\mathcal{E}},\theta)\in
M^{\mathcal{L}}_{\mathrm{Dol}}|~{}~{}\mathrm{det}({\mathcal{E}})\simeq{\mathcal{N}}\in\mathrm{Pic}^{d}(C),\mathrm{trace}(\theta)=0\\}$

admits a Hitchin map

$\widecheck{h}^{\mathcal{L}}:\widecheck{M}^{\mathcal{L}}_{\mathrm{Dol}}\to\widehat{A}^{\mathcal{L}}:=\bigoplus_{i=2}^{n}H^{0}\left(C,{\Omega_{\mathcal{L}}}^{\otimes
i}\right)$

and a fiberwise $\Gamma=\mathrm{Pic}^{0}(C)[n]$ action by tensor product.
Taking the $\Gamma$-quotient recovers the ${\mathcal{L}}$-twisted
$\mathrm{PGL}_{n}$ Hitchin map

$\widehat{h}^{\mathcal{L}}:\widehat{M}^{\mathcal{L}}_{\mathrm{Dol}}=\widecheck{M}^{\mathcal{L}}_{\mathrm{Dol}}/\Gamma\to\widehat{A}^{\mathcal{L}}.$

An observation in [23, Section 4] is that, for a fixed closed point $p\in C$,
the ${\mathcal{L}}$-twisted and ${\mathcal{L}}(p)$-twisted $\mathrm{SL}_{n}$
Dolbeault moduli spaces can be related via critical loci and vanishing cycles,
which we recall in the following.

By viewing a ${\mathcal{L}}$-twisted Higgs bundle naturally as a
${\mathcal{L}}(p)$-twisted Higgs bundle, we have the natural embedding
$i:\widecheck{M}^{\mathcal{L}}_{\mathrm{Dol}}\hookrightarrow\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}$
which induces the commutative diagram

(12)
${\widecheck{M}^{\mathcal{L}}_{\mathrm{Dol}}}$${\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}}$${\widehat{A}^{\mathcal{L}}}$${\widehat{A}^{{\mathcal{L}}(p)}.}$$\scriptstyle{i}$$\scriptstyle{\widecheck{h}^{\mathcal{L}}}$$\scriptstyle{\widecheck{h}^{{\mathcal{L}}(p)}}$

We recall the following theorem from [23].

######  Theorem 2.4 ([23] Theorem 4.5).

There exists a regular function
$g:\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}\to{\mathbb{A}}^{1}$
factorized as $g=\nu\circ\widecheck{h}^{{\mathcal{L}}(p)}$ with
$\nu:\widehat{A}^{{\mathcal{L}}(p)}\to{\mathbb{A}}^{1}$ such that

$\varphi_{g}\simeq\mathrm{IC}_{\widecheck{M}^{{\mathcal{L}}}_{\mathrm{Dol}}}={\mathbb{Q}}_{\widecheck{M}^{{\mathcal{L}}}_{\mathrm{Dol}}}[\mathrm{dim}\widecheck{M}^{{\mathcal{L}}}_{\mathrm{Dol}}].$

Since the embedding
$i:\widecheck{M}^{\mathcal{L}}_{\mathrm{Dol}}\hookrightarrow\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}$
is $\Gamma$-equivariant, taking $\Gamma$-quotients in the diagram (12) yields
the following commutative diagram:

(13)
${\widehat{M}^{\mathcal{L}}_{\mathrm{Dol}}}$${\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}}$${\widehat{A}^{\mathcal{L}}}$${\widehat{A}^{{\mathcal{L}}(p)}.}$$\scriptstyle{\widehat{i}}$$\scriptstyle{\widehat{h}^{\mathcal{L}}}$$\scriptstyle{\widehat{h}^{{\mathcal{L}}(p)}}$

######  Proposition 2.5.

The vanishing cycle complex $\varphi_{\widehat{g}}$ associated with the
regular function

$\widehat{g}:=\nu\circ\widehat{h}^{{\mathcal{L}}(p)}:\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}\to{\mathbb{A}}^{1}$

satisfies

$\varphi_{\widehat{g}}\simeq\mathrm{IC}_{\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}}}={\mathbb{Q}}_{\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}}}\left[\mathrm{dim}\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}}\right].$

###### Proof.

We reduce Proposition 2.5 to Theorem 2.4. Consider the $\Gamma$-quotient map
$r:\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}\to\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}$.
The direct image
$r_{*}{\mathbb{Q}}_{\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}}$ admits a
natural $\Gamma$-equivariant structure, whose invariant part recovers

(14)
$\left(r_{*}{\mathbb{Q}}_{\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}}\right)^{\Gamma}\simeq{\mathbb{Q}}_{\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}}.$

Since $g=r\circ\widehat{g}$, we have

$\displaystyle\varphi_{\widehat{g}}$
$\displaystyle=\varphi_{\widehat{g}}\left({\mathbb{Q}}_{\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}}\left[\mathrm{dim}\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}\right]\right)$
$\displaystyle\simeq\varphi_{\widehat{g}}\left(r_{*}{\mathbb{Q}}_{\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}}\left[\mathrm{dim}\widecheck{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}\right]\right)^{\Gamma}$
$\displaystyle\simeq(r_{*}\varphi_{g})^{\Gamma}$
$\displaystyle\simeq{\mathbb{Q}}_{\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}}}\left[\mathrm{dim}\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}}\right].$

Here the first equation follows by definition, the second uses (14), the third
follows from the base change, and the last is given by Theorem 2.4. ∎

Now we formulate a sheaf-theoretic enhancement of Conjecture 2.3.

######  Theorem 2.6.

There exists an effective line bundle ${\mathcal{L}}$ such that the class

$\mathrm{ch}_{k}({\mathcal{U}}^{\mathcal{L}})\in
H^{2k}(C\times\widehat{M}^{\mathcal{L}}_{\mathrm{Dol}},{\mathbb{Q}})$

has strong perversity $k$ with respect to

$\mathbf{h}^{\mathcal{L}}:=\mathrm{id}\times\widehat{h}^{\mathcal{L}}:C\times\widehat{M}^{\mathcal{L}}_{\mathrm{Dol}}\to
C\times\widehat{A}^{\mathcal{L}}.$

Here ${\mathcal{U}}^{\mathcal{L}}$ be the universal $\mathrm{PGL}_{n}$-bundle
over $C\times\widehat{M}_{\mathrm{Dol}}$.

Notice that the perversity bound here is stronger than the trivial one of
$2k$. We prove in the rest of this section that Theorem 2.6 indeed implies
Conjecture 2.3; therefore it implies Conjecture 0.1. We prove Theorem 2.6 in
Sections 3 and 4.

######  Remark 2.7.

Note that although for our purposes, we only need _one_ effective line bundle
${\mathcal{L}}$ satisfying Theorem 2.6, our proof actually shows that it is
true for _any_ effective ${\mathcal{L}}$.

###  2.3. Theorem 2.6 implies Conjecture 2.3

We start with the following claim.

Claim. For a point $p\in C$, if the class

$\mathrm{ch}_{k}({\mathcal{U}}^{{\mathcal{L}}(p)})\in
H^{2k}(C\times\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}},{\mathbb{Q}})$

has strong perversity $k$ with respect to
$\mathbf{h}^{{\mathcal{L}}(p)}:C\times\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}\to
C\times\widehat{A}^{{\mathcal{L}}(p)}$, then the class

$\mathrm{ch}_{k}({\mathcal{U}}^{{\mathcal{L}}})\in
H^{2k}(C\times\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}},{\mathbb{Q}})$

has strong perversity $k$ with respect to
$\mathbf{h}^{{\mathcal{L}}}:C\times\widehat{M}^{\mathcal{L}}_{\mathrm{Dol}}\to
C\times\widehat{A}^{\mathcal{L}}$.

###### Proof of Claim.

We consider the commutative diagram obtained from (13):

(15)
${C\times\widehat{M}^{\mathcal{L}}_{\mathrm{Dol}}}$${C\times\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}}$${C\times\widehat{A}^{\mathcal{L}}}$${C\times\widehat{A}^{{\mathcal{L}}(p)}.}$$\scriptstyle{\mathbf{i}}$$\scriptstyle{\mathbf{h}^{\mathcal{L}}}$$\scriptstyle{\mathbf{h}^{{\mathcal{L}}(p)}}$

Here $\mathbf{i}=\mathrm{id}\times\widehat{i}$. By definition, we have
$\mathbf{i}^{*}{\mathcal{U}}^{{\mathcal{L}}(p)}={\mathcal{U}}^{{\mathcal{L}}}$.
Therefore

$\mathbf{i}^{*}\mathrm{ch}_{k}({\mathcal{U}}^{{\mathcal{L}}(p)})=\mathrm{ch}_{k}({\mathcal{U}}^{\mathcal{L}}).$

We know from Proposition 2.5 that the vanishing cycle complex associated with
the regular function

$\mathbf{g}:=\widehat{g}\circ\mathrm{pr}_{M}:C\times\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}\to\widehat{M}^{{\mathcal{L}}(p)}_{\mathrm{Dol}}\to{\mathbb{A}}^{1}$

satisfies

$\varphi_{\mathbf{g}}\simeq\mathrm{IC}_{C\times\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}}}={{\mathbb{Q}}}_{C\times\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}}}\left[\mathrm{dim}~{}(C\times\widehat{M}^{{\mathcal{L}}}_{\mathrm{Dol}})\right].$

Our claim then follows from Proposition 1.4 (applied to the diagram (15), the
class $\mathrm{ch}_{k}({\mathcal{U}}^{{\mathcal{L}}(p)})$, and the map
$\mathbf{g}$). ∎

As a consequence of the claim, the statement of Theorem 2.6 holds for
${\mathcal{L}}\simeq{\mathcal{O}}_{C}$, _i.e._ , the class

$\mathrm{ch}_{k}({\mathcal{U}})\in
H^{2k}(C\times\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$

has strong perversity $k$ with respect to

$\mathbf{h}:=\mathrm{id}\times\widehat{h}:C\times\widehat{M}_{\mathrm{Dol}}\to
C\times\widehat{A}.$

Finally, to deduce the cohomological statement, we note that since
$\mathbf{h}$ is trivial at the $C$-factor, its induced perverse filtration can
be described as:

(16)
$P_{k}H^{*}(C\times\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})=H^{*}(C,{\mathbb{Q}})\otimes
P_{k}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}}).$

Choose a homogeneous basis of $H^{*}(C,{\mathbb{Q}})$:

$\Sigma=\\{\sigma_{0},\sigma_{1},\dots,\sigma_{2g+2}\\}$

with Poincaré-dual basis $\\{\sigma^{\vee}_{0},\dots,\sigma^{\vee}_{2g+2}\\}$.

We may express $\mathrm{ch}_{k}({\mathcal{U}})$ as

(17)
$\mathrm{ch}_{k}({\mathcal{U}})=\sum_{\sigma\in\Sigma}\sigma^{\vee}\otimes
c_{k}(\sigma)\in H^{*}(C,{\mathbb{Q}})\otimes
H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}}).$

Since $\mathrm{ch}_{k}({\mathcal{U}})$ has strong perversity $k$, Lemma 1.1
implies that

$\mathrm{ch}_{k}({\mathcal{U}})\cup-:P_{s}H^{*}(C\times\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})\to
P_{s+k}H^{*}(C\times\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}}).$

Applying this operator to $\sigma\otimes
P_{s}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})\subset
H^{*}(C\times\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$ with $\sigma\in\Sigma$,
we obtain from (16) and (17) that

$c_{k}(\sigma)\cup-:P_{s}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})\to
P_{s+k}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}}).$

In particular, for any class $\gamma\in H^{*}(C,{\mathbb{Q}})$ we have

$c_{k}(\gamma):P_{s}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})\to
P_{s+k}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$

which further yields

$\prod_{i}c_{k_{1}}(\gamma_{i})=\left(\prod_{i}c_{k_{1}}(\gamma_{i})\right)\cup
1\in P_{\Sigma_{i}k_{i}}H^{*}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}}).$

Here we used $1\in P_{0}H^{0}(\widehat{M}_{\mathrm{Dol}},{\mathbb{Q}})$ in the
last equation.

This completes the proof that Theorem 2.6 implies Conjecture 2.3. ∎

##  3\. Global Springer theory

In this section, we review Yun’s global Springer theory [37, 38, 39] and use
it to deduce Theorem 2.6, assuming a support theorem (Theorem 3.2). Global
Springer theory was previously used to study perverse filtrations for affine
Springer fibers in terms of Chern classes [31, 32]. In our setting, we require
a partial extension of Yun’s results from the elliptic locus to the entire
Hitchin base.

###  3.1. Notations

We fix ${\mathcal{L}}$ to be an effective line bundle of sufficiently large
degree ($\mathrm{deg}\left({\mathcal{L}}\right)>2g$ is enough for us). Since
from now on we only concern the ${\mathcal{L}}$-twisted moduli spaces, we will
use $M,\widecheck{M}$, and $\widehat{M}$ to denote the ${\mathcal{L}}$-twisted
$\mathrm{GL}_{n}$, $\mathrm{SL}_{n}$, and $\mathrm{PGL}_{n}$ Dolbeault moduli
spaces
$M^{\mathcal{L}}_{\mathrm{Dol}},\widecheck{M}^{\mathcal{L}}_{\mathrm{Dol}}$,
and $\widehat{M}^{\mathcal{L}}_{\mathrm{Dol}}$ respectively. For the same
reason, we will uniformly use the term _Higgs bundles_ to call
${\mathcal{L}}$-twisted Higgs bundles.

From now on we let $G=\mathrm{PGL}_{n}$, $\mathfrak{g}$ its Lie algebra,
$B\subset G$ the Borel subgroup induced by upper triangular matrices with
$\mathfrak{b}$ the Lie algebra, $T\subset B$ the maximal torus given by
diagonal matrices, and $W\simeq\mathfrak{S}_{n}$ the Weyl group. We denote by
${\mathbb{X}}^{*}(T)$ the character group of $T$; it is isomorphic to
${\mathbb{Z}}^{n-1}$ as an abelian group.

###  3.2. Parabolic moduli stacks

Let $\widehat{{\mathfrak{M}}}$ be the moduli stack of $G$-Higgs bundles on
$C$; we do not impose any stability condition on $\widehat{{\mathfrak{M}}}$ so
that it is only a (singular) algebraic stack. The stable locus of
$\widehat{{\mathfrak{M}}}$ is a nonsingular Deligne–Mumford substack

$\widehat{M}\hookrightarrow\widehat{{\mathfrak{M}}}.$

Yun’s global Springer theory [37] constructs an algebraic stack
$\widehat{{\mathfrak{M}}}^{\mathrm{par}}$ over
$C\times\widehat{{\mathfrak{M}}}$:

(18) $\pi:\widehat{{\mathfrak{M}}}^{\mathrm{par}}\to
C\times\widehat{{\mathfrak{M}}}$

which is a global analog of the Grothendieck simultaneous resolution. There
are two equivalent constructions of $\widehat{{\mathfrak{M}}}^{\mathrm{par}}$
given in [37, Section 2.1].

The first is to construct $\widehat{{\mathfrak{M}}}^{\mathrm{par}}$ as the
moduli stack of parabolic Higgs bundles, which are quadruples
$(x,{\mathcal{E}},\theta,{\mathcal{E}}_{x}^{B})$, with
$({\mathcal{E}},\theta)$ is a $G$-Higgs bundle, $x\in C$ a closed point, and
${\mathcal{E}}_{x}^{B}$ a $B$-reduction of ${\mathcal{E}}$ at $x$, satisfying
the constraint that $\theta$ is compatible with ${\mathcal{E}}_{x}^{B}$; see
[37, Definition 2.1.1]. Then the morphism $\pi$ is given by forgetting the
$B$-reduction:

$\pi(x,{\mathcal{E}},\theta,{\mathcal{E}}_{x}^{B})=(x,({\mathcal{E}},\theta))\in
C\times\widehat{{\mathfrak{M}}}.$

The second construction is via the Grothendieck simultaneous resolution
$\pi_{G}:[\mathfrak{b}/B]\to[\mathfrak{g}/G]$. More precisely, let
$\rho_{\mathcal{L}}$ be the ${\mathbb{G}}_{m}$-torsor over $C$ associated with
the line bundle $\Omega_{\mathcal{L}}$. Denote by
$[\mathfrak{g}/G]_{\mathcal{L}}$ (resp. $[\mathfrak{b}/B]_{\mathcal{L}}$) the
family of $[\mathfrak{g}/G]$ (resp. $[\mathfrak{b}/B]$) over the curve $C$
twisted by the torsor $\rho_{\mathcal{L}}$. We have a tautological evaluation
map

(19)
${C\times\widehat{{\mathfrak{M}}}}$${{[\mathfrak{g}/G]_{\mathcal{L}}}}$${C}$$\scriptstyle{\mathrm{ev}}$

which is a natural $C$-morphism; after base change to a closed point $x\in C$,
the map $\mathrm{ev}$ sends a $G$-Higgs bundle to the evaluation of its Higgs
field at $x$. The morphism (18) is then induced by the base change of the
Grothendieck simultaneous resolution along the evaluation map (see [37, Lemma
2.1.2]):

${\widehat{{\mathfrak{M}}}^{\mathrm{par}}}$${{[\mathfrak{b}/B]_{\mathcal{L}}}}$${C\times\widehat{{\mathfrak{M}}}}$${{[\mathfrak{g}/G]_{\mathcal{L}}}.}$$\scriptstyle{\mathrm{ev}^{p}}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{G}}$$\scriptstyle{\mathrm{ev}}$

The parabolic Hitchin system

$\widehat{h}^{p}:\widehat{{\mathfrak{M}}}^{\mathrm{par}}\to
C\times\widehat{A}$

is the composition of $\pi:\widehat{{\mathfrak{M}}}^{\mathrm{par}}\to
C\times\widehat{{\mathfrak{M}}}$ and the morphism
$\mathbf{h}:C\times\widehat{{\mathfrak{M}}}\to C\times\widehat{A}$ induced by
the standard (stacky) Hitchin map
$\widehat{h}:\widehat{{\mathfrak{M}}}\to\widehat{A}$.111Recall that we always
use the ${\mathcal{L}}$-twisted version.

In general the moduli stack $\widehat{{\mathfrak{M}}}^{\mathrm{par}}$ is
singular, and the parabolic Hitchin map is not proper. In the next section we
will impose a stability condition and then restrict to the stable locus.

###  3.3. Stable loci

We define the _stable locus_ of the moduli of parabolic $G$-Higgs bundles to
be

$\widehat{M}^{\mathrm{par}}:=(C\times\widehat{M})\times_{C\times\widehat{{\mathfrak{M}}}}\widehat{{\mathfrak{M}}}^{{\mathrm{par}}};$

equivalently, it fits into the Cartesian diagrams

(20)
${\widehat{M}^{\mathrm{par}}}$${\widehat{{\mathfrak{M}}}^{\mathrm{par}}}$${{[\mathfrak{b}/B]_{\mathcal{L}}}}$${C\times\widehat{M}}$${C\times\widehat{{\mathfrak{M}}}}$${{[\mathfrak{g}/G]_{\mathcal{L}}}.}$$\scriptstyle{\pi}$$\scriptstyle{\mathrm{ev}^{p}}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{G}}$$\scriptstyle{\mathrm{ev}}$

We use the same notation as for the stacky case to denote the parabolic
Hitchin map restricted to the stable locus:

(21) $\widehat{h}^{p}:\widehat{M}^{\mathrm{par}}\to C\times\widehat{A}.$

######  Proposition 3.1.

  1. (i)

The moduli stack $\widehat{M}^{\mathrm{par}}$ is nonsingular and
Deligne–Mumford; and

  2. (ii)

the parabolic Hitchin map (21) is proper.

###### Proof.

The Deligne–Mumford part of (i) follows from the proof of [37, Proposition
2.5.1 (3)]. Indeed, the left vertical arrow in the diagram (20) is the
pullback of $\pi_{G}$, therefore it is schematic and of finite type. This
implies that $\widehat{M}^{\mathrm{par}}$ is Deligne–Mumford since
$C\times\widehat{M}$ is Deligne–Mumford.

To prove the smoothness part of (i), we use the evaluation map (19). Recall
that [23, Proposition 4.1] (which was proven via deformation theory) shows
that the $C$-morphism $\mathrm{ev}$ is smooth after restricting over the
stable locus $C\times\widehat{M}$. Hence by the Cartesian diagrams of (20),
the evaluation map

$\mathrm{ev}:\widehat{M}^{\mathrm{par}}\to{[\mathfrak{b}/B]_{\mathcal{L}}}$

is also smooth. Since the target is a nonsingular algebraic stack, so is the
source.

(ii) follows directly from (20) and that
$\widehat{h}:\widehat{M}\to\widehat{A}$ is proper. ∎

As a consequence of Proposition 3.1, the direct image complex

$R{\widehat{h}^{p}}_{*}~{}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}\in
D^{b}_{c}(C\times\widehat{A})$

satisfies the decomposition theorem [1].

######  Theorem 3.2 (Support theorem for parabolic Hitchin map).

The decomposition theorem for the parabolic Hitchin map has full support,
_i.e._ , any non-trivial simple perverse summand of

${}^{\mathfrak{p}}{\mathcal{H}}^{i}\left(R{\widehat{h}^{p}}_{*}~{}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}\right),\quad\forall
i\in{\mathbb{Z}}$

has support $C\times\widehat{A}$.

We will postpone the proof of Theorem 3.2 to Section 4.

###  3.4. Proof of Theorem 2.6

We first prove Theorem 2.6 assuming Theorem 3.2.

There are three ingredients of global Springer theory

$\pi:\widehat{M}^{\mathrm{par}}\to C\times\widehat{M}$

which are important in the proof of Theorem 2.6. We summarize them as follows.

  1. (A)

_(Splitting of the universal $G$-bundle)_ As explained in the second bullet
point of [37, Construction 6.1.4], the Chern roots of $\pi^{*}{\mathcal{U}}$
are given by $c_{1}(L(\xi))$ where $L(\xi)$ is the tautological line bundle on
$\widehat{M}^{\mathrm{par}}$ associated with $\xi\in{\mathbb{X}}^{*}(T)$.

More precisely, $\pi^{*}{\mathcal{U}}$ is the universal $G$-bundle on
$\widehat{M}^{\mathrm{par}}$ whose fiber over
$(x,{\mathcal{E}},\theta,{\mathcal{E}}^{B}_{x})$ is ${\mathcal{E}}_{x}$; the
$B$-reduction ${\mathcal{E}}_{x}^{B}$ yields a $T$-torsor over
$\widehat{M}^{\mathrm{par}}$ which induces for each
$\xi\in{\mathbb{X}}^{*}(T)$ a line bundle $L(\xi)$.

  2. (B)

_(Strong perversity for $c_{1}(L(\xi))$)_ By [38, Lemma 3.2.3], there exists a
Zariski dense open subset of $C\times\widehat{A}$ over which the operator

$c_{1}(L(\xi)):R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}\to
R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}[2]$

has strong perversity $1$ for any $\xi\in{\mathbb{X}}^{*}(T)$. Since
$c_{1}(L(\xi))$ automatically has strong perversity $2$, showing it has strong
perversity $1$ is equivalent to showing the induced morphism of perverse
cohomology sheaves

${}^{p}{\mathcal{H}}^{i}\left(R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}\right)\rightarrow{}^{p}{\mathcal{H}}^{i}\left(R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}[2]\right)$

vanishes for each $i$. Once we have Theorem 3.2, these sheaves have full
support over the entire base, so this vanishing (and thus strong perversity
$1$) extends over the total base $C\times\widehat{A}$ as well. In fact, [38,
Lemma 3.2.3] was proven in exactly this way using a support theorem [38,
Section 4.6.2] for the elliptic locus.

  3. (C)

_(Springer’s Weyl group action)_ By the Cartesian diagrams (20), we may
pullback Springer’s sheaf-theoretic Weyl group action from the Grothendieck
simultaneous resolution; in particular, the object
$R\pi_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}$ admits a canonical
$W$-action whose invariant part recovers the trivial local system

$\left(R\pi_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}\right)^{W}={\mathbb{Q}}_{C\times\widehat{M}}.$

By taking global cohomology we have

$H^{*}(\widehat{M}^{\mathrm{par}},{\mathbb{Q}})=H^{*}(C\times\widehat{M},{\mathbb{Q}})\oplus\left(\mathrm{variant~{}~{}part~{}~{}of~{}~{}}W\right).$

Now we prove Theorem 2.6. We first prove its parabolic version.

Claim. The class

$\mathrm{ch}_{k}\left(\pi^{*}{\mathcal{U}}\right)\in
H^{2k}(\widehat{M}^{\mathrm{par}},{\mathbb{Q}})$

has strong perversity $k$ with respect to the parabolic Hitchin system (21).

###### Proof of Claim.

By (A), the Chern character $\mathrm{ch}_{k}(\pi^{*}{\mathcal{U}})$ can be
expressed in terms of $c_{1}(L(\xi))$. Hence the claim follows from Lemma 1.2
and the strong perversity of $c_{1}(L(\xi))$ given by (B). ∎

Next, we reduce Theorem 2.6 to the claim above via the following lemma.

######  Lemma 3.3.

For a class $\gamma\in H^{l}(C\times\widehat{M},{\mathbb{Q}})$, if
$\pi^{*}\gamma\in H^{l}(\widehat{M}^{\mathrm{par}},{\mathbb{Q}})$ has strong
perversity $k$ with respect to $\widehat{h}^{p}:\widehat{M}^{\mathrm{par}}\to
C\times\widehat{A}$, then $\gamma$ has strong perversity $k$ with respect to
${\mathbf{h}}:C\times\widehat{M}\to C\times\widehat{A}$.

###### Proof.

This is a consequence of (C). We write the class $\gamma$ as a map

(22)
$\gamma:{\mathbb{Q}}_{C\times\widehat{M}}\to{\mathbb{Q}}_{C\times\widehat{M}}[l],$

whose pullback

(23)
$\pi^{*}\gamma:{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}\to{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}[l]$

recovers the class $\pi^{*}\gamma\in
H^{l}(\widehat{M}^{\mathrm{par}},{\mathbb{Q}})$. By the projection formula
$\pi_{*}\pi^{*}\gamma=|W|\cdot\gamma$, we may recover the morphism (22) from
(23) by derived pushing forward to $C\times\widehat{M}$ and taking the
$W$-invariant part.

Now by the assumption we know that the action of $\pi^{*}\gamma$ on the object
$R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}$ satisfies

(24) $\pi^{*}\gamma:{{}^{\mathfrak{p}}\tau_{\leq
i}}R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}\to{{}^{\mathfrak{p}}\tau_{\leq
i+(k-l)}}\left(R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}[l]\right).$

Since we have

$R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}=R\mathbf{h}_{*}\left(R\pi_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}\right),$

the ingredient (C) produces a natural $W$-action on this object whose
invariant part recovers $R\mathbf{h}_{*}{\mathbb{Q}}_{C\times\widehat{M}}$;
the operator

(25) $\gamma:R\mathbf{h}_{*}{\mathbb{Q}}_{C\times\widehat{M}}\to
R\mathbf{h}_{*}{\mathbb{Q}}_{C\times\widehat{M}}[l]$

is then recovered from the $W$-invariant part of the action of $\pi^{*}\gamma$
on $R\widehat{h}^{p}_{*}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}$. In
particular, the desired property concerning (25) follows from the
$W$-invariant part of (24). ∎

Thus we have completed the proof of Theorem 2.6. ∎

##  4\. Parabolic support theorem

In the last section, we prove Theorem 3.2. This is the parabolic version of
the Chaudouard–Laumon support theorem [2]. For the proof, we ultimately reduce
it to a relative dimension bound which we establish in Section 4.4.

###  4.1. Review of support theorems

We start with a review of support theorems for Hitchin systems.

For a proper morphism $f:X\to Y$ with $X,Y$ nonsingular Deligne–Mumford
stacks, the decomposition theorem [1] yields

$Rf_{*}{\mathbb{Q}}_{X}\simeq\bigoplus_{i}{{}^{\mathfrak{p}}{\mathcal{H}}^{i}(Rf_{*}{\mathbb{Q}}_{X})[-i]}$

with ${{}^{\mathfrak{p}}{\mathcal{H}}^{i}}(Rf_{*}{\mathbb{Q}}_{X})$ semisimple
perverse sheaves on $Y$; we say that a closed $Z\subset Y$ is a support of $f$
if it is a support of a simple summand of some
${{}^{\mathfrak{p}}{\mathcal{H}}^{i}}(Rf_{*}{\mathbb{Q}}_{X})$. A particularly
interesting case is that $f$ has full support, that is, $Y$ is the only
support of $f$; in this case the cohomology of any closed fiber of $f$ is
governed by the nonsingular fibers.

The study of supports for Hitchin systems was initiated by B.C. Ngô which is
crucial in his proof of the fundamental lemma of the Langlands program [30].
He determines all the supports for the Hitchin system (including the
${\mathcal{L}}$-twisted cases) after restricting to the elliptic locus of the
Hitchin base; that is the subset formed by integral spectral curves.

After Ngô’s work, Chaudouard–Laumon [2] observed that, if we consider the
moduli space of ${\mathcal{L}}$-twisted $\mathrm{GL}_{n}$ stable Higgs bundles
with $\mathrm{deg}({\mathcal{L}})>0$, then Ngô’s support theorem can be
extended to the total Hitchin base; in particular they showed that the
${\mathcal{L}}$-twisted $\mathrm{GL}_{n}$ Hitchin system has full support.
Chaudouard–Laumon’s idea was extended to the $\mathrm{SL}_{n}$ case [4], the
endoscopic moduli spaces [23], and singular cases involving strictly
semistable Higgs bundles [24, 25]. See also [8, 27] concerning the supports
over the open subset of reduced spectral curves for the untwisted (_i.e._
${\mathcal{L}}=0$) Hitchin system.

The idea of Chaudouard–Laumon [2] is to show that each support of the
${\mathcal{L}}$-twisted Hitchin system has generic point lying in the elliptic
locus; this is achieved by combining two constraints: (I) the support
inequality for a weak abelian fibration, and (II) $\delta$-regularity for
spectral curves. We describe them in more detail.

  1. (I)

_(Support inequaltiy)_ The Hitchin system admits the structure of a weak
abelian fibration, that is, there is a commutative group scheme $P$ over the
Hitchin base acting on the moduli space which satisfies certain properties.
Then an argument generalizing the Goresky–MacPherson inequality leads to a
codimension estimate for the supports. More concretely, it says that any
support $Z$ has codimension bounded above by the $\delta$-function of the
group $P$. This part was already carried out in Ngô [30, Section 7]; see [24,
Section 1] for a summary.

  2. (II)

_( $\delta$-regularity)_ In the case of type $A$ and for the stable locus, the
group scheme is obtained from the multi-degree 0 relative Picard (or, for
$\mathrm{SL}_{n}$, Prym) variety associated with the family of spectral
curves. Then we have the Severi inequality, referred to as _$\delta$
-regularity_, for the spectral curves. We refer to [4, Theorem 5.4.4] for the
precise statement.

Using (I) and (II), one can deduce that no support is allowed to have generic
point lying outside the open subset of integral curves; this was explained in
[2, Section 11] for $\mathrm{GL}_{n}$, in [4, Section 6.2] for
$\mathrm{SL}_{n}$, and in [24, Section 4.5] for a general complete linear
system in a del Pezzo surface. The argument is to combine the two inequalities
above to deduce a numerical contradiction if there is a support that appears
outside the elliptic locus.

###  4.2. Parabolic Hitchin systems

Now we focus on the ${\mathcal{L}}$-twisted parabolic Hitchin system (21). Let
$\widehat{A}^{\mathrm{ell}}\subset\widehat{A}$ be the open subset
parameterizing integral spectral curves (the elliptic locus). Following Ngô’s
method, Yun [38, 39] proved a parabolic support theorem over
$C\times\widehat{A}^{\mathrm{ell}}$ and determined all the supports of (21) in
$C\times\widehat{A}^{\mathrm{ell}}$. More precisely, by [39, Section 2] any
strict subset of $C\times\widehat{A}^{\mathrm{ell}}$, which is a support of
$\widehat{h}^{\mathrm{par}}|_{C\times\widehat{A}^{\mathrm{ell}}}$, is a
component of the _endoscopic loci_ governed by the endoscopic theory of $G$.
As we only consider the special case $G=\mathrm{PGL}_{n}$, there are no
nontrivial endoscopic loci and the restricted Hitchin map
$\widehat{h}^{\mathrm{par}}|_{C\times\widehat{A}^{\mathrm{ell}}}$ has full
support.

To prove Theorem 3.2, it suffices to show that there is no support of (21)
whose generic point lying outside $C\times\widehat{A}^{\mathrm{ell}}$.

Since stability for a $\mathrm{PGL}_{n}$ Higgs bundle is described by its
corresponding vector bundle (see (9)), it is more convenient to work with the
$\mathrm{SL}_{n}$ moduli spaces. We consider the following Cartesian diagram

(26)
${\widecheck{M}^{\mathrm{par}}}$${\widehat{M}^{\mathrm{par}}}$${C\times\widecheck{M}}$${C\times\widehat{M}.}$$\scriptstyle{(-)/\Gamma}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{(-)/\Gamma}$

Here the horizontal maps are given by the natural quotient maps by
$\Gamma=\mathrm{Pic}^{0}(C)[n]$. To describe the map $\pi^{\prime}$ at the
left-side of the diagram (see [37, Example 2.2.5]), we recall that
$\widecheck{M}$ parameterizes traceless Higgs bundles with fixed determinant

$({\mathcal{E}},\theta),\quad\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\Omega_{\mathcal{L}},\quad\mathrm{rk}({\mathcal{E}})=n,~{}~{}\mathrm{det}({\mathcal{E}})\simeq{\mathcal{N}},~{}~{}\mathrm{trace}(\theta)=0$

with respect to slope stability. Similarly $\widecheck{M}^{\mathrm{par}}$
parameterizes

$(x,{\mathcal{E}}={\mathcal{E}}_{0}\supset{\mathcal{E}}_{1}\supset\cdots\supset{\mathcal{E}}_{n}={\mathcal{E}}_{0}(-x),\theta)$

where $x\in C$, $({\mathcal{E}},\theta)\in\widecheck{M}$, each
${\mathcal{E}}_{i}$ in the flag is of rank $n$ with
${\mathcal{E}}_{i}/{\mathcal{E}}_{i+1}$ a length 1 skyscraper sheaf supported
at $x$, the Higgs field preserves the flag
$\theta({\mathcal{E}}_{i})\subset{\mathcal{E}}_{i}\otimes\Omega_{\mathcal{L}}$,
and the map $\pi^{\prime}$ in the diagram is the forgetful map. We note that
the stability condition for a parabolic Higgs bundle is determined by the
stability of the underlying Higgs bundle.

Following [37, Example 2.2.5] we may also describe
$\widecheck{M}^{\mathrm{par}}$ via spectral curves. If we present a point in
$\widecheck{M}$ as a 1-dimensional sheaf ${\mathcal{F}}_{\alpha}$ supported on
a spectral curve $C_{\alpha}\subset\mathrm{Tot}(\Omega_{\mathcal{L}})$ with
$p_{\alpha}:C_{\alpha}\to C$ the projection, then for any $x\in C$, a closed
point in the fiber $\pi^{\prime-1}(x,{\mathcal{F}}_{\alpha})$ is represented
by

${\mathcal{F}}_{\alpha}={\mathcal{F}}_{0}\supset{\mathcal{F}}_{1}\supset\cdots\supset{\mathcal{F}}_{n}={\mathcal{F}}_{0}\otimes
p_{\alpha}^{*}{\mathcal{O}}_{C}(-x),\quad\mathrm{lengh}({\mathcal{F}}_{i}/{\mathcal{F}}_{i+1})=1;$

see [37, (2.4)].

Now we consider the $\mathrm{SL}_{n}$ parabolic Hitchin system

(27) $\widecheck{h}^{p}:\widecheck{M}^{\mathrm{par}}\to
C\times\widecheck{M}\to C\times\widehat{A}.$

From the diagram (26), it recovers the $\mathrm{PGL}_{n}$ Hitchin system
$\widehat{h}^{p}:\widehat{M}^{\mathrm{par}}\to C\times\widehat{A}$ by taking
the quotient of the source by $\Gamma$. In particular, we have

$R{\widehat{h}^{p}}_{*}~{}{\mathbb{Q}}_{\widehat{M}^{\mathrm{par}}}=\left(R{\widecheck{h}^{p}}_{*}~{}{\mathbb{Q}}_{\widecheck{M}^{\mathrm{par}}}\right)^{\Gamma}\in
D_{c}^{b}(C\times\widehat{A}).$

Hence in order to prove Theorem 3.2, it suffices to show that there is no
support of (27) with generic point lying outside
$C\times\widehat{A}^{\mathrm{ell}}$. In the next two sections, we adapt the
strategy of Chaudouard–Laumon [2] and de Cataldo [4] (for the
$\mathrm{SL}_{n}$-version) to this parabolic setting, and verify the parabolic
analogs of (I) and (II) of Section 4.1.

###  4.3. Weak abelian fibrations and $\delta$-regularity

A general setup for support theorems was given in [24, Theorem 1.8]. We now
check that all the assumptions there are satisfied by (27).

We first construct the $(C\times\widehat{A})$-group scheme $P$ which acts on
$\widecheck{M}^{\mathrm{par}}$. Recall that in [4] de Cataldo showed that
$\widecheck{M}$ admits a weak abelian fibration structure
$(\widecheck{P},\widecheck{M},\widehat{A})$.222We refer to [24, Section 1] for
an introduction to weak abelian fibrations. Here the $\widehat{A}$-group
scheme $\widecheck{P}$ which acts on $\widecheck{M}$ is given by the identity
component of the relative Prym variety associated with the spectral curves [4,
(43)]. For a spectral curve $p_{\alpha}:C_{\alpha}\to C$ with a Higgs bundle
given by ${\mathcal{F}}_{\alpha}$ supported on $C_{\alpha}$, the Prym action
is induced by tensor product

${\mathcal{Q}}\cdot{\mathcal{F}}_{\alpha}={\mathcal{Q}}\otimes{\mathcal{F}}_{\alpha},\quad{\mathcal{Q}}\in\mathrm{Prym}^{0}(C_{\alpha}/C)=\widecheck{P}_{[C_{\alpha}]},~{}\quad[C_{\alpha}]\in\widehat{A}.$

We denote by $P$ the $(C\times\widehat{A})$-group scheme obtained by the
pullback of $\widecheck{P}$. The $P$-action on $C\times\widecheck{M}$ can be
lifted to $\widecheck{M}^{\mathrm{par}}$:

${\mathcal{Q}}\cdot(x,{\mathcal{F}}_{\alpha}={\mathcal{F}}_{0}\supset{\mathcal{F}}_{1}\supset\cdots\supset{\mathcal{F}}_{n})=(x,{\mathcal{Q}}\otimes{\mathcal{F}}_{\alpha}={\mathcal{Q}}\otimes{\mathcal{F}}_{0}\supset{\mathcal{Q}}\otimes{\mathcal{F}}_{1}\supset\cdots\supset{\mathcal{Q}}\otimes{\mathcal{F}}_{n}),$

since the stability condition does not rely on the flag.

######  Proposition 4.1.

The triple $(P,\widecheck{M},C\times\widehat{A})$ forms an weak abelian
fibration, _i.e._ , it satisfies (i,ii,ii) of [24, Section 1.1].

###### Proof.

(i) is clear from (20), and (iii) only depends on the group scheme $P$ which
follows from the corresponding property for $\widecheck{P}$ proved in [4,
Theorem 4.7.2]. To prove (ii): since the $P$-action on
$\widecheck{M}^{\mathrm{par}}$ is a lifting of the $P$-action on
$C\times\widecheck{M}$, and the latter has affine stabilizers by [4], we
obtain that the $P$-stabilizer for any point
$z\in\widecheck{M}^{\mathrm{par}}$ is contained in the (affine) $P$-stabilizer
for $\pi(z)\in C\times\widecheck{M}$. This proves (ii). ∎

From this proposition, in order to show the support inequality ((I) of Section
4.1) for $(P,\widecheck{M},C\times\widehat{A})$, by [24, Theorem 1.8], it
suffices to estabish the relative dimension bound

(28)
$\tau_{>2\mathbf{d}}\left(R\widecheck{h}^{p}_{*}{\mathbb{Q}}_{\widecheck{M}^{\mathrm{par}}}\right)=0,\quad\mathbf{d}:=\mathrm{dim}\widecheck{M}^{\mathrm{par}}-\mathrm{dim}(C\times\widehat{A})$

with $\tau_{>*}$ the standard truncation functor.

Once we have (28), we can combine the support inequality with
$\delta$-regularity of $\widecheck{P}$ [4, corollary 5.4.4] to prove Theorem
3.2 as follows. If $Z\subset C\times\widehat{A}$ is an irreducible support of
$\widecheck{h}^{p}$, the projection
$W=\mathrm{pr}_{\widehat{A}}(Z)\subset\hat{A}$ also satisfies the support
inequality

$\mathrm{codim}_{\widehat{A}}W\leq\mathrm{codim}_{C\times\widehat{A}}Z\leq\delta_{Z}=\delta_{W}.$

Here we used in the second inequality the support inequality for
$\widecheck{h}^{p}$, and used in the last equality that the $\delta$-function
on $C\times\widehat{A}$ is pulled back from $\widehat{A}$. The argument of [4,
Section 6.2] then shows that the generic point of $W$ lies in
$\widehat{A}^{\mathrm{ell}}$. Hence the generic point of $Z$ lies in
$C\times\widehat{A}^{\mathrm{ell}}$ as we desired.

In conclusion, this reduces Theorem 3.2 to (28).

###  4.4. Proof of the relative dimension bound (28)

We need to show that each closed fiber of the morphism

$\widecheck{h}^{p}:\widecheck{M}^{\mathrm{par}}\rightarrow C\times\widehat{A}$

has dimension

$\mathbf{d}=\mathrm{dim}\widecheck{M}^{\mathrm{par}}-\mathrm{dim}(C\times\widehat{A})=\frac{n(n-1)}{2}\deg({\mathcal{L}})+(n^{2}-1)(g-1).$

Here the last equation is obtained by the dimension formula of [4, Proposition
2.4.6] directly:

$\mathbf{d}=\mathrm{dim}\widecheck{M}-\mathrm{dim}\widehat{A}=\frac{n(n-1)}{2}\deg({\mathcal{L}})+(n^{2}-1)(g-1).$

First, we note that the morphism $\widecheck{h}^{p}$ is surjective since the
usual Hitchin map $\widecheck{h}:\widecheck{M}\to\widehat{A}$ is surjective.
Furthermore, as $\widecheck{h}^{p}$ is equivariant with respect to the scaling
${\mathbb{G}}_{m}$-action on the Higgs fields, it suffices to bound from above
the dimension of the fiber over $(x,0)\in C\times\widehat{A}$ for each point
$x\in C$ by $\mathbf{d}$, since upper semicontinuity will force all other
fibers to have the same dimension. We fix the point $p$ from now on.

Using the assumption $\deg({\mathcal{L}})>2g$, we may express ${\mathcal{L}}$
as ${\mathcal{O}}_{C}(D)$ with $D=x_{0}+x_{1}+\dots+x_{t}$ an effective
reduced divisor containing $x=x_{0}$. In particular, a Higgs bundle
$({\mathcal{E}},\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\Omega_{\mathcal{L}})$
can be viewed as a meromorphic (un-twisted) Higgs bundles with at most simple
poles along $D$:

$({\mathcal{E}},\theta),\quad\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\Omega_{C}^{1}(D).$

We will control the fiber dimension using an analogous bound for the nilpotent
cone for _strongly parabolic_ Higgs bundles for the pair $(C,D)$.

Recall that, given $D$ as above, a strongly parabolic Higgs bundle consists of
an $\mathrm{SL}_{n}$ Higgs bundle of degree $d$:

$({\mathcal{E}},\theta),\quad\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\Omega^{1}_{C}(D),~{}~{}~{}\mathrm{rk}({\mathcal{E}})=n,~{}~{}\mathrm{det}({\mathcal{E}})\simeq{\mathcal{N}},~{}~{}\mathrm{trace}(\theta)=0,$

along with a flag at each point $x_{i}\in D$:

${\mathcal{E}}|_{x_{i}}=F_{x_{i},0}\supset F_{x_{i},1}\supset
F_{x_{i},2},\supset\cdots\supset
F_{x_{i},n}=0,\quad\mathrm{dim}F_{x_{i},j}/F_{x-i,j+1}=1,$

such that the Higgs field satisfies $\theta(F_{x_{i},j})\subset
F_{x_{i},j+1}$.333One may compare this with the (non-strong) parabolic
condition $\theta(F_{x,j})\subset F_{x,j}$ we used earlier in this paper for
the global Springer theory.

Let $\widecheck{M}^{\mathrm{spar}}(D)$ denote the moduli space of strongly
parabolic Higgs bundles associated with the pair $(C,D)$, such that the
underlying twisted Higgs bundle is stable. Let

$A(D)=\bigoplus_{i=2}^{n}H^{0}(C,\Omega_{\mathcal{L}}^{\otimes
i}(-D))\subset\widehat{A}$

denote the linear subspace consisting of sections which vanish along the
points of $D$. There is a Hitchin map for the strongly parabolic Higgs moduli
space

$\widecheck{h}^{\mathrm{spar}}:\widecheck{M}^{\mathrm{spar}}(D)\rightarrow
A(D)$

which is proper, surjective, and Lagrangian with respect to a natural
holomorphic symplectic form [14]. In particular, the dimension of the zero
fiber is given by

$\displaystyle\dim\widecheck{M}^{\mathrm{spar}}(D)_{0}=\dim A(D)$
$\displaystyle=\sum_{j=2}^{n}\left((j-1)(\deg(\Omega_{L}))+g-1\right)$
$\displaystyle=\frac{n(n-1)}{2}\deg({\mathcal{L}})+(n^{2}-1)(g-1).$

See also [36, Theorem 6.9] for a direct proof of the above dimension
formula.444The formula above has $g$ less than the dimension formula obtained
in [36, Theorem 6.9], since we consider the $\mathrm{SL}_{n}$ case where we
fixed the determinant on $C$.

We now consider the following diagram relating the two types of parabolic
Higgs moduli spaces $\widecheck{M}^{\mathrm{spar}}(D)$ and
$\widecheck{M}^{\mathrm{par}}$:

(29)
${\widecheck{M}^{\mathrm{spar}}(D)_{0}}$${\widecheck{M}^{\mathrm{spar}}(D)}$${\widecheck{M}^{\mathrm{par}}_{(x,0)}}$${{\widecheck{M}^{\mathrm{par}}}|_{x\times
A(D)}}$${\widecheck{M}^{\mathrm{par}}}$${\\{(x,0)\\}}$${x\times
A(D)}$${C\times\widehat{A}}$$\scriptstyle{q_{0}}$$\scriptstyle{q}$$\scriptstyle{\widecheck{h}^{p}}$

where all squares are Cartesian.

We first claim that the natural map $q$ (sending a strong parabolic Higgs
bundle to a parabolic Higgs bundle) is surjective. Indeed, given a point
$z\in\widecheck{M}^{\mathrm{par}}$ lying over $C\times{A}(D)$, choosing a
point in its preimage $q^{-1}(z)$ only consists of fixing a flag at each point
of $x_{i}$ with $i>0$, preserved by the Higgs field — since the characteristic
polynomial already has zero roots at all points of $D$, the Higgs field is
automatically strongly parabolic.

By base change, this implies that $q_{0}$ is surjective as well, so we get the
desired dimension upper bound

$\dim{\widecheck{M}^{\mathrm{par}}}_{(x,0)}\leq\dim\widecheck{M}^{\mathrm{spar}}(D)_{0}=\frac{n(n-1)}{2}\deg({\mathcal{L}})+(n^{2}-1)(g-1),$

which completes the proof. ∎

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