# Hitchin fibrations, abelian surfaces,
and the P=W conjecture

Mark Andrea de Cataldo  Stony Brook University<EMAIL_ADDRESS>,  Davesh Maulik  Massachusetts Institute of Technology<EMAIL_ADDRESS>and
Junliang Shen  Yale University<EMAIL_ADDRESS>
###### Abstract.

We study the topology of Hitchin fibrations via abelian surfaces. We establish
the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus
and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology
generated by even tautological classes. Furthermore, we show that all
tautological generators lie in the correct pieces of the perverse filtration
as predicted by the P=W conjecture. In combination with recent work of Mellit,
this reduces the full conjecture to the multiplicativity of the perverse
filtration.

Our main technique is to study the Hitchin fibration as a degeneration of the
Hilbert–Chow morphism associated with the moduli space of certain torsion
sheaves on an abelian surface, where the symmetries induced by Markman’s
monodromy operators play a crucial role.

###### Contents

  1. 0 Introduction
  2. 1 Perverse filtrations
  3. 2 Moduli of one-dimensional sheaves
  4. 3 Splittings of perverse filtrations
  5. 4 Topology of Hitchin fibrations

##  0\. Introduction

###  0.1. Perverse filtrations

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

Let $\pi:X\rightarrow Y$ be a proper morphism with $X$ a nonsingular algebraic
variety. The perverse $t$-structure on the constructible derived category
$D_{c}^{b}(Y)$ with rational coefficients ${\mathbb{Q}}$ induces a finite and
increasing filtration on the rational cohomology groups
$H^{*}(X,{\mathbb{Q}})$

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

called the _perverse filtration_ associated with $\pi$. The filtration (1) is
governed by the topology of the morphism $\pi:X\rightarrow Y$. See Section 1
for a brief review of the subject.

Every cohomology class on $X$ is indexed by the perverse filtration (1). We
say that a class $\alpha\in H^{d}(X,{\mathbb{Q}})$ has _perversity_ $k$ if
$\alpha=0$, or

$\alpha\in P_{k}H^{d}(X,{\mathbb{Q}})\;\mathrm{and}\;\alpha\notin
P_{k-1}H^{d}(X,{\mathbb{Q}}).$

The purpose of this paper is to study perverse filtrations associated with
Hitchin fibrations in view of the _P=W conjecture_ by de Cataldo, Hausel, and
Migliorini [13]. Our method is to use _symmetries_ induced by the monodromy of
moduli spaces of sheaves on abelian surfaces.

###  0.2. The P=W conjecture

Let $C$ be an irreducible nonsingular projective curve of genus $g\geq 2$.
There are two moduli spaces which are attached to the curve $C$, the reductive
Lie group $\mathrm{GL}_{r}$, and an integer $\chi$ with
$\mathrm{gcd}(r,\chi)=1$. They are the twisted versions of Simpson’s Dolbeault
and Betti moduli spaces [49]; see [30] for the non-abelian Hodge theory in the
twisted case.

The first moduli space ${\mathcal{M}}_{\mathrm{Dol}}$ parametrizes stable
Higgs bundles on $C$

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

with $\mathrm{rank}({\mathcal{E}})=r$ and $\chi({\mathcal{E}})=\chi$. The
variety ${\mathcal{M}}_{\mathrm{Dol}}$ admits a projective morphism with
connected fibers

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

sending $({\mathcal{E}},\theta)\in{\mathcal{M}}_{\mathrm{Dol}}$ to the
characteristic polynomial $\mathrm{char}(\theta)\in\Lambda$. The proper
morphism (2), called the _Hitchin fibration_ , is Lagrangian with respect to
the canonical holomorphic symplectic form on ${\mathcal{M}}_{\mathrm{Dol}}$
given by the hyper-Kähler metric on ${\mathcal{M}}_{\mathrm{Dol}}$; see [32].
The second moduli space is the (twisted) character variety
${\mathcal{M}}_{B}$. It can be described (see [30], or [47] for a an
alternative description) as parametrizing isomorphism classes of irreducible
local systems

$\rho:\pi_{1}(C\setminus\\{p\\})\to\mathrm{GL}_{r}$

where $\rho$ sends a loop around a chosen point $p$ to
$\xi^{\chi}_{r}I_{r}\in\mathrm{GL}_{r}$, with $\xi_{r}$ a primitive root of
unity. The character variety ${\mathcal{M}}_{B}$ is affine.

In [48], Simpson constructed a diffeomorphism between the (un-twisted) moduli
spaces ${\mathcal{M}}_{\mathrm{Dol}}$ and ${\mathcal{M}}_{B}$, called the non-
abelian Hodge theory; see [30] for the twisted case. A striking prediction,
suggested by the parallel between the Relative Hard Lefschetz [7] and Curious
Hard Lefschetz [29] Theorems, was formulated by de Cataldo, Hausel, and
Migliorini [13]; it predicts that the perverse filtration of
${\mathcal{M}}_{\mathrm{Dol}}$ with respect to the Hitchin fibration (2)
matches the weight filtration of the mixed Hodge structure on
$\mathcal{M}_{B}$ under the identification

$H^{\ast}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{\ast}({\mathcal{M}}_{B},{\mathbb{Q}})$

induced by Simpson’s (twisted) non-abelian Hodge theory.

######  Conjecture 0.1 ([13] P=W).

We have

$P_{k}H^{\ast}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=W_{2k}H^{\ast}({\mathcal{M}}_{B},{\mathbb{Q}})=W_{2k+1}H^{\ast}({\mathcal{M}}_{B},{\mathbb{Q}}),\quad
k\geq 0.$

Conjecture 0.1 establishes a surprising connection between the topology of
Hitchin fibrations and the Hodge theory of character varieties. It was proven
in [13] in the case of $r=2$ for any genus $g\geq 2$. See also [14, 46, 52]
for certain parabolic cases, and [45, 27] for a compact analog concerning
Lagrangian fibrations on projective holomorphic symplectic manifolds.

The first main result of this paper is a proof of Conjecture 0.1 for curves of
genus $2$ and arbitrary rank $r\geq 1$.

######  Theorem 0.2.

The P=W Conjecture 0.1 holds when $C$ has genus $g=2$.

For an irreducible nonsingular curve $C$ of genus $g\geq 2$, it is a general
fact that $P=W$ for $\mathrm{GL}_{r}$ implies $P=W$ for $\mathrm{PGL}_{r}$;
see (70) and (90). Hence we conclude immediately from Theorem 0.2 that the
$P=W$ conjecture holds for $\mathrm{PGL}_{r}$ when the curve $C$ has genus
$g=2$.

As explained in [13, Section 1], the curious Poincaré duality and the Curious
Hard Lefschetz conjectures [29, Conjectures 4.2.4 and 4.2.7] for character
varieties are consequences of the P=W Conjecture 0.1. Moreover, by [9] and
[42, Section 9.3], the P=W Conjecture 0.1 implies the correspondence between
Gopakumar–Vafa invariants and Pandharipande–Thomas invariants [42, Conjecture
3.13] for the local Calabi–Yau 3-fold $T^{*}C\times{\mathbb{C}}$ in the curve
class $r[C]$. Hence Theorem 0.2 verifies all these conjectures for genus 2
curves.111As we discuss later, the curious Poincaré and Lefschetz conjectures
have been recently shown by Mellit [43] to hold for all genera.

###  0.3. Tautological classes

Assume that $C$ has genus $g\geq 2$ and $r\geq 1$. A set of generators for the
cohomology rings identified by the non-abelian Hodge diffeomorphism

(3)
$H^{\ast}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{\ast}({\mathcal{M}}_{B},{\mathbb{Q}})$

is described in [37] by tautological classes.

Let

$p_{C}:C\times{\mathcal{M}}_{\mathrm{Dol}}\to C,\quad
p_{\mathcal{M}}:C\times{\mathcal{M}}_{\mathrm{Dol}}\to{\mathcal{M}}_{\mathrm{Dol}}$

be the projections. We say that a triple

$({\mathcal{U}}^{\alpha},\theta)=({\mathcal{U}},\theta,\alpha)$

is a twisted universal family over $C\times{\mathcal{M}}_{\mathrm{Dol}}$, if
$({\mathcal{U}},\theta)$ is a universal family and

$\alpha=p_{C}^{\ast}\alpha_{1}+p_{\mathcal{M}}^{\ast}\alpha_{2}\in
H^{2}(C\times{\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),$

with $\alpha_{1}\in H^{2}(C,{\mathbb{Q}})$ and $\alpha_{2}\in
H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$.

For a twisted universal family $({\mathcal{U}}^{\alpha},\theta)$, we define
the twisted Chern character $\mathrm{ch}^{\alpha}({\mathcal{U}})$ as

$\mathrm{ch}^{\alpha}({\mathcal{U}})=\mathrm{ch}({\mathcal{U}})\cup\mathrm{exp}(\alpha)\in
H^{*}(C\times{\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),$

and we denote by

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

its degree $k$ part. The class $\mathrm{ch}^{\alpha}({\mathcal{U}})$ is called
_normalized_ if

$\mathrm{ch}^{\alpha}_{1}({\mathcal{U}})|_{p\times{\mathcal{M}}_{\mathrm{Dol}}}=0\in
H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),\qquad\mathrm{ch}^{\alpha}_{1}({\mathcal{U}})|_{C\times
q}=0\in H^{2}(C,{\mathbb{Q}}),$

with $p\in{\mathcal{M}}_{\mathrm{Dol}}$ and $q\in C$ points. Since two
universal families differ by the pull-back of a line bundle on
${\mathcal{M}}_{\mathrm{Dol}}$, a straightforward calculation shows that
normalized classes on $C\times{\mathcal{M}}_{\mathrm{Dol}}$ exist and are
uniquely determined. We introduce the tautological classes associated with the
normalized class $\mathrm{ch}^{\alpha}({\mathcal{U}})$ as follows.

For any $\gamma\in H^{i}(C,{\mathbb{Q}})$, let $c(\gamma,k)$ denote the
_tautological class_

(4)
$c(\gamma,k):=\int_{\gamma}\mathrm{ch}_{k}^{\alpha}({\mathcal{U}})={p_{{\mathcal{M}}}}_{\ast}(p_{C}^{\ast}\gamma\cup\mathrm{ch}^{\alpha}_{k}({\mathcal{U}}))\in
H^{i+2k-2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}).$

Markman showed in [37] that the classes $c(\gamma,k)$ generate the cohomology
ring (3) as a ${\mathbb{Q}}$-algebra. Furthermore, Shende proved in [47] (cf.
Lemma 4.6) that

(5)
$c(\gamma,k)\in{{}^{k}\mathrm{Hdg}^{i+2k-2}({\mathcal{M}}_{B})},\quad\forall~{}\gamma\in
H^{i}(C,{\mathbb{Q}}),$

where

${}^{k}\mathrm{Hdg}^{d}({\mathcal{M}}_{B})=W_{2k}H^{d}({\mathcal{M}}_{B},{\mathbb{Q}})\cap
F^{k}H^{d}({\mathcal{M}}_{B},{\mathbb{C}})\cap\bar{F}^{k}H^{d}({\mathcal{M}}_{B},{\mathbb{C}}),$

with $F^{*}$ and $W_{*}$ the Hodge and weight filtrations, respectively. It
follows that the rational cohomology $H^{*}({\mathcal{M}}_{B},{\mathbb{Q}})$
is split of Hodge–Tate type, and there is a canonical decomposition of the
graded vector spaces (3) induced by the mixed Hodge theory of
${\mathcal{M}}_{B}$,

(6)
$H^{\ast}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{\ast}({\mathcal{M}}_{B},{\mathbb{Q}})=\bigoplus_{k,d}{{}^{k}\mathrm{Hdg}^{d}({\mathcal{M}}_{B})}.$

The P=W conjecture, which can be restated as predicting that

(7)
$P_{k}H^{\ast}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=\bigoplus_{k^{\prime}\leq
k}{{}^{k^{\prime}}\mathrm{Hdg}^{*}({\mathcal{M}}_{B})},$

is equivalent to the following statements.

######  Conjecture 0.3 (Equivalent version of P=W).

There exists a splitting
$G_{*}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$ of the perverse
filtration associated with the Hitchin fibration (2) satisfying the following
two properties.

  1. (a)

(Tautological classes) Every tautological class $c(\gamma,k)$ has perversity
$k$ and in fact

$c(\gamma,k)\in
G_{k}H^{\ast}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),\quad\forall k\geq
0,\,\forall\gamma\in H^{*}(C,{\mathbb{Q}}).$

  2. (b)

(Multiplicativity) The perverse decomposition is multiplicative, _i.e._ ,

$\cup:G_{k}H^{d}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})\times
G_{k^{\prime}}H^{d^{\prime}}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})\to
G_{k+k^{\prime}}H^{d+d^{\prime}}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}).$

We recall that the multiplicativity of the weight filtration

$\cup:W_{k}H^{d}(X,{\mathbb{Q}})\times
W_{k^{\prime}}H^{d^{\prime}}(X,{\mathbb{Q}})\to
W_{k+k^{\prime}}H^{d+d^{\prime}}(X,{\mathbb{Q}})$

is standard from mixed Hodge theory. However, the perverse filtration
associated with a proper flat morphism is not multiplicative in general; see
[10, Exercise 5.6.8]. It is mysterious why the perverse filtration associated
with the Hitchin fibration (2) should be multiplicative, as is predicted by
the P=W conjecture. In fact, one consequence of this paper is that the full
P=W conjecture can be reduced to the multiplicativity of the perverse
filtration; see Theorem 0.6. This approach of analyzing the P=W conjecture via
Conjecture 0.3 goes back to [13] where it is applied to prove the case of rank
$2$.

###  0.4. P=W for tautological classes

Theorem 0.2 is concerned with genus $g=2$. In this section, we state our
results for genus $g\geq 2$.

####  0.4.1. Even tautological classes

We consider the ${\mathbb{Q}}$-subalgebra

$R^{*}({\mathcal{M}}_{\mathrm{Dol}})\subset
H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{*}({\mathcal{M}}_{B},{\mathbb{Q}})$

generated by all the _even_ tautological classes

$c(\gamma,k),\quad k\geq 0,\quad\gamma\in H^{0}(C,{\mathbb{Q}})\oplus
H^{2}(C,{\mathbb{Q}}).$

Our next theorem establishes the P=W conjecture in arbitrary genus and rank,
but restricted to this subalgebra.

######  Theorem 0.4.

The P=W conjecture holds, for any genus $g\geq 2$ and rank $r\geq 1$ for
$R^{*}({\mathcal{M}}_{\mathrm{Dol}})$, _i.e._

$\begin{split}P_{k}H^{\ast}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})\cap
R^{*}({\mathcal{M}}_{\mathrm{Dol}})&=W_{2k}H^{\ast}({\mathcal{M}}_{B},{\mathbb{Q}})\cap
R^{*}({\mathcal{M}}_{\mathrm{Dol}})\\\
&=W_{2k+1}H^{\ast}({\mathcal{M}}_{B},{\mathbb{Q}})\cap
R^{*}({\mathcal{M}}_{\mathrm{Dol}}).\end{split}$

In fact, in the proof of Theorem 0.4, we obtain a multiplicative decomposition

$R^{*}({\mathcal{M}}_{\mathrm{Dol}})=\bigoplus_{k,d}G_{k}R^{d}({\mathcal{M}}_{\mathrm{Dol}})$

that splits the restricted perverse filtration
$P_{k}H^{\ast}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})\cap
R^{*}({\mathcal{M}}_{\mathrm{Dol}})$ and such that

$G_{k}R^{d}({\mathcal{M}}_{\mathrm{Dol}})={{}^{k}\mathrm{Hdg}^{d}}({\mathcal{M}}_{B})\cap
R^{d}({\mathcal{M}}_{\mathrm{Dol}}).$

We refer to Section 4.6 for more details.

####  0.4.2. Odd tautological classes

The following theorem concerns odd tautological classes

(8) $c(\gamma,k)\in
H^{2k-1}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),\quad\forall k\geq
0,\,\forall\gamma\in H^{1}(C,{\mathbb{Q}}).$

######  Theorem 0.5.

Any odd tautological class (8) has perversity $k$.

Theorems 0.4 and 0.5 provide strong evidence for the P=W Conjecture 0.1 (and
for its reformulation Conjecture 0.3) for any genus $g\geq 2$. In particular,
we obtain that all tautological generators $c(\gamma,k)$ satisfy the $P=W$
match:

(9) $c(\gamma,k)\in
P_{k}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),\quad\forall~{}\gamma\in
H^{*}(C,{\mathbb{Q}}).$

####  0.4.3. Multiplicativity is equivalent to P=W

The P=W Conjecture 0.1 implies immediately that the perverse filtration
$P_{*}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$ is multiplicative,
_i.e._

(10) $\cup:P_{k}H^{d}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})\times
P_{k^{\prime}}H^{d^{\prime}}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})\to
P_{k+k^{\prime}}H^{d+d^{\prime}}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}).$

The following theorem shows that the converse is also true. It is a corollary
of Theorems 0.4 and 0.5, and of the Curious Hard Lefschetz conjecture recently
established by Mellit [43].

######  Theorem 0.6.

The P=W Conjecture 0.1 is equivalent to the multiplicativity (10) of the
perverse filtration.

In fact, we deduce from (5), (9), and (10) that

$W_{2i}H^{*}({\mathcal{M}}_{B},{\mathbb{Q}})\subset
P_{i}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),\quad\forall i\geq 0,$

which further implies Conjecture 0.1 by [43, Theorem 1.5.3] and the second
paragraph following [43, Corollary 1.5.4].

######  Remark 0.7.

All the results of this paper hold for arbitrary rank $r\geq 1$. The case
$r=1$ is classical and easy: the Dolbeault moduli space is the cotangent
bundle of a Jacobian of some degree of the curve and the Betti moduli space is
a product of $\mathbb{G}_{m}^{2g}$. In this paper, starting with the
assumption $\beta^{2}\geq 6$ in Section 2.1, we focus on the case of rank
$r\geq 2$, which corresponds to $\beta^{2}\geq 8$. The case of rank one could
be dealt with also by using the methods of this paper, but at the price of
some easy modifications that we deemed a distraction.

###  0.5. Idea of the proof

The key idea in proving our results is to first study perverse filtrations for
certain _compact_ hyperkähler manifolds, motivated in part by earlier work of
the third author and Yin [45]. That is, we consider an abelian surface $A$
with ample curve class $\beta\in H_{2}(A,{\mathbb{Z}})$, and prove analogs of
our main result for the moduli space ${\mathcal{M}}_{\beta,A}$ of one-
dimensional sheaves with support in the class $\beta$; see Theorem 2.1 for the
precise statement. The advantage of the compact geometry is that we can use
results of Markman [41] on monodromy operators for ${\mathcal{M}}_{\beta,A}$.
These are symmetries, arising from parallel transport and Fourier-Mukai
transforms, that do not appear in the Hitchin setting and which heavily
constrain tautological classes of universal families. We show that these
operators relate perverse filtrations and tautological classes for different
choices of $\beta$ and, in particular, allow us to pass from the case of
imprimitive $\beta$ to primitive $\beta$, which can be studied directly using
[52, 46].

In order to apply this to Hitchin fibrations, we consider the degeneration to
the normal cone of an embedding of a genus $g$ curve $C$ into an abelian
surface

$j_{C}:C\hookrightarrow A,$

and study the specialization map on cohomology of the associated moduli
spaces. In general, there is a great deal of information loss in this
specialization because the family is non-proper. Nevertheless, we are able to
show its compatibility with tautological classes and perverse splittings.
Finally we use these compatibilities to deduce our main theorems for such a
curve $C$; this implies our results hold for all curves in view of [15].

###  0.6. Outline of paper

We briefly outline the contents of this paper. In Section $1$, we recall some
basic facts about perverse filtrations and earlier results of [52, 46] for
Hilbert schemes of points on the special abelian surface $E\times E^{\prime}$.
In Section $2$, we pass to the setting of abelian surfaces. One key result
here is the splitting Theorem 2.1. After recalling Markman’s work on monodromy
operators, we prove the result for primitive classes $\beta$ by a direct
analysis, and then we combine the primitive case with Markman’s results to
establish the general case. In Section $3$, we formulate and prove Theorem
3.8, which is a strengthened version of the splitting of Theorem 2.1 and which
is more robust for specialization arguments. Finally, in section $4$, we study
specialization maps for our degeneration, and we use them to prove the main
theorems.

While this paper is written in logical order, some readers may prefer to start
from §4, and refer backwards as needed.

###  0.7. Acknowledgements

We are grateful to Tamás Hausel, Jochen Heinloth, Luca Migliorini, Vivek
Shende, David Vogan, Qizheng Yin, Zhiwei Yun, and Zili Zhang for helpful
discussions. M.A. de Cataldo is partially supported by NSF grants DMS 1600515
and 1901975. D. Maulik is partially supported by NSF FRG grant DMS-1159265. J.
Shen is supported by the NSF grant DMS 2134315.

##  1\. Perverse filtrations

###  1.1. Overview

We begin with the definition and some relevant properties of the perverse
filtration associated with a proper surjective morphism $\pi:X\rightarrow Y$.
Some references are [7, 17, 19, 50]. Throughout this section, for simplicity,
we assume $X$ and $Y$ are nonsingular.

Following [14, 52, 46], we discuss the perverse filtration for the Hilbert
scheme of $n$-points on an abelian surface $E\times E^{\prime}$ product of two
elliptic curves, induced by the natural second projection morphism

$E\times E^{\prime}\to E^{\prime}.$

This example plays a crucial role in Section 2 concerning moduli of sheaves on
abelian surfaces.

###  1.2. Perverse sheaves

Let $D_{c}^{b}(Y)$ denote the bounded derived category of
${\mathbb{Q}}$-constructible sheaves on $Y$, and let
${\mathbb{D}}:D_{c}^{b}(Y)^{\mathrm{op}}\to D_{c}^{b}(Y)$ be the Verdier
duality functor. The full subcategories

${}^{\mathfrak{p}}D_{\leq 0}^{b}(Y)$ $\displaystyle=\left\\{{\mathcal{E}}\in
D_{c}^{b}(Y):\dim\mathrm{Supp}({\mathcal{H}}^{i}({\mathcal{E}}))\leq-i\right\\},$
${}^{\mathfrak{p}}D_{\geq 0}^{b}(Y)$ $\displaystyle=\left\\{{\mathcal{E}}\in
D_{c}^{b}(Y):\dim\mathrm{Supp}({\mathcal{H}}^{i}({\mathbb{D}}{\mathcal{E}}))\leq-i\right\\}$

give rise to the _perverse $t$-structure_ on $D_{c}^{b}(Y)$, whose heart is
the abelian category of perverse sheaves,

$\mathrm{Perv}(Y)\subset D_{c}^{b}(Y).$

For $k\in{\mathbb{Z}}$, let ${}^{\mathfrak{p}}\tau_{\leq k}$ be the truncation
functor associated with the perverse $t$-structure. Given an object
${\mathcal{C}}\in D_{c}^{b}(Y)$, there is a natural morphism

(11) $^{\mathfrak{p}}\tau_{\leq k}{\mathcal{C}}\rightarrow{\mathcal{C}}.$

For the morphism $\pi:X\to Y$, we obtain from (11) the morphism

${}^{\mathfrak{p}}\tau_{\leq k}R\pi_{\ast}{\mathbb{Q}}_{X}\rightarrow
R\pi_{\ast}{\mathbb{Q}}_{X},$

which further induces a morphism of (hyper-)cohomology groups

(12) $H^{d-(\dim X-{R})}\Big{(}Y,\,^{\mathfrak{p}}\tau_{\leq
k}(R\pi_{\ast}{\mathbb{Q}}_{X}[\dim X-{R}])\Big{)}\rightarrow
H^{d}(X,{\mathbb{Q}}).$

Here

(13) ${R}=\dim X\times_{Y}X-\dim X$

is the _defect of semismallness_. The $k$-th piece of the perverse filtration
(1)

$P_{k}H^{d}(X,{\mathbb{Q}})\subset H^{d}(X,{\mathbb{Q}})$

is defined to be the image of (12).222Here the shift $[\dim X-{R}]$ is to
ensure that the perverse filtration is concentrated in the degrees $[0,2{R}]$.

We say that a graded vector space decomposition

$H^{*}(X,{\mathbb{Q}})=\bigoplus_{k,d}G_{k}H^{d}(X,{\mathbb{Q}})$

_splits_ the perverse filtration, if

$P_{k}H^{d}(X,{\mathbb{Q}})=\bigoplus_{i\leq
k}G_{i}H^{d}(X,{\mathbb{Q}}),\quad\forall k,d.$

###  1.3. The decomposition theorem

Perverse filtrations can be described through the decomposition theorem [7,
17]. By applying the decomposition theorem to the morphism $\pi:X\to Y$, we
obtain an isomorphism

(14) $R\pi_{*}{\mathbb{Q}}_{X}[\dim
X-{R}]\simeq\bigoplus_{i=0}^{2{R}}\mathcal{P}_{i}[-i]\in D^{b}_{c}(Y)$

with $\mathcal{P}_{i}\in\mathrm{Perv}(Y)$. The perverse filtration can be
identified as

$P_{k}H^{d}(X,{\mathbb{Q}})=\mathrm{Im}\Big{\\{}H^{d-(\dim
X-{R})}(Y,\bigoplus_{i=0}^{k}\mathcal{P}_{i}[-i])\to
H^{d}(X,{\mathbb{Q}})\Big{\\}}.$

In general, the isomorphism (14) in the decomposition theorem is not
canonical. Once we fix such an isomorphism $\phi$, we obtain a splitting
$G_{\ast}H^{\ast}(X,{\mathbb{Q}})$ of the perverse filtration,

$P_{k}H^{d}(X,{\mathbb{Q}})=\bigoplus_{i\leq k}G_{i}H^{d}(X,{\mathbb{Q}}).$

Here

$G_{i}H^{d}(X,{\mathbb{Q}})=\mathrm{Im}\Big{\\{}H^{d-(\dim
X-{R})}(Y,\mathcal{P}_{i}[-i])\to H^{d}(X,{\mathbb{Q}})\Big{\\}}$

with

$H^{d-(\dim X-{R})}(Y,\mathcal{P}_{i}[-i])\to H^{d}(X,{\mathbb{Q}})$

the morphism induced by $\phi^{-1}$.

###  1.4. Perverse filtration for projective bases

We review another description of the perverse filtration associated with

$\pi:X\to Y,$

when $Y$ is projective.

We fix $\eta$ to be an ample class on $Y$, and we consider

$L=\pi^{*}\eta\in H^{2}(X,{\mathbb{Q}}).$

The class $L$ acts on $H^{*}(X,{\mathbb{Q}})$ as an nilpotent operator via cup
product

$L:H^{*}(X,{\mathbb{Q}})\xrightarrow{\cup L}H^{*}(X,{\mathbb{Q}}).$

The following proposition shows that the filtration (1) is completely
described by an ample class on the base. It is stated in the special case
needed in this paper.

######  Proposition 1.1 ([17] Proposition 5.2.4.).

Assume that

$\dim X=2\dim Y=2R.$

Then we have

$P_{k}H^{m}(X,{\mathbb{Q}})=\sum_{i\geq
1}\left(\mathrm{Ker}(L^{R+k+i-m})\cap\mathrm{Im}(L^{i-1})\right)\cap
H^{m}(X,{\mathbb{Q}}).$

###  1.5. Abelian surfaces and Hilbert schemes

Let $A=E\times E^{\prime}$ be an abelian surface with $E$ and $E^{\prime}$
elliptic curves, and let $n\in{\mathbb{Z}}^{\geq 0}$. The natural projection,

$p:A\to E^{\prime}$

induces a morphism

$p_{n}:A^{[n]}\to{E^{\prime}}^{(n)}$

from the Hilbert scheme of $n$ points on $A$ to the symmetric product of the
elliptic curve $E^{\prime}$. We briefly review the construction [46] of a
_canonical_ splitting

(15)
$H^{\ast}(A^{[n]},{\mathbb{Q}})=\bigoplus_{i,d}\widetilde{G}_{i}H^{d}(A^{[n]},{\mathbb{Q}})$

of the perverse filtration associated with $p_{n}$, which culminates with the
explicit formula (18).

We start with a canonical decomposition

(16)
$H^{*}(A,{\mathbb{Q}})=\bigoplus_{i,d}\widetilde{G}_{i}H^{d}(A,{\mathbb{Q}})$

with $\widetilde{G}_{i}H^{d}(A,{\mathbb{Q}})$ the Künneth factor

$H^{i}(E,{\mathbb{Q}})\otimes H^{d-i}(E^{\prime},{\mathbb{Q}})\subset
H^{d}(A,{\mathbb{Q}}).$

Since $p:A\to E^{\prime}$ is a trivial fibration, (16) splits the perverse
filtration associated with $p$. Assume that we already have decompositions of
$H^{*}(X_{1},{\mathbb{Q}})$ and $H^{*}(X_{2},{\mathbb{Q}})$, a decomposition
of $H^{\ast}(X_{1}\times X_{2},{\mathbb{Q}})$ can be constructed by using the
Künneth decomposition. In particular, we obtain the direct sum decomposition
of $H^{*}(A^{n},{\mathbb{Q}})$ with summands

$\widetilde{G}_{k}H^{*}(A^{n},{\mathbb{Q}})=\left\langle\alpha_{1}\boxtimes\cdots\boxtimes\alpha_{n};~{}~{}\alpha\in\widetilde{G}_{k_{i}}H^{*}(A,{\mathbb{Q}}),~{}~{}\sum_{i}{k_{i}}=k\right\rangle,$

whose $\mathfrak{S}_{n}$-invariant part induces a decomposition

$H^{*}(A^{(n)},{\mathbb{Q}})=\bigoplus\widetilde{G}_{i}H^{d}(A^{(n)},{\mathbb{Q}}).$

In turn, by using Künneth decompositions again, this gives us canonical
decompositions for

$H^{\ast}(A^{(n_{1})}\times A^{(n_{2})}\times\cdots\times
A^{(n_{k})},{\mathbb{Q}}),\quad n_{i}\geq 1.$

Finally, the cohomology of the Hilbert scheme $A^{[n]}$ is related to the
cohomology of symmetric products by [25, 16]. We employ exponential notation
for partitions $\nu:=(\nu_{i})_{i=1}^{l}$, with $\nu_{i}>0$ and
$\sum_{i}\nu_{i}=n,$ of a positive integer $n$. Namely, we let $a_{k}$ be the
number of times the integer $k$ appears in the partition, so that the
expression

$1^{a_{1}}2^{a_{2}}\cdots n^{a_{n}}$

recovers the partition $\nu$. We use $A^{(\nu)}$ to denote the variety
$A^{(a_{1})}\times A^{(a_{2})}\times\cdots\times A^{(a_{n})}$. The cohomology
group $H^{d}(A^{[n]},{\mathbb{Q}})$ admits a canonical decomposition

(17)
$H^{d}(A^{[n]},{\mathbb{Q}})=\bigoplus_{\nu}H^{d+2l(\nu)-2n}(A^{(\nu)},{\mathbb{Q}})$

where $\nu$ runs through all partitions of $n$ and $l(\nu)$ is the length of
$\nu$. The desired canonical splitting (15) is then defined by setting

(18)
$\widetilde{G}_{k}H^{d}(A^{[n]},{\mathbb{Q}})=\bigoplus_{\nu}\widetilde{G}_{k+l(\nu)-n}H^{d+2l(\nu)-2n}(A^{(\nu)},{\mathbb{Q}})$

to be the sub-vector space of $H^{d}(A^{[n]},{\mathbb{Q}})$ under the
identification (17). By [52, Proposition 4.12], this decomposition splits the
perverse filtration associated with $p_{n}$.333We will give another
interpretation of the splitting $\tilde{G}_{*}H^{*}(A^{[n]},{\mathbb{Q}})$ in
Corollary 3.7.

In view of [52, Proposition 2.1], by using the Künneth decomposition again, we
obtain a canonical splitting

(19) $H^{\ast}(A\times
A^{[n]},{\mathbb{Q}})=\bigoplus_{i,d}\widetilde{G}_{i}H^{d}(A\times
A^{[n]},{\mathbb{Q}})$

of the perverse filtration associated with

$p\times p_{n}:A\times A^{[n]}\to E^{\prime}\times E^{\prime(n)}.$

The following theorem obtained in [52, 46] is concerned with tautological
classes and multiplicativity in the context of Hilbert schemes.

######  Theorem 1.2.

Let ${\mathcal{I}}_{n}$ be the universal ideal sheaf on $A\times A^{[n]}$.

  1. (a)

(Tautological classes) We have

$\mathrm{ch}_{k}({\mathcal{I}}_{n})\in\widetilde{G}_{k}H^{2k}(A\times
A^{[n]},{\mathbb{Q}}).$

  2. (b)

(Multiplicativity) The decomposition (15) is multiplicative, i.e.

$\cup:\widetilde{G}_{i}H^{d}(A^{[n]},{\mathbb{Q}})\times\widetilde{G}_{i^{\prime}}H^{d^{\prime}}(A^{[n]},{\mathbb{Q}})\to\widetilde{G}_{i+i^{\prime}}H^{d+d^{\prime}}(A^{[n]},{\mathbb{Q}}).$

###### Proof.

Theorem 1.2 (b) was proven in [52]. Although the multiplicativity was shown
only for the perverse filtration in [52], the same proof works for the
decomposition (15). This was explained in [46, Proposition 1.8]. Theorem 1.2
(a) follows directly from the proof of [46, Theorem 0.1] and the proof of [46,
Theorem 3.3], where it is explained how to treat perverse decompositions
instead of perverse filtrations. ∎

##  2\. Moduli of one-dimensional sheaves

###  2.1. Overview and main results

Throughout this section, we assume $A$ is an abelian surface and fix
$\chi\in\mathbb{Z}$. Let $\beta\in H_{2}(A,{\mathbb{Z}})$ be an ample curve
class with $\beta^{2}\geq 6$; we only consider classes $\beta$ for which the
vector $(0,\beta,\chi)\in H^{\mathrm{ev}}(A,{\mathbb{Z}})$ is primitive.
Finally, we assume that we are given a polarization $H$ that is generic with
respect to the vector $(0,\beta,\chi)$. We will typically omit $\chi$ and $H$
from our notation.

Let ${\mathcal{M}}_{\beta,A}$ be the moduli space which parametrizes one-
dimensional sheaves satisfying

(20) $[\mathrm{supp}({\mathcal{F}})]=\beta,\quad\chi({\mathcal{F}})=\chi,$

that are Gieseker-stable with respect to our generic polarization. In the
paper, by “support”, we mean Fitting support (see [34]), and the square
bracket denotes the associated homology class. Equivalently, these are
Gieseker-stable sheaves on $A$ with Mukai vector $(0,\beta,\chi)$. By [51],
the moduli space ${\mathcal{M}}_{\beta,A}$ is a non-empty, nonsingular
projective variety of dimension $\beta^{2}+2$.

Let

$\pi_{A}:A\times{\mathcal{M}}_{\beta,A}\to
A,\quad\pi_{\mathcal{M}}:A\times{\mathcal{M}}_{\beta,A}\to{\mathcal{M}}_{\beta,A}$

be the projections. If there exists a universal family ${\mathcal{F}}_{\beta}$
on $A\times{\mathcal{M}}_{\beta,A}$, then, in analogy to Section 0.3, we
define the twisted Chern character associated with a class

(21) $\alpha=\pi_{A}^{\ast}\alpha_{1}+\pi_{\mathcal{M}}^{\ast}\alpha_{2}\in
H^{2}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{C}})$

by setting

$\mathrm{ch}^{\alpha}({\mathcal{F}}_{\beta})=\mathrm{ch}({\mathcal{F}}_{\beta})\cup\mathrm{exp}(\alpha)\in
H^{*}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{C}})$

(the use of ${\mathbb{C}}$-coefficients is for later use in Theorem 2.1), and
we denote its degree $2k$ part by

$\mathrm{ch}_{k}^{\alpha}({\mathcal{F}}_{\beta})\in
H^{2k}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{C}}).$

In general, a universal family may not exist; however, as explained in [38,
Section 3.1], there exists a _universal class_

$[{\mathcal{F}}_{\beta}]\in K_{\mathrm{top}}(A\times{\mathcal{M}}_{\beta,A}),$

which is well-defined up to pullback of a topological line bundle on
${\mathcal{M}}_{\beta,A}$.444Here $K_{\mathrm{top}}(X)$ denotes the
Grothendieck ring of topological complex vector bundles on $X$ [1]. As before,
given a choice of $\alpha$ as in (21), we can take its associated twisted
Chern character

$\mathrm{ch}^{\alpha}([{\mathcal{F}}_{\beta}])\in
H^{*}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{C}}),$

which we still denote by $\mathrm{ch}^{\alpha}({\mathcal{F}}_{\beta})$ for
notational convenience.

The moduli space ${\mathcal{M}}_{\beta,A}$ admits a natural Hilbert–Chow
morphism

(22) $\pi_{\beta}:{\mathcal{M}}_{\beta,A}\to
B,\quad\pi_{\beta}({\mathcal{F}})=[\mathrm{supp}({\mathcal{F}})]$

where $B$ is the Chow variety parametrizing effective one-cycles in the class
$\beta$; see [34, 35]. The purpose of Section 2 is to prove the following
theorem, which can be viewed as an analog of Conjecture 0.3 for abelian
surfaces.

######  Theorem 2.1.

There exists a choice of universal twisted class
$\mathrm{ch}^{\alpha}({\mathcal{F}}_{\beta})$ with $\alpha$ of type (21), and
a splitting
$\widetilde{G}_{\ast}H^{\ast}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$ of the
perverse filtration associated with the Hilbert-Chow morphism $\pi_{\beta}$
(22), satisfying the following properties.

  1. (a)

(Tautological classes) For any $\gamma\in H^{*}(A,{\mathbb{C}})$, we have

$\int_{\gamma}\mathrm{ch}^{\alpha}_{k}({\mathcal{F}}_{\beta})\in\widetilde{G}_{k-1}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}}).$

  2. (b)

(Multiplicativity) We have

$\cup:\widetilde{G}_{i}H^{d}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\times\widetilde{G}_{i^{\prime}}H^{d^{\prime}}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\to\widetilde{G}_{i+i^{\prime}}H^{d+d^{\prime}}({\mathcal{M}}_{\beta,A},{\mathbb{C}}).$

Theorem 2.1 will be deduced from Theorem 1.2 and Markman’s results on the
monodromy of holomorphic symplectic varieties [39, 41]. The strategy of the
argument is to use Theorem 1.2 and direct computations to study the case of
primitive curve classes for special abelian varieties. To pass to imprimitive
curve classes (which form a distinct deformation class of Lagrangian
fibrations), we require a mild extension of Markman’s work to the setting of
complex coefficients.

In the next three sections, we review Markman’s work, and in the subsequent
three sections, we prove Theorem 2.1.

###  2.2. Mukai lattices

Let

$H^{\ast}(A,{\mathbb{Z}})=S_{A}^{+}\oplus S_{A}^{-}$

be the even/odd decomposition of the cohomology $H^{\ast}(A,{\mathbb{Z}})$
with

$S_{A}^{+}=H^{0}(A,{\mathbb{Z}})\oplus H^{2}(A,{\mathbb{Z}})\oplus
H^{4}(A,{\mathbb{Z}}),\quad S_{A}^{-}=H^{1}(A,{\mathbb{Z}})\oplus
H^{3}(A,{\mathbb{Z}}).$

The Mukai pairing on $S_{A}^{+}$ is given by

(23) $\langle a,b\rangle=\int_{A}{(a_{1}\cup b_{1}-a_{0}\cup b_{2}-a_{2}\cup
b_{0})},$

where $a=(a_{0},a_{1},a_{2})$ and $b=(b_{0},b_{1},b_{2})$ are elements in
$S_{A}^{+}$. We have

$(S_{A}^{+},\langle~{}~{}~{},~{}~{}~{}\rangle)\cong U^{\oplus 4}$

with $U$ the parabolic plane. For a coherent sheaf ${\mathcal{F}}$ on $A$, its
Mukai vector

$v({\mathcal{F}})=\mathrm{ch}({\mathcal{F}})\cup\sqrt{\mathrm{td}_{A}}\in
S_{A}^{+}$

coincides with the Chern character $\mathrm{ch}({\mathcal{F}})$.

Throughout Section 2, we assume $v\in S_{A}^{+}$ is a primitive vector such
that the moduli space $M_{v,A}$ of Gieseker-stable sheaves ${\mathcal{F}}$ on
$A$ (with respect to a very general polarization $H$) with Mukai vector
$v({\mathcal{F}})=v$ is non-empty. By [51, Theorem 0.1], the variety $M_{v,A}$
is nonsingular and projective of dimension

$\mathrm{dim}(M_{v,A})=\langle v,v\rangle+2,$

and it is deformation equivalent to

$M_{(1,0,-n),A}=A^{[n]}\times\hat{A}$

where $\hat{A}$ is the dual abelian variety of $A$ and $2n=\langle
v,v\rangle$.

A universal family on

$A\times M_{(1,0,-n),A}=A\times A^{[n]}\times\hat{A}$

is

(24)
${\mathcal{E}}_{n}=\pi_{12}^{\ast}{\mathcal{I}}_{n}\otimes\pi_{13}^{\ast}{\mathcal{P}},$

with ${\mathcal{I}}_{n}$ the universal ideal sheaf, ${\mathcal{P}}$ the
normalized Poincaré line bundle on $A\times\hat{A}$, and $\pi_{ij}$ the
corresponding projections.

###  2.3. $\mathrm{Spin}$ representations

The even cohomology $S_{A}^{+}$ together with the Mukai pairing is a
unimodular lattice. Consider the corresponding Spin group for this lattice,
denoted by $\mathrm{Spin}(S_{A}^{+})$. We recall certain representations of
this group involved in Markman’s work following [41, Section 3].

Let $H^{*}(A,{\mathbb{Z}})$ denote the total cohomology of $A$ equipped with
the pairing

(25) $(a,b)_{A}=\int_{A}\tau(a)\cup b,$

where $\tau$ acts on $H^{i}(A,{\mathbb{Z}})$ by $(-1)^{i(i-1)/2}$. The group
$\mathrm{Spin}(S_{A}^{+})$ admits an action by parity-preserving isometries on
$H^{*}(A,{\mathbb{Z}})$

$\mathrm{Spin}(S_{A}^{+})\rightarrow\mathrm{Aut}(H^{*}(A,{\mathbb{Z}})),$

which extends the natural representation on $S_{A}^{+}$.

We consider $V$ to be the lattice

$V=H^{1}(A,{\mathbb{Z}})\oplus H^{1}(\hat{A},{\mathbb{Z}})$

carrying a symmetric bilinear form via the canonical identification

$H^{1}(\hat{A},{\mathbb{Z}})=H^{1}(A,{\mathbb{Z}})^{*}.$

There exists a representation

$\mathrm{Spin}(S_{A}^{+})\rightarrow\mathrm{Aut}(V)$

by isometries which we can extend to an action of $\mathrm{Spin}(S_{A}^{+})$
on $\bigwedge^{k}V=H^{k}(A\times\hat{A},{\mathbb{Z}})$ for each $k$. This
action will play a role in the proof of Theorem 2.8.

These constructions are linear-algebraic and make sense for the groups
$\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{K}})$ with general coefficients

${\mathbb{K}}={\mathbb{Z}},{\mathbb{Q}},{\mathbb{R}},\mathrm{or}~{}~{}{\mathbb{C}}.$

###  2.4. Monodromy representations

In this section, we recall Markman’s monodromy operators [41] for abelian
surfaces. They play a crucial role in the proof of Theorem 2.1. We refer to
[39] for a parallel theory concerning $K3$ surfaces.

Let $H^{\ast}(A)$ be the cohomology $H^{\ast}(A,{\mathbb{K}})$ with
coefficients ${\mathbb{K}}={\mathbb{Z}}~{}~{}\mathrm{or}~{}~{}{\mathbb{C}}$.
Assume that

$g:H^{*}(A)\to H^{*}(A)$

is a parity-preserving homomorphism induced by an element (cf. Section 2.3)

$g\in\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{K}})$

which fixes a primitive Mukai vector $v\in S_{A}^{+}$. Assume we are given a
generic polarization $H$ with respect to $v$. We explain how to construct a
graded ${\mathbb{K}}$-algebra automorphism

$\gamma_{g,v}:H^{*}(M_{v,A},{\mathbb{K}})\to H^{*}(M_{v,A},{\mathbb{K}}).$

For a projective variety $X$, we define

$l:\bigoplus_{i}H^{2i}(X,{\mathbb{Q}})\rightarrow\bigoplus_{i}H^{2i}(X,{\mathbb{Q}})$

to be the universal polynomial map which takes the exponential Chern character
to its total Chern class,

$l(r+a_{1}+a_{2}+\cdots)=1+a_{1}+(a_{1}^{2}/2-a_{2})+\cdots,$

and we define

$(-)^{\vee}:\bigoplus_{i}H^{\mathrm{2i}}(X,{\mathbb{Z}})\to\bigoplus_{i}H^{2i}(X,{\mathbb{Z}}),\quad\omega\mapsto\omega^{\vee}$

to be the dualizing automorphism which acts by $(-1)^{i}$ on the even
cohomology $H^{2i}(X,{\mathbb{Z}})$.

Let $[{\mathcal{E}}_{v}]$ and $[{\mathcal{E}}^{\prime}_{v}]$ be universal
classes on $A\times M_{v,A}$, in the sense of Section 2.1. We denote by
$x_{g,v}$ the class

$x_{g,v}=\pi_{12}^{\ast}\left[(\mathrm{id}\otimes
g)\mathrm{ch}({\mathcal{E}}_{v})\right]^{\vee}\cup\pi_{23}^{\ast}\mathrm{ch}({\mathcal{E}}^{\prime}_{v})\in
H^{*}(M_{v,A}\times A\times M_{v,A},{\mathbb{K}}),$

where $\pi_{ij}$ are the projections from the product $M_{v,A}\times A\times
M_{v,A}$ to the corresponding factors. Via the formalism of correspondences,
to define an endomorphism of cohomology, it suffices to give a cohomology
class in the product; in our case, the morphism $\gamma_{g,v}$ is defined by
the class ()

(26)
$\gamma_{g,v}([{\mathcal{E}}_{v}],[{\mathcal{E}}^{\prime}_{v}])=\left(\left[l({\pi_{13}}_{\ast}x_{g,v})\right]^{-1}\right)_{\mathrm{deg}~{}2d_{M}}\in
H^{2d_{M}}(M_{v,A}\times M_{v,A},{\mathbb{K}})$

where $d_{M}=\mathrm{dim}(M_{v,A})$, and $-_{\mathrm{deg}~{}~{}k}$ denotes the
degree $k$ part of a cohomology class $-$.

Markman proves the following statements regarding
$\gamma_{g,v}([{\mathcal{E}}_{v}],[{\mathcal{E}}^{\prime}_{v}])$ when
${\mathbb{K}}={\mathbb{Z}}$

  1. (a)

$\gamma_{g,v}:=\gamma_{g,v}([{\mathcal{E}}_{v}],[{\mathcal{E}}^{\prime}_{v}])$
is independent of the choice of universal classes $[{\mathcal{E}}_{v}]$ and
$[{\mathcal{E}}^{\prime}_{v}]$.

  2. (b)

$\gamma_{g,v}$ is a graded ${\mathbb{K}}$-algebra automorphism.

Consider the subgroup

$\mathrm{Spin}(S_{A}^{+})_{v}\subset\mathrm{Spin}(S_{A}^{+})$

preserving $v$. By [41, Corollary 8.4], every $\gamma_{g,v}$ with
$g\in\mathrm{Spin}(S_{A}^{+})_{v}$ is a graded ring automorphism of
$H^{\ast}(M_{v,A},{\mathbb{Z}})$ and the relation

(27) $\gamma_{g_{1},v}\circ\gamma_{g_{2},v}=\gamma_{g_{1}g_{2},v}$

holds. Therefore, we have a group homomorphism (Markman’s monodromy operators)

(28)
$\mathrm{mon}:\mathrm{Spin}(S_{A}^{+})_{v}\to\mathrm{Aut}(H^{*}(M_{v,A},{\mathbb{Z}})),\quad\mathrm{mon}(g)=\gamma_{g,v}.$

The following theorem is proved by Markman for ${\mathbb{Z}}$ coefficients in
[39, 41]. The extension of (28) to ${\mathbb{C}}$-coefficients is crucial in
the proof of Theorem 2.1.

######  Theorem 2.2.

For $g\in\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{C}})_{v}$, the class
$\gamma_{g,v}:=\gamma_{g,v}([{\mathcal{E}}_{v}],[{\mathcal{E}}^{\prime}_{v}])$
is independent of the choice of universal classes $[{\mathcal{E}}_{v}]$ and
$[{\mathcal{E}}^{\prime}_{v}]$. The morphism (28) extends to a group
homomorphism

$\mathrm{mon}:\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{C}})_{v}\to\mathrm{Aut}(H^{*}(M_{v,A},{\mathbb{C}})).$

Moreover, for a fixed universal class $[{\mathcal{E}}_{v}]$ and any
$g\in\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{C}})_{v}$, there exists $\alpha\in
H^{2}(M_{v,A},{\mathbb{C}})$ so that we have

(29)
$(g\otimes\gamma_{g,v})\mathrm{ch}({\mathcal{E}}_{v})=\mathrm{ch}({{\mathcal{E}}}_{v})\cup\pi_{M}^{*}\mathrm{exp}(\alpha)$

where $\pi_{M}:A\times M_{v,A}\to M_{v,A}$ is the second projection.

###### Proof.

We deduce this theorem from the case of integral coefficients using the
Zariski-density of integral points

$\mathrm{Spin}(S_{A}^{+})_{v}\subset\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{C}})_{v},$

which follows from the Borel Density Theorem [8] applied to the simple group

$\mathrm{Spin}(S_{A}^{+})_{v}\cong\mathrm{Spin}(v^{\perp}),\quad
v^{\perp}\subset S_{A}^{+}.$

The first claim, on the independence of universal classes, expresses a
Zariski-closed condition on the element $g$, so that it can be deduced from
(a) above; see [39, Lemma 3.11].

For the remaining claims, as in the proof of [41, Corollary 8.4], we can
reduce the theorem to the case $\mathrm{rank}(v)>0$. It follows that the class
$\alpha$ in the equation (29) is uniquely determined by

$\mathrm{rank}(v)\cdot\pi_{M}^{\ast}\alpha=c_{1}([{\mathcal{E}}_{v}])-\left[(g\otimes\gamma_{g,v})\mathrm{ch}([{\mathcal{E}}_{v}])\right]_{\mathrm{deg}~{}2}.$

The operator $\gamma_{g,v}$ (26) arises as a grading-preserving element in

$\mathrm{End}(H^{*}(M_{v,A},{\mathbb{C}})).$

We need to show that for any
$g\in\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{C}})_{v}$, we have that:

  1. (i)

$\gamma_{g,v}$ is an automorphism of graded ${\mathbb{C}}$-algebras, and

  2. (ii)

the equation (29) holds.

We first show (ii). By [41, Corollary 8.4], for every integral point
$g\in\mathrm{Spin}(S_{A}^{+})$, the homomorphism

$g\otimes\gamma_{g,v}:H^{*}(A\times M_{v,A},{\mathbb{C}})\to H^{*}(A\times
M_{v,A},{\mathbb{C}})$

sends a universal class to a universal class in the sense of [41, Definition
3.4]. It follows that the equation (29) holds for any integral $g$. Since this
equation expresses a Zariski-closed condition with respect to the parameter
$g$, we obtain (29) by the density of integral points in
$\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{C}})_{v}$.

As to (i), we deduce, again, from Zariski-density, that $\gamma_{g,v}$ is a
graded ${\mathbb{C}}$-algebra homomorphism for any
$g\in\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{C}})_{v}$. The endomorphism
$\gamma_{g,v}$ being an automorphism is not a Zariski closed condition.
However, to prove $\gamma_{g,v}\in\mathrm{Aut}(H^{*}(M_{v,A},{\mathbb{C}}))$,
it suffices to show that

(30) $\gamma_{g,v}\circ\gamma_{g^{-1},v}=\mathrm{id},\quad
g\in\mathrm{Spin}(S_{A}^{+}\otimes{\mathbb{C}})_{v}.$

For integral $g$, the equation (30) follows from (27),

$\gamma_{g,v}\circ\gamma_{g^{-1},v}=\gamma_{g\cdot
g^{-1},v}=\gamma_{\mathrm{id},v}=\mathrm{id}.$

Since (30) is a Zariski closed condition, we conclude that it holds for any
$g$, and (i) follows. ∎

###  2.5. Fourier–Mukai transforms

Sections 2.5 to 2.7 are devoted to proving Theorem 2.1. We first treat the
special case where, if we let $E$ and $E^{\prime}$ be two elliptic curves,
then we set

(31) $A=E\times E^{\prime},\quad\beta=\sigma+nf,$

where $\sigma=[\mathrm{pt}\times E^{\prime}]$ and $f=[E\times\mathrm{pt}]$ are
the classes of a section and a fiber with respect to the elliptic fibration
$p:A\to E^{\prime}$. Note that $\beta^{2}=2n$.

By [51, Theorem 3.15], we have an isomorphism of moduli spaces

(32) ${\mathcal{M}}_{\beta,A}\cong M_{v_{n},A}(=A^{[n]}\times\hat{A}),\quad
v_{n}=(1,0,-n)$

induced by a relative Fourier–Mukai transform with respect to the elliptic
fibration $p:A\to E^{\prime}$. Our strategy is to compare a universal rank 1
torsion free sheaf on $A\times M_{v_{n},A}$ with a universal 1-dimensional
sheaf on $A\times{\mathcal{M}}_{\beta,A}$ under the isomorphism (32). Then we
reduce the special case (31) of Theorem 2.1 to Theorem 1.2.

The proof of [51, Theorem 3.15] shows that the support of a sheaf
${\mathcal{F}}\in{\mathcal{M}}_{\beta,A}$ is the union of a section and
several fibers (with multiplicities). Hence the Hilbert–Chow morphism (22) is
exactly the morphism

(33) $p_{n}\times q:A^{[n]}\times\hat{A}\to E^{\prime(n)}\times E.$

Here we use the identification between $\hat{A}$ and $A$ for the abelian
surface $A=E\times E^{\prime}$, and $q$ is the projection to the first factor.
Under the identification

$A\times A=(E\times E)\times(E^{\prime}\times E^{\prime}),$

the kernel for the relative Fourier–Mukai transform associated with the
elliptic fibration $p:A\to E^{\prime}$ is given by

(34)
${\mathcal{P}}^{\prime}_{E}={\mathcal{P}}_{E}\boxtimes{\mathcal{O}}_{\Delta_{E^{\prime}}}$

with

${\mathcal{P}}_{E}={\mathcal{O}}_{E\times E}(\Delta_{E}-o_{E}\times E-E\times
o_{E})$

the normalized Poincaré line bundle. Recall the universal family
${\mathcal{E}}_{n}$ on $A\times M_{v_{n},A}$ defined in (24). A universal
family ${\mathcal{F}}_{\beta}$ of 1-dimensional Gieseker-stable sheaves on
$A\times{\mathcal{M}}_{\beta,A}$ is induced by ${\mathcal{E}}_{n}$ and
${\mathcal{P}}_{E}^{\prime}$ under the isomorphism (32) as follows.

For $\chi$ given in (20), we define

$N={\mathcal{O}}_{A}\left(\chi\cdot[E\times
o_{E^{\prime}}]\right)=p^{*}{\mathcal{O}}_{E^{\prime}}(\chi\cdot
o_{E^{\prime}})\in\mathrm{Pic}(A).$

By [51, Theorem 3.15] together with its proof, every closed point in
${\mathcal{M}}_{\beta,A}$ is given by

$\left(\phi_{{\mathcal{P}}^{\prime}_{E}}(I_{Z}\otimes L)\otimes
N\right)[1]\in\mathrm{Coh}(A)$

where $\phi_{{\mathcal{P}}^{\prime}_{E}}$ is the Fourier–Mukai transform with
kernel (34), and $I_{Z}\otimes L\in M_{v_{n},A}=A^{[n]}\times\hat{A}$ is a
uniquely determined rank one torsion free sheaf on $A$. In particular, the
sheaf

(35)
${\mathcal{F}}_{\beta}={R\pi_{13}}_{\ast}\left(\pi_{12}^{\ast}(\pi_{1}^{*}N\otimes{{\mathcal{P}}^{\prime}_{E}})\otimes^{{\mathbb{L}}}\pi_{23}^{\ast}{\mathcal{E}}_{n}\right)[1]\in\mathrm{Coh}(A\times{\mathcal{M}}_{\beta,A})$

is a universal family on $A\times M_{\beta,A}$, where $\pi_{i},\pi_{ij}$ are
projections from $A\times A\times M_{v_{n},A}$ to the corresponding factors,
and we identify $A\times M_{v_{n},A}$ with $A\times{\mathcal{M}}_{\beta,A}$
via (32).

For notational convenience, we define the shifted normalized universal family

(36)
${\mathcal{F}}^{n}_{\beta}={R\pi_{13}}_{\ast}\left(\pi_{12}^{\ast}{{\mathcal{P}}^{\prime}_{E}}\otimes^{{\mathbb{L}}}\pi_{23}^{\ast}{\mathcal{E}}_{n}\right)={\mathcal{F}}_{\beta}\otimes\pi_{A}^{\ast}N^{-1}[-1]$

on $A\times{\mathcal{M}}_{\beta,A}$, whose restriction over every closed point
${\mathcal{F}}\in{\mathcal{M}}_{\beta,A}$ recovers the shifted 1-dimensional
coherent sheaf $\left({\mathcal{F}}\otimes N^{-1}\right)[-1]$ on A.

We use the universal family (35) and the class $\alpha$ (39) to serve as the
ones in Theorem 2.1 for the special case (31). Next, we construct a
decomposition of the perverse filtration

(37)
$H^{\ast}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})=\bigoplus_{i,d}\widetilde{G}_{i}H^{d}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})$

which serves as the splitting in Theorem 2.1.

A canonical splitting of the perverse filtration associated with

$q:\hat{A}=A\to E$

is given by

$G^{\prime}_{k}H^{d}(\hat{A},{\mathbb{Q}})=\bigoplus_{i+k=d}H^{d-k}(E,{\mathbb{Q}})\otimes
H^{k}(E^{\prime},{\mathbb{Q}}).$

We define (37) as the decomposition

(38)
$\widetilde{G}_{k}H^{\ast}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})=\bigoplus_{i+j=k}\widetilde{G}_{i}H^{\ast}(A^{[n]},{\mathbb{Q}})\otimes
G^{\prime}_{j}H^{\ast}(\hat{A},{\mathbb{Q}}),$

where $\widetilde{G}_{\ast}H^{\ast}(A^{[n]},{\mathbb{Q}})$ was defined by
(17). Since $\widetilde{G}_{\ast}H^{\ast}(A^{[n]},{\mathbb{Q}})$ splits the
perverse filtration (18) associated with

$p_{n}:A^{[n]}\to E^{\prime(n)},$

the filtration (38) splits the perverse filtration associated with the
Hilbert–Chow morphism (33).

The following theorem verifies Theorem 2.1 for the special case (31), where we
can choose the twisting class to be

(39) $\alpha=-\pi_{A}^{*}c_{1}(N)\in
H^{2}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}}).$

######  Theorem 2.3.

The decomposition (38) is multiplicative, and, with $\alpha$ as in (39), we
have

(40)
$\int_{\gamma}\mathrm{ch}^{\alpha}_{k}({\mathcal{F}}_{\beta})\in\widetilde{G}_{k-1}H^{\ast}({\mathcal{M}}_{\beta,A},{\mathbb{Q}}),\quad\;\forall
k\geq 0,\forall\gamma\in H^{\ast}(A,{\mathbb{Q}}).$

###### Proof.

The multiplicativity of (38) follows directly from Theorem 1.2 (b) and the
Künneth formula. We need to prove (40), which, by the very definition of
$\alpha$ and ${\mathcal{F}}^{n}_{\beta}$ and by the fact that a cohomological
shift only changes the signs of Chern classes, is equivalent to

(41)
$\int_{\gamma}\mathrm{ch}_{k}({\mathcal{F}}^{n}_{\beta})\in\widetilde{G}_{k-1}H^{\ast}({\mathcal{M}}_{\beta,A},{\mathbb{Q}}),\quad\forall
k\geq 0,\;\forall\gamma\in H^{\ast}(A,{\mathbb{Q}}).$

We consider the product $A\times{\mathcal{M}}_{\beta,A}$ and the
multiplicative splitting of its cohomology

(42)
$\widetilde{G}_{k}H^{\ast}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}}):=H^{\ast}(A,{\mathbb{Q}})\otimes\widetilde{G}_{k}H^{\ast}({\mathcal{M}}_{\beta,A},{\mathbb{Q}}).$

This decomposition splits the perverse filtration associated with the morphism

$\mathrm{id}\times\pi_{\beta}:A\times{\mathcal{M}}_{\beta,A}\to A\times B.$

The statement (41) is equivalent to the following:

(43) $\mathrm{ch}({\mathcal{F}}^{n}_{\beta})\in\bigoplus_{k\geq
1}\widetilde{G}_{k-1}H^{2k}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}}).$

For a nonsingular projective variety $X$ with a decomposition
$G_{*}H^{*}(X,{\mathbb{Q}})$ on its cohomology, we say that a class $\alpha\in
H^{*}(X,{\mathbb{Q}})$ is _balanced_ with respect to this decomposition if

$\alpha\in\bigoplus_{k}G_{k}H^{2k}(X,{\mathbb{Q}}).$

We prove (43) by the following 3 steps.

_Step 1._ We consider the new decomposition

(44)
$\begin{split}\overline{G}_{k}H^{*}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}})&:=\bigoplus_{i+j=k}H^{i}(E,{\mathbb{Q}})\otimes
H^{*}(E^{\prime},{\mathbb{Q}})\otimes\widetilde{G}_{j}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})\\\
&=\bigoplus_{i+j=k}\widetilde{G}_{i}H^{\ast}(A\times
A^{[n]},{\mathbb{Q}})\otimes
G^{\prime}_{j}H^{\ast}(\hat{A},{\mathbb{Q}}).\end{split}$

of the cohomology $H^{\ast}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}})$. It
is a multiplicative decomposition splitting the perverse filtration associated
with the morphism

$p\times\pi_{\beta}:A\times{\mathcal{M}}_{\beta,A}\to E^{\prime}\times B.$

We first show that the class

$\mathrm{ch}({\mathcal{E}}_{n})\in H^{\ast}(A\times
M_{A,v_{n}},{\mathbb{Q}})=H^{*}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}})$

is balanced with respect to the splitting (44). Here the last identification
is given by (32).

Recall the expression (24) of the universal family ${\mathcal{E}}_{n}$. By
Theorem 1.2 (a), the class

$\mathrm{ch}({\mathcal{I}}_{n})\in H^{\ast}(A\times A^{[n]},{\mathbb{Q}})$

is balanced with respect to the splitting (19). Hence via pulling back along
the projection

$\pi_{12}:A\times{\mathcal{M}}_{\beta,A}=A\times A^{[n]}\times\hat{A}\to
A\times A^{[n]},$

we obtain that the class

$\begin{split}\pi_{12}^{\ast}\mathrm{ch}({\mathcal{I}}_{n})=\mathrm{ch}({\mathcal{I}}_{n})\boxtimes
1_{\hat{A}}&\in H^{\ast}(A\times A^{[n]},{\mathbb{Q}})\otimes
H^{0}(\hat{A},{\mathbb{Q}})\\\ &\subset
H^{*}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}})\end{split}$

is also balanced with respect to (44). A direct calculation shows that the
class $\pi_{13}^{\ast}c_{1}({\mathcal{P}})$ is balanced. Hence we conclude
from the multiplicativity of (44) that the class

$\mathrm{ch}({\mathcal{E}}_{n})=\pi_{12}^{\ast}\mathrm{ch}({\mathcal{I}}_{n})\otimes\pi_{13}^{\ast}\mathrm{ch}({\mathcal{P}})=\pi_{12}^{\ast}\mathrm{ch}({\mathcal{I}}_{n})\otimes\pi_{13}^{\ast}\mathrm{exp}(c_{1}({\mathcal{P}}))$

is balanced.

_Step 2._ We further consider the multiplicative decomposition

(45) $\widetilde{G}_{k}H^{*}(A\times
A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}}):=H^{*}(A,{\mathbb{Q}})\otimes\overline{G}_{k}H^{*}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}})$

of $H^{*}(A\times A\times{\mathcal{M}}_{\beta,A})$, which splits the perverse
filtration associated with the morphism

$\mathrm{id}\times p\times\pi_{\beta}:A\times
A\times{\mathcal{M}}_{\beta,A}\to A\times E^{\prime}\times B.$

We show that

(46)
$\mathrm{ch}(\pi_{12}^{\ast}{{\mathcal{P}}^{\prime}_{E}}\otimes^{{\mathbb{L}}}\pi_{23}^{\ast}{\mathcal{E}}_{n})\in\bigoplus_{k\geq
1}\widetilde{G}_{k-1}H^{2k}(A\times
A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}}).$

Here $\pi_{ij}$ are the projections from $A\times
A\times{\mathcal{M}}_{\beta,A}$ to the corresponding factors; compare the
notation of (36).

In fact, by Step 1, we obtain that the class

$\pi_{23}^{*}\mathrm{ch}({\mathcal{E}}_{n})=1_{A}\boxtimes\mathrm{ch}({\mathcal{E}}_{n})\in
H^{0}(A,{\mathbb{Q}})\otimes
H^{*}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}})$

is balanced with respect to (45). The relative Poincaré sheaf (34) can be
expressed as

(47)
$\pi_{13}^{\ast}\mathrm{ch}({\mathcal{P}}^{\prime}_{E})=\pi_{13}^{\ast}\mathrm{ch}({\mathcal{P}}_{E})\cup\pi_{13}^{*}\mathrm{ch}({\mathcal{O}}_{\Delta_{E^{\prime}}}).$

The class $\pi_{12}^{\ast}\mathrm{ch}({\mathcal{P}}_{E})$ is balanced via a
direct calculation, and

$\pi_{12}^{*}\mathrm{ch}({\mathcal{O}}_{\Delta_{E}^{\prime}})=\pi_{12}^{*}c_{1}({\mathcal{O}}_{\Delta_{E^{\prime}}})\in\widetilde{G}_{0}H^{2}(A\times
A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}}).$

Hence by (47) and the multiplicativity of (45), we get

$\pi_{12}^{*}\mathrm{ch}({\mathcal{P}}^{\prime}_{E})\in\bigoplus_{k\geq
1}\widetilde{G}_{k-1}H^{2k}(A\times A\times{\mathcal{M}}_{\beta,A}).$

This yields (46) since

$\mathrm{ch}(\pi_{12}^{\ast}{{\mathcal{P}}^{\prime}_{E}}\otimes^{{\mathbb{L}}}\pi_{23}^{\ast}{\mathcal{E}}_{n})=\pi_{12}^{*}\mathrm{ch}({\mathcal{P}}^{\prime}_{E})\cup\pi_{23}^{\ast}\mathrm{ch}({\mathcal{E}}_{n}).$

_Step 3._ We finish the proof of (43).

Recall the expression (36) of ${\mathcal{F}}^{n}_{\beta}$. Appying the
Grothendieck–Riemann-Roch formula to the projection

$\pi_{13}:A\times A\times{\mathcal{M}}_{\beta,A}\to
A\times{\mathcal{M}}_{\beta,A},$

we obtain that

$\mathrm{ch}({\mathcal{F}}^{n}_{\beta})={\pi_{13}}_{\ast}\mathrm{ch}(\pi_{12}^{\ast}{{\mathcal{P}}^{\prime}_{E}}\otimes^{{\mathbb{L}}}\pi_{23}^{\ast}{\mathcal{E}}_{n}).$

Equivalently, the class $\mathrm{ch}({\mathcal{F}}^{n}_{\beta})$ corresponds
to the Künneth factor of the class

$\mathrm{ch}(\pi_{12}^{\ast}{{\mathcal{P}}^{\prime}_{E}}\otimes^{{\mathbb{L}}}\pi_{23}^{\ast}{\mathcal{E}}_{n})$

in the subspace

$H^{4}(A,{\mathbb{Q}})\otimes
H^{*}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}})\subset H^{*}(A\times
A\times{\mathcal{M}}_{\beta,A},{\mathbb{Q}}).$

Here $H^{4}(A,{\mathbb{Q}})$ is spanned by the point class in the second
factor of the product $A\times A\times{\mathcal{M}}_{\beta,A}$. Hence (43)
follows from (46). ∎

###  2.6. Perverse filtrations and
$H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})$

In this section, we assume that $A$ is any abelian surface and $\beta$ is an
ample curve class satisfying $\beta^{2}=v_{n}^{2}=2n$. Recall from Section 2.1
that ${\mathcal{M}}_{\beta,A}$ is the moduli space of Gieseker-stable sheaves
with respect to the primitive Mukai vector

(48) $v_{\beta}=(0,\beta,\chi)\in S_{A}^{+}$

with dimension

$\mathrm{dim}({\mathcal{M}}_{\beta,A})=v_{\beta}^{2}+2=2n+2.$

Our goal is to prove Proposition 2.5, which is a variant of Proposition 1.1
for ${\mathcal{M}}_{\beta,A}$. More precisely, we introduce a _canonical_
class

(49) $L_{\beta}\in H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})$

to characterize the perverse filtration
$P_{\star}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})$ associated with the
morphism

$\pi_{\beta}:{\mathcal{M}}_{\beta,A}\to B.$

Let ${\mathcal{F}}_{0}\in{\mathcal{M}}_{\beta,A}$ be a general point on the
moduli space. We consider the morphisms defined by considering determinants of
coherent sheaves

$\mathrm{det}:{\mathcal{M}}_{\beta,A}\to\hat{A},\quad{\mathcal{F}}\mapsto\mathrm{det}({\mathcal{F}})\otimes\mathrm{det}({\mathcal{F}}_{0})^{\vee}$

and

$\hat{\mathrm{det}}:{\mathcal{M}}_{\beta,A}\to
A,\quad{\mathcal{F}}\mapsto\mathrm{det}(\phi_{\mathcal{P}}({\mathcal{F}}))\otimes\mathrm{det}(\phi_{\mathcal{P}}({\mathcal{F}}))^{\vee}.$

Here $\phi_{\mathcal{P}}:D^{b}\mathrm{Coh}(A)\to D^{b}\mathrm{Coh}(\hat{A})$
is the Fourier–Mukai transform induced by the normalized Poincaré line bundle
${\mathcal{P}}$ on $A\times\hat{A}$. By [51, (4.1) and Theorem 4.1], the
Albanese morphism for ${\mathcal{M}}_{\beta,A}$ with respect to the reference
point ${\mathcal{F}}_{0}$ is

$\mathrm{alb}=\hat{\mathrm{det}}\times\mathrm{det}:{\mathcal{M}}_{\beta,A}\to
A\times\hat{A}.$

The closed fiber over the origin $0\in A\times\hat{A}$, denoted by

$K_{\beta,A}=\mathrm{alb}^{-1}(0)\subset{\mathcal{M}}_{\beta,A},$

is a holomorphic symplectic vairiety of Kummer type; see [51, Theorem 0.2]. It
parametrizes 1-dimensional sheaves ${\mathcal{F}}\in{\mathcal{M}}_{\beta,A}$
satisfying

$\mathrm{det}({\mathcal{F}})=\mathrm{det}({\mathcal{F}}_{0})\in\hat{A},\quad\mathrm{det}(\phi_{\mathcal{P}}({\mathcal{F}}))=\mathrm{det}(\phi_{\mathcal{P}}({\mathcal{F}}))\in\hat{\hat{A}}=A.$

The restriction of the Hilbert–Chow morphism (22) to the subvariety
$K_{\beta,A}$ is a Lagrangian fibration

$\pi^{\prime}_{\beta}:K_{\beta,A}\to|{\mathcal{O}}_{A}(\beta)|={\mathbb{P}}^{n-1}.$

Here $|{\mathcal{O}}_{A}(\beta)|$ denotes the linear system associated with
the divisor

$\mathrm{supp}({\mathcal{F}}_{0})\subset A.$

We consider the following commutative diagram,

(50) ${K_{\beta,A}\times
A\times\hat{A}}$${{\mathcal{M}}_{\beta,A}}$${{\mathbb{P}}^{n-1}\times
A}$${B.}$$\scriptstyle{h}$$\scriptstyle{\pi^{\prime}_{\beta}\times\mathrm{pr}_{A}}$$\scriptstyle{\pi_{\beta}}$$\scriptstyle{h^{\prime}}$

Here the horizontal arrows are defined by

$h({\mathcal{F}},a,L)={t_{a}}_{*}{\mathcal{F}}\otimes L,\quad
h^{\prime}(C,a)={t_{a}}_{\ast}C$

with $t_{a}:A\xrightarrow{\simeq}A$ the translation by $a$. Since both $h$ and
$h^{\prime}$ are finite and surjective, the map

$h^{*}:H^{k}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})\to H^{k}(K_{\beta,A}\times
A\times\hat{A},{\mathbb{Q}})$

is injective by the projection formula. Moreover, it is an isomorphism when
$k=2$ since

$\mathrm{dim}\,H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})=\mathrm{dim}\,H^{2}(A^{[n]}\times\hat{A},{\mathbb{Q}})=\mathrm{dim}\,H^{2}(K_{\beta,A}\times
A\times\hat{A},{\mathbb{Q}})$

by Göttsche’s formula [24] for the Betti numbers of the Hilbert schemes and
the generalized Kummer varieties.

By [51, Theorem 0.1 and (1.6)], the correspondence induced by
$\mathrm{ch}({\mathcal{F}}_{\beta})\in H^{*}(A\times{\mathcal{M}}_{\beta,A})$
together with the restriction map

$H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})\to
H^{2}(K_{\beta,A},{\mathbb{Q}})$

gives an isometry between $v_{\beta}^{\perp}\subset
S_{A}^{+}\otimes{\mathbb{Q}}$ and $H^{2}(K_{\beta,A},{\mathbb{Q}})$ (endowed
with the Beauville-Bogomolov-Fujiki form)

$\theta:v_{\beta}^{\perp}\xrightarrow{\simeq}H^{2}(K_{\beta,A},{\mathbb{Q}}).$

Therefore, the pullback $h^{*}$ induces an isometry (see also [51, Theorem
4.1.(3) eq. (4.3)])

(51)
$\varphi:H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})\xrightarrow{\simeq}v_{\beta}^{\perp}\oplus
H^{2}(A\times\hat{A},{\mathbb{Q}}),$

which, by [39, Section 8], is ${\rm
Spin}(S_{A}^{+}\otimes{\mathbb{Q}})_{v_{\beta}}$-equivariant.

######  Remark 2.4.

The equivariance means that Markman’s monodromy operators
$\gamma_{g,v_{\beta}}$ (28) acting on
$H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})$ are identified, via $\varphi$,
with the action of $g$ on the right-hand side of (51) described in Section
2.3. We may say that $\varphi$ is compatible with $g$ and
$\gamma_{g,v_{\beta}}$.

We consider the following classes

(52) $\begin{split}l_{\beta}&=\theta\left((0,0,1)\in
v_{\beta}^{\perp}\right)\in H^{2}(K_{\beta,A},{\mathbb{Q}}),\\\
L_{\beta}&=\varphi^{-1}\left((0,0,1)+(\beta\boxtimes 1_{\hat{A}})\right)\in
H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{Q}}).\end{split}$

######  Proposition 2.5.

We have

$P_{k}H^{m}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})=\left(\sum_{i\geq
1}\mathrm{Ker}\left(L_{\beta}^{(n+1)+k+i-m}\right)\cap\mathrm{Im}(L_{\beta}^{i-1})\right)\cap
H^{m}({\mathcal{M}}_{\beta,A},{\mathbb{Q}}).$

###### Proof.

We denote by

$\pi_{K}=\pi^{\prime}_{\beta}\times\mathrm{pr}_{A}:K_{\beta,A}\times
A\times\hat{A}\to{\mathbb{P}}^{n-1}\times A$

the left vertical map of the diagram (50).

By [40, Example 3.1], we have

$l_{\beta}={\pi^{\prime}_{\beta}}^{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n-1}}(1)\in
H^{2}(K_{\beta,A},{\mathbb{Q}}).$

Hence

$h^{*}L_{\beta}=l_{\beta}\boxtimes
1_{A\times\hat{A}}+1_{K_{\beta,A}}\boxtimes(\beta\otimes 1_{\hat{A}})\in
H^{2}(K_{\beta,A}\times A\times\hat{A},{\mathbb{Q}})$

is the pullback of an ample class on the base ${\mathbb{P}}^{n-1}\times A$ via
the morphism $\pi_{K}$. Since $h$ and $h^{\prime}$ are finite, Proposition 2.5
is deduced immediately from the diagram (50) and the following lemma. ∎

######  Lemma 2.6.

We consider a commutative diagram

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

where all the varieties are nonsingular and irreducible, the horizontal
morphisms are finite and surjective, and the vertical morphisms are proper. We
assume

$\mathrm{dim}(X)=\mathrm{dim}(X^{\prime})=2\mathrm{dim}(Y)=2\mathrm{dim}(Y^{\prime})=2R$

for some positive integer $R$, and that there is a class $\alpha\in
H^{2}(X^{\prime},{\mathbb{Q}})$ satisfying

$h^{*}\alpha=f^{*}L$

with $L\in H^{2}(Y,{\mathbb{Q}})$ an ample class on $Y$. Then the perverse
filtration $P_{\star}H^{*}(X^{\prime},{\mathbb{Q}})$ associated with
$f^{\prime}$ can be expressed as

(53) $P_{k}H^{m}(X^{\prime},{\mathbb{Q}})=\left(\sum_{i\geq
1}\mathrm{Ker}(\alpha^{R+k+i-m})\cap\mathrm{Im}(\alpha^{i-1})\right)\cap
H^{m}(X^{\prime},{\mathbb{Q}}).$

###### Proof.

The composition of the following maps is multiplication by the degree of $h$

${\mathbb{Q}}_{X^{\prime}}\to
h_{*}h^{*}{\mathbb{Q}}_{X^{\prime}}=h_{*}{\mathbb{Q}}_{X}\to{\mathbb{Q}}_{X^{\prime}},$

where the first map is the adjunction and the last map is the trace map. This
composition is nonzero since $h$ has non-zero degree, so it yields the
splitting

(54)
$h_{*}{\mathbb{Q}}_{X}={\mathbb{Q}}_{X^{\prime}}\oplus{\mathcal{G}},\quad{\mathcal{G}}\in\mathrm{D}^{b}_{c}(X^{\prime}),$

which further induces the embedding

(55) $h^{*}:H^{*}(X^{\prime},{\mathbb{Q}})\hookrightarrow
H^{*}(X,{\mathbb{Q}})$

as a direct summand. After pushing forward (54) along
$f^{\prime}:X^{\prime}\to Y^{\prime}$, we obtain

${{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\mathrm{Rf^{\prime}}_{\ast}h_{\ast}{\mathbb{Q}}_{X})={{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\mathrm{Rf^{\prime}}_{\ast}{\mathbb{Q}}_{X^{\prime}})\oplus{{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\mathrm{Rf^{\prime}}_{\ast}{\mathcal{G}}).$

On the other hand, by the $t$-exactness of the finite morphism $h^{\prime}$,
we have

${{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\mathrm{Rf^{\prime}}_{\ast}h_{\ast}{\mathbb{Q}}_{X})={{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(h^{\prime}_{*}\mathrm{Rf}_{\ast}{\mathbb{Q}}_{X})=h^{\prime}_{*}{{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\mathrm{Rf}_{\ast}{\mathbb{Q}}_{X}).$

Hence the pullback (55) induced by (54) is filtered strict with respect to the
perverse filtrations associated with $f$ and $f^{\prime}$, and we have

(56) $P_{k}H^{d}(X,{\mathbb{Q}})\cap\mathrm{Im}(h^{*})\subset
P_{k}H^{d}(X^{\prime},{\mathbb{Q}}),\quad\forall k,d.$

We deduce (53) from (56) and Proposition 1.1 (applied to $f:X\to Y$). ∎

###  2.7. Proof of Theorem 2.1

In this section, we complete the proof of Theorem 2.1 by combining the tools
developed in the previous sections.

Let $A$ be an abelian surface and $\beta$ be an ample curve class. A _perverse
package_ associated with the pair $(A,\beta)$ is defined to be a triple

$([{\mathcal{F}}_{\beta}],\alpha_{\beta},L_{\beta}),$

where $[{\mathcal{F}}_{\beta}]$ is a universal class on
$A\times{\mathcal{M}}_{\beta,A}$, $L_{\beta}\in
H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$ is the class introduced in
Section 2.6, and $\alpha\in
H^{2}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{C}})$ is of the type (21).

An isomorphism between the perverse packages associated with $(A,\beta)$ and
$(A^{\prime},\beta^{\prime})$ is the following:

  1. (i)

An isomorphism of graded ${\mathbb{C}}$-vector spaces

$g:H^{*}(A,{\mathbb{C}})\xrightarrow{\simeq}H^{*}(A^{\prime},{\mathbb{C}}).$

  2. (ii)

An isomorphism of graded ${\mathbb{C}}$-algebras

$\phi:H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\xrightarrow{\simeq}H^{*}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}}).$

  3. (iii)

The isomorphisms $g$ and $\phi$ satisfy that

$\begin{split}(g\otimes\phi)\mathrm{ch}^{\alpha_{\beta}}({\mathcal{F}}_{\beta})&=\mathrm{ch}^{\alpha_{\beta^{\prime}}}({\mathcal{F}}_{\beta^{\prime}}),\\\
\phi(L_{\beta})&=L_{\beta^{\prime}}.\end{split}$

Isomorphisms of perverse packages form an equivalence relation. We call two
pairs $(A,\beta)$ and $(A^{\prime},\beta^{\prime})$ _perversely isomorphic_ if
there is a perverse package associated with $(A,\beta)$ isomorphic to a
perverse package associated with $(A^{\prime},\beta^{\prime})$.

######  Proposition 2.7.

Assume that $(A,\beta)$ is perversely isomorphic to
$(A^{\prime},\beta^{\prime})$. If Theorem 2.1 holds for $(A,\beta)$, then it
also holds for $(A^{\prime},\beta^{\prime})$.

###### Proof.

By Proposition 2.5, the condition

$\phi(L_{\beta})=L_{\beta^{\prime}}$

implies that the perverse filtrations on
$H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$ and
$H^{*}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}})$ are identified
via $\phi$. Let

$H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})=\bigoplus_{k,d}\widetilde{G}_{k}H^{d}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$

be the splitting of the perverse filtration for ${\mathcal{M}}_{\beta,A}$
satisfying

$\int_{\gamma}\mathrm{ch}^{\alpha_{\beta}}_{k}({\mathcal{F}}_{\beta})\in\widetilde{G}_{k-1}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}}).$

Then the decomposition

$\
\widetilde{G}_{k}H^{d}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}})=\phi(\widetilde{G}_{k}H^{d}({\mathcal{M}}_{\beta,A},{\mathbb{C}}))$

also splits the perverse filtration for
${\mathcal{M}}_{\beta^{\prime},A^{\prime}}$, and the corresponding
tautological classes lie in its correct pieces by condition (iii) in the
definition of isomorphism of perverse package. ∎

If $(A,\beta)$ is perversely isomorphic to $(A^{\prime},\beta^{\prime})$, then
condition (ii) implies that

(57)
$\beta^{2}=\mathrm{dim}({\mathcal{M}}_{\beta,A})-2=\mathrm{dim}({\mathcal{M}}_{\beta^{\prime},A^{\prime}})-2=\beta^{\prime
2}.$

The following theorem, which we deduce from Theorem 2.2, establishes the
converse to (57). In other words, perverse equivalence classes associated with
pairs $(A,\beta)$ are larger than the equivalence classes given by the
deformation types of the _fibrations_ $\pi_{\beta}:{\mathcal{M}}_{\beta,A}\to
B$; see Remark 2.9.

######  Theorem 2.8.

Any two pairs $(A,\beta)$ and $(A^{\prime},\beta^{\prime})$ satisfying

$\beta^{2}=\beta^{\prime 2}$

are perversely isomorphic.

###### Proof.

As we explain at the end of the proof, it will be sufficient to find an
isomorphism of graded ${\mathbb{C}}$-vector spaces

(58) $g:H^{*}(A,{\mathbb{C}})\to H^{*}(A^{\prime},{\mathbb{C}})$

satisfying the following two properties:

  1. (a)

$g$ preserves the intersection pairing;

  2. (b)

$g$ satisfies the following identities

(59) $\begin{split}g(0,0,1)&=(0,0,1),\\\ g(1,0,0)&=(1,0,0),\\\
g(0,\beta,0)&=(0,\beta^{\prime},0),\end{split}$

and a graded ${\mathbb{C}}$-algebra isomorphism

(60) $\phi:H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\to
H^{*}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}})$

satisfying condition (iii) in the definition of isomorphism of perverse
packages.

We construct $(g,\phi)$ in two steps.

We note that $g_{1},g_{2}$, and $g=g_{2}\circ g_{1}$ constructed below
automatically satisfy condition (a) by construction.

Firstly, since the Mukai vectors (48) $v_{\beta}$ and $v_{\beta^{\prime}}$ are
primitive with the same Mukai length,

$v_{\beta}^{2}=v_{\beta^{\prime}}^{2}\in{\mathbb{Z}},$

we can apply [41, Theorem 8.3] (which follows from Yoshioka [51]) to find
isometries

$g_{1}:H^{*}(A,{\mathbb{Z}})\to
H^{*}(A^{\prime},{\mathbb{Z}}),\quad\phi_{1}:H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})\to
H^{*}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{Q}})$

such that $g_{1}(v_{\beta})=v_{\beta^{\prime}}$ and

(61)
$(g_{1}\otimes\phi_{1})\mathrm{ch}({\mathcal{F}}_{\beta})=\mathrm{ch}({\mathcal{F}}_{\beta^{\prime}})$

with $[{\mathcal{F}}_{\beta^{\prime}}]$ a universal class on
$A\times{\mathcal{M}}_{\beta,A}$. Moreover, the morphism $\phi_{1}$ satisfies
the evident variant of Remark 2.4, _i.e._ , the morphism

$\varphi_{\beta^{\prime}}^{-1}\phi_{1}\varphi_{\beta}:v_{\beta}^{\perp}\oplus
H^{2}(A\times\hat{A},{\mathbb{Q}})\to v_{\beta^{\prime}}^{\perp}\oplus
H^{2}(A^{\prime}\times\hat{A^{\prime}},{\mathbb{Q}})$

is induced by the isomtery $g_{1}:H^{*}(A,{\mathbb{Z}})\to
H^{*}(A^{\prime},{\mathbb{Z}})$; see [39, Proposition 8.5]. We may say that
the identifications $\varphi$ (51) for $\beta$ and $\beta^{\prime}$ are
compatible with $g_{1}$ and $\phi_{1}$.

If $\beta$ and $\beta^{\prime}$ have different divisibility, then $g_{1}$
cannot satisfy the condition (b) above, so that we proceed as follows. We
choose
$g_{2}\in\mathrm{SO}(S_{A^{\prime}}^{+}\otimes{{\mathbb{Q}}})_{v_{\beta^{\prime}}}$
such that the restriction

$g_{2}\circ g_{1}|_{S_{A}^{+}}:S_{A}^{+}\otimes{{\mathbb{Q}}}\rightarrow
S_{A^{\prime}}^{+}\otimes{{\mathbb{Q}}}$

is grading-preserving and satisfies condition (b). We can lift $g_{2}$ to an
element of
$\mathrm{Spin}(S_{A^{\prime}}^{+}\otimes{\mathbb{C}})_{v_{\beta^{\prime}}}$,
also denoted by $g_{2}$, and take the associated isomorphisms

$\begin{split}g_{2}:&H^{*}(A^{\prime},{\mathbb{C}})\rightarrow
H^{*}(A^{\prime},{\mathbb{C}})\qquad\mbox{ (cf. Section \ref{spin8reps})}\\\
\phi_{2}=\gamma_{g_{2},v_{\beta^{\prime}}}:&H^{*}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}})\rightarrow
H^{*}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}})\quad\mbox{(cf.
Section \ref{Monodromy})}.\end{split}$

By Theorem 2.2, we have

(62)
$(g_{2}\otimes\phi_{2})\mathrm{ch}({\mathcal{F}}_{\beta^{\prime}})=\mathrm{ch}^{\pi_{\mathcal{M}}^{*}a}({\mathcal{F}}_{\beta^{\prime}}),\quad{\exists}a\in
H^{2}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}}).$

We set $g=g_{2}\circ g_{1}$ and $\phi=\phi_{2}\circ\phi_{1}$. Then (61) and
(62) imply that

$(g\otimes\phi)\mathrm{ch}({\mathcal{F}}_{\beta})=\mathrm{ch}^{\pi_{\mathcal{M}}^{*}a}({\mathcal{F}}_{\beta^{\prime}}).$

In particular, for any class $\alpha_{\beta}\in
H^{2}(A\times{\mathcal{M}}_{\beta,A},{\mathbb{C}})$ of type (21), we have

$(g\otimes\phi)\mathrm{ch}^{\alpha_{\beta}}({\mathcal{F}}_{\beta})=\mathrm{ch}^{\alpha_{\beta^{\prime}}}({\mathcal{F}}_{\beta^{\prime}}),\quad{\rm
with}\;\alpha_{\beta^{\prime}}=\pi_{\mathcal{M}}^{*}a+(g\otimes\phi)\alpha_{\beta},$

where the class $\alpha_{\beta^{\prime}}$ is also of type (21). The first
condition appearing in (iii) is thus met.

It remains to verify that $\phi(L_{\beta})=L_{\beta^{\prime}}$. In view of the
compatibility of the $\varphi$’s for $\beta$ and $\beta^{\prime}$ with
$g_{1},\phi_{1}$, and of $\varphi$ for $\beta^{\prime}$ with $g_{2},\phi_{2}$
(cf. Remark 2.4), and in view of the construction of $g_{1}$ and $g_{2}$, the
isomorphism $\phi$ sends $(0,0,1)\in v_{\beta}^{\perp}$ to $(0,0,1)\in
v_{\beta^{\prime}}^{\perp}$, and sends $\beta\boxtimes 1_{\hat{A}}\in
H^{2}(A\times\hat{A},{\mathbb{C}})$ to $\beta^{\prime}\boxtimes
1_{\hat{A^{\prime}}}\in H^{2}(A^{\prime}\times\hat{A^{\prime}},{\mathbb{C}})$.
In particular, we have $\phi(L_{\beta})=L_{\beta^{\prime}}$, and the desired
condition (3) is met in its entirety. This completes the proof of Theorem 2.8.
∎

Theorem 2.1 follows from Theorem 2.3, Proposition 2.7, and Theorem 2.8 (with
$n:=\beta^{2}/2$, $v_{n}$ and $\beta$). In Section 3.5, we prove a
strengthened version of Theorem 2.1 to the effect that the decomposition
$\tilde{G}_{*}H^{*}({\mathcal{M}}_{A,\beta},{\mathbb{C}})$ is the one induced
by a variant of a construction of Deligne’s.

######  Remark 2.9.

It was shown in [15] that perverse filtrations behave well under deformations.
However, for the pairs $(A,\beta)$ and $(A^{\prime},\beta^{\prime})$ such that
$\beta$ and $\beta^{\prime}$ have different divisibilities in
$H^{2}(A,{\mathbb{Z}})$ and $H^{2}(A^{\prime},{\mathbb{Z}})$, the fibrations
$\pi_{\beta}$ and $\pi_{\beta^{\prime}}$ are not deformation equivalent.
Therefore, in order to reduce the proof of Theorem 2.1 for any pair
$(A,\beta)$ to that for a special pair

$A=E\times E^{\prime},\quad\beta=\sigma+nf,$

it is essential to consider an equivalence relation weaker than deformation
equivalences of fibrations $\pi_{\beta}$. This is our motivation for
introducing equivalences of perverse packages.

##  3\. Splittings of perverse filtrations

###  3.1. Overview

In this section, we study splittings of the perverse filtration associated
with a proper surjective morphism $\pi:X\to Y$ with $X$ and $Y$ nonsingular.
As an application, we strengthen Theorem 2.1 by requiring that the
decomposition given by said theorem

(63)
$H^{\ast}({\mathcal{M}}_{\beta,A},{\mathbb{C}})=\bigoplus_{i,d}\widetilde{G}_{i}H^{d}({\mathcal{M}}_{\beta,A},{\mathbb{C}}).$

is induced by a “Lefschetz class” via the mechanisms introduced in [21, 11,
20]; see Theorem 3.8. As discussed in Remark 4.5, this is crucial in the study
of the specialization morphism (82) in Section 4.

Throughout, we work with ${\mathbb{Q}}$-coefficients except for Section 3.5,
where we need to switch to ${\mathbb{C}}$-coefficients in order to apply
results in Section 2.7. However, we note that all discussions in Sections
3.2–3.4 remain valid if we replace ${\mathbb{Q}}$ by ${\mathbb{C}}$.

###  3.2. Lefschetz classes

We consider the perverse filtration associated with $\pi:X\to Y$,

(64) $P_{0}H^{\ast}(X,{\mathbb{Q}})\subset
P_{1}H^{\ast}(X,{\mathbb{Q}})\subset\dots\subset
P_{2{R}}H^{\ast}(X,{\mathbb{Q}})=H^{\ast}(X,{\mathbb{Q}}),$

where ${R}$ is the defect of semismallness of $\pi$ (13). The action of a
class $\eta\in H^{2}(X,{\mathbb{Q}})$ on the cohomology
$H^{*}(X,{\mathbb{Q}})$ via cup product satisfies

(65) $\eta:P_{k}H^{i}(X,{\mathbb{Q}})\to P_{k+2}H^{i+2}(X,{\mathbb{Q}}),$

which further induces an action on

${\mathbb{H}}=\bigoplus_{\star,\bullet\geq
0}\mathrm{Gr}_{\star}^{P}H^{\bullet}(X,{\mathbb{Q}})=\bigoplus_{\star,\bullet\geq
0}\left(P_{\star}H^{\bullet}(X,{\mathbb{Q}})/P_{\star-1}H^{\bullet}(X,{\mathbb{Q}})\right).$

We say that $\eta\in H^{2}(X,{\mathbb{Q}})$ is a _$\pi$ -Lefschetz class_ if
its induced action on ${\mathbb{H}}$ satisfies the hard Lefschetz-type
condition in the sense of [11, Assumption 2.3.1], _i.e._ , the actions

(66)
$\eta^{k}:\mathrm{Gr}^{P}_{{R}-k}H^{*}(X,{\mathbb{Q}})\xrightarrow{\simeq}\mathrm{Gr}^{P}_{{R}+k}H^{*}(X,{\mathbb{Q}}),\quad\forall
k\geq 0,$

are isomorphisms. As a typical example, the relative Hard Lefschetz Theorem
[7] with respect to $\pi:X\to Y$ implies that a relatively ample class for
$\pi$ is a $\pi$-Lefschetz class. The following lemma gives examples of
Lefschetz classes other than relatively ample classes.

######  Lemma 3.1.

Let $A$ be an abelian variety of dimension $m$. Any class $\eta\in
H^{2}(A,{\mathbb{Q}})$ satisfying

(67) $\eta^{m}\neq 0$

is a Lefschetz class with respect to $\pi:A\to\mathrm{pt}$.

###### Proof.

Let $W=H^{1}(A,{\mathbb{Q}})$. We identify $H^{k}(A,{\mathbb{Q}})$ with
$\wedge^{k}W$, and therefore $\eta\in\wedge^{2}W$. The condition (67) implies
that $\eta$ defines a (constant) symplectic form on $W^{*}$. In particular, we
can write $\eta=\sum_{i=1}^{m}e_{i}\wedge f_{i}$ under a basis
$e_{1},\dots,e_{m},f_{1},\dots,f_{m}$ of $W$. Hence the pair
$(W\otimes{\mathbb{C}},\eta)$ is the tensor product of $m$ copies of the
symplectic plane $({\mathbb{C}}^{2},e\wedge f)$, for which the statement is
clear. ∎

Recall the Hitchin fibration $h:{\mathcal{M}}_{\mathrm{Dol}}\to\Lambda$. The
following proposition is a numerical criterion for Lefschetz classes with
respect to $h$.

######  Proposition 3.2.

Let $F$ be a closed fiber of $h$. A class $\eta\in
H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$ is a Lefschetz class with
respect to $h$ if and only if the following cap product with the fundamental
class of the connected fiber $F$ (counting components with multiplicities) is
non-trivial:

(68) $0\neq\eta^{\mathrm{dim(F)}}\cap[F]\in
H_{0}(F,{\mathbb{Q}})\simeq{\mathbb{Q}}.$

###### Proof.

We denote by $\hat{{\mathcal{M}}}_{\mathrm{Dol}}$ the corresponding moduli
space of stable $\mathrm{PGL}_{r}$-Higgs bundles with

$\hat{h}:\hat{{\mathcal{M}}}_{\mathrm{Dol}}\to\hat{\Lambda}$

the Hitchin fibration. By [30, 37], we have

$H^{2}(\hat{{\mathcal{M}}}_{\mathrm{Dol}},{\mathbb{Q}})={\mathbb{Q}}\cdot\alpha$

where $\alpha$ is the first Chern class of an $\hat{h}$-ample line bundle on
$\hat{{\mathcal{M}}}_{\mathrm{Dol}}$, and therefore is a $\hat{h}$-Lefschetz
class. By the discussion in [13, Section 2.4], we have an isomorpism

(69)
$H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{*}(\hat{{\mathcal{M}}}_{\mathrm{Dol}},{\mathbb{Q}})\otimes
H^{*}(\mathrm{Pic}^{0}(C),{\mathbb{Q}})$

satisfying that

(70)
$P_{k}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=\bigoplus_{i+j=k}P_{i}H^{*}(\hat{{\mathcal{M}}}_{\mathrm{Dol}},{\mathbb{Q}})\otimes
H^{j}(\mathrm{Pic}^{0}(C),{\mathbb{Q}}).$

Under the identification

$H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{2}(\hat{{\mathcal{M}}}_{\mathrm{Dol}},{\mathbb{Q}})\oplus
H^{2}(\mathrm{Pic}^{0}(C),{\mathbb{Q}})={\mathbb{Q}}\cdot\alpha\oplus
H^{2}(\mathrm{Pic}^{0}(C)),$

induced by (69), we can express any class $\eta\in
H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$ as

(71) $\eta=\mu{\alpha\otimes 1+1\otimes\xi},\quad\mu\in{\mathbb{Q}},~{}\xi\in
H^{2}(\mathrm{Pic}^{0}(C),{\mathbb{Q}}).$

Lemma 3.1 combined with [11, Appendix] implies that the class $\eta$ is
Lefschetz if and only if

(72) $\mu\neq 0,\quad\xi^{g}\neq 0.$

Therefore, it suffices to show that (72) is equivalent to the condition (68).

We first consider traceless Higgs bundles

${\mathcal{M}}^{0}_{\mathrm{Dol}}=\\{({\mathcal{E}},\theta)\in{\mathcal{M}}_{\mathrm{Dol}}:\mathrm{trace}(\theta)=0\\}\subset{\mathcal{M}}_{\mathrm{Dol}}.$

Recall from [13, Proposition 2.4.3] that there is a finite morphism

$q:\check{{\mathcal{M}}}_{\mathrm{Dol}}\times\mathrm{Pic}^{0}(C)\to{\mathcal{M}}^{0}_{\mathrm{Dol}}$

with $\check{{\mathcal{M}}}_{\mathrm{Dol}}$ the corresponding Dolbeault moduli
space of stable $\mathrm{SL}_{r}$-Higgs bundles. The preimage of a closed
fiber $F\subset{\mathcal{M}}^{0}_{\mathrm{Dol}}$ of the restricted Hitchin
fibration $h|_{{\mathcal{M}}^{0}_{\mathrm{Dol}}}$ is the product

$q^{-1}(F)=\check{F}\times\mathrm{Pic}^{0}(C),$

where $\check{F}$ is the corresponding closed fiber of the $\mathrm{SL}_{n}$
Hitchin fibration. The pullback of a class (71) along

$\check{F}\times\mathrm{Pic}^{0}(C)\hookrightarrow\check{{\mathcal{M}}}_{\mathrm{Dol}}\times\mathrm{Pic}^{0}(C)\xrightarrow{q}{\mathcal{M}}^{0}\hookrightarrow{\mathcal{M}}_{\mathrm{Dol}}$

is of the type

$\check{\eta}=\mu\check{\alpha}\otimes 1+1\otimes\xi\in
H^{2}(\check{F},{\mathbb{Q}})\oplus
H^{2}(\mathrm{Pic}^{0}(C),{\mathbb{Q}})\subset
H^{2}(\check{F}\times\mathrm{Pic}^{0}(C),{\mathbb{Q}})$

with $\check{\alpha}$ an ample class on $\check{F}$. Since
$q:\check{F}\times\mathrm{Pic}^{0}(C)\to F$ is finite and surjective, the
condition (68) is equivalent to

$0\neq\check{\eta}^{\mathrm{dim}(\check{F}\times\mathrm{Pic}^{0}(C))}\cap[\check{F}]=\check{\eta}^{\mathrm{dim}(F)}\cap[\check{F}],$

which, in turn, is equivalent to (72). This completes the proof. ∎

###  3.3. The first Deligne splitting

Given a projective morphism $\pi:X\to Y$ and a $\pi$-ample line bundle on $X$,
Deligne [21] constructs three splittings of the direct image
$R\pi_{*}{\mathbb{Q}}_{X}$ (resp. $R\pi_{*}\mathrm{IC}_{X}$ if $X$ is
singular), which induce splittings of the perverse filtration on the (resp.
intersection) cohomology groups $H^{*}(X,{\mathbb{Q}})$. In this paper, we
need the variant [11] of this construction where one starts with a
$\pi$-Lefschetz class $\eta\in H^{2}(X,{\mathbb{Q}})$, _i.e._ one that does
not necessarily satisfy the relative Hard Lefschetz Theorem in the derived
category $D^{b}_{c}(Y)$, but nonetheless satisfies the cohomological version
(66). We need only the first of the resulting three splittings, which we name
the _first Deligne splitting_ ([21, 20, 11]).

According to [11], the first Deligne splitting of the perverse filtration (64)

$H^{*}(X,{\mathbb{Q}})=\bigoplus_{i}G_{i}H^{*}(X,{\mathbb{Q}})$

associated with the $\pi$-Lefschetz class $\eta$ (_i.e._ (66) holds) can be
described using _only_ the action of $\eta$ on $H^{*}(X,{\mathbb{Q}})$. We
explain this description explicitly as follows.

For $i\geq 0$, we let
$\mathrm{Gr}^{P}_{i}:=P_{i}H^{*}(X,{\mathbb{Q}})/P_{i-1}H^{*}(X,{\mathbb{Q}})$
denote the graded spaces, which are subquotients of cohomology, and let
$G_{i}:=G_{i}H^{*}(X,{\mathbb{Q}})$ denote the corresponding image of
$\mathrm{Gr}^{P}_{i}$ via the first Deligne splitting, i.e. these are the
summands in the last paragraph, which are subspaces of cohomology that we want
to characterize.

For $0\leq k\leq{R}$, let

$\mathrm{Gr}^{P}_{k}\supseteq
Q^{\prime}_{k}:=\ker\\{\eta^{{R}-k+1}:\mathrm{Gr}^{P}_{k}\to\mathrm{Gr}^{P}_{2{R}-k+2}\\}$

be the associated graded-primitive spaces (here $\eta$ acts on the graded
spaces of the perverse filtration on cohomology). Let $Q_{k}\subseteq G_{k}$
be the image of $Q^{\prime}_{k}$ via the first Deligne splitting. We have
$G_{k}=\sum_{i\geq 0}\eta^{i}Q_{k-2i}$ for $0\leq k\leq{R}$, and
$G_{{R}+\kappa}=\eta^{k}G_{{R}-\kappa}$ (here $\eta$ acts on cohomology).

It follows that, in order to have a complete description of the first Deligne
splitting that involves solely the action of $\eta$ in cohomology, we only
need to describe $Q_{k}\subseteq H^{*}(X,{\mathbb{Q}})$ in such terms.

We describe $Q_{k}$ following [20, Section 2.7]. Note that its context is the
one of the fist Deligne splitting arising from working in the derived
category, but the proof works verbatim in the present context of cohomology
acted upon by a $\pi$-Lefschetz class. By (65), we have

$\eta^{j}P_{k}H^{*}(X,{\mathbb{Q}})\subset P_{k+2j}H^{*}(X,{\mathbb{Q}}).$

Let $\Phi_{0}(\eta)$ be the composition of the morphisms

$\Phi_{0}(\eta):P_{k}H^{*}(X,{\mathbb{Q}})\xrightarrow{\eta^{{R}-k+1}}P_{2{R}-k+2}H^{*}(X,{\mathbb{Q}})\to\mathrm{Gr}_{2{R}-k+2}^{P}H^{*}(X,{\mathbb{Q}}),$

where the first map is cup product and the second map is the projection to the
graded piece. We obtain the sub-vector space

$\mathrm{Ker}(\Phi_{0}(\eta))\subset P_{k}H^{*}(X,{\mathbb{Q}}).$

The morphisms $\Phi_{m}(\eta)$ are defined inductively for $m\geq 0$ as
follows:

$\Phi_{m}(\eta):\mathrm{Ker}(\Phi_{m-1}(\eta))\xrightarrow{\eta^{{R}-k+m}}P_{2{R}-k+2m}H^{*}(X,{\mathbb{Q}})\to\mathrm{Gr}_{2{R}-k+2m}^{P}H^{*}(X,{\mathbb{Q}}).$

Therefore for fixed $k$, we obtain a sequence of sub-vector spaces

$\cdots\subset\mathrm{Ker}(\Phi_{1}(\eta))\subset\mathrm{Ker}(\Phi_{0}(\eta))\subset
P_{k}H^{*}(X,{\mathbb{Q}}).$

According to [20, Proposition 2.7.1], we have the desired description

$Q_{k}=Q_{k}(\eta)=\mathrm{Ker}(\Phi_{k}(\eta))\subset
P_{k}H^{*}(X,{\mathbb{Q}}),\quad\forall 0\leq k\leq{R}.$

The description of the first Deligne splitting yields immediately the
following comparison lemma, which expresses a kind of functoriality for the
first Deligne splitting.

######  Lemma 3.3.

Let $X_{i}$ $(i=1,2)$ be nonsingular varieties with proper surjective
morphisms $f_{i}:X_{i}\to Y_{i}$, and let $P_{\star}H^{*}(X_{i},{\mathbb{Q}})$
be the corresponding perverse filtrations. Let

$\phi:H^{*}(X_{1},{\mathbb{Q}})\to H^{*}(X_{2},{\mathbb{Q}})$

be a morphism of graded ${\mathbb{Q}}$-algebras satisfying

  1. (a)

$\phi(P_{k}H^{d}(X_{1},{\mathbb{Q}}))\subset P_{k}H^{d}(X_{2},{\mathbb{Q}})$;

  2. (b)

$\phi(\eta_{1})=\eta_{2}$, with $\eta_{i}\in H^{2}(X_{i},{\mathbb{Q}})$ an
$f_{i}$-Lefschetz class for $i=1,2$.

Then we have

$\phi(G_{k}H^{*}(X_{1},{\mathbb{Q}}))\subset
G_{k}H^{*}(X_{2},{\mathbb{Q}}),\quad\forall k\geq 0$

with $H^{*}(X_{i},{\mathbb{Q}})=\oplus_{k\geq
0}G_{k}H^{*}(X_{i},{\mathbb{Q}})$ the first Deligne splitting associated with
$\eta_{i}$, $i=1,2$.

###### Proof.

It follows from the description of the first Deligne splittings

$H^{*}(X_{i},{\mathbb{Q}})=\bigoplus_{0\leq k\leq r}\bigoplus_{j\geq
0}\left(\eta^{j}_{i}Q_{k-2j}(\eta_{i})\bigoplus\eta^{r-k+j}_{i}Q_{k-2j}(\eta_{i})\right)\quad
i=1,2$

summarized above, and the fact that the sub-vector spaces

$\mathrm{Ker}(\Phi_{k}(\eta_{i}))=Q_{k}(\eta_{i})\subset
P_{k}H^{*}(X_{i},{\mathbb{Q}})$

are preserved under the $P$-filtered morphism $\phi$. ∎

######  Remark 3.4.

Since cohomologically, the Hitchin fibration

$h:{\mathcal{M}}_{\mathrm{Dol}}\to\Lambda,$

behaves like the product of the fibrations
$\hat{h}:\hat{{\mathcal{M}}}_{\mathrm{Dol}}\to\hat{\Lambda}$ and
$\mathrm{Pic}^{0}(C)\to\mathrm{pt}$ via the ring isomorphism

$H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{*}(\hat{{\mathcal{M}}}_{\mathrm{Dol}},{\mathbb{Q}})\otimes
H^{*}(\mathrm{Pic}^{0}(C),{\mathbb{Q}})$

given in [13, Section 2.4], we see from the proof of Proposition 3.2 (cf.
(72)) together with [11, Appendix] that the first Deligne splitting associated
with any $h$-Lefschetz class, i.e. a class of the form $\mu\alpha\otimes
1+1\otimes\xi$ with $\mu\neq 0$ and $\xi^{g}\neq 0$, has the form

(73)
$G_{k}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=\bigoplus_{i+j=k}G_{i}H^{*}(\hat{{\mathcal{M}}}_{\mathrm{Dol}},{\mathbb{Q}})\otimes
H^{j}(\mathrm{Pic}^{0}(C),{\mathbb{Q}}),$

where $G_{i}H^{*}(\hat{{\mathcal{M}}}_{\mathrm{Dol}},{\mathbb{Q}})$ is the
first Deligne splitting associated with the $\hat{h}$-Lefschetz class

$\alpha\in H^{2}(\hat{{\mathcal{M}}}_{\mathrm{Dol}},{\mathbb{Q}}).$

######  Remark 3.5.

In particular, the splitting (73) does not depend on the choice of an
$h$-Lefschetz class.

For every genus $g\geq 2$, we expect, and actually prove in the $g=2$ case,
that (73) serves as the splitting in Conjecture 0.3.

###  3.4. Semismall maps and Hilbert schemes

In this section, we study the situation where our morphism $\pi:X\to Y$ can be
factored as

$X\xrightarrow{f}Z\xrightarrow{g}Y$

with $f:X\to Z$ semismall and surjective; in particular, $f$ is generically
finite and $\dim(X)=\dim(Z)$. We further assume that there are closed
irreducible sub-varieties $Z_{i}\subset Z$ such that the decomposition theorem
for $f$ takes the form of a canonical finite direct sum decomposition

(74)
$Rf_{*}{\mathbb{Q}}_{X}=\bigoplus_{i}\mathrm{IC}_{Z_{i}}[-\mathrm{dim}(X)],$

where each $\mathrm{IC}_{Z_{i}}$ is the (perverse) intersection cohomology
complex of $Z_{i}$. Note that we have the following identity concerning
intersection cohomology groups $\mathrm{IH}^{d}(Z_{i},{\mathbb{Q}})=H^{d-\dim
Z_{i}}(Z_{i},\mathrm{IC}_{Z_{i}})$. One of the $Z_{i}$ is the total variety
$Z$. The restriction of $g$ to each $Z_{i}$ yields the morphism

$g_{i}:Z_{i}\to Y_{i}\subset Y.$

We deduce from (74) a canonical decomposition of the cohomology of $X$

(75)
$H^{d}(X,{\mathbb{Q}})=\bigoplus_{i}\mathrm{IH}^{d-c_{i}}(Z_{i},{\mathbb{Q}}),{\quad
c_{i}=\dim X-\dim Z_{i}={\rm codim}_{X}Z_{i}.}$

Let ${R}$ be the defect of semismallness of $\pi$ and, for each $i$, let
${R_{i}}$ be the defect of semismallness of $g_{i}:Z_{i}\to Y_{i}$.

######  Proposition 3.6.

Let $\alpha\in H^{2}(Z,{\mathbb{Q}})$ satisfy that, for every $i$, the
restriction

$\alpha_{i}=\alpha|_{Z_{i}}\in H^{2}(Z_{i},{\mathbb{Q}})$

is a $g_{i}$-Lefschetz class with associated first Deligne splitting

$\mathrm{IH}^{*}(Z_{i},{\mathbb{Q}})=\bigoplus_{k}G_{k}\mathrm{IH}^{*}(Z_{i},{\mathbb{Q}}).$

Then $\eta=f^{*}\alpha\in H^{2}(X,{\mathbb{Q}})$ is a $\pi$-Lefschetz class,
whose associated first Deligne splitting is, under the identification (75),
given by

(76)
${G_{k}H^{d}(X,{\mathbb{Q}})=\bigoplus_{i}G_{k-R+{R_{i}}}\mathrm{IH}^{d-c_{i}}(Z_{i},{\mathbb{Q}}).}$

###### Proof.

Recall that we have defined the perverse filtration $P$ on
$H^{*}(X,{\mathbb{Q}})$ concentrated in the interval $[0,{2R}]$, and
similarly, for every $i$, the perverse filtration $P$ on
$\rm{IH}^{*}(Z_{i},{\mathbb{Q}})$ is concentrated in the interval
$[0,{2R_{i}}]$. It follows that, according to (74) and (75), the direct
summand $\rm{IH}^{*}(Z_{i},{\mathbb{Q}})$ contributes to
$H^{*}(X,{\mathbb{Q}})$ in perversities in the interval ${[R-R_{i},R+R_{i}]}$.

We apply the decomposition theorem to the composition

$X\to Z\to Y$

and obtain that the perverse filtration $P_{\star}H^{*}(X,{\mathbb{Q}})$ can
be expressed in terms of the perverse filtrations
$P_{\star}\mathrm{IH}^{*}(Z_{i},{\mathbb{Q}})$ under (75), _i.e._ ,

$P_{k}H^{d}(X,{\mathbb{Q}})=\bigoplus_{i}P_{k-{R}+{R_{i}}}\mathrm{IH}^{d-c_{i}}(Z_{i},{\mathbb{Q}}).$

By [17, Remark 4.4.3], the action of $\eta=f^{*}\alpha$ on the l.h.s. of (75),
is the direct sum of the actions of the classes $\alpha_{i}=\alpha_{|Z_{i}}$
on the summands of the r.h.s..

Since every $\alpha_{i}$ is a $g_{i}$-Lefschetz class, we deduce that $\eta$
is a $\pi$-Lefschetz class with associated first Deligne splitting
decomposition (76). ∎

Next, we show that, given a fibered abelian surface

$p:A=E\times E^{\prime}\to E^{\prime}$

as in Section 1.5, the splitting (15) of the perverse filtration on
$H^{*}(A^{[n]},{\mathbb{Q}})$ associated with the natural morphism
$p_{n}:A^{[n]}\to E^{\prime(n)}$, is the first Deligne splitting induced by a
$p_{n}$-Lefschetz class.

The morphism $p_{n}:A^{[n]}\to E^{\prime(n)}$ admits the natural factorization

(77) $A^{[n]}\xrightarrow{f}A^{(n)}\xrightarrow{g}E^{\prime(n)}$

where the Hilbert–Chow morphism $f$ is semismall [25]. There are canonical
morphisms

$\kappa_{\nu}:A^{(\nu)}\to A^{(n)}$

together with a canonical stratification of $A^{(n)}$ indexed by the
partitions $\nu$ of $n$,

$A^{(n)}=\bigcup_{\nu}A_{\nu},\quad\overline{A_{\nu}}=\mathrm{Im}(\kappa_{\nu})\subset
A^{(n)}.$

Note that the resulting morphism $\kappa_{\nu}:A^{(\nu)}\to\overline{A_{\nu}}$
is the normalization of the target, so that

(78) $H^{*}(A^{(\nu)},{\mathbb{Q}})={\rm
IH}^{*}(\overline{A_{\nu}},{\mathbb{Q}}).$

By [25, Theorem 3], the decomposition theorem associated with $f$ takes the
form analogous to (74),

$Rf_{*}{\mathbb{Q}}_{A^{[n]}}=\bigoplus_{\nu}\mathrm{IC}_{\overline{A_{\nu}}}[-2n],$

and the restriction of $g:A^{(n)}\to E^{\prime(n)}$ to each
$\overline{A_{\nu}}$ is described by the commutative diagrams

(79)
${A^{(\nu)}}$${\overline{A_{\nu}}}$${A^{(n)}}$${{E^{\prime}}^{(\nu)}}$${\overline{{E^{\prime}_{\nu}}}}$${E^{\prime(n)}.}$$\scriptstyle{\kappa_{\nu}}$$\scriptstyle{p_{\nu}}$

We consider the $p$-ample class

$\alpha=\mathrm{pt}\otimes 1_{E^{\prime}}\in H^{2}(E,{\mathbb{Q}})\otimes
H^{0}(E^{\prime},{\mathbb{Q}})\subset H^{2}(A,{\mathbb{Q}})$

of $p:A\to E^{\prime}$ which induces a Lefschetz class

(80) $\alpha^{(n)}\in H^{2}(A^{(n)},{\mathbb{Q}})$

with respect to $A^{(n)}\to E^{\prime(n)}$. It further induces a Lefschetz
class

$\alpha^{(\nu)}\in H^{2}(A^{(\nu)},{\mathbb{Q}})$

with respect to $A^{(\nu)}\to E^{\prime(\nu)}$ for every partition $\nu$ of
$n$. By the diagram (79), the pullback of $\alpha^{(n)}$ to every $A^{(\nu)}$
via $\kappa_{\nu}$ coincides with $\alpha^{(\nu)}$.

By keeping in mind that ${\rm codim}_{A^{(n)}}\overline{A_{\nu}}=2n-2l(\nu)$,
that the defects of semismallness of the morphisms $p_{n}$ and $p_{\nu}$ are
$n$ and $l(\nu)$, respectively, and the identity (78), we obtain the following
corollary by applying Proposition 3.6 to the factorization (77).

######  Corollary 3.7.

The decomposition (15) is the first Deligne splitting induced by the
$(p_{n}:A^{[n]}\to E^{\prime(n)})$-Lefschetz class ($f$ as in (77),
$\alpha^{(n)}$ as in (80))

$\eta_{A^{[n]}}=f^{*}\alpha^{(n)}\in H^{2}(A^{[n]},{\mathbb{Q}}).$

###  3.5. A strengthened version of Theorem 2.1

We study the decomposition (63) constructed for Theorem 2.1. In the special
case (31), we have

${\mathcal{M}}_{\beta,A}\cong A^{[n]}\times\hat{A},\qquad A=E\times
E^{\prime},$

with the morphism $\pi_{\beta}:{\mathcal{M}}_{A,\beta}\to B$ given by the
morphism $p_{n}\times q$ in (33). Corollary 3.7 implies that the decomposition
(38) is the first Deligne splitting associated with the $(p_{n}\times
q)$-Lefschetz class

$\eta_{A,\beta}=\eta_{A^{[n]}}\boxtimes
1_{\hat{A}}+1_{A^{[n]}}\boxtimes\left(1_{E}\boxtimes\mathrm{pt}\right)\in
H^{2}(A^{[n]}\times\hat{A},{\mathbb{Q}})=H^{2}({\mathcal{M}}_{\beta,A},{\mathbb{Q}}).$

By Theorem 2.8, a pair $(A^{\prime},\beta^{\prime})$ with $\beta^{\prime
2}=2n$ is perversely isomorphic to the special pair $(A,\beta)$ given by (31).
So there is a graded isomorphism

$\phi:H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\xrightarrow{\simeq}H^{*}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}})$

of ${\mathbb{C}}$-algebras preserving the corresponding perverse filtrations,
and we obtain a decomposition (63) as

(81)
$\widetilde{G}_{k}H^{d}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}})=\phi(\widetilde{G}_{k}H^{d}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}}));$

see the proof of Proposition 2.7. Hence

$\eta_{\beta^{\prime},A^{\prime}}=\phi(\eta_{\beta,A})\in
H^{2}({\mathcal{M}}_{\beta^{\prime},A^{\prime}},{\mathbb{C}})$

is a $\pi_{\beta^{\prime}}$-Lefschetz class, and Lemma 3.3 implies that the
decomposition (81) is the first Deligne splitting associated with
$\eta_{\beta^{\prime},A^{\prime}}$.

This gives the following strengthened version of Theorem 2.1, which, we
stress, is about any pair $(A,\beta)$ with $v_{\beta}=(0,\beta,\chi)$
primitive, not just the special cases (31).

######  Theorem 3.8.

Theorem 2.1 holds for a splitting
$\widetilde{G}_{\ast}H^{\ast}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$ given by
the first Deligne splitting associated with a suitable $\pi_{\beta}$-Lefschetz
class $\eta_{\beta}$, where $\pi_{\beta}:{\mathcal{M}}_{\beta,A}\to B$ is the
morphism (22).

##  4\. Topology of Hitchin fibrations

###  4.1. Overview

Throughout the section, we assume $C$ is a nonsingular projective integral
curve of genus $g$ embedded into an abelian surface $A$. We study a kind of
specialization morphism

(82) $\mathrm{sp}^{!}:H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{Q}})\to
H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$

where ${\mathcal{M}}_{\mathrm{Dol}}$ is the moduli of stable Higgs bundles of
rank $r$ and Euler characteristic $\chi$, and

$\beta=r[C]\in H_{2}(A,{\mathbb{Z}}).$

Then we deduce Theorems 0.2, 0.4, and 0.5 from Theorem 2.1 via the properties
of the morphism (82) which we establish hereafter.

###  4.2. Deformation to the normal cone

The moduli space ${\mathcal{M}}_{\mathrm{Dol}}$ of stable Higgs bundles with
rank $r$ and Euler characteristic $\chi$ can be realized as the moduli space
of $1$-dimensional Gieseker-stable sheaves ${\mathcal{F}}$ on the cotangent
bundle surface $T^{*}C$ with (cf. see [6])

$[\mathrm{supp}({\mathcal{F}})]=r[C]\in
H_{2}(T^{*}C,{\mathbb{Z}}),\quad\chi({\mathcal{F}})=\chi.$

In the following, we describe ${\mathcal{M}}_{\mathrm{Dol}}$ as the “limit” of
a trivial family of ${\mathcal{M}}_{\beta,A}$. A similar construction using
$K3$ surfaces was considered in [22].

Assume $T={\mathbb{P}}^{1}$ and $T^{\circ}={\mathbb{P}}^{1}\setminus
0={\mathbb{C}}$. Let

(83) ${\mathcal{S}}=\mathrm{Bl}_{C\times 0}(A\times T)\setminus(A\times 0)\to
T$

be the total space of the deformation to the normal cone associated with the
embedding $j_{C}:C\hookrightarrow A$. The central fiber of (83) is

${\mathcal{S}}_{0}=T^{*}C\to 0\in T,$

and the restriction over $T^{\circ}$ is a trivial fibration

$A\times T^{\circ}\to T^{\circ}\subset T.$

We associate to $\beta$ a family of homology classes

(84) $\beta_{t}=r[C]\in H_{2}({\mathcal{S}}_{t},{\mathbb{Z}}).$

Let ${\mathcal{M}}\rightarrow T$ be the (coarse) relative moduli space which
parametrizes, for each $t\in T$, pure one-dimensional Gieseker-stable sheaves
${\mathcal{F}}$ on ${\mathcal{S}}_{t}$ with $\chi({\mathcal{F}})=\chi$ and
such that the support of ${\mathcal{F}}$ is a proper subscheme in the class
$\beta_{t}$. Similarly, let ${\mathcal{B}}\rightarrow T$ denote the component
of the relative Hilbert scheme which parametrizes Cartier divisors in
${\mathcal{S}}_{t}$ with proper support in the class $\beta_{t}$. Following
[44, Section 5.3], we have a proper morphism

${{\mathcal{M}}}$${{\mathcal{B}}}$${T}$$\scriptstyle{h_{T}}$

over the base $T$, which, on the level of points, sends a sheaf
${\mathcal{F}}$ on ${\mathcal{S}}_{t}$ to its Fitting support, viewed as an
element in ${\mathcal{B}}_{t}$. Clearly both ${\mathcal{M}}$ and
${\mathcal{B}}$ become trivial families when restricted over $T^{\circ}$,

${{\mathcal{M}}_{\beta,A}\times T^{\circ}}$${B\times
T^{\circ}}$${T^{\circ}}$.$\scriptstyle{h_{T^{\circ}}}$

The fibers over $0\in T$ recover the moduli space
${\mathcal{M}}_{\mathrm{Dol}}$, the Hitchin base $\Lambda$, and the Hitchin
fibration (2).

The following lemma seems standard; we include a brief proof since we are not
aware of an adequate reference.

######  Lemma 4.1.

  1. (i)

Both ${\mathcal{M}}$ and ${\mathcal{B}}$ are irreducible varieties and smooth
over $T$.

  2. (ii)

There exists a universal class

$[{\mathcal{F}}_{T}]\in
K_{\mathrm{top}}({\mathcal{M}}\times_{T}{\mathcal{S}})$

whose restriction to each fiber gives a universal class for
${\mathcal{M}}_{t}\times{\mathcal{S}}_{t}$.

###### Proof.

For part (i), we first argue for ${\mathcal{B}}$. For notational convenience,
we define

$\tilde{g}=\mathrm{dim}({\mathcal{B}}_{t})=r^{2}(g-1)+1,\quad\forall~{}t\in
T.$

We observe that the closure $\overline{\mathcal{B}}^{\circ}$ in
${\mathcal{B}}$ of

${\mathcal{B}}^{\circ}=B\times T^{\circ}$

is irreducible of dimension $\tilde{g}+1$, so that the intersection

$\overline{{\mathcal{B}}^{\circ}}\cap{\mathcal{B}}_{0}\subset{\mathcal{B}}_{0}$

must have at least dimension $\tilde{g}$ at every closed point. This
intersection is non-empty since it contains the divisor $r[C]$ which clearly
deforms to the generic fiber. Since ${\mathcal{B}}_{0}$ is irreducible of
dimension $\tilde{g}$, we have
${\mathcal{B}}_{0}\subset\overline{{\mathcal{B}}^{\circ}}$. Therefore
${\mathcal{B}}$ is irreducible. This implies the flatness of ${\mathcal{B}}$
over the nonsingular curve $T$. Furthermore, since its fibers are all
nonsingular, the flatness further implies that the morphism ${\mathcal{B}}\to
T$ is smooth.

The same argument applies for ${\mathcal{M}}$. The only thing to check is that
there exists a point in the central fiber ${\mathcal{M}}_{0}$ which deforms to
the generic fiber. By the smoothness of ${\mathcal{B}}/T$, we can choose a
nonsingular curve $Z_{0}\subset{\mathcal{S}}_{0}$ which deforms to
$Z_{t}\subset{\mathcal{S}}_{t}$. Any line bundle on $Z_{0}$ with Euler
characteristic $\chi$ can be deformed as well and this gives the desired point
in ${\mathcal{M}}_{0}$.

For part (ii), we follow the same argument from [38, Section 3.1]. We denote
by $\mathfrak{M}$ the relative moduli stack of stable sheaves on
${\mathcal{S}}$, which is a $\mathbb{G}_{m}$-gerbe over ${\mathcal{M}}$. It
suffices to show that this gerbe is _topologically_ trivializable, so that we
can pull back the topological $K$-theory class of the universal family on
$\mathfrak{M}\times_{T}{\mathcal{S}}$ to
${\mathcal{M}}\times_{T}{\mathcal{S}}$ via a section. To construct a
trivialization, we take the nonsingular 3-fold
${\mathcal{S}}^{\prime}=\mathrm{Bl}_{C\times 0}(A\times T)$ which contains
${\mathcal{S}}$ as an open subset,

${\mathcal{S}}\subset{\mathcal{S}}^{\prime}.$

Let ${\mathcal{M}}_{{\mathcal{S}}^{\prime}}$ be the (coarse) moduli space of
stable sheaves on ${\mathcal{S}}^{\prime}$ with Chern character given by
$w=\mathrm{ch}({i_{t}}_{\ast}{\mathcal{F}})\in
H^{*}({\mathcal{S}}^{\prime},{\mathbb{Q}})$, where

$i_{t}:{\mathcal{S}}_{t}\xrightarrow{\simeq}A\times
t\subset{\mathcal{S}}^{\prime},\quad t\neq 0$

is the inclusion of a nonsingular fiber, and ${\mathcal{F}}\in{\mathcal{M}}$
is a coherent sheaf supported on ${\mathcal{S}}_{t}$. The moduli space
${\mathcal{M}}_{{\mathcal{S}}^{\prime}}$ contains ${\mathcal{M}}$ as an open
subvariety, therefore it is enough to show that the corresponding
$\mathbb{G}_{m}$-gerbe
$\mathfrak{M}_{{\mathcal{S}}^{\prime}}\rightarrow{\mathcal{M}}_{{\mathcal{S}}^{\prime}}$
is topologically trivial. As in [38, Section 3.1], a trivialization can be
constructed with a topological K-theory class in ${\mathcal{S}}^{\prime}$
whose pairing with $w$ is $1$. For this, we can use the pullback to
${\mathcal{S}}^{\prime}$ of a class on $A$, whose pairing with $(0,r[C],\chi)$
is $1$, via the natural projection

${\mathcal{S}}^{\prime}=\mathrm{Bl}_{C\times 0}(A\times T)\to A.\qed$

###  4.3. Specializations

Specialization morphisms with respect to perverse filtrations have been
studied systematically in [15, 12]. We provide a concrete description of the
specialization morphism in our setting as follows.

As before, we assume $T={\mathbb{P}}^{1}$ and
$T^{\circ}={\mathbb{P}}^{1}\setminus 0$. Let $f:W\to T$ be a smooth morphism,
whose restriction over $T^{\circ}$ is a trivial product

$f^{\circ}:W^{\circ}=W_{t}\times T^{\circ}\rightarrow
T^{\circ},\quad\forall~{}t\neq 0,$

with $W_{t}$ proper. By Deligne’s Global Invariant Cycle Theorem, see [19,
Theorem 1.7.1] for example, the restriction

$\mathrm{res}_{t}:H^{*}(W,{\mathbb{Q}})\to
H^{*}(W^{\circ},{\mathbb{Q}})=H^{*}(W_{t},{\mathbb{Q}})$

is surjective for $t\neq 0$. Let

$\alpha_{1},\alpha_{2}\in H^{*}(W,{\mathbb{Q}})$

be two liftings of a class $\alpha\in H^{*}(W_{t},{\mathbb{Q}})$ with $t\neq
0$. Since $H^{*}(W^{\circ},{\mathbb{Q}})=H^{*}(W_{t},{\mathbb{Q}})$, the long
exact sequences for the pairs $(W,W_{t})$ and $(W,W^{\circ})$ are isomorphic.
Since $W$ is nonsingular, the relative cohomology for $(W,W^{\circ})$ can be
identified with the Borel-Moore homology of $W_{0}$. It follows that

(85) $\alpha_{1}-\alpha_{2}={i_{0}}_{*}\gamma,$

for some class $\gamma$ in the Borel–Moore homology of $W_{0}$. Here
$i_{0}:W_{0}\hookrightarrow W$ is the closed embedding of the central fiber,
and we identify the Borel–Moore homology and the cohomology of the nonsingular
$W$ in the equation (85). The excess intersection formula [23, Corollary 6.3],
together with the triviality of the normal bundle $N_{W_{0}/W}$, implies that
we have

$\mathrm{res}_{0}(\alpha_{1})-\mathrm{res}_{0}(\alpha_{2})=i_{0}^{*}{i_{0}}_{*}\gamma=c_{1}(N_{W_{0}/W})\cap\gamma=0.$

We define the _specialization morphism_ 666In the literature, often, one
defines a specialization morphism $\mathrm{sp}^{*}$ in the opposite direction,
as it is dictated by the morphism $\sigma:i_{0}^{*}\to\psi_{f}$ with
$\psi_{f}$ nearby cycle-functor. In the present set-up, since $W_{0}$ is not
proper, such an arrow $sp^{*}$ does not exist; by dualizing $\sigma$, one sees
that an arrow in the opposite direction always exists, but if $W_{0}$ is
singular, for example, this arrow is not a familiar object.

(86) $\mathrm{sp}^{!}:H^{*}(W_{t},{\mathbb{Q}})\rightarrow
H^{*}(W_{0},{\mathbb{Q}}),\quad t\neq 0$

as

(87) $\mathrm{sp^{!}}(\alpha)=\mathrm{res}_{0}(\tilde{\alpha})\in
H^{*}(W_{0},{\mathbb{Q}})$

where $\tilde{\alpha}\in H^{*}(W,{\mathbb{Q}})$ is any lifting of $\alpha$.
The discussion above implies that the class (87) does not depend on the
lifting $\tilde{\alpha}$. Since $\mathrm{res}_{0}$ is a homomorphism of
${\mathbb{Q}}$-algebras, so is $\mathrm{sp}^{!}$ (86).

######  Example 4.2.

Let ${\mathcal{S}}\to T$ be the relative surface (83) associated with the
embedding $j_{C}:C\hookrightarrow A$. We have the specialization morphism

(88) $\mathrm{sp}^{!}:H^{*}(A,{\mathbb{Q}})\to
H^{*}(T^{*}C,{\mathbb{Q}})=H^{*}(C,{\mathbb{Q}}).$

By the definition of $\mathrm{sp}^{!}$, the morphism (88) is given by the
pullback along

$T^{*}C\hookrightarrow{\mathbb{P}}(T^{*}C\oplus{\mathcal{O}}_{C})\hookrightarrow\mathrm{Bl}_{C\times
0}(A\times T)\to A\times T\to A.$

In particular, we have

$\mathrm{sp}^{!}(\gamma)=j_{C}^{*}\gamma\in
H^{*}(T^{*}C,{\mathbb{Q}})=H^{*}(C,{\mathbb{Q}}),\quad\forall~{}\gamma\in
H^{*}(A,{\mathbb{Q}}).$

Now we discuss the interaction between $\mathrm{sp}^{!}$ and perverse
filtrations. Let $W/T$ be as above. We consider a commutative diagram

${W}$${V}$${T}$$\scriptstyle{h}$

where the morphism $h:W\to V$ is proper of relative dimension $d$. For every
$t\in T$, there is a perverse filtration $P_{\star}H^{*}(W_{t},{\mathbb{Q}})$
associated with the morphism $h_{t}:W_{t}\to V_{t}$.

######  Proposition 4.3.

The specialization morphism (86) preserves777In fact, the proof shows more,
namely that $\mathrm{sp}^{!}$ is filtered strict. the perverse filtrations,
_i.e._ ,

$\mathrm{sp}^{!}\left(P_{k}H^{*}(W_{t},{\mathbb{Q}})\right)\subset
P_{k}H^{*}(W_{0},{\mathbb{Q}}),\quad\forall k.$

###### Proof.

We show that there exist splittings of the perverse filtrations

$H^{*}(W_{t},{\mathbb{Q}})=\bigoplus_{i}G_{i}H^{*}(W_{t},{\mathbb{Q}}),\quad\forall~{}t\in
T$

such that the splitting is constant for $t\in T^{o}$, and

$\mathrm{sp}^{!}\left(G_{i}H^{*}(W_{t},{\mathbb{Q}})\right)\subset
G_{i}H^{*}(W_{0},{\mathbb{Q}}).$

We apply the decomposition theorem [7] to the morphism $h:W\to V$, and fix an
isomorphism

$\phi:Rh_{\ast}{\mathbb{Q}}_{W}[\dim{W}-d]\xrightarrow{\simeq}\bigoplus_{i=0}^{2d}{\mathcal{P}}_{V}^{i}[-i].$

Here, the ${\mathcal{P}}_{V}^{i}$ are semisimple perverse sheaves on $V$. By
the discussion in Section 1.3, the isomorphism $\phi$ induces a splitting of
the perverse filtration on $H^{*}(W,{\mathbb{Q}})$ associated with $h$. By the
smoothness of $f$ and the compatibility of vanishing cycles with the derived
pushforward $\mathrm{Rh}_{*}$, we have
$\varphi(\mathrm{Rh}_{\ast}{\mathbb{Q}}_{W})=0$. As a result, [15, Corollary
3.1.6] shows that the restriction of ${\mathcal{P}}_{V}^{i}$ to a closed fiber
over $t\in T$ is a shifted perverse sheaf,

${\mathcal{P}}_{V,t}^{i}[-1]=\mathrm{res}_{t}^{*}{\mathcal{P}}_{V}^{i}[-1]\in\mathrm{Perv}(V_{t}),\quad\forall~{}t\in
T.$

Therefore, the restriction $\phi_{t}$ of the decomposition $\phi$ induces
splittings of the perverse filtration

$H^{*}(W_{t},{\mathbb{Q}})=\bigoplus_{i\geq
0}\phi_{t}(H^{*}(V_{t},{\mathcal{P}}_{V,t}^{i})),\quad\forall~{}t\in T,$

and we conclude from the definition of $\mathrm{sp}^{!}$ that

$\mathrm{sp}^{!}\left(\phi_{t}(H^{k}(V_{t},{\mathcal{P}}_{V,t}^{i}))\right)\subset\phi_{0}(H^{k}(V_{0},{\mathcal{P}}_{V,0}^{i})).$

This completes the proof. ∎

Now we consider the family

$h_{T}:{\mathcal{M}}\to{\mathcal{B}}$

over $T$ constructed in Section 4.2, especially Lemma 4.1. Proposition 4.3
implies that the specialization morphism

$\mathrm{sp}^{!}:H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\to
H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})$

preserves the perverse filtrations. Let
$\widetilde{G}_{*}H^{*}({\mathcal{M}}_{A,\beta},{\mathbb{Q}})$ be a splitting
given by Theorem 3.8. In particular, it is the first Deligne splitting induced
by a $(\pi_{\beta}=h_{t\neq 0})$-Lefschetz class888Note that this class was
denoted by $\eta_{\beta^{\prime},A^{\prime}}$ in Section 3.5, since $A$ was
special there.

$\eta_{\beta,A}\in H^{2}({\mathcal{M}}_{A,\beta},{\mathbb{C}}).$

######  Proposition 4.4.

The class

$\eta_{0}=\mathrm{sp}^{!}\left(\eta_{\beta,A}\right)\in
H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})$

is a Lefschetz class for the Hitchin fibration
$h:{\mathcal{M}}_{\mathrm{Dol}}\to\Lambda$. Assume further that

$H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})=\bigoplus_{k,j}\widetilde{G}_{k}H^{j}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})$

is the first Deligne splitting associated with $\eta_{0}$. Then we have (cf.
Theorem 3.8)

(89)
$\mathrm{sp}^{!}\left(\widetilde{G}_{k}H^{d}({\mathcal{M}}_{A,\beta},{\mathbb{C}})\right)\subset\widetilde{G}_{k}H^{d}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}}).$

###### Proof.

Let $F_{\beta,A}\subset{\mathcal{M}}_{\beta,A}$ be a closed fiber of the
morphism $\pi_{\beta}:{\mathcal{M}}_{\beta,A}\to B$. We have

$\mathrm{dim}(F_{\beta,A})=\frac{1}{2}\mathrm{dim}({\mathcal{M}}_{\beta,A})=\tilde{g}.$

Since the fiber class

$[F_{\beta,A}]\in H^{2\tilde{g}}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$

lies in $P_{0}H^{2\tilde{g}}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$ by
Proposition 1.1, and $\eta_{\beta,A}$ is a Lefschetz class, we obtain that

$\eta_{\beta,A}^{\tilde{g}}\cap{[F_{\beta,A}]}=\eta_{\beta,A}^{\tilde{g}}\cup[F_{\beta,A}]\neq
0$

by the hard Lefschetz condition. Hence after specialization, we have

${\eta_{0}^{\mathrm{dim(F_{0})}}}\cap[F_{0}]=\eta_{\beta,A}^{\tilde{g}}\cap{[F_{\beta,A}]}\neq
0$

with $F_{0}$ a closed fiber of $h:{\mathcal{M}}_{\mathrm{Dol}}\to\Lambda$. We
conclude from Proposition 3.2 that $\eta_{0}$ is a Lefschetz class with
respect to $h$. The inclusion (89) then follows from Lemma 3.3 (applied to
$\phi=\mathrm{sp}^{!}$). ∎

######  Remark 4.5.

Proposition 4.4 implies that
$\mathrm{sp}^{!}(G_{*}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}}))$ splits the
restricted perverse filtration
$P_{*}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})\cap\mathrm{Im(sp^{!})}$,
_i.e._ ,

$P_{k}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})\cap\mathrm{Im(sp^{!})}=\bigoplus_{i\leq
k}\mathrm{sp}^{!}\left(\widetilde{G}_{i}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\right).$

In general, for an arbitrary splitting
$G_{*}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$ of the perverse filtration
$P_{*}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$, it may not be true that

$\mathrm{sp}^{!}\left({G}_{i}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\right)\cap\mathrm{sp}^{!}\left({G}_{j}H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\right)=\\{0\\},\quad\forall
i\neq j.$

It is thus crucial, in our approach to the results of this paper, to realize
the splitting

$H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})=\bigoplus_{k,j}\widetilde{G}_{k}H^{j}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$

as the first Deligne splitting associated with a Lefschetz class as in Theorem
3.8.

###  4.4. Normalized classes

Our purpose is to apply the specialization morphism $\mathrm{sp}^{!}$ to the
tautological classes. We first discuss some properties of the normalized
classes introduced in Section 0.3.

Following [29, 30, 47], the cohomology

(90)
$H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{*}({\mathcal{M}}_{B},{\mathbb{Q}})$

can be understood by the associated $\mathrm{PGL}_{n}$-character variety
$\hat{{\mathcal{M}}}_{\mathrm{B}}$. More precisely, let

$H^{*}({\mathcal{M}}_{B},{\mathbb{Q}})\cong
H^{*}(\hat{{\mathcal{M}}}_{\mathrm{B}},{\mathbb{Q}})\otimes
H^{*}(({\mathbb{C}}^{*})^{2g},{\mathbb{Q}})$

be the isomorphism established in [30, Section 1] and [29, Theorem 2.2.12].
Every class $w\in H^{i}(\hat{{\mathcal{M}}}_{\mathrm{B}},{\mathbb{Q}})$ can be
naturally viewed as a class

$w=w\otimes 1\in H^{i}({\mathcal{M}}_{B},{\mathbb{Q}}).$

The weights of the tautological classes associated with the universal
$\mathrm{PGL}_{r}$-bundle ${\mathcal{T}}$ on $C\times{\mathcal{M}}_{B}$
(induced by the universal $\mathrm{GL}_{r}$-bundle) were calculated in [47],

(91)
$\int_{\gamma}c_{k}({\mathcal{T}})\in{{}^{k}\mathrm{Hdg}^{i+2k-2}({\mathcal{M}}_{B})},\quad{\forall
k\geq 0},\forall~{}\gamma\in H^{i}(C,{\mathbb{Q}}).$

The following lemma deduces (5) from (91).

######  Lemma 4.6.

A twisted universal family $({\mathcal{U}}^{\alpha},\theta)$ on
$C\times{{\mathcal{M}}}_{\mathrm{Dol}}$ satisfies

(92)
$\int_{\gamma}\mathrm{ch}^{\alpha}_{k}({\mathcal{U}})\in{{}^{k}\mathrm{Hdg}^{i+2k-2}({\mathcal{M}}_{B})},\quad\forall~{}\gamma\in
H^{i}(C,{\mathbb{Q}}),~{}~{}\forall~{}k\geq 0,$

if and only if $\mathrm{ch}^{\alpha}({\mathcal{U}})$ is normalized.

###### Proof.

We define

${{}^{k}\mathrm{Hdg}^{d}(C\times{\mathcal{M}}_{B})}=\bigoplus_{i+j=d}H^{i}(C,{\mathbb{Q}})\otimes{{}^{k}\mathrm{Hdg}^{j}({\mathcal{M}}_{B})}.$

Then (92) is equivalent to

$\mathrm{ch}^{\alpha}({\mathcal{U}})\in\bigoplus_{k}{{}^{k}\mathrm{Hdg}^{2k}(C\times{\mathcal{M}}_{B})}.$

By a direct calculation using Chern roots, we have (recall that $r$ is the
rank)

$\mathrm{ch}^{\alpha}({\mathcal{U}})=\mathrm{ch}({\mathcal{T}})\cup\mathrm{exp}\left(\frac{c_{1}({\mathcal{U}})}{r}+\alpha\right).$

By using the universal relations between Chern character and total Chern
class, we note that (91) is equivalent to

$\mathrm{ch}({\mathcal{T}})\in\bigoplus_{k}{{}^{k}\mathrm{Hdg}^{2k}(C\times{\mathcal{M}}_{B})}.$

By [47], we have

$H^{0}({\mathcal{M}}_{B},{\mathbb{Q}})={{}^{0}\mathrm{Hdg}^{0}({\mathcal{M}}_{B})},\quad
H^{2}({\mathcal{M}}_{B},{\mathbb{Q}})={{}^{2}\mathrm{Hdg}^{2}({\mathcal{M}}_{B})},$

which implies

${{}^{1}\mathrm{Hdg}^{2}(C\times{\mathcal{M}}_{B})}=H^{1}(C,{\mathbb{Q}})\otimes
H^{1}({\mathcal{M}}_{B},{\mathbb{Q}}).$

Therefore, we obtain that (92) holds if and only if

$\mathrm{ch}_{1}^{\alpha}({\mathcal{U}})=c_{1}({\mathcal{U}})+r\alpha\in
H^{1}(C,{\mathbb{Q}})\otimes H^{1}({\mathcal{M}}_{B},{\mathbb{Q}}).$

This is equivalent to the condition that the class
$\mathrm{ch}({\mathcal{U}}^{\alpha})$ is normalized. ∎

Next, in parallel to Lemma 4.6, we give a suffcient criterion involving the
perverse filtration for a class $\mathrm{ch}^{\alpha}({\mathcal{U}})$ to be
normalized.

######  Lemma 4.7.

Suppose we have a twisted universal family $({\mathcal{U}}^{\alpha},\theta)$
such that, for every $k\geq 0$ and every $\gamma\in H^{2i}(C,{\mathbb{Q}})$,
the tautological class

$\int_{\gamma}\mathrm{ch}^{\alpha}_{k}({\mathcal{U}})\in
H^{2i+2k-2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),$

has perversity $k$. Then $\mathrm{ch}^{\alpha}({\mathcal{U}})$ is normalized.

###### Proof.

We have

$P_{1}H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=0,\quad
P_{0}H^{0}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{0}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}).$

The first vanishing follows from the decomposition (69), and the corresponding
vanishing for $\hat{{\mathcal{M}}}_{\mathrm{Dol}}$ due to the fact that its
second cohomology is generated by a relative ample class which must therefore
have perversity $2$.101010To see that a relative ample class has perversity 2,
we apply [13, Theorem 1.4.8] together with the fact that an ample class does
not vanish over a general fiber of the Hitchin morphism. The second equality
is clear in view of the isomorphism
$H^{0}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})\cong
H^{0}(\Lambda,{\mathbb{Q}})$ via pull-back from the base of the Hitchin
fibration (2). Since by assumption, any Künneth factor of
$\mathrm{ch}_{1}^{\alpha}({\mathcal{U}})$ in
$H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$ has perversity $1$, we
reach the desired conclusion

$\mathrm{ch}_{1}^{\alpha}({\mathcal{U}})\in H^{1}(C,{\mathbb{Q}})\otimes
H^{1}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}).\qed$

######  Remark 4.8.

In fact, we see from the proofs of Lemmas 4.6 and 4.7 that both lemmas hold if
we use ${\mathbb{C}}$-coefficients for the cohomology groups and we allow
$\alpha$ to be a ${\mathbb{C}}$-class. In particular, a
${\mathbb{C}}$-normalized class $\mathrm{ch}^{\alpha}({\mathcal{U}})$ is
unique and rational.

###  4.5. Specializations and tautological classes

Assume $j_{C}:C\hookrightarrow A$ is the closed embedding of the curve in an
abelian surface. Let ${\mathcal{S}}\to T$ and ${\mathcal{M}}\to T$ be the
family of surfaces and the relative moduli space of 1-dimensional sheaves
introduced in Section 4.2. In the following, we construct a family of
cohomology classes on ${\mathcal{M}}$ over $T$ whose restriction to every
closed fiber ${\mathcal{M}}_{t}$ is a twisted tautological class.

For our purpose, we take a relative compactification
${\mathcal{S}}\subset{\mathcal{S}}^{\prime}$ over $T$ as in the proof of Lemma
4.1 (ii). Then ${\mathcal{M}}$ is the coarse moduli space of Gieseker-stable
1-dimensional sheaves ${\mathcal{F}}$ on ${\mathcal{S}}^{\prime}$ satisfying
that $\mathrm{supp}({\mathcal{F}})$ is proper and contained in
${\mathcal{S}}_{t}\subset{\mathcal{S}}_{t}^{\prime}$ with the numerical
conditions

$\chi({\mathcal{F}})=\chi,\quad[\mathrm{supp}({\mathcal{F}})]=\beta_{t}.$

Here recall that $\beta_{t}$ is the class (84). By the proof of Lemma 4.1
(ii), there is a universal $K$-theory class $[{\mathcal{F}}_{T}]$ on
${\mathcal{S}}^{\prime}\times_{T}{\mathcal{M}}$.

Let

$\pi_{{\mathcal{S}}^{\prime}}:{\mathcal{S}}^{\prime}\times_{T}{\mathcal{M}}\to{\mathcal{S}}^{\prime},\quad\pi_{\mathcal{M}}:{\mathcal{S}}^{\prime}\times_{T}{\mathcal{M}}\to{\mathcal{M}}$

be the projections. We have the Chern character

$\mathrm{ch}({{\mathcal{F}}_{T}})\in
H^{*}({\mathcal{S}}^{\prime}\times_{T}{\mathcal{M}},{\mathbb{C}})$

defined by the universal class $[{\mathcal{F}}_{T}]$. For any class of the
type

(93)
$\tilde{\alpha}=\pi_{\mathcal{S}}^{*}\tilde{\alpha}_{1}+\pi_{\mathcal{M}}^{*}\tilde{\alpha}_{2}\in
H^{2}({\mathcal{S}}^{\prime}\times_{T}{\mathcal{M}},{\mathbb{C}})$

with $\tilde{\alpha}_{1}\in
H^{2}({\mathcal{S}}^{\prime},{\mathbb{C}}),~{}\tilde{\alpha}_{2}\in
H^{2}({\mathcal{M}},{\mathbb{C}})$, we consider the relative twisted class

(94)
$\mathrm{ch}^{\tilde{\alpha}}({\mathcal{F}}_{T})=\mathrm{exp}(\tilde{\alpha})\cup\mathrm{ch}({\mathcal{F}}_{T}),$

whose degree $2k$ parts

$\mathrm{ch}_{k}^{\tilde{\alpha}}({\mathcal{F}}_{T})\in
H^{2k}({\mathcal{S}}^{\prime}\times_{T}{\mathcal{M}},{\mathbb{C}})$

induce a relative tautological class

(95)
$\int_{\tilde{\gamma}}\mathrm{ch}_{k}^{\tilde{\alpha}}({\mathcal{F}}_{T}):={\pi_{\mathcal{M}}}_{*}\left((\pi_{{\mathcal{S}}^{\prime}}^{*}\tilde{\gamma}\cup\mathrm{ch}_{k}^{\tilde{\alpha}}({\mathcal{F}}_{T}))\cap[{\mathcal{S}}^{\prime}\times_{T}{\mathcal{M}}]\right)\in
H_{*}^{\mathrm{BM}}({\mathcal{M}},{\mathbb{C}})=H^{*}({\mathcal{M}},{\mathbb{C}})$

for any $\tilde{\gamma}\in H^{*}({\mathcal{S}}^{\prime},{\mathbb{C}})$. Here
we use the Poincaré duality in the last identity, and we require the
properness of $\pi_{\mathcal{M}}$ for the pushforward functor
${\pi_{\mathcal{M}}}_{*}:H^{\mathrm{BM}}_{*}({\mathcal{S}}^{\prime}\times_{T}{\mathcal{M}},{\mathbb{C}})\to
H^{\mathrm{BM}}_{*}({\mathcal{M}},{\mathbb{C}})$.

Now we check that (95) is the desired cohomology class on ${\mathcal{M}}$
which gives a family of twisted tautological classes over the base $T$.

For any closed point $t\in T$, base change [23, Proposition 1.7] implies that
the restriction of (95) to a closed fiber ${\mathcal{M}}_{t}$ is a twisted
universal class of the form

(96)
$\int_{\gamma_{t}}\mathrm{res}_{t}\left(\mathrm{ch}_{k}^{\tilde{\alpha}}\left({\mathcal{F}}_{T}\right)\right)\in
H^{*}({\mathcal{M}}_{t},{\mathbb{C}}),\quad\gamma_{t}\in
H^{*}({\mathcal{S}}^{\prime}_{t},{\mathbb{C}}).$

When $t\neq 0$, we have ${\mathcal{S}}_{t}={\mathcal{S}}^{\prime}_{t}$, and
therefore (96) recovers a twisted universal class on
${\mathcal{M}}_{\beta,A}$,

$\int_{\gamma_{t}}\mathrm{ch}_{k}^{\alpha}({\mathcal{F}}_{\beta})\in
H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}}).$

We now calculate the class (96) for $t=0$. By [31, Theorem 4.6.5], there
exists a coherent sheaf ${\mathcal{F}}_{0}$ supported on
$T^{*}C\times{\mathcal{M}}_{\mathrm{Dol}}$ which represents a universal class
$[{\mathcal{F}}_{0}]$ on
${\mathcal{S}}^{\prime}\times{\mathcal{M}}_{\mathrm{Dol}}$. Moreover, the
coherent sheaf ${\mathcal{F}}_{0}$ is supported on the universal spectral
curve ${\mathcal{C}}\subset T^{*}C\times{\mathcal{M}}_{\mathrm{Dol}}$ which is
proper over ${\mathcal{M}}_{\mathrm{Dol}}$. Hence by [3, Property 2.1], we
have

(97)
$\mathrm{ch}_{k}({\mathcal{F}}_{0})\cap[{\mathcal{S}}_{0}^{\prime}\times{\mathcal{M}}_{\mathrm{Dol}}]=\sum_{i}c_{i}[Z_{i}]\in\mathrm{CH}_{*}({\mathcal{S}}^{\prime}_{0}\times{\mathcal{M}}_{\mathrm{Dol}})_{\mathbb{Q}}$

where $c_{i}\in{\mathbb{Q}}$ and $Z_{i}$ are closed subvarieties in
$T^{*}C\times{\mathcal{M}}_{\mathrm{Dol}}$ which are proper over
${\mathcal{M}}_{\mathrm{Dol}}$ via the composition

(98) $Z_{i}\hookrightarrow
T^{*}C\times{\mathcal{M}}_{\mathrm{Dol}}\xrightarrow{p_{\mathcal{M}}}{\mathcal{M}}_{\mathrm{Dol}}.$

In particular, the expression (97) together with the properties for $Z_{i}$
implies that the class

$\int_{\gamma_{0}}\mathrm{res}_{0}\left(\mathrm{ch}_{k}^{\tilde{\alpha}}\left({\mathcal{F}}_{T}\right)\right)\in
H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})$

only depends on the restrictions of the classes $\gamma_{0}$ to
$H^{2}(T^{*}C,{\mathbb{C}})$ and $\mathrm{res}_{0}(\tilde{\alpha})$ to
$H^{2}(T^{*}C,{\mathbb{C}})\oplus
H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})$.

Note that a class of the type

(99)
${p_{\mathcal{M}}}_{*}(\omega\cup\mathrm{ch}_{k}({\mathcal{F}}_{0})):=\sum_{i}c_{i}\cdot{p_{{\mathcal{M}}}}_{*}(\omega\cap[Z_{i}])\in
H^{\mathrm{BM}}_{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$

is well-defined for $\omega\in
H^{*}(T^{*}C\times{\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})$ due to the
properness of (98). We see from the discussion above that the restriction of
the class (95) to ${\mathcal{M}}_{0}(={\mathcal{M}}_{\mathrm{Dol}})$ is given
by the following:

(100)
$\mathrm{res}_{0}\left(\int_{\tilde{\gamma}}\mathrm{ch}_{k}^{\tilde{\alpha}}({\mathcal{F}}_{T})\right)=\int_{\mathrm{sp}^{!}(\gamma_{t})}\mathrm{ch}_{k}^{\alpha_{0}}({\mathcal{F}}_{0})\in
H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}}).$

Here $\mathrm{sp}^{!}(\gamma_{t})\in
H^{*}({\mathcal{S}}_{0},{\mathbb{C}})=H^{*}(T^{*}C,{\mathbb{C}})$,
$\alpha_{0}\in H^{2}(T^{*}C,{\mathbb{C}})\oplus
H^{2}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})$, and the class on the r.h.s.
of (100) is defined by (99).

The following proposition shows that the tautological classes (4) on the
Dolbeault side ${\mathcal{M}}_{\mathrm{Dol}}$ are obtained by specializing
certain other tautological classes on the compact geometry side
${\mathcal{M}}_{\beta,A}$.

######  Proposition 4.9.

Let

$\int_{\gamma}\mathrm{ch}_{k}^{\alpha}({\mathcal{F}}_{\beta})\in
H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})$

be the classes of Theorem 2.1 with $\gamma\in H^{*}(A,{\mathbb{Q}})$ a
rational class on $A$, then we have

$\mathrm{sp}^{!}\left(\int_{\gamma}\mathrm{ch}_{k}^{\alpha}({\mathcal{F}}_{\beta})\right)=c(k-1,j_{C}^{*}\gamma)\in
H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}).$

###### Proof.

By the definition of $\mathrm{sp}^{!}$ and (100), we have

$\mathrm{sp}^{!}\left(\mathrm{res}_{t}\left(\int_{\tilde{\gamma}}\mathrm{ch}_{k}^{\tilde{\alpha}}({\mathcal{F}}_{T})\right)\right)=\mathrm{res}_{0}\left(\int_{\tilde{\gamma}}\mathrm{ch}_{k}^{\tilde{\alpha}}({\mathcal{F}}_{T})\right)=\int_{\mathrm{sp}^{!}(\gamma_{t})}\mathrm{ch}_{k}^{\alpha_{0}}({\mathcal{F}}_{0})$

for any $\tilde{\alpha}$ of the type (93).

A direct calculation by applying the Grothendieck–Riemann–Roch formula to the
natural projection

$\mathrm{pr}:T^{*}C\times{\mathcal{M}}_{\mathrm{Dol}}\to
C\times{\mathcal{M}}_{\mathrm{Dol}}$

(see the paragraph before [41, Remark 8]) together with Example 4.2 yields

$\int_{\mathrm{sp}^{!}(\gamma)}\mathrm{ch}_{k}^{\alpha_{0}}({\mathcal{F}}_{0})=\int_{j_{C}^{*}\gamma}\mathrm{ch}_{k-1}^{\alpha_{0}}({\mathcal{U}}).$

Here ${\mathcal{U}}=\mathrm{pr}_{*}{\mathcal{F}}_{0}$ is a universal family on
$C\times{\mathcal{M}}_{\mathrm{Dol}}$.

In conclusion, we obtain that

$\mathrm{sp}^{!}\left(\int_{\gamma}\mathrm{ch}_{k}^{\alpha}({\mathcal{F}}_{\beta})\right)=\int_{j_{C}^{*}\gamma}\mathrm{ch}_{k-1}^{\alpha_{0}}({\mathcal{U}}).$

Moreover, we know from Proposition 4.4 that the twisted class

$\int_{j_{C}^{*}\gamma}\mathrm{ch}_{k-1}^{\alpha_{0}}({\mathcal{U}}),\quad\forall\gamma\in
H^{*}(A,{\mathbb{Q}}),~{}\forall k\geq 1$

has perversity $k-1$. Hence Lemma 4.7 and Remark 4.8 imply, by recalling (4),
that

$\int_{j_{C}^{*}\gamma}\mathrm{ch}_{k-1}^{\alpha_{0}}({\mathcal{U}})=c(k-1,j_{C}^{*}\gamma).\qed$

###  4.6. Proofs of Theorems 0.2, 0.4, and 0.5

Since the perverse filtration with respect to the Hitchin fibration
$h:{\mathcal{M}}_{\mathrm{Dol}}\to\Lambda$ is locally constant when we deform
the curve $C$ [15], it suffices to prove all the three theorems for a curve
which can be embedded in an abelian surface

$j_{C}:C\hookrightarrow A.$

After having done so, we can apply the specialization morphism

$\mathrm{sp}^{!}:H^{*}({\mathcal{M}}_{\beta,A},{\mathbb{C}})\to
H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{C}})$

introduced in Section 4.3. The following results are immediate consequences of
Theorem 3.8, Proposition 4.4, and Proposition 4.9:

  1. (i)

We have

(101)
$c(\gamma,k)\in\widetilde{G}_{k}H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),\quad\forall\gamma\in\mathrm{Im}({j_{C}^{*}})\subset
H^{*}(C,{\mathbb{Q}}).$

  2. (ii)

The restriction of the decomposition

$H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})=\bigoplus_{k,d}\widetilde{G}_{k}H^{d}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$

to the subalgebra of $H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$
generated by the classes (101) is multiplicative.

We first prove Theorem 0.2. The Abel-Jacobi morphism embeds a genus 2 curve
$C$ into its Jacobian

$j_{C}:C\hookrightarrow\mathrm{Jac}(C)=A.$

Hence the restriction morphism $j_{C}^{*}$ is surjective, and Theorem 0.2
follows from (i) and (ii) above.

The proof of Theorem 0.4 is similar. For any embedding $j_{C}$, the image of
$j_{C}^{*}$ always contains the sub-vector space

$H^{0}(C,{\mathbb{Q}})\oplus H^{2}(C,{\mathbb{Q}})\subset
H^{*}(C,{\mathbb{Q}}).$

Hence the subalgebra

$R^{*}({\mathcal{M}}_{\mathrm{Dol}})\subset
H^{*}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}})$

is contained in the subalgebra generated by the classes (101), and we again
conclude Theorem 0.4 by (i) and (ii).

Finally we treat the odd classes

$c(\gamma,k)\in
H^{2k-1}({\mathcal{M}}_{\mathrm{Dol}},{\mathbb{Q}}),\quad\gamma\in
H^{1}(C,{\mathbb{Q}})$

and prove Theorem 0.5.

When the curve $C$ has genus $\geq 3$, the restriction

(102) $j_{C}^{*}:H^{1}(A,{\mathbb{Q}})\to H^{1}(C,{\mathbb{Q}})$

is not surjective. We know from (i) that $c(\gamma,k)$ has perversity $k$ for
any $\gamma$ lying in the image of (102). Since the monodromy group of the
moduli space ${\mathcal{M}}_{g}$ of nonsingular genus g curves is the full
symplectic group $\mathrm{Sp}_{2g}$ by [2], the sub-vector space

$\mathrm{Im}\left({j_{C}^{*}}:H^{1}(A,{\mathbb{Q}})\to
H^{1}(C,{\mathbb{Q}})\right)\subset H^{1}(C,{\mathbb{Q}})$

generates the total cohomology $H^{1}(C,{\mathbb{Q}})$ via the action of the
monodromy group. We deduce Theorem 0.5 from [15]. ∎

## References

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