# Perverse filtration for generalized Kummer varieties of fibered surfaces

Zili Zhang  School of Mathematical Sciences, Tongji University
<EMAIL_ADDRESS>
###### Abstract.

Let $A\to C$ be a proper surjective morphism from a smooth connected quasi-
projective commutative group scheme of dimension 2 to a smooth curve. The
construction of generalized Kummer varieties gives a proper morphism
$A^{[[n]]}\to C^{((n))}$. We show that the perverse filtration associated with
this morphism is multiplicative.

###### Contents

  1. 1 Introduction
    1. 1.1 Perverse filtrations
    2. 1.2 The P=W conjecture
    3. 1.3 Generalized Kummer varieties of fibered surfaces
    4. 1.4 Acknowledgements
  2. 2 Perverse filtrations
  3. 3 Hilbert schemes of fibered surfaces
    1. 3.1 Partitions
    2. 3.2 Products and symmetric products
    3. 3.3 Perverse filtration of Hilbert schemes
  4. 4 Generalized Kummer varieties for fibered surfaces
    1. 4.1 Fibered commutative group schemes of dim 2
    2. 4.2 Perverse filtration for generalized Kummer varieties

##  1\. Introduction

###  1.1. Perverse filtrations

Let $f:X\to Y$ be a proper morphism between smooth quasi-projective varieties.
Let ${}^{\mathfrak{p}}\tau_{\leq k}$ be the perverse truncations of the
bounded derived category of constructible sheaves $D^{b}_{c}(Y)$. Applying
${}^{\mathfrak{p}}\tau_{\leq k}$ to the pushforward $Rf_{*}{\mathbb{Q}}_{X}$
gives an increasing filtration in $H^{*}(X,{\mathbb{Q}})$

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

The filtration (1) is called the perverse filtration associated with the
morphism $f$. A perverse filtration is called _multiplicative_ if the cup
product

$P_{k}H^{d}(X,{\mathbb{Q}})\cup P_{l}H^{d}(X,{\mathbb{Q}})\subset
P_{k+l}H^{d+e}(X,{\mathbb{Q}}),~{}~{}~{}~{}k,l,d,e\geq 0.$

We say that a perverse filtration _admits a strongly multiplicative splitting_
if there is a direct sum decomposition

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

splitting the perverse filtration in the sense that

$P_{k}H^{*}(X,{\mathbb{Q}})=\bigoplus_{i=0}^{k}G_{i}H^{*}(X,{\mathbb{Q}}),$

such that the cup product

$G_{i}H^{d}(X,{\mathbb{Q}})\cup G_{j}H^{d}(X,{\mathbb{Q}})\subset
G_{i+j}H^{d+e}(X,{\mathbb{Q}}),~{}~{}~{}~{}i,j,d,e\geq 0.$

###  1.2. The P=W conjecture

Let $C$ be a smooth projective curve. There are two moduli spaces associated
with $C$ and the structure group $GL(n,{\mathbb{C}})$. They are the Betti
moduli $M_{B}$ which parametrizes all (twisted)
$GL(n,{\mathbb{C}})$-representations of the fundamental group $\pi_{1}(C)$ and
the Dolbeault moduli $M_{D}$ which parametrizes all semi-stable Higgs bundles
of degree $d$ rank $n$. The Dolbeault moduli $M_{D}$ admits a natural proper
morphism, called Hitchin map, to an affine space $\pi:M_{D}\to{\mathbb{A}}$.
Simpson proves that there is a _canonical_ diffeomorphism between $M_{B}$ and
$M_{D}$, and hence there is a canonical isomorphism of cohomology groups

(2) $H^{*}(M_{D},{\mathbb{Q}})=H^{*}(M_{B},{\mathbb{Q}}).$

In [6], de Cataldo, Hausel, and Migliorini conjectured that under the
identification (2), the perverse filtration associated with $\pi$ coincides
with the mixed Hodge-theoretic weight filtration, _i.e._

######  Conjecture 1.1.

[6]

$P_{k}H^{*}(M_{D},{\mathbb{Q}})=W_{2k}H^{*}(M_{B},{\mathbb{Q}})=W_{2k+1}H^{*}(M_{B},{\mathbb{Q}}),k\geq
0.$

Conjecture 1.1 is referred to as the $P=W$ conjecture. Since the Hodge-
theoretic weight filtration for arbitrary varieties is always multiplicative,
the $P=W$ conjecture suggests that the perverse filtration associated with
Hitchin maps are multiplicative. In fact, de Cataldo, Maulik, and Shen give a
complete description of the ring generators of the cohomology ring
$H^{*}(M_{B},{\mathbb{Q}})=H^{*}(M_{D},{\mathbb{Q}})$, and show that the $P=W$
conjecture is equivalent to the multiplicativity of the perverse filtration
associated with the Hitchin map $\pi$; see [7, Proposition 1.6]. However, the
multiplicativity of perverse filtration does not hold for general morphisms;
see [13, Example 1.5].

Since the Doubeault moduli spaces are hyperKähler manifolds and the Hitchin
maps are Lagrangian fibrations, the $P=W$ conjecture suggests that the
multiplicativity of perverse filtrations holds for Lagrangian fibrations. Shen
and Yin verified for compact hyperKähler varieties and Lagrangian fibrations
in [9]. For non-compact cases, not many cases are known. The Hitchin map of
Dolbeault moduli of rank 2 Higgs bundles is studied in [6], and over genus 2
curve is studied in [7]. The multiplicativity of perverse filtrations for
Hilbert schemes of fibered surfaces is studied in [10, 13, 14]. In this
article, we give a concrete description of the perverse filtrations associated
with generalized Kummer varieties of quasi-projective fibered surfaces, and
show that the perverse filtration is multiplicative and admits a natural
strongly multiplicative splitting.

###  1.3. Generalized Kummer varieties of fibered surfaces

Let $A$ be a smooth connected (not necessarily compact) commutative group
scheme of dimension 2. Denote $A^{[n]}$ the Hilbert scheme of $n$ points on
$A$. Let $A^{(n)}$ be the $n$-fold symmetric product of $A$. There is natural
summation map $A^{(n)}\to A$. The kernel of the composition of Hilbert-Chow
morphism and the summation

$A^{[n]}\to A^{(n)}\to A$

is called the generalized Kummer variety, denoted as $A^{[[n]]}$111Throughout
this paper, we use the term “generalized Kummer variety $A^{[[n]]}$” without
assuming $A$ to be compact. . $A^{[[n]]}$ is a smooth variety of dimension
$2n-2$. When $A$ admits a proper surjective morphism to a curve $C$, there is
a natural Lagrangian fibration constructed as follows. Consider the
composition

(3) $A^{[[n]]}\hookrightarrow A^{[n]}\to A^{(n)}\to C^{(n)}.$

Let $C^{((n))}$ be the image of (3). Then $C^{((n))}$ is a variety of
dimension $n-1$ (Proposition 4.7), and the morphism $A^{[[n]]}\to C^{((n))}$
is a proper morphism. Our main result is

######  Theorem 1.2 (Theorem 4.13).

Let $f:A\to C$ be a proper surjective morphism from a connected quasi-
projective commutative group scheme $A$ of dimension 2 to a quasi-projective
curve $C$. Then the perverse filtration associated with the induced morphism
$h^{\prime}:A^{[[n]]}\to C^{((n))}$ is multiplicative.

There are three main ingredients: (1) the classification of fibered group
surfaces (Proposition 4.1), (2) the cup product formula ([4, Theorem 32]), and
(3) the description of the perverse filtration associated with $A^{[[n]]}\to
C^{((n))}$ (Proposition 4.9).

This article is organized as follows. In Section 2, we recall the definition
and properties of perverse filtrations on the cohomology groups and compactly
supported cohomology associated with general proper morphisms between smooth
varieties. In Section 3, we recall the properties of Hilbert schemes of points
of fibered surfaces. In Section 4, we first classify quasi-projective smooth
commutative group schemes of dimension 2 properly fibered over smooth curves.
Then we calculate the perverse filtrations of the generalized Kummer varieties
of fibered surfaces, and prove that they are multiplicative.

###  1.4. Acknowledgements

I am grateful to Michel Brion for help on non-compact 2D algebraic groups, and
for pointing out references. I also thank Lie Fu and Shizhang Li for helpful
discussions. I am partially supported by the Fundamental Research Funds for
the Central Universities.

##  2\. Perverse filtrations

Let $f:X\to Y$ be a proper morphism between smooth quasi-projective varieties.
The perverse $t$-structure on $D^{b}_{c}(Y)$, the bounded derived category of
constructible sheaves on $Y$, gives a morphism

${}^{\mathfrak{p}}\tau_{\leq k}\left(Rf_{*}{\mathbb{Q}}_{X}[\dim
X-r(f)]\right)\to Rf_{*}{\mathbb{Q}}_{X}[\dim X-r(f)],$

where

$r(f)=\dim X\times_{Y}X-\dim X$

is the defect of semismallness of $f$. Applying hypercohomology yields a
morphism in cohomology groups

(4) $\begin{split}{\mathbb{H}}^{d-\dim
X+r(f)}\left({{}^{\mathfrak{p}}\tau_{\leq k}}(Rf_{*}{\mathbb{Q}}_{X}[\dim
X-r(f)])\right)&\to\\\ {\mathbb{H}}^{d-\dim
X+r(f)}(Rf_{*}{\mathbb{Q}}_{X}[\dim
X-&r(f)])=H^{d}(X,{\mathbb{Q}}).\end{split}$

We define the subspace $P_{k}^{f}H^{d}(X,{\mathbb{Q}})$ to be the image of
(4)222The shift $[\dim X-r(f)]$ is to ensure that the perverse filtration
starts at 0.. We will omit the reference to the morphism $f$ when no confusion
arises. Then we have an increasing filtration

$P_{0}H^{d}(X,{\mathbb{Q}})\subset
P_{1}H^{d}(X,{\mathbb{Q}})\subset\cdots\subset
H^{d}(X,{\mathbb{Q}}),~{}~{}~{}~{}d\geq 0.$

called the perverse filtration associated with morphism $f$. It is
concentrated at the interval $[0,2r]$, _i.e._ $P_{-1}H^{d}(X,{\mathbb{Q}})=0$
and $P_{2r}H^{d}(X,{\mathbb{Q}})=H^{d}(X,{\mathbb{Q}})$ for all $d$. For a
nonzero class $\alpha\in H^{*}(X,{\mathbb{Q}})$, we denote
${\mathfrak{p}}^{f}(\alpha)=k$ if $\alpha\in P_{k}H^{*}(X,{\mathbb{Q}})$ and
$\alpha\not\in P_{k-1}H^{*}(X,{\mathbb{Q}})$. We say that the perverse
filtration is multiplicative if

$P_{k}H^{d}(X,{\mathbb{Q}})\cup P_{l}H^{e}(X,{\mathbb{Q}})\subset
P_{k+l}H^{d+e}(X,{\mathbb{Q}}),~{}~{}~{}~{}k,l,d,e\geq 0.$

Suppose the perverse filtration $P_{\bullet}H^{*}(X,{\mathbb{Q}})$ admits a
splitting

$P_{k}H^{*}(X,{\mathbb{Q}})=\bigoplus_{i=0}^{k}G_{i}H^{*}(X,{\mathbb{Q}})$

satisfying

(5) $G_{i}H^{d}(X,{\mathbb{Q}})\cup G_{j}H^{e}(X,{\mathbb{Q}})\subset
G_{i+j}H^{d+e}(X,{\mathbb{Q}}).$

Then we say that the perverse filtration $P_{\bullet}H^{*}(X,{\mathbb{Q}})$
admits a strongly multiplicative splitting, or the morphism $f$ admits a
strongly multiplicative perverse decomposition. A class $\beta\in
G_{i}H^{d}(X,{\mathbb{Q}})$ for some $i$ is called _pure of perversity $i$
with respect to the splitting $G$_. It follows from the definition that if $f$
admits a strongly multiplicative perverse decomposition, then the perverse
filtration associated with $f$ is multiplicative, but not vice versa.

We say that a basis $\\{\beta_{i}\\}$ of $H^{*}(X,{\mathbb{Q}})$ is adapted to
the perverse filtration $P_{\bullet}H^{*}(X,{\mathbb{Q}})$ if

$P_{k}H^{*}(X,{\mathbb{Q}})=\left\langle\beta_{i}\mid\beta_{i}\in
P_{k}H^{*}(X,{\mathbb{Q}})\right\rangle.$

Similarly, the perverse filtration can also be defined on the compactly
supported cohomology by taking $R\Gamma_{c}$ in (4). Then the natural
transformation of functors $R\Gamma_{c}\to R\Gamma$ implies that the forgetful
map $\iota:H^{*}_{c}(X,{\mathbb{Q}})\to H^{*}(X,{\mathbb{Q}})$ preserves the
perverse filtration, _i.e._ $\iota P_{k}H^{*}_{c}(X,{\mathbb{Q}})\subset
P_{k}H^{*}(X,{\mathbb{Q}})$.

######  Proposition 2.1.

Let $f:X\to Y$ be a proper morphism between smooth quasi-projective varieties.
Then there exists a basis $\\{\beta_{i}\\}$ of $H^{*}(X,{\mathbb{Q}})$ adapted
to $P_{\bullet}H^{*}(X,{\mathbb{Q}})$ and a basis $\\{\beta^{i}\\}$ adapted to
$P_{\bullet}H^{*}_{c}(X,{\mathbb{Q}})$, such that

  1. (1)

$\\{\beta_{i}\\}$ and $\\{\beta^{i}\\}$ are dual with respect to the Poincaré
pairing, _i.e._

$\langle\beta_{i},\beta^{j}\rangle_{X}=\delta_{ij}.$

  2. (2)

${\mathfrak{p}}(\beta_{i})+{\mathfrak{p}}(\beta^{i})=2r(f)$.

In particular, if ${\mathfrak{p}}(\alpha)+{\mathfrak{p}}(\beta)<2r(f)$ for
some $\alpha\in H^{*}_{c}(X,{\mathbb{Q}})$ and $\beta\in
H^{*}_{c}(X,{\mathbb{Q}})$, then $\langle\alpha,\beta\rangle_{X}=0$.

###### Proof.

Fix a (non-canonical) decomposition

$Rf_{*}{\mathbb{Q}}[\dim
X-r(f)]\cong{\mathcal{P}}_{0}\bigoplus\cdots\bigoplus{\mathcal{P}}_{2r(f)}[-2r(f)],$

where ${\mathcal{P}}_{i}$ are perverse sheaves. Then [14, Proposition 4.1] and
[14, Remark 2.9] produce the desired bases $\\{\beta_{i}\\}$ and
$\\{\beta^{i}\\}$. ∎

The relation of push-forward and pull-back of perverse filtrations is
described as follows.

######  Proposition 2.2.

Let $f:X\to A$ and $g:Y\to B$ be two proper morphism between smooth quasi-
projective varieties. Let $h:X\to Y$ be a proper morphism. Then for any
integer $m$, the following are equivalent.

  1. (1)

${\mathfrak{p}}^{f}(h^{*}\beta)\leq{\mathfrak{p}}^{g}(\beta)+m$ for all
$\beta\in H^{*}(Y,{\mathbb{Q}})$.

  2. (2)

${\mathfrak{p}}^{g}(h_{*}\alpha)\leq{\mathfrak{p}}^{f}(\alpha)+m+2r(g)-2r(f)$
for all $\alpha\in H^{*}_{c}(X,{\mathbb{Q}})$.

###### Proof.

By adjunction, for $\alpha\in H^{*}_{c}(X,{\mathbb{Q}})$, $\beta\in
H^{*}(Y,{\mathbb{Q}})$, we have

$\langle h_{*}\alpha,\beta\rangle_{Y}=\langle\alpha,h^{*}\beta\rangle_{X},$

where $\langle-,-\rangle$ denotes the Poincaré paring.
We prove $(1)\Rightarrow(2)$. Suppose $\alpha\in
P_{k}H^{*}_{c}(X,{\mathbb{Q}})$. Then for any $\beta\in
P_{2r(f)-k-m-1}H^{*}(Y,{\mathbb{Q}})$, ${\mathfrak{p}}(h^{*}\beta)\leq
2r(f)-k-1$. By Proposition 2.1, $\langle\alpha,h^{*}\beta\rangle_{X}=0$, and
hence $\langle h_{*}\alpha,\beta\rangle_{Y}=0$. Let $\\{\beta^{i}\\}$ be the
basis obtained in Proposition 2.1 for the morphism $g:Y\to B$. Suppose
${\mathfrak{p}}^{g}(f_{*}\alpha)>k+m+2r(g)-2r(f)$, then at least one
$\beta^{i}$ with ${\mathfrak{p}}(\beta^{i})>k+m+2r(g)-2r(f)$ would have
nonzero coefficient in the expansion of $f_{*}\alpha$. Then by Proposition
2.1, ${\mathfrak{p}}(\beta_{i})<2r(f)-k-m$ and $\beta_{i}$ would have nonzero
paring with $f_{*}\alpha$, contradiction. The proof of the reverse direction
is similar. ∎

######  Proposition 2.3.

[13, Proposition 2.1] Let $f_{1}:X_{1}\to Y_{1}$ and $f_{2}:X_{2}\to Y_{2}$ be
proper morphisms between smooth quasi-projective varieties. Then

${\mathfrak{p}}^{f_{1}\times
f_{2}}(\alpha_{1}\boxtimes\alpha_{2})={\mathfrak{p}}^{f_{1}}(\alpha_{1})+{\mathfrak{p}}^{f_{2}}(\alpha_{2}).$

Let $\Gamma:X\to X\times Y$ be the graph of $f:X\to Y$. Since $\Gamma$ is a
proper morphism, the Gysin push-forward is defined as Borel-Moore homology

$\Gamma_{*}:H^{*}(X,{\mathbb{Q}})\cong H^{BM}_{*}(X,{\mathbb{Q}})\to
H^{BM}_{*}(X\times Y,{\mathbb{Q}})\cong H^{*}(X\times Y,{\mathbb{Q}}).$

Equivalently,

(6)
$\begin{split}\Gamma_{*}:H^{*}(X,{\mathbb{Q}})\xrightarrow{a}&H^{*}(X,{\mathbb{Q}})\otimes
H^{*}_{c}(X,{\mathbb{Q}})\\\ \xrightarrow{\textup{id}\otimes
f_{*}}&H^{*}(X,{\mathbb{Q}})\otimes
H^{*}_{c}(Y,{\mathbb{Q}})\xrightarrow{\textup{id}\otimes\iota}H^{*}(X,{\mathbb{Q}})\otimes
H^{*}(Y,{\mathbb{Q}}),\end{split}$

where $a$ is the adjoint map of the cup product

$\cup:H^{*}(X,{\mathbb{Q}})\times H^{*}(X,{\mathbb{Q}})\to
H^{*}(X,{\mathbb{Q}})$

and $\iota$ is the forgetful map

$\iota:H^{*}_{c}(Y,{\mathbb{Q}})\to H^{*}(Y,{\mathbb{Q}}).$

We have the following perversity estimation of the graph of a proper morphism.

######  Proposition 2.4.

Let $f:X\to A$ be a proper morphism between smooth quasi-projective varieties.
Let $g:Y\to B$ be a proper morphism between smooth connected quasi-projective
varieties. Let $h:X\to Y$ be a proper morphism satisfying the following two
properties.

  1. (1)

The pull-back $h^{*}:H^{*}(Y,{\mathbb{Q}})\to H^{*}(X,{\mathbb{Q}})$ preserves
the perverse filtrations, i.e.
${\mathfrak{p}}^{f}(h^{*}(\alpha))\leq{\mathfrak{p}}^{g}(\alpha)$ holds for
all $\alpha$.

  2. (2)

The perverse filtration associated with the morphism $f:X\to A$ is
multiplicative.

Let $\Gamma:X\to X\times Y$ be the graph of $h$. Then the push-forward along
the closed embedding $\Gamma$ increases the perversity by $2r(g)$, i.e. for
any class $\gamma\in H^{*}(X,{\mathbb{Q}})$,

${\mathfrak{p}}^{f\times
g}(\Gamma_{*}(\gamma)\leq{\mathfrak{p}}^{f}(\gamma)+2r(g).$

###### Proof.

Let $\\{\alpha_{i}\\}$ and $\\{\alpha^{i}\\}$ be dual bases of
$H^{*}(X,{\mathbb{Q}})$ and $H^{*}_{c}(X,{\mathbb{Q}})$ adapted to the
perverse filtration associated with $f$ obtained in Proposition 2.1. By (6),
we have

$\Gamma_{*}(\gamma)=\sum_{i}\gamma\alpha_{i}\otimes\iota f_{*}\alpha^{i}.$

Then by Proposition 2.2, condition (1) implies

${\mathfrak{p}}^{g}(h_{*}\alpha^{i}))\leq{\mathfrak{p}}^{f}(\alpha^{i})+2r(g)-2r(f).$

Now by Proposition 2.3, condition (2) and Proposition 2.1,

$\begin{split}{\mathfrak{p}}^{f\times g}(\gamma\alpha_{i}\otimes\iota
f_{*}\alpha^{i})\leq&{\mathfrak{p}}^{f}(\gamma)+{\mathfrak{p}}^{f}(\alpha_{i})+{\mathfrak{p}}^{f}(\alpha^{i})+2r(g)-2r(f)\\\
\leq&{\mathfrak{p}}^{f}(\gamma)+2r(g).\end{split}$

So ${\mathfrak{p}}^{f\times
g}(\Gamma_{*}(\gamma))\leq{\mathfrak{p}}^{f}(\gamma)+2r(g)$. ∎

As an application, we have the following generalization of [13, Proposition
3.8] and [13, Proposition 5.5].

######  Proposition 2.5.

Let $f:X\to Y$ be a proper morphism between smooth quasi-projective varieties.
Let $f^{n}:X^{n}\to Y^{n}$ be the induced morphism. Let $\Delta_{n}:X\to
X^{n}$ be the diagonal embedding. Suppose that the perverse filtration
associated with $f$ is multiplicative. Then

${\mathfrak{p}}^{f^{n}}(\Delta_{n*}(\alpha))\leq{\mathfrak{p}}^{f}(\alpha)+2(n-1)r(f).$

###### Proof.

The $n=2$ case follows from taking $f=g$ and $h=\textup{id}$ in Proposition
2.4. The general $n$ follows from an induction argument using

$\Delta_{n}:X\xrightarrow{\Delta_{n-1}}X^{n-1}\xrightarrow{\Delta_{2}\times\textup{id}^{n-2}}X^{n}.$

∎

For later use, we also include the following easy fact.

######  Proposition 2.6.

Let $X$, $Y$ and $Z$ be quasi-projective varieties. Let $f:X\to Y$ be a proper
morphism and $g:Y\to Z$ be a finite surjective morphism. Then the perverse
filtration associated with $f$ is identical with the one associated with
$g\circ f$, _i.e._

$P^{f}_{k}H^{*}(X,{\mathbb{Q}})=P^{g\circ f}_{k}H^{*}(X,{\mathbb{Q}}).$

###### Proof.

Since an finite morphism is $t$-exact, $g_{*}$ commutes perverse truncations
${}^{\mathfrak{p}}\tau_{\leq k}$ and we have the following commutative
diagram.

(7) ${{}^{\mathfrak{p}}\tau_{\leq k}R(g\circ
f)_{*}{\mathbb{Q}}_{X}}$${g_{*}{{}^{\mathfrak{p}}\tau_{\leq
k}}Rf_{*}{\mathbb{Q}}_{X}}$${R(g\circ
f)_{*}{\mathbb{Q}}_{X}}$${g_{*}Rf_{*}{\mathbb{Q}}_{X}.}$$\scriptstyle{\cong}$

By the definition of perverse filtrations (4), the diagram (7) implies

$P^{f}_{k-\dim X+r(g\circ f)}H^{*}(X,{\mathbb{Q}})=P^{g\circ f}_{k-\dim
X+r(f)}H^{*}(X,{\mathbb{Q}}).$

It remains to check $r(f)=r(g\circ f)$, which follows from the standard
argument of stratification by fiber dimensions. ∎

##  3\. Hilbert schemes of fibered surfaces

###  3.1. Partitions

We recall some notations of partitions in this section. We say
$\nu=(\nu_{1},\nu_{2},\cdots,\nu_{l})$ is a partition of an integer $n$ if
$\nu_{1}\geq\cdots\geq\nu_{l}>0$ and $n=\nu_{1}+\cdots+\nu_{l}$. The integer
$l$ is called the length of the partition $\nu$, also denoted as $l(\nu)$. The
greatest common divisor $\gcd(\nu)$ is defined as
$\gcd(\nu_{1},\cdots,\nu_{l})$. For a given partition $\nu$, denote by $a_{i}$
the number of times that $i$ appears in the partition. Then
$n=a_{1}+2a_{2}+\cdots+na_{n}$ and $l=a_{1}+\cdots+a_{n}$. We also write
$\nu=1^{a_{1}}\cdots n^{a_{n}}$.

###  3.2. Products and symmetric products

Let $X$ be a smooth quasi-projective variety. We denote by $X^{n}$ the
$n$-fold Cartesian product and by $X^{(n)}=X^{n}/\mathfrak{S}_{n}$ the
symmetric product, where $\mathfrak{S}_{n}$ is the symmetric group. The
elements in $X^{(n)}$ are denoted as $x_{1}+\cdots+x_{n}$, where $x_{i}\in X$.
Let $\nu=1^{a_{1}}\cdots n^{a_{n}}$ be a partition of $n$. Denote by $X^{\nu}$
or $X^{l(\nu)}$ the Cartesian product $X^{a_{1}}\times\cdots\times X^{a_{n}}$.
Denote by $X^{(\nu)}$ the product of symmetric products
$X^{(a_{1})}\times\cdots\times X^{(a_{n})}$. Let
$\mathfrak{S}_{\nu}=\mathfrak{S}_{a_{1}}\times\cdots\times\mathfrak{S}_{a_{n}}$.
Then $X^{(\nu)}=X^{\nu}/\mathfrak{S}_{\nu}$. By a theorem of Grothendieck,

(8)
$H^{*}(X^{(n)},{\mathbb{Q}})=H^{*}(X^{n},{\mathbb{Q}})^{\mathfrak{S}_{n}}=\textup{Sym}^{n}H^{*}(X,{\mathbb{Q}}).$

We also have a closed embedding

$\displaystyle\iota^{(\nu)}:$ $\displaystyle X^{(\nu)}$ $\displaystyle\to
X^{(n)}$
$\displaystyle\displaystyle\left(\sum_{j=1}^{a_{1}}x_{1j},\cdots,\sum_{j=1}^{a_{n}}x_{nj}\right)$
$\displaystyle\mapsto\sum_{i=1}^{n}\sum_{j=1}^{a_{i}}ix_{ij}.$

Let $f:X\to Y$ be a proper morphism between smooth quasi-projective varieties.
Denote by $f^{n}:X^{n}\to Y^{n}$ and $f^{(n)}:X^{(n)}\to Y^{(n)}$ the induced
maps. By Proposition 2.3, The perverse filtrations associated with $f^{n}$ is

(9)
$P_{k}^{f^{n}}H^{*}(X^{n},{\mathbb{Q}})=\left\langle\alpha_{1}\boxtimes\cdots\boxtimes\alpha_{n}\mid\sum_{i=1}^{n}{\mathfrak{p}}(\alpha_{i})\leq
k.\right\rangle$

Taking $\mathfrak{S}_{n}$-invariant part of (9) yields the filtration
associated with $f^{(n)}$.

Let $\nu$ be a partition, then we also have the induced maps
$f^{\nu}:X^{\nu}\to Y^{\nu}$ and $f^{(\nu)}:X^{(\nu)}\to Y^{(\nu)}$. The
perverse filtration associated with $f^{\nu}$ and $f^{(\nu)}$ are computed
easily by Proposition 2.3 and (8).

###  3.3. Perverse filtration of Hilbert schemes

Let $f:S\to C$ be a proper surjective morphism from a smooth quasi-projective
surface $S$ to a smooth curve $C$. Following the notation in Section 3.1 and
3.2, we have the diagram

${S^{[n]}}$${S^{\nu}}$${S^{(\nu)}}$${S^{(n)}}$${C^{\nu}}$${C^{(\nu)}}$${C^{(n)}.}$$\scriptstyle{\pi}$$\scriptstyle{h}$$\scriptstyle{/\mathfrak{S}_{\nu}}$$\scriptstyle{f^{\nu}}$$\scriptstyle{\iota^{(\nu)}_{S}}$$\scriptstyle{f^{(\nu)}}$$\scriptstyle{f^{(n)}}$$\scriptstyle{/\mathfrak{S}_{\nu}}$$\scriptstyle{\iota^{(\nu)}_{C}}$

######  Theorem 3.1.

[5, Theorem 2, Theorem 4] Let $S$ be a smooth quasi-projective surface. Then
there is a canonical decomposition

(10) $R\pi_{*}{\mathbb{Q}}_{S^{[n]}}[2n]=\bigoplus_{\nu\vdash
n}\iota^{(\nu)}_{S*}{\mathbb{Q}}_{S^{(\nu)}}[2l(\nu)].$

In particular, there is a decomposition on the cohomology.

(11) $H^{*}(S^{[n]},{\mathbb{Q}})=\bigoplus_{\nu\vdash
n}H^{*}(S^{(\nu)},{\mathbb{Q}})[2l(\nu)-2l].$

The perverse filtration associated with $h:S^{[n]}\to C^{(n)}$ is calculated
in [13].

######  Proposition 3.2.

[13, Corollary 4.14] Under the canonical isomorphism (11), the perverse
filtration

(12) $P_{k}^{h}H^{*}(S^{[n]},{\mathbb{Q}})=\bigoplus_{\nu\vdash
n}P_{k+l(\nu)-n}^{f^{(\nu)}}H^{*}(S^{(\nu)},{\mathbb{Q}})[2l(\nu)-2n].$

######  Definition 3.3.

Let $\alpha_{\nu}\in H^{*}(A^{(\nu)},{\mathbb{Q}})$ be a class. Denote by
$\alpha_{\nu}^{[n]}\in H^{*}(A^{[n]},{\mathbb{Q}})$ its image in
$H^{*}(A^{[n]},{\mathbb{Q}})$ via the decomposition (11).

######  Corollary 3.4.

Let $\alpha_{\nu}\in H^{*}(A^{(\nu)},{\mathbb{Q}})$ be a cohomology class.
Then

${\mathfrak{p}}^{h}(\alpha_{\nu}^{[n]})={\mathfrak{p}}^{f^{(\nu)}}(\alpha_{\nu})+n-l(\nu).$

Let $\alpha=\sum_{\nu}\alpha_{\nu}^{[n]}$. Then

${\mathfrak{p}}^{h}(\alpha)=\max_{\nu}\left\\{{\mathfrak{p}}^{h}\left(\alpha_{\nu}^{[n]}\right)\right\\}=\max_{\nu}\left\\{{\mathfrak{p}}^{f^{(\nu)}}(\alpha_{\nu})+n-l(\nu)\right\\}.$

The following result is a slight generalization of [13, Theorem 4.18] and [13,
Theorem 5.6].

######  Theorem 3.5.

Let $f:S\to C$ be proper surjective morphism from a smooth quasi-projective
surface $S$ with trivial canonical bundle to a smooth curve $C$. Let
$h:S^{[n]}\to C^{(n)}$ be the induced morphism. Suppose further that $S$
admits a smooth compactification $\bar{S}$ such that the restriction
$H^{*}(\bar{S})\to H^{*}(S)$ is surjective. Then the perverse filtration
associated with $h$ is multiplicative.

###### Proof.

The projective case and the five families of Hitchin system ([13, Section
5.1]) case are treated in [13, Theorem 4.18] and [13, Theorem 5.6],
respectively. Their proofs are similar, which are based on the following three
ingredients.

  1. (1)

The cup product formula for Hilbert schemes $S^{[n]}$ with trivial canonical
bundle. [8, Theorem 3.2] for projective case and [13, Proposition 4.10] for
quasi-projective case.

  2. (2)

The description of the perverse filtration Proposition 3.2.

  3. (3)

The diagonal estimation [13, Proposition 3.8] for projective case and [13,
Proposition 5.5] for the five families of Hitchin systems.

In our generality, the cup product formula and the description of the perverse
filtration still hold. The diagonal estimation is Proposition 2.5. ∎

##  4\. Generalized Kummer varieties for fibered surfaces

###  4.1. Fibered commutative group schemes of dim 2

We say a fibration $f:A\to C$ of quasi-projective varieties satisfying the
condition ($\dagger$) if the following holds.

($\dagger$) $f:A\to C$ is a proper surjective morphism from a connected smooth
quasi-projective commutative group scheme $A$ of dimension 2 to a quasi-
projective curve $C$.

The following proposition classifies fibrations satisfying ($\dagger$).

######  Proposition 4.1.

Let $f:A\to C$ be a fibration satisfying ($\dagger$). Then the fibration $f$
factors as $A\to B\to C$, satisfying one of the following cases.

  1. (1)

$A$ is an abelian surface, $B$ is an elliptic curve, and $C$ is
${\mathbb{P}}^{1}$ or an elliptic curve. $A\to B$ is a group homomorphism and
$B\to C$ is a finite morphism.

  2. (2)

$A=E\times{\mathbb{C}}$, and $B=C={\mathbb{C}}$ where $E$ is an elliptic
curve. $A\to B$ is the natural projection and $B\to C$ is a finite morphism.

  3. (3)

$A=(E\times{\mathbb{C}}^{*})/\Gamma$ and $B=C={\mathbb{C}}^{*}$, where $E$ is
an elliptic curve and $\Gamma$ is a finite cyclic group. $A\to B$ is an
equivariant $\Gamma$-quotient of the natural projection
$E\times{\mathbb{C}}^{*}\to{\mathbb{C}}^{*}$ and $B\to C$ is a finite
morphism.

###### Proof.

Let $A\to B\to C$ be the Stein factorization of $f$, _i.e._ $f^{\prime}:A\to
B$ has connected fibers and $g:B\to C$ is a finite morphism. Since $f:A\to C$
is surjective and $g:B\to C$ is finite, $B$ is a curve. By [12, Tag 03H0,
Theorem 39.47.4], $B$ is the relative normalization of $C$ in $A$, and hence
is normal [12, Tag 0BAK, Theorem 29.53.13]. So $B$ is a smooth curve.

When $B$ is compact, $A$ is an abelian variety. Then the genus $g(B)\leq 1$.
Since the generic fiber of any fibration $A\to{\mathbb{P}}^{1}$ is
disconnected (to be proved in Lemma 4.2), $B$ must be an elliptic curve. Then
$f^{\prime}:A\to B$ is a group homomorphism after a choice of the origin in
$B$, [2, Proposition V.12].

When $B$ is non-compact, it is an affine curve and $f^{\prime}:A\to B$ is the
affinization of $A$ [3, Section 3.2] and $f^{\prime}$ is a group homomorphism.
So $B={\mathbb{C}}$ or ${\mathbb{C}}^{*}$, and $A$ is an extension of $B$ with
an elliptic curve $E$,

(13) $1\to E\to A\to B\to 1.$

When $B={\mathbb{C}}$, by Chevalley’s theorem [3, Theorem 2], there exists a
unique exact sequence of algebraic groups

$1\to X\xrightarrow{s}A\to Y\to 1,$

where $X$ is a smooth affine group scheme and $Y$ is proper. So
$X={\mathbb{C}}$ and $Y$ is an elliptic curve. Since any non-trivial additive
group endomorphism of ${\mathbb{C}}$ (as ${\mathbb{C}}$-varieties) is an
isomorphism, the composition

${\mathbb{C}}\xrightarrow{s}A\xrightarrow{f^{\prime}}{\mathbb{C}}$

is an isomorphism. Therefore the exact sequence (13) splits and
$A=E\times{\mathbb{C}}$.

When $B={\mathbb{C}}^{*}$, a similar argument gives a composition
${\mathbb{C}}^{*}\to A\to{\mathbb{C}}^{*}$. Since any non-trivial endomorphism
of ${\mathbb{C}}^{*}$ is given by $z\to z^{n}$ for some nonzero integer $n$.
We may assume $n>0$; precomposing the isomorphism $z\to z^{-1}$ otherwise. The
exact sequence (13) splits after a base change $z\to z^{n}$. Therefore,
$f:A\to{\mathbb{C}}^{*}$ is an equivariant quotient of the projection
$E\times{\mathbb{C}}^{*}\to{\mathbb{C}}^{*}$ by a finite cyclic group
${\mathbb{Z}}/n{\mathbb{Z}}$. ∎

######  Lemma 4.2.

Let $A$ be an abelian surface. Then any surjective morphism
$f:A\to{\mathbb{P}}^{1}$ has disconnected generic fiber.

###### Proof.

By [1, Lemma 1.1], any irreducible component of any fiber is elliptic curve,
and any such two elliptic curves are algebraically equivalent. Since all
fibers are algebraic equivalent, the numbers of irreducible components of
fibers must be constant. If the generic fiber were connected, then
$f:A\to{\mathbb{P}}^{1}$ would be smooth. Then the decomposition theorem

$\begin{split}Rf_{*}{\mathbb{Q}}_{A}=&R^{0}f_{*}{\mathbb{Q}}_{A}\bigoplus
R^{1}f_{*}{\mathbb{Q}}_{A}[-1]\bigoplus R^{2}f_{*}{\mathbb{Q}}_{A}[-2]\\\
=&{\mathbb{Q}}_{{\mathbb{P}}^{1}}\bigoplus{\mathbb{Q}}_{{\mathbb{P}}^{1}}^{\oplus
2}[-1]\bigoplus{\mathbb{Q}}_{{\mathbb{P}}^{1}}[-2]\end{split}$

would imply $H^{*}(A,{\mathbb{Q}})=H^{*}({\mathbb{P}}^{1},{\mathbb{Q}})\oplus
H^{*}({\mathbb{P}}^{1},{\mathbb{Q}})^{\oplus 2}[-1]\oplus
H^{*}({\mathbb{P}}^{1},{\mathbb{Q}})[-2]$, which is a contradiction. ∎

######  Proposition 4.3.

Let $f:A\to C$ be a fibration satisfying $(\dagger)$. Then the perverse
filtration associated with $f$ admits a strongly multiplicative splitting.

###### Proof.

It suffices to prove for the three cases in Proposition 4.1. By Proposition
2.6, it suffices to prove for $A\to E$, $E\times{\mathbb{C}}\to{\mathbb{C}}$
and $(E\times{\mathbb{C}}^{*})/\Gamma\to{\mathbb{C}}^{*}/\Gamma$.

Since $A\to E$ is a smooth morphism, we have

$Rf_{*}{\mathbb{Q}}_{A}={\mathbb{Q}}_{E}\bigoplus{\mathbb{Q}}^{2}_{E}[-1]\bigoplus{\mathbb{Q}}_{E}[-2],$

and hence the perverse numbers are

$\dim$ | $P_{0}$ | $\textup{Gr}_{1}^{P}$ | $\textup{Gr}_{2}^{P}$
---|---|---|---
$H^{0}$ | 1 | 0 | 0
$H^{1}$ | 2 | 2 | 0
$H^{2}$ | 1 | 4 | 1
$H^{3}$ | 0 | 2 | 2
$H^{4}$ | 0 | 0 | 1

Let $\alpha,\beta\in P_{0}H^{1}(A,{\mathbb{Q}})$, $\gamma,\delta\in
H^{1}(A,{\mathbb{Q}})$ form a basis of $H^{1}(A,{\mathbb{Q}})$ adapted to the
perverse filtration. Since the perverse filtration of $f$ is multiplicative
[13, Proposition 4.17], we have

(14)
${\mathfrak{p}}(\alpha\beta)=0,~{}~{}~{}~{}{\mathfrak{p}}(\alpha\gamma),{\mathfrak{p}}(\alpha\delta),{\mathfrak{p}}(\beta\gamma),{\mathfrak{p}}(\beta\delta)\leq
1,~{}~{}~{}~{}{\mathfrak{p}}(\gamma\delta)\leq 2.$

By the ring structure of $H^{*}(A,{\mathbb{Q}})$, they form basis of
$H^{2}(A,{\mathbb{Q}})$. Comparing with the perverse numbers, we see that the
equalities of (14) must hold. They give a splitting of the perverse filtration
$P_{\bullet}H^{2}(A,{\mathbb{Q}})$.

$H^{2}(A,{\mathbb{Q}})=\langle\alpha\beta\rangle\bigoplus\langle\alpha\gamma,\alpha\delta,\beta\gamma,\beta\delta\rangle\bigoplus\langle\gamma\delta\rangle.$

The same argument works for $H^{3}(A,{\mathbb{Q}})$ and it is straightforward
to see that the induced splitting on the exterior algebra generated by
$\alpha,\beta,\gamma,\delta$ is strongly multiplicative.

The perverse filtration of $E\times{\mathbb{C}}\to{\mathbb{C}}$ coincides with
the cohomological degree, which is obviously strongly multiplicative.

Since the $\Gamma$-action on $H^{*}(E\times{\mathbb{C}}^{*},{\mathbb{Q}})$ is
trivial, the perverse filtration associated with
$(E\times{\mathbb{C}}^{*})/\Gamma\to{\mathbb{C}}^{*}/\Gamma$ is identical to
the one of $E\times{\mathbb{C}}^{*}\to{\mathbb{C}}^{*}$, and hence is strongly
multiplicative. ∎

######  Proposition 4.4.

Let $f:A\to B$ be a fibration satisfying $(\dagger)$. Then the pull-back of
the multiplication map

$m:A^{n}=A\times\cdots\times A\to A$

preserves the perverse filtration, where the perverse filtration on the left
side is associated with the natural map $A^{n}\to B^{n}$.

###### Proof.

We prove by induction on the degree of cohomology classes. For a class
$\alpha\in H^{1}(A,{\mathbb{Q}})$, we have

$m^{*}(\alpha)=\alpha\otimes 1\cdots\otimes
1+1\otimes\alpha\otimes\cdots\otimes 1+\cdots+1\otimes\cdots\otimes
1\otimes\alpha.$

So ${\mathfrak{p}}(m^{*}(\alpha))={\mathfrak{p}}(\alpha)$. Suppose the
proposition is proved for degree less than $k$, then for degree $k$,
$H^{k}(A,{\mathbb{Q}})$ is spanned by classes of the form of $\beta\gamma$,
where $\beta,\gamma$ have lower degrees. By the strong multiplicativity, we
have

${\mathfrak{p}}(m^{*}(\beta\gamma))={\mathfrak{p}}(m^{*}(\beta)m^{*}(\gamma))={\mathfrak{p}}(m^{*}(\beta)){\mathfrak{p}}(m^{*}(\gamma))={\mathfrak{p}}(\beta){\mathfrak{p}}(\gamma)={\mathfrak{p}}(\beta\gamma).$

∎

######  Proposition 4.5.

Let $f:A\to C$ be a fibration satisfying $(\dagger)$. Consider the diagram

${A^{a-1}}$${A^{a}}$${C^{a-1}}$${C^{a}.}$$\scriptstyle{f^{a-1}}$$\scriptstyle{\delta}$$\scriptstyle{f^{a}}$

where

$\delta(x_{1},\cdots,x_{a-1})=(x_{1},\cdots,x_{a-1},-m(x_{1},\cdots,x_{a-1})).$

Then the map $\delta_{*}:H^{*}(A^{a-1},{\mathbb{Q}})\to
H^{*}(A^{a},{\mathbb{Q}})$ increases the perversity by at most 2, i.e. for any
class $\alpha\in H^{*}(A^{a-1},{\mathbb{Q}})$,

${\mathfrak{p}}^{f^{a}}(\delta_{*}(\alpha))\leq{\mathfrak{p}}^{f^{a-1}}(\alpha)+2.$

###### Proof.

We apply Proposition 2.4, with $f:A^{a-1}\to B^{a-1}$, $g:A\to B$, and
$h=-m:A^{a-1}\to A$. Condition (1) follows from Proposition 4.4 and condition
(2) follows from Proposition 2.3 and [13, Proposition 4.17]. ∎

######  Remark 4.6.

Under the hypothesis of the strong multiplicativity, it is straightforward to
check that for any pure class $\alpha$ in the perverse decomposition, the
perversity is increased by exactly 2.

###  4.2. Perverse filtration for generalized Kummer varieties

Let $X$ be a connected quasi-projective commutative group scheme. Then the
summation map $m:X^{n}\to X$ descends to the symmetric product $m:X^{(n)}\to
X$ and we define

(15) $X^{((n))}=\ker\left(X^{(n)}\xrightarrow{m}X\right).$

For a given partition $\nu=1^{a_{1}}\cdots n^{a_{n}}$ of $n$, we define
$X^{((\nu))}$ as the base change

${X^{((\nu))}}$${X^{((n))}}$${X^{(\nu)}}$${X^{(n)}.}$$\scriptstyle{\iota^{((\nu))}}$$\scriptstyle{\iota^{(\nu)}}$

Let

(16) $X^{\nu}_{0}=\left\\{(x_{ij})_{1\leq i\leq n,1\leq j\leq a_{i}}\mid
x_{ij}\in X,\,\sum_{i=1}^{n}\sum_{j=1}^{a_{i}}ix_{ij}=0\right\\}\subset
X^{\nu},$

then $X^{\nu}_{0}$ admits a natural $\mathfrak{S}_{\nu}$-action and the
quotient is $X^{((\nu))}$.

When $\dim X=2$, we define the generalized Kummer variety $X^{[[n]]}$ by the
Cartesian diagram

${X^{[[n]]}}$${X^{[n]}}$${X^{((n))}}$${X^{(n)},}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$

Let $f:A\to C$ be a fibration satisfying ($\dagger$). Then by Proposition 4.1,
$f$ factors as

$A\xrightarrow{f^{\prime}}B\xrightarrow{g}C$

where $B$ is a smooth group scheme of dimension 1, $f^{\prime}$ is a
homomorphism of algebraic groups and $g$ is a finite morphism. Then there is a
natural map

$f^{\prime((n))}:A^{((n))}\to B^{((n))}.$

By abuse of notation, denote by $C^{((n))}$ the image of the composition

$A^{((n))}\hookrightarrow A^{(n)}\xrightarrow{f^{(n)}}C^{(n)}.$

and denote

$f^{((n))}:A^{((n))}\xrightarrow{f^{\prime((n))}}B^{((n))}\xrightarrow{g^{((n))}}C^{((n))}.$

######  Proposition 4.7.

Let $f:A\to C$ be a fibration satisfying ($\dagger$). Then
$g^{((n))}:B^{((n))}\to C^{((n))}$ is a finite surjective morphism. In
particular, $\dim C^{((n))}=n-1$ and the perverse filtration associated with
$h^{\prime}:A^{[[n]]}\to C^{((n))}$ is identical to the one associated with
$h^{\prime}_{B}:A^{[[n]]}\to B^{((n))}$.

###### Proof.

Since $B\to C$ is finite, the composition

$B^{((n))}\hookrightarrow B^{(n)}\xrightarrow{g^{(n)}}C^{(n)}$

is finite. By definition, $g^{((n))}$ is the morphism onto its image and hence
is finite surjective. By (15), $\dim B^{((n))}=n-1$. So $\dim C^{((n))}=n-1$.
The identification of perverse filtration follows from Proposition 2.6. ∎

Using the notations introduced above, we have the commutative diagram

${A^{[n]}}$${A^{\nu}}$${A^{(\nu)}}$${A^{(n)}}$${A^{[[n]]}}$${A^{\nu}_{0}}$${A^{((\nu))}}$${A^{((n))}}$${B^{\nu}_{0}}$${B^{((\nu))}}$${B^{((n))}}$$\scriptstyle{\pi}$$\scriptstyle{/\mathfrak{S}_{\nu}}$$\scriptstyle{\iota^{(\nu)}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{h^{\prime}_{B}}$$\scriptstyle{/\mathfrak{S}_{\nu}}$$\scriptstyle{f^{\prime\nu}_{0}}$$\scriptstyle{f^{\prime((\nu))}}$$\scriptstyle{\iota^{((\nu))}_{A}}$$\scriptstyle{f^{\prime((n))}}$$\scriptstyle{/\mathfrak{S}_{\nu}}$$\scriptstyle{\iota^{((\nu))}_{B}}$

By proper base change and Theorem 3.1, we have

(17)
$\begin{split}Rh^{\prime}_{B*}{\mathbb{Q}}_{A^{[[n]]}}[2n]=&Rf^{\prime((n))}_{*}\bigoplus_{\nu}\iota_{A*}^{((\nu))}{\mathbb{Q}}_{A^{((\nu))}}[2l(\nu)]\\\
=&\bigoplus_{\nu}\iota_{B*}^{((\nu))}Rf^{\prime((\nu))}_{*}{\mathbb{Q}}_{A^{((\nu))}}[2l(\nu)]\\\
=&\bigoplus_{\nu}\iota_{B*}^{((\nu))}(Rf^{\prime\nu}_{0*}{\mathbb{Q}}_{A^{\nu}_{0}})^{\mathfrak{S}_{\nu}}[2l(\nu)]\end{split}$

To understand the perverse filtration associated with

$f^{\prime\nu}_{0}:A^{\nu}_{0}\to B^{\nu}_{0},$

together with its $\mathfrak{S}_{\nu}$-action, we construct a non-canonical
isomorphism $A^{\nu}_{0}\cong A[\gcd(\nu)]\times A^{l(\nu)-1}$ to calculate
the Betti numbers of $A^{\nu}_{0}$, and then use them to find the perverse
filtration of $f^{\prime\nu}_{0}\times f^{\prime}$, which is canonical and
compatible with the $\mathfrak{S}_{\nu}$-action and the description of the cup
product in $H^{*}(A^{[[n]]},{\mathbb{Q}})$.

We introduce some notation first. Denote

$\vec{x}=(x_{ij})_{1\leq i\leq n,1\leq j\leq a_{i}}\in A^{\nu},$

and

$\vec{v}=(v_{ij})_{1\leq i\leq n,1\leq j\leq a_{i}}\in{\mathbb{Z}}^{\nu},$

where $v_{ij}=i$. Then by (16),

$A^{\nu}_{0}=\\{\vec{x}\in A^{\nu}\mid\vec{v}\cdot\vec{x}=0.\\}$

Let

$\vec{v}^{\prime}=\frac{1}{\gcd(\nu)}\vec{v},$

then $\vec{v}\cdot\vec{x}=0$ is equivalent to $\vec{v}^{\prime}\cdot\vec{x}\in
A[\gcd(\nu)]$, or

(18) $A_{0}^{\nu}=\bigsqcup_{\sigma\in A[\gcd(\nu)]}\\{\vec{x}\in
A^{\nu}\mid\vec{v}^{\prime}\cdot\vec{x}=\sigma\\}.$

We denote $\tilde{A}^{\nu}_{\sigma}=\\{\vec{x}\in
A^{\nu}\mid\vec{v}^{\prime}\cdot\vec{x}=\sigma\\}$ for short.

######  Proposition 4.8.

There is a non-canonical isomorphism of cohomology groups

$H^{*}(A^{\nu}_{0},{\mathbb{Q}})\cong\bigoplus_{\sigma\in
A[\gcd(\nu)]}H^{*}(A,{\mathbb{Q}})^{\otimes l(\nu)-1}.$

In particular,

(19) $\dim H^{*}(A^{\nu}_{0},{\mathbb{Q}})=|A[\gcd(\nu)]|\cdot(\dim
H^{*}(A,{\mathbb{Q}}))^{l(\nu)-1}.$

###### Proof.

Since $\vec{v}^{\prime}$ is primitive in the lattice ${\mathbb{Z}}^{\nu}$, we
may extend $\vec{v}^{\prime}$ to be a basis of ${\mathbb{Z}}^{\nu}$.
Equivalently, there is an invertible $l(\nu)\times l(\nu)$ matrix $M$ with
integer entries whose first row is $\vec{v}^{\prime}$. We define $M:A^{\nu}\to
A^{\nu}$ formally by the rule of linear transformations. Then
$\vec{v}^{\prime}\cdot\vec{x}=\sigma$ for a $\gcd(\nu)$-torsion point $\sigma$
if and only if the first entry of $M(x)$ is $\sigma$. Therefore, $M$ maps
$A^{\nu}_{0}$ isomorphically to $A[\gcd(\nu)]\times A^{l(\nu)-1}$, _i.e._

${A^{\nu}_{0}}$${A[\gcd(\nu)]\times
A^{l(\nu)-1}}$${A^{\nu}}$${A^{\nu}}$$\scriptstyle{M}$$\scriptstyle{M}$

By Künneth formula,

$H^{*}(A^{\nu}_{0},{\mathbb{Q}})\cong\bigoplus_{\sigma\in
A[\gcd(\nu)]}H^{*}(A,{\mathbb{Q}})^{\otimes l(\nu)-1}.$

∎

Since the isomorphism is is non-canonical, and it is difficult to understand
the $\mathfrak{S}_{\nu}$-action on the right side, we use the identity of
dimensions (19) to construct a canonical decomposition.

######  Proposition 4.9.

Let $f^{\prime}:A\to B$ be the morphism obtained in Proposition 4.1. Then
there is a canonical $\mathfrak{S}_{\nu}$-equivariant isomorphism

$H^{*}(A_{0}^{\nu}\times A,{\mathbb{Q}})=\bigoplus_{\sigma\in
A[\gcd(\nu)]}H^{*}(A^{\nu},{\mathbb{Q}})$

where $\mathfrak{S}_{\nu}$ acts on the factor
$H^{*}(A_{0}^{\nu},{\mathbb{Q}})$ on the left side. Under this isomorphism, we
have

$P_{k}^{f^{\prime\nu}_{0}}H^{*}(A_{0}^{\nu}\times
A,{\mathbb{Q}})=\bigoplus_{\sigma\in
A[\gcd(\nu)]}P^{f^{\prime\nu}}_{k}H^{*}(A^{\nu},{\mathbb{Q}}).$

In particular, we have

$P_{k}^{f^{\prime(\nu)}_{0}}H^{*}(A_{0}^{(\nu)}\times
A,{\mathbb{Q}})=\bigoplus_{\sigma\in
A[\gcd(\nu)]}P^{f^{\prime(\nu)}}_{k}H^{*}(A^{(\nu)},{\mathbb{Q}}).$

###### Proof.

Consider the diagram

${A^{\nu}_{0}\times A}$${A^{\nu}}$${B^{\nu}_{0}\times
B}$${B^{\nu},}$$\scriptstyle{\Sigma}$$\scriptstyle{f^{\prime\nu}_{0}\times
f}$$\scriptstyle{f^{\prime\nu}}$$\scriptstyle{\Sigma}$

where $\Sigma((\vec{x}),a)=(x_{ij}+a)_{1\leq i\leq n,1\leq j\leq a_{i}}$. Then
$\Sigma$ is an étale covering and is $\mathfrak{S}_{\nu}$-equivariant. By
(18), the map $\Sigma$ is a disjoint union of étale coverings
$\Sigma_{\sigma}:\tilde{A}_{\sigma}^{\nu}\times A\to A^{\nu}$ for all
$\sigma\in A[\gcd(\nu)]$. We have

(20) $\Sigma_{*}{\mathbb{Q}}_{A^{\nu}_{0}\times A}=\bigoplus_{\sigma\in
A[\gcd(\nu)]}\Sigma_{\sigma*}{\mathbb{Q}}_{\tilde{A}_{\sigma}^{\nu}\times
A}=\bigoplus_{\sigma\in A[\gcd(\nu)]}({\mathbb{Q}}_{A^{\nu}}\oplus
F_{\sigma}),$

for some sheaves $F_{\sigma}$. By taking the hyper-cohomology, we have

(21) $H^{*}(A_{0}^{\nu}\times A,{\mathbb{Q}})=\bigoplus_{\sigma\in
A[\gcd(\nu)]}H^{*}(A^{\nu},{\mathbb{Q}})\oplus{\mathbb{H}}(F_{\sigma}).$

The summands $H^{*}(A^{\nu},{\mathbb{Q}})$ is canonical since
${\mathbb{Q}}_{A^{\nu}}$ is canonical in the decomposition (20). Proposition
4.8 and Künneth formula yields an identification of Betti numbers

(22) $\dim H^{*}(A^{\nu}_{0}\times A,{\mathbb{Q}})=|A[\gcd(\nu)]|\cdot\dim
H^{*}(A,{\mathbb{Q}})^{\otimes l(\nu)}.$

Comparing 21 and (22), we see that ${\mathbb{H}}(F_{\sigma})=0$ for all
$\sigma$. Now we use the definition (4) to calculate the perverse filtration
associated with $f^{\prime\nu}_{0}$. Since $\Sigma$ is finite, and hence is
$t$-exact, applying ${}^{{\mathfrak{p}}}\tau_{\leq k}Rf^{\prime\nu}_{*}$ to
(20) yields

$\Sigma_{*}{{}^{\mathfrak{{\mathfrak{p}}}}}\tau_{\leq
k}R(f^{\prime\nu}_{0}\times f)_{*}{\mathbb{Q}}_{A^{\nu}_{0}\times
A}=\bigoplus_{\sigma\in
A[\gcd(\nu)]}\left({{}^{\mathfrak{{\mathfrak{p}}}}}\tau_{\leq
k}Rf^{\prime\nu}_{*}{\mathbb{Q}}_{A^{\nu}}\oplus{{}^{\mathfrak{{\mathfrak{p}}}}}\tau_{\leq
k}Rf^{\prime\nu}_{*}F_{\sigma}\right).$

The hypercohomology yields

${{\mathbb{H}}({{}^{\mathfrak{{\mathfrak{p}}}}}\tau_{\leq
k}R(f^{\prime\nu}_{0}\times f)_{*}{\mathbb{Q}}_{A^{\nu}_{0}\times
A})}$${\displaystyle\bigoplus_{\sigma\in
A[\gcd(\nu)]}{\mathbb{H}}\left({{}^{\mathfrak{{\mathfrak{p}}}}}\tau_{\leq
k}Rf^{\prime\nu}_{*}{\mathbb{Q}}_{A^{\nu}}\oplus{{}^{\mathfrak{{\mathfrak{p}}}}}\tau_{\leq
k}Rf^{\prime\nu}_{*}F_{\sigma}\right)}$${{\mathbb{H}}(R(f^{\prime\nu}_{0}\times
f)_{*}{\mathbb{Q}}_{A^{\nu}_{0}\times A})}$${\displaystyle\bigoplus_{\sigma\in
A[\gcd(\nu)]}{\mathbb{H}}\left(Rf^{\prime\nu}_{*}{\mathbb{Q}}_{A^{\nu}}\oplus
Rf^{\prime\nu}_{*}F_{\sigma}\right)}$${H^{*}(A^{\nu}_{0}\times
A,{\mathbb{Q}})}$${\displaystyle\bigoplus_{\sigma\in
A[\gcd(\nu)]}H^{*}(A^{\nu},{\mathbb{Q}}).}$

By the definition of perverse filtrations (4), we have an isomorphism

(23) $P_{k}^{f^{\prime\nu}_{0}\times f^{\prime}}H^{*}(A_{0}^{\nu}\times
A,{\mathbb{Q}})=\bigoplus_{\sigma\in
A[\gcd(\nu)]}P_{k}^{f^{\prime\nu}}H^{*}(A^{\nu},{\mathbb{Q}}).$

The isormophism (23) is $\mathfrak{S}_{\nu}$-equivariant since
$\Sigma_{\sigma}$ is for all $\sigma\in A[\gcd(\nu)]$, and is canonical since
the summmands

$H^{*}(A^{\nu},{\mathbb{Q}})\to H^{*}(A_{0}^{\nu}\times A,{\mathbb{Q}})$

are canonical for all $\sigma$ by the decomposition theorem. ∎

######  Theorem 4.10.

Let $f:A\to C$ be a fibration satisfying $(\dagger)$. Let
$h^{\prime}:A^{[[n]]}\to C^{((n))}$ be the induced morphism. Then there is a
canonical decomposition

$H^{*}(A^{[[n]]}\times A,{\mathbb{Q}})=\bigoplus_{\nu}\bigoplus_{\sigma\in
A[\gcd(\nu)]}H^{*}(A^{(\nu)},{\mathbb{Q}})[2l(\nu)-2n].$

Under this identification, we have

$P_{k}^{h^{\prime}\times f}H^{*}(A^{[[n]]}\times
A,{\mathbb{Q}})=\bigoplus_{\nu}\bigoplus_{\sigma\in
A[\gcd(\nu)]}P_{k+l(\nu)-n}^{f^{(\nu)}}H^{*}(A^{(\nu)},{\mathbb{Q}})[2l(\nu)-2n].$

###### Proof.

By (17) and Künneth formula,

(24) $\begin{split}R(h^{\prime}_{B}\times
f^{\prime})&{}_{*}{\mathbb{Q}}_{A^{[[n]]}\times A}[2n]\\\
=&\bigoplus_{\nu}\left(\iota_{B}^{((\nu))}\times\textup{id}\right)_{*}\left(R(f^{\prime\nu}_{0}\times
f^{\prime})_{*}{\mathbb{Q}}_{A^{\nu}_{0}\times
A}\right)^{\mathfrak{S}_{\nu}}[2l(\nu)]\end{split}$

By Proposition 4.9, we have

$\begin{split}H^{*}({A^{[[n]]}\times
A},{\mathbb{Q}})[2n]=&\bigoplus_{\nu}H^{*}(A^{\nu}_{0}\times
A,{\mathbb{Q}})^{\mathfrak{S}_{\nu}}[2l(\nu)].\\\
=&\bigoplus_{\nu,\sigma}H^{*}(A^{\nu},{\mathbb{Q}})^{\mathfrak{S}_{\nu}}[2l(\nu)]\\\
=&\bigoplus_{\nu,\sigma}H^{*}(A^{(\nu)},{\mathbb{Q}})[2l(\nu)].\end{split}$

To calculate the perverse filtration associated with $h^{\prime}_{B}\times
f^{\prime}$, we apply the morphism of functors
${\mathbb{H}}\circ^{\mathfrak{p}}\tau_{\leq k}\to{\mathbb{H}}$ to (24). Since
$h^{\prime}_{B}\times f^{\prime}$ is a flat morphism of relative dimension
$n$, the left side calculates

$P_{k+n}^{h^{\prime}_{B}\times f^{\prime}}H^{*}(A^{[[n]]},{\mathbb{Q}})[2n].$

Since $f^{\prime\nu}_{0}\times f^{\prime}$ is a flat morphism of relative
dimension $l(\nu)$, the right side calculates

$\begin{split}&\bigoplus_{\nu,\sigma}P_{k+l(\nu)}^{f^{\prime\nu}_{0}\times
f^{\prime}}H^{*}(A^{\nu}_{0}\times
A,{\mathbb{Q}})^{\mathfrak{S}_{\nu}}[2l(\nu)]\\\
=&\bigoplus_{\nu,\sigma}P_{k+l(\nu)}^{f^{\prime(\nu)}_{0}\times
f^{\prime}}H^{*}(A^{(\nu)}_{0}\times A,{\mathbb{Q}})[2l(\nu)]\\\
=&\bigoplus_{\nu,\sigma}P_{k+l(\nu)}^{f^{\prime(\nu)}}H^{*}(A^{(\nu)},{\mathbb{Q}})[2l(\nu)],\textup{
(by Proposition \ref{aaa}).}\end{split}$

Therefore, we have the identification of perverse filtrations

$P_{k}^{h^{\prime}_{B}\times
f^{\prime}}H^{*}(A^{[[n]]},{\mathbb{Q}})=\bigoplus_{\nu,\sigma}P_{k+l(\nu)-n}^{f^{\prime(\nu)}}H^{*}(A^{(\nu)},{\mathbb{Q}})[2l(\nu)-2n].$

By Proposition 2.3 and Proposition 4.7, the perverse filtration associated
with $h^{\prime}_{B}\times f^{\prime}$ is identical to the one associated with
$h^{\prime}\times f$. By Proposition 2.6, the finiteness of
$g^{(\nu)}:B^{(\nu)}\to C^{(n)}$ implies that the perverse filtration
associated with $f^{\prime(\nu)}$ is identical with $f^{(\nu)}$. We have

$P_{k}^{h^{\prime}\times
f}H^{*}(A^{[[n]]},{\mathbb{Q}})=\bigoplus_{\nu,\sigma}P_{k+l(\nu)-n}^{f^{(\nu)}}H^{*}(A^{(\nu)},{\mathbb{Q}})[2l(\nu)-2n].$

∎

To state the cup product of generalized Kummer varieties, we introduce the
following notations.

######  Definition 4.11.

Let $\nu$ be a partition of $n$, and $\alpha_{\nu}\in
H^{*}(A^{(\nu)},{\mathbb{Q}})$ be a cohomology class. Let $\sigma\in A[n]$ be
an $n$-torsion of $A$. Denote by $\alpha_{\nu,\sigma}$ the image of
$\alpha_{\nu}$ in $H^{*}(A^{[[n]]}\times A,{\mathbb{Q}})$ via the
decomposition

$H^{*}(A^{[[n]]}\times
A,{\mathbb{Q}})=\bigoplus_{\nu,\sigma}H^{*}(A^{(\nu)},{\mathbb{Q}})[2l(\nu)-2n]$

if $\sigma\in A[\gcd(\nu)]$ and 0 otherwise.

When $\nu$ runs through all partitions of $n$, $\alpha_{\nu}$ runs through a
basis of $H^{*}(A^{(\nu)},{\mathbb{Q}})$ for each $\nu$, and $\sigma$ runs
through $A[\gcd(\nu)]$, the classes $\alpha_{\nu,\sigma}$ form a basis of the
cohomology group $H^{*}(A^{[[n]]}\times A,{\mathbb{Q}})$. It suffices to
describe the cup product on this basis.

######  Theorem 4.12.

[4, Theorem 32] Let $A$ be a smooth connected quasi-projective333The theorem
is originally stated for abelian surfaces, but the proof only uses the cup
product of relative cohomology with coefficient in locally constant sheaves on
each locally closed strata indexed by partitions of $n$, which does not depend
on the compactness. commutative group scheme of dimension 2. Let
$\alpha_{\nu}\in H^{*}(A^{(\nu)},{\mathbb{Q}})$ and $\beta_{\mu}\in
H^{*}(A^{(\mu)},{\mathbb{Q}})$ be two cohomology classes. Following the
notation in Definition 3.3, let

(25)
$\alpha_{\nu}^{[n]}\cdot\beta_{\mu}^{[n]}=\sum_{\lambda}\gamma_{\lambda}^{[n]}$

be the cup product in $H^{*}(A^{[n]},{\mathbb{Q}})$. Then the product in
$H^{*}(A^{[[n]]}\times A,{\mathbb{Q}})$ is calculated as

$\alpha_{\nu,\sigma}\cdot\beta_{\mu,\tau}=\sum_{\lambda}\gamma_{\lambda,\sigma\tau}.$

######  Theorem 4.13.

Let $f:A\to C$ be a proper surjective morphism from a connected quasi-
projective commutative group scheme $A$ of dimension 2 to a quasi-projective
curve $C$. Then the perverse filtration associated with the induced morphism
$h^{\prime}:A^{[[n]]}\to C^{((n))}$ is multiplicative.

###### Proof.

Let $\alpha\in H^{*}(A^{(\nu)},{\mathbb{Q}})$. Then by Proposition 3.2 and
Theorem 4.10, we have

(26) ${\mathfrak{p}}^{h^{\prime}\times
f}(\alpha_{\nu,\sigma})={\mathfrak{p}}^{f^{(\nu)}}(\alpha_{\nu})+n-l(\nu)={\mathfrak{p}}^{h}(\alpha_{\nu}^{[n]})$

for any $\sigma\in A[\gcd(\nu)]$. By Proposition 3.5, the perverse filtration
associated with $\pi$ is multiplicative

${\mathfrak{p}}^{h}(\alpha_{\nu}^{[n]}\cdot\beta_{\mu}^{[n]})\leq{\mathfrak{p}}^{h}(\alpha_{\nu}^{[n]})+{\mathfrak{p}}^{h}(\beta_{\mu}^{[n]}).$

By (25) and Corollary 3.4,

${\mathfrak{p}}^{h}(\gamma_{\lambda}^{[n]})\leq{\mathfrak{p}}^{h}(\alpha_{\nu}^{[n]})+{\mathfrak{p}}^{h}(\beta_{\mu}^{[n]})$

holds for each class $\gamma_{\lambda}^{[n]}$ occurring in (25). By (26),

${\mathfrak{p}}^{h^{\prime}\times
f}(\gamma_{\lambda,\sigma\tau})\leq{\mathfrak{p}}^{h^{\prime}\times
f}(\alpha_{\nu,\sigma})+{\mathfrak{p}}^{h^{\prime}\times f}(\beta_{\mu,\tau})$

holds for each non-zero $\gamma_{\lambda,\sigma\tau}$. So

(27) ${\mathfrak{p}}^{h^{\prime}\times
f}(\alpha_{\nu,\sigma}\cdot\beta_{\mu,\tau})\leq{\mathfrak{p}}^{h^{\prime}\times
f}(\alpha_{\nu,\sigma})+{\mathfrak{p}}^{h^{\prime}\times
f}(\beta_{\mu,\tau}).$

Therefore, the perverse filtration associated with $h^{\prime}\times f$ is
multiplicative. $f:A\to C$ is flat implies ${\mathfrak{p}}^{f}(1)=0$. Then by
Proposition 2.3 and (27),

$\begin{split}{\mathfrak{p}}^{h^{\prime}}(\alpha\beta)={\mathfrak{p}}^{h^{\prime}\times
f}(\alpha\beta\boxtimes 1)={\mathfrak{p}}^{h^{\prime}\times f}(\alpha\boxtimes
1\cdot\beta\boxtimes 1)\\\ \leq{\mathfrak{p}}^{h^{\prime}\times
f}(\alpha\boxtimes 1)+{\mathfrak{p}}^{h^{\prime}\times f}(\beta\boxtimes
1)={\mathfrak{p}}^{h^{\prime}}(\alpha)+{\mathfrak{p}}^{h^{\prime}}(\beta).\end{split}$

∎

######  Corollary 4.14.

Let $f:A\to C$ be a fibration satisfying the condition of Theorem 4.13. Then
the perverse filtration associated with the induced morphism
$h^{\prime}:A^{[[n]]}\to C^{((n))}$ admits a natural strongly multiplicative
splitting in the sense of (5).

###### Proof.

By Proposition 4.3, the perverse filtration associated with $f:A\to C$ admits
a strongly multiplicative splitting. Such a splitting induces a splitting of
the perverse filtration associated with $f^{(\nu)}:A^{(\nu)}\to C^{(\nu)}$ for
any partition $\nu$, and a splitting of the perverse filtration associated
with $h:A^{[n]}\to C^{(n)}$; see [10, Section 2.2]. Now Proposition 4.9 and
Theorem 4.10 produce splittings of the perverse filtrations associated with
$f^{\prime(\nu)}\times f^{\prime}:A^{(\nu)}_{0}\times A\to B^{(\nu)}_{0}\times
B$ and $h^{\prime}\times f:A^{[[n]]}\times A\to C^{((n))}\times C$,
respectively. It follows directly from the construction that if
$\alpha_{\nu}\in H^{*}(A^{(n)},{\mathbb{Q}})$ is a pure class with respect to
the splitting, then $\alpha_{\nu}^{[n]}$ and $\alpha_{\nu,\sigma}$ are both
pure in the corresponding splittings, and the equation (26) still holds. By
[10, Proposition 2.8], the perverse filtration associated with $h$ is strongly
multiplicative, the inequalities in the proof of Theorem 4.13 are equations
for pure classes $\alpha_{\nu}^{[n]}$, $\beta_{\nu}^{[n]}$,
$\alpha_{\nu,\sigma}$, and $\beta_{\nu,\tau}$. Therefore, the splitting we
constructed is strongly multiplicative. ∎

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