# Perverse filtrations and Fourier transforms

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

We study the interaction between Fourier–Mukai transforms and perverse
filtrations for a certain class of dualizable abelian fibrations.
Multiplicativity of the perverse filtration and the “Perverse $\supset$ Chern”
phenomenon for these abelian fibrations are immediate consequences of our
theory. We also show that our class of fibrations include families of
compactified Jacobians of integral locally planar curves.

Applications include the following: (a) we prove the motivic decomposition
conjecture for this class (including compactified Jacobian fibrations), which
generalizes Deninger–Murre’s theorem for abelian schemes; (b) we provide a new
proof of the $P=W$ conjecture for $\mathrm{GL}_{r}$; (c) we prove half of the
$P=C$ conjecture concerning refined BPS invariants for the local
${\mathbb{P}}^{2}$; (d) we show that the perverse filtration for the
compactified Jacobian associated with an integral locally planar curve is
multiplicative, which generalizes a result of Oblomkov–Yun for homogeneous
singularities.

Our techniques combine Arinkin’s autoduality for coherent categories, Ngô’s
support theorem for the decomposition theorem, Adams operations in operational
$K$-theory, and Corti–Hanamura’s theory of relative Chow motives.

###### Contents

  1. 0 Introduction
  2. 1 Dualizable abelian fibrations
  3. 2 Multiplicativity and perversity
  4. 3 Compactified Jacobians
  5. 4 Twisted compactified Jacobians
  6. 5 Moduli of one-dimensional sheaves on surfaces

##  0\. Introduction

###  0.1. Overview

Throughout, we work over the complex numbers ${\mathbb{C}}$. For a proper
morphism $f:X\to Y$ between nonsingular varieties, the perverse truncation
functor [8] filters the derived direct image $Rf_{*}{\mathbb{Q}}_{X}\in
D^{b}_{c}(Y)$, which yields an increasing filtration on the cohomology either
for the total space $X$ or for a closed fiber. Such a filtration is called the
perverse filtration, which encodes important topological invariants of the map
$f:X\to Y$.

In recent years, perverse filtrations have played a crucial role in the study
of integrable systems, enumerative geometry, and geometric representation
theory. For many interesting abelian fibrations (_e.g._ Hitchin systems), the
associated perverse filtration is discovered to satisfy the following two
mysterious properties:

  1. (i)

it is _multiplicative_ with respect to the cup-product, and

  2. (ii)

the location of a tautological class in the filtration is determined by its
_Chern grading_ of the universal family.

Indeed, the $P=W$ conjecture of de Cataldo–Hausel–Migliorini for
$\mathrm{GL}_{r}$ is equivalent to (i, ii) for the $\mathrm{GL}_{r}$-Hitchin
system [11, 12], and was proven recently using special properties of Hitchin
moduli spaces [36, 24]. Similar phenomena have also been found or conjectured
for Lagrangian fibrations associated with compact hyper-Kähler manifolds [49,
12], compactified Jacobians associated with torus knots [45, 46], and moduli
of $1$-dimensional stable sheaves on ${\mathbb{P}}^{2}$ [30]. Note that, for
general proper morphisms, the perverse filtration will typically not be
multiplicative even if the singularities of the fibers are mild [10, Exercise
5.6.8].

The purpose of this paper is to initiate a framework to study systematically
the perverse filtrations associated with abelian fibrations, which gives a
uniform explanation for (i, ii) that applies in different geometric settings.
We show that both (i) and (ii) are essentially consequences of the _duality_
between derived categories of coherent sheaves and the _support theorem_ in
the decomposition theorem package. Our work also shows the motivic nature of
the decomposition theorem associated with abelian fibrations.

###  0.2. Dualizable abelian fibrations

In Section 1.4, we introduce a class of proper morphisms, called _dualizable
abelian fibrations_. They are modelled on Hitchin systems and compactified
Jacobian fibrations.

Roughly, for nonsingular varieties $M$ and $B$, we say that a proper morphism
$\pi:M\to B$ is a dualizable abelian fibration if it is an abelian fibration
with a dual fibration $\pi^{\vee}:M^{\vee}\to B$, satisfying the following
properties:

  1. (a)

(Duality) the bounded derived categories of coherent sheaves on $M$ and
$M^{\vee}$ are related by Fourier–Mukai transforms with certain nice
properties similar to those of dual abelian schemes;

  2. (b)

(Support) every simple perverse summand in the decomposition theorem for
$R\pi_{*}{\mathbb{Q}}_{M}$ has full support $B$.

We refer to Section 1.4 for more precise statements concerning (a, b). The
Chern character of the Fourier–Mukai kernel of (a) induces a Fourier transform
in cohomology:

$\mathfrak{F}=\sum_{k}\mathfrak{F}_{k}:H^{*}(M^{\vee},{\mathbb{Q}})\to
H^{*}(M,{\mathbb{Q}}),\quad\mathfrak{F}_{k}(H^{d}(M^{\vee},{\mathbb{Q}}))\subset
H^{d+2k-2g}(M,{\mathbb{Q}}),$

where $g$ is the relative dimension $\dim M-\dim B$. The Fourier inverse
$\mathfrak{F}^{-1}=\sum_{k}{\mathfrak{F}}^{-1}_{k}$ behaves similarly. In
order for the Fourier operators to have interesting homological consequences,
we further impose a vanishing condition on the relative product
$M^{\vee}\times_{B}M^{\vee}$. We say that a dualizable abelian fibration as
above satisfies _the Fourier vanishing (FV)_ , if

(FV) $\quad\quad{\mathfrak{F}}^{-1}_{i}\circ{\mathfrak{F}}_{j}=0,\quad
i+j<2g.$

######  Theorem 0.1.

Let $\pi:M\to B$ be a dualizable abelian fibration satisfying (FV). Then

  1. (i)

(Multiplicativity) the perverse filtration associated with $\pi$ is
multiplicative, _i.e._ ,

$P_{k}H^{d}(M,{\mathbb{Q}})\times
P_{k^{\prime}}H^{d^{\prime}}(M,{\mathbb{Q}})\xrightarrow{~{}~{}\cup~{}~{}}P_{k+k^{\prime}}H^{d+d^{\prime}}(M,{\mathbb{Q}});$

  2. (ii)

(Perverse $\supset$ Chern111The term “Perverse $\supset$ Chern” here means
that the perversity of a class $\mathfrak{F}_{k}(\alpha)$ is bounded by the
Chern grading $k$ associated with the Fourier transform.) for any class
$\alpha\in H^{*}(M^{\vee},{\mathbb{Q}})$, we have

$\mathfrak{F}_{k}(\alpha)\in P_{k}H^{*}(M,{\mathbb{Q}}).$

In fact, we prove a stronger version of Theorem 0.1 which is of a motivic
nature and specializes to sheaf-theoretic statements; see Theorem 2.4.

Before discussing applications, we make some remarks on the condition (FV).
For an abelian scheme $\pi:A\to B$ with its dual $\pi^{\vee}:A^{\vee}\to B$,
the Fourier transform ${\mathfrak{F}}$ was first considered by Beauville [7].
It induces a canonical (motivic) decomposition, called the _Beauville
decomposition_ , of the cohomology or the Chow ring of the total space $A$;
see Section 1.2 for more discussions on this construction. Once we obtain the
Beauville decomposition, the statements (i, ii) above for the abelian scheme
$\pi:A\to B$ are immediate consequences. A key step in Beauville’s theory is
the observation that the operators ${\mathfrak{F}}_{k}$ provide _projectors_ ,
which follows from the vanishing

${\mathfrak{F}}^{-1}_{i}\circ{\mathfrak{F}}_{j}=0,\quad i+j\neq 2g.$

Clearly, this vanishing is stronger than (FV) in the case of abelian schemes.
The reason why we do not pose this stronger condition is that it is hard to
verify when the abelian fibration has singular fibers, and furthermore, it is
unclear if the vanishing for $i+j>2g$ holds in general.

On the other hand, the weaker condition (FV) is already sufficient to deduce
the desired properties (i, ii) of Theorem 0.1. Moreover, the condition (FV) is
natural: in view of Arinkin’s work [5, 6], we will explain in Section 3 that
(FV) holds essentially for the _same reason_ as the fact that the Poincaré
line bundle and the Fourier–Mukai transforms can be extended over the singular
fibers for certain abelian fibrations. The following theorem is our main
source of applications.

######  Theorem 0.2.

Let $C\to B$ be a flat family of integral projective curves which admits a
section. If the associated compactified Jacobian fibration
$\pi:\overline{J}_{C}\to B$ satisfies that

  1. (i)

every curve in the family has at worst planar singularities, and

  2. (ii)

the total space $\overline{J}_{C}$ is nonsingular,

then $\pi:\overline{J}_{C}\to B$ is a (self-)dualizable abelian fibration
which satisfies (FV).

By promoting the Fourier theory from schemes to gerbes, we also prove a
version of Theorems 0.1 and 0.2 (as well as Theorem 0.3 below) for certain
twisted compactified Jacobian fibrations which are associated with families of
curves admitting no section; see Corollary 4.5.

###  0.3. Applications

We discuss in this section some applications of Theorem 0.1, which explain and
unify some common features discovered in non-abelian Hodge theory, enumerative
geometry, and representation theory.

####  0.3.1. The motivic decomposition conjecture

We first note that as a by-product of our proof of Theorem 0.1, we verify in
Corollary 2.5 the motivic decomposition conjecture of Corti–Hanamura [15] for
a dualizable abelian fibration $\pi:M\to B$ satisfying (FV). This extends the
motivic decomposition of abelian schemes by Deninger–Murre [16]. Combined with
Theorem 0.2, we obtain the following.

######  Theorem 0.3.

Let $\pi:\overline{J}_{C}\to B$ be a compactified Jacobian fibration as in
Theorem 0.2. Then the decomposition of
$R\pi_{*}{\mathbb{Q}}_{\overline{J}_{C}}$ into (shifted) semisimple perverse
sheaves admits a motivic lifting.

We refer to Corollary 2.5 for the precise statement. This verifies the motivic
decomposition conjecture [15] for $\pi:\overline{J}_{C}\to B$, where the
projectors are provided by Arinkin’s Fourier–Mukai theory. See _e.g._ [14, 3]
for other recent developments of the motivic decomposition conjecture, and
[13] for an unconditional construction of the (homological) motivic
decomposition.

####  0.3.2. The $P=W$ conjecture

One of our motivations and applications comes from the $P=W$ conjecture in
non-abelian Hodge theory, which we briefly review.

For a projective curve $\Sigma$ of genus $g\geq 2$ and two coprime integers
$r,n$, the non-abelian Hodge correspondence yields a diffeomorphism between
the Dolbeault moduli space $M_{\mathrm{Dol}}$ which parameterizes rank $r$ and
degree $n$ stable Higgs bundles on $\Sigma$, and the Betti moduli space
$M_{B}$ which is the corresponding character variety. This further induces

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

In 2010, de Cataldo, Hausel, and Migliorini [11] proposed a relationship
between the topology of the Hitchin system $h:M_{\mathrm{Dol}}\to B$ and the
Hodge theory of $M_{B}$ via (1); more precisely, they conjectured that the
perverse filtration associated with the Hitchin system is matched with the
double indexed weight filtration for the character variety:

(2) $\text{``}P=W\text{''},\quad
P_{k}H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}})=W_{2k}H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}}).$

This conjecture, known as the $P=W$ conjecture, has now been proven by the
first two authors [36] and Hausel–Mellit–Minets–Schiffmann [24] independently
via different methods.

######  Theorem 0.4 ([36, 24]).

The $P=W$ conjecture (2) holds.

As we sketch now, our work in this paper provides a new proof of Theorem 0.4.

The $P=W$ conjecture can be decomposed into three identities:

(3)
$P_{k}H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}})=C_{k}H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}})=C_{k}H^{*}(M_{B},{\mathbb{Q}})=W_{2k}H^{*}(M_{B},{\mathbb{Q}});$

Here $C_{\bullet}$ stands for the Chern filtration defined via the
tautological classes of the moduli space.

The second and third identities have been established earlier by work of
Markman [33] Hausel–Thaddeus [25] and Shende [50] respectively. As a result,
the $P=W$ conjecture is reduced to the first identity — the “Perverse = Chern”
phenomenon for the Hitchin system. Indeed, both existing proofs [36, 24]
proceed by establishing the first identity using special features of the
moduli space of Higgs bundles.

Moreover, in [39] Mellit proved the curious Hard Lefschetz for the character
variety $M_{B}$, which further reduces the full $P=W$ (2) to

(4) $P_{k}H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}})\supset
C_{k}H^{*}(M_{\mathrm{Dol}},{\mathbb{Q}}).$

The present work grew out of an attempt to understand the geometric nature of
(4). The surprising interaction between the perverse filtration and the Chern
classes predicts the existence of a good theory of Fourier transforms for
certain abelian fibrations with singular fibers. The discovery of an analogous
conjecture for the ${\mathbb{P}}^{2}$ geometry (see Section 0.3.3) further
suggests that (4) should be a phenomenon beyond Hitchin systems. This leads us
to Theorem 0.1.

Conversely, we will show that (4) is a consequence of our Fourier theory; in
combination with an argument from [24] to reduce to integral spectral curves,
as we explain in Section 5.4, this yields Theorem 0.4.

######  Remark 0.5.

The above proof of $P=W$ suggests that, over the locus of integral spectral
curves, (4) is a property of compactified Jacobian fibrations associated with
curves with planar singularities and does not rely on the representation
theory perspectives of [36, 24]. This approach gives a possible avenue for
studying $P=W$ phenomena beyond the Higgs setting. More precisely, for any
dualizable abelian fibration as in Theorem 0.1 we may define the Chern
filtration $C_{k}H^{*}(M,{\mathbb{Q}})$ as the span of the classes

${\mathfrak{F}}_{j}(\alpha)\in H^{*}(M,{\mathbb{Q}}),\quad j\leq
k,\quad\alpha\in H^{*}(M^{\vee},{\mathbb{Q}}).$

This generalizes the Chern filtration defined in the $\mathrm{GL}_{r}$ case,
and does not rely on Markman’s generation result [33]. Therefore, in view of
(3), the $P=W$ phenomenon is reduced to understanding the interaction between
the Fourier transform and the weight filtration on the Betti side.
Speculatively, for other Hitchin moduli spaces, this interaction may be
related to the Betti geometric Langlands correspondence.

####  0.3.3. Enumerative geometry of local ${\mathbb{P}}^{2}$

A phenomenon similar to (4) was discovered in [30] for the refined BPS
invariants of local ${\mathbb{P}}^{2}$. It has a different origin from $P=W$,
since the non-abelian Hodge correspondence is not relevant here.

Fix coprime integers $r,\chi$ with $r\geq 3$. We consider the moduli
$M_{r,\chi}$ of $1$-dimensional stable sheaves $F$ on ${\mathbb{P}}^{2}$ with
$[\mathrm{supp}(F)]=rH,\,\chi(F)=\chi$. Here $H$ is the hyperplane class. The
moduli space $M_{r,\chi}$ admits a proper Hilbert–Chow morphism

$h:M_{r,\chi}\to{\mathbb{P}}H^{0}({\mathbb{P}}^{2},{\mathcal{O}}_{{\mathbb{P}}^{2}}(r)),\quad
F\mapsto\mathrm{supp}(F)$

where $\mathrm{supp}(-)$ stands for the Fitting support. This proper map
induces a perverse filtration on the cohomology of $M_{r,\chi}$. Analogously
to the tautological classes [25] for the moduli of Higgs bundles, Pi and the
second author [47] introduced the tautological class

$c_{k}(j)\in H^{2(k+j-1)}(M_{r,\chi},{\mathbb{Q}})$

given by integrating the normalized $\mathrm{ch}_{k+1}$ of a universal family
over $H^{j}\in H^{2j}({\mathbb{P}}^{2},{\mathbb{Q}})$. Moreover, it was shown
in [47] that the first $3r-7$ tautological classes

$c_{k}(j)\in H^{\leq 2(r-2)}(M_{r,\chi},{\mathbb{Q}})$

generate the total cohomology as a ${\mathbb{Q}}$-algebra, and there is no
relation in degrees $\leq 2(r-2)$. This allows us to consider the Chern
filtration $C_{k}H^{\leq 2(r-2)}(M_{r,\chi},{\mathbb{Q}})$ spanned by

$\prod_{i=1}^{s}c_{k_{i}}(j_{i})\in H^{\leq
2(r-2)}(M_{r,\chi},{\mathbb{Q}}),\quad\sum_{i=1}^{s}k_{i}\leq k.$

The following conjecture was proposed in [30], inspired by the original $P=W$
conjecture:

(5) $\text{``}P=C\text{''},\quad P_{k}H^{\leq
2(r-2)}(M_{r,\chi},{\mathbb{Q}})=C_{k}H^{\leq
2(r-2)}(M_{r,\chi},{\mathbb{Q}}).$

Perverse filtrations associated with moduli of $1$-dimensional sheaves have
roots in enumerative geometry [27, 29, 37, 35]. In particular, the dimensions
of the left-hand side of (5) calculate the refined BPS invariants of the local
Calabi–Yau threefold $\mathrm{Tot}(K_{{\mathbb{P}}^{2}})$ for the curve class
$rH$. The $P=C$ conjecture (5) offers a geometric explanation to an asymptotic
product formula for the refined BPS invariants of the local ${\mathbb{P}}^{2}$
calculated via Pandharipande–Thomas theory. We refer to [30, Section 0] for an
introduction.

The following theorem, which will be proven in Section 5.3, is a consequence
of Theorem 0.1. It verifies half of the conjecture (5).

######  Theorem 0.6.

We have

$P_{k}H^{\leq 2(r-2)}(M_{r,\chi},{\mathbb{Q}})\supset C_{k}H^{\leq
2(r-2)}(M_{r,\chi},{\mathbb{Q}}).$

Consequently, the conjecture (5) is equivalent to its numerical version [30,
Conjecture 0.1]:

$\dim\mathrm{Gr}^{P}_{i}H^{i+j}(M_{r,\chi},{\mathbb{Q}})=\dim\mathrm{Gr}^{C}_{i}H^{i+j}(M_{r,\chi},{\mathbb{Q}}),\quad
i+j\leq 2(r-2).$

We refer to Section 5.3.2 for more discussions on the $P=C$ conjecture (5) and
its consequences.

####  0.3.4. Compactified Jacobians

For an integral projective curve $C_{0}$ with planar singularities, the
compactified Jacobian $\overline{J}_{C_{0}}$ is an integral projective
variety. These varieties behave like local analogues of Hitchin systems; in
particular, the singular cohomology $H^{*}(\overline{J}_{C_{0}},{\mathbb{Q}})$
admits a canonical perverse filtration

(6) $P_{0}H^{*}(\overline{J}_{C_{0}},{\mathbb{Q}})\subset
P_{1}H^{*}(\overline{J}_{C_{0}},{\mathbb{Q}})\subset\cdots\subset
H^{*}(\overline{J}_{C_{0}},{\mathbb{Q}})$

by [38, 40]. When $C_{0}$ is a rational curve with a unique planar singular
point, it was proposed by Shende that $\overline{J}_{C_{0}}$ should serve as
the Dolbeault side of a $P=W$ conjecture with some “Betti moduli space”
related to the link of the singularity. This proposal was motivated by the
connection between algebro-geometric invariants associated with a planar
singularity and topological invariants associated with the corresponding link;
we refer to [34, 44, 43, 51, 53, 9] for relevant work and recent developments.
Such a $P=W$ conjecture is wide open to the best of our knowledge;
nevertheless, it predicts that the perverse filtration (6) has to be
multiplicative with respect to the cup-product. For special singularities
$x^{p}=y^{q}$ with $(p,q)=1$, the multiplicativity was proven by Oblomkov–Yun
[45, 46] using tautological generators and representations of the rational
Cherednik algebra.

We establish the multiplicativity in full generality as a consequence of the
sheaf-theoretic version of Theorem 0.1 together with Theorem 0.2. This
provides evidence for the $P=W$ conjecture for compactified Jacobians.

######  Theorem 0.7.

For an integral projective curve $C_{0}$ with planar singularities, the
perverse filtration (6) is multiplicative with respect to the cup-product,
_i.e._ ,

$P_{k}H^{d}(\overline{J}_{C_{0}},{\mathbb{Q}})\times
P_{k^{\prime}}H^{d^{\prime}}(\overline{J}_{C_{0}},{\mathbb{Q}})\xrightarrow{~{}~{}\cup~{}~{}}P_{k+k^{\prime}}H^{d+d^{\prime}}(\overline{J}_{C_{0}},{\mathbb{Q}}).$

In [48], Rennemo constructed a decomposition

$H^{*}(\overline{J}_{C_{0}},{\mathbb{Q}})=\bigoplus_{k,d}D_{k}H^{d}(\overline{J}_{C_{0}},{\mathbb{Q}})$

using natural operators on the cohomology of the Hilbert schemes of points
$C^{[k]}_{0}$, and showed that it splits the perverse filtration. He asked in
[48, Question 1.4] whether this decomposition is multiplicative with respect
to the cup-product. Theorem 0.7 implies that $D_{\leq k}$ is multiplicative,
which answers affirmatively a weaker version of Rennemo’s question. However,
so far we do not know if the multiplicativity holds for a natural splitting of
the perverse filtration. See also [38, Conjecture 2.17] for a natural
(conjectural) splitting of the perverse filtration.

###  0.4. Outline of paper

We briefly outline the contents of this paper. In Section 1, we review
Beauville’s work on abelian schemes and define the class of dualizable abelian
fibrations which generalize it. In Section 2, we prove our first main result
Theorem 0.1, giving multiplicativity and Perverse $\supset$ Chern statements
for dualizable abelian fibrations. Here we use the work of Corti–Hanamura on
relative Chow motives which gives motivic enhancements of these results. In
Section 3, we prove Theorem 0.2; the key input is Arinkin’s theory of
Fourier–Mukai transforms for compactified Jacobians of integral locally planar
curves, combined with an argument using Adams operations. In Section 4 we use
twisted sheaves to extend this to families of curves without a section. In
Section 5, we explain the proof of Theorem 0.6 and give the details of the new
proof of Theorem 0.4.

###  0.5. Acknowledgements

We would like to thank Giuseppe Ancona, Daniel Huybrechts, Max Lieblich, Weite
Pi, and Bin Wang for useful discussions. We are especially grateful to Dima
Arinkin for conversations on his work. J.S. was supported by the NSF grant
DMS-2301474. Q.Y. was supported by the NSFC grants 11831013 and 11890661.

##  1\. Dualizable abelian fibrations

###  1.1. Overview

In this section, we first review the Beauville decomposition for an abelian
scheme as a motivating example. Then we introduce the notion of dualizable
abelian fibrations which is a suitable framework for generalizing Beauville’s
theory.

###  1.2. The Beauville decomposition

To motivate our method, we start with a brief review of the Beauville
decomposition [7, 16]. We focus on the homological version for convenience.

Let $\pi:A\to B$ be an abelian scheme of relative dimension $g$ over a
nonsingular base variety $B$. The “multiplication by $N$” map $[N]:A\to A$
induces a canonical decomposition of the cohomology

(7) $H^{*}(A,{\mathbb{Q}})=\bigoplus_{i}H^{*}_{(i)}(A,{\mathbb{Q}}),$

which we call the (homological) Beauville decomposition. Here

$H^{*}_{(i)}(A,{\mathbb{Q}})=\\{\alpha\in
H^{*}(A,{\mathbb{Q}})\,|\,[N]^{*}\alpha=N^{i}\alpha\\}$

is an eigenspace of the pullback map $[N]^{*}:H^{*}(A,{\mathbb{Q}})\to
H^{*}(A,{\mathbb{Q}})$. The decomposition (7) is motivic (_i.e._ induced by
projectors) which splits the Leray filtration associated with the fibration
$\pi:A\to B$. For our purpose, we would like to extend (7) to abelian
fibrations with singular fibers. We give another description of (7) using
duality and Fourier transforms.

We consider the dual abelian scheme $\pi^{\vee}:A^{\vee}\to B$, and denote by
${\mathcal{L}}$ the normalized Poincaré line bundle over
$A^{\vee}\times_{B}A$. Then the correspondence given by the Chern character

$\mathrm{ch}({\mathcal{L}})=\exp(c_{1}({\mathcal{L}}))$

induces a Fourier transform

$\mathfrak{F}:H^{*}(A^{\vee},{\mathbb{Q}})\to
H^{*}(A,{\mathbb{Q}}),\quad\alpha\mapsto
p_{2*}(p_{1}^{*}\alpha\cup\mathrm{ch}({\mathcal{L}}))$

with $p_{i}$ the natural projections. The inverse transform
$\mathfrak{F}^{-1}$ is induced by $(-1)^{g}\mathrm{ch}({\mathcal{L}}^{\vee})$.
Using the Fourier transform $\mathfrak{F}$ and its inverse
$\mathfrak{F}^{-1}$, we obtain a canonical decomposition of the cohomology
$H^{*}(A,{\mathbb{Q}})$ which recovers the decomposition (7). Indeed, for any
class $\alpha\in H^{d}(A,{\mathbb{Q}})$, we have

$\mathfrak{F}^{-1}(\alpha)=\sum_{i}\alpha_{(i)}^{\vee},\quad\alpha^{\vee}_{(i)}\in
H^{d+2g-2i}(A^{\vee},{\mathbb{Q}}).$

This yields

(8)
$\alpha=\mathfrak{F}\left(\sum_{i}\alpha^{\vee}_{(i)}\right)=\sum_{i}\mathfrak{F}\left(\alpha^{\vee}_{(i)}\right),\quad\mathfrak{F}\left(\alpha^{\vee}_{(i)}\right)\in
H^{d}_{(i)}(A,{\mathbb{Q}}).$

As was explained in [16], the decomposition (8) is motivic with projectors
given by the Chern character of the normalized Poincaré line bundle
${\mathcal{L}}$. Consequently, the Beauville decomposition (7) is governed by
the normalized Poincaré line bundle ${\mathcal{L}}$.

###  1.3. Terminology and notation

We fix some terminology and notation in this section to prepare for further
discussions.

####  1.3.1. Abelian fibrations

We say that $\pi:M\to B$ is an abelian fibration if both $M$ and $B$ are
nonsingular and irreducible, $\pi$ is proper with equidimensional fibers
(hence flat), and $M$ contains an open subset $P$ which is a smooth
commutative $B$-group scheme $\pi:P\to B$ whose restriction to a certain open
subset

$\pi_{U}:P_{U}\to U\subset B$

is an abelian scheme.

We say that $\pi^{\vee}:M^{\vee}\to B$ is dual to the abelian fibration
$\pi:M\to B$, if $\pi^{\vee}:M^{\vee}\to B$ is an abelian fibration and there
exists an open subset $U\subset B$ over which $\pi$ and $\pi^{\vee}$ form dual
abelian schemes. Denote by $P^{\vee}\subset M^{\vee}$ the open commutative
$B$-group scheme associated with $\pi^{\vee}$. We say that

${{\mathcal{P}}}\in D^{b}\mathrm{Coh}(M^{\vee}\times_{B}M)$

is a _Poincaré complex_ , if the restriction of ${\mathcal{P}}$ to
$M_{U}^{\vee}\times_{U}M_{U}$ recovers the normalized Poincaré line bundle
${\mathcal{L}}$. Here $M_{U},M_{U}^{\vee}$ are dual abelian schemes over some
open $U\subset B$.

####  1.3.2. Coherent derived categories

Here all morphisms are assumed to be proper. For two morphisms

$f_{1}:X_{1}\to B,\quad f_{2}:X_{2}\to B$

with $X_{1},X_{2},B$ nonsingular, an object ${\mathcal{K}}\in
D^{b}\mathrm{Coh}(X_{1}\times_{B}X_{2})$ induces a Fourier–Mukai transform

$\mathrm{FM}_{{\mathcal{K}}}:D^{b}\mathrm{Coh}(X_{1})\to
D^{b}\mathrm{Coh}(X_{2}),\quad F\mapsto
p_{2*}(p_{1}^{*}F\otimes{\mathcal{K}}).$

Here all functors are derived and $p_{i}$ are the natural projections from
$X_{1}\times_{B}X_{2}$.

For $f_{3}:X_{3}\to B$ with $X_{3}$ nonsingular, the composition of two
objects

${\mathcal{K}}\in
D^{b}\mathrm{Coh}(X_{1}\times_{B}X_{2}),\quad{\mathcal{K}}^{\prime}\in
D^{b}\mathrm{Coh}(X_{2}\times_{B}X_{3})$

is defined under certain flatness assumptions. For example, if $f_{1}$ and
$f_{3}$ are both flat then

(9)
${\mathcal{K}}^{\prime}\circ{\mathcal{K}}:=p_{13*}\delta^{*}({\mathcal{K}}\boxtimes{\mathcal{K}}^{\prime})\in
D^{b}\mathrm{Coh}(X_{1}\times_{B}X_{3}),$

where $p_{13}$ is the projection to the first and the third factors and

$\delta^{*}:D^{b}\mathrm{Coh}((X_{1}\times_{B}X_{2})\times(X_{2}\times_{B}X_{3}))\to
D^{b}\mathrm{Coh}(X_{1}\times_{B}X_{2}\times_{B}X_{3})$

is the pullback along the regular embedding
$X_{1}\times_{B}X_{2}\times_{B}X_{3}\hookrightarrow(X_{1}\times_{B}X_{2})\times(X_{2}\times_{B}X_{3})$
base-changed from $\Delta_{X_{2}}:X_{2}\hookrightarrow X_{2}\times
X_{2}$.222Without flatness (or derived algebraic geometry), the object
${\mathcal{K}}^{\prime}\circ{\mathcal{K}}$ defined above may be unbounded or
ill-behaved. In this case the composition of the two Fourier–Mukai transforms

$\mathrm{FM}_{\mathcal{K}}:D^{b}\mathrm{Coh}(X_{1})\to
D^{b}\mathrm{Coh}(X_{2}),\quad\mathrm{FM}_{{\mathcal{K}}^{\prime}}:D^{b}\mathrm{Coh}(X_{2})\to
D^{b}\mathrm{Coh}(X_{3})$

is induced by the Fourier–Mukai kernel
${\mathcal{K}}^{\prime}\circ{\mathcal{K}}$.

######  Remark 1.1.

We also note that when descended to $K$-theory, compositions make sense
without flatness and for arbitrary objects

${\mathcal{K}}\in K_{*}(X_{1}\times_{B}X_{2}),\quad{\mathcal{K}}^{\prime}\in
K_{*}(X_{2}\times_{B}X_{3}).$

Here $K_{*}(-)$ stands for the Grothendieck group of coherent sheaves. We
define

(10)
${\mathcal{K}}^{\prime}\circ{\mathcal{K}}:=p_{13*}\delta^{!}({\mathcal{K}}\boxtimes{\mathcal{K}}^{\prime})\in
K_{*}(X_{1}\times_{B}X_{3}),$

where

$\delta^{!}:K_{*}((X_{1}\times_{B}X_{2})\times(X_{2}\times_{B}X_{3}))\to
K_{*}(X_{1}\times_{B}X_{2}\times_{B}X_{3})$

is the refined Gysin pullback with respect to the regular embedding
$\Delta_{X_{2}}:X_{2}\hookrightarrow X_{2}\times X_{2}$; see [4, Section 3].
The compatibility between (9) and (10) (when the former is well-defined) is
straightforward.

For any nonsingular $B$-scheme $X$, the identity
$\mathrm{id}_{D^{b}\mathrm{Coh}(X)}:D^{b}\mathrm{Coh}(X)\to
D^{b}\mathrm{Coh}(X)$ is induced by the structure sheaf of the relative
diagonal ${\mathcal{O}}_{\Delta_{X/B}}\in D^{b}\mathrm{Coh}(X\times_{B}X)$. We
say that two objects

${\mathcal{K}}\in
D^{b}\mathrm{Coh}(X_{1}\times_{B}X_{2}),\quad{\mathcal{K}}^{\prime}\in
D^{b}\mathrm{Coh}(X_{2}\times_{B}X_{1})$

are _inverse_ , if

${\mathcal{K}}^{\prime}\circ{\mathcal{K}}\simeq{\mathcal{O}}_{\Delta_{X_{1}/B}},\quad{\mathcal{K}}\circ{\mathcal{K}}^{\prime}\simeq{\mathcal{O}}_{\Delta_{X_{2}/B}}.$

Consequently, the Fourier–Mukai transforms $\mathrm{FM}_{{\mathcal{K}}}$ and
$\mathrm{FM}_{{\mathcal{K}}^{\prime}}$ are both derived equivalences which are
inverse to each other.

We say that $F\in D^{b}\mathrm{Coh}(X)$ is of codimension $c$, if the union of
the support of each nontrivial cohomology ${\mathcal{H}}^{i}(F)$ is of
codimension $c$.

####  1.3.3. Constructible derived categories

Let $f:X\to B$ be a proper morphism between nonsingular varieties with
equidimensional fibers. By the decomposition theorem [8], we have

$f_{*}{\mathbb{Q}}_{X}\simeq\bigoplus_{i}{{}^{\mathfrak{p}}}{\mathcal{H}}^{i}(f_{*}{\mathbb{Q}}_{X})[-i]\in
D^{b}_{c}(B)$

where $D^{b}_{c}(-)$ stands for the bounded derived category of constructible
sheaves and all functors are derived. Each perverse cohomology on the right-
hand side is a semisimple perverse sheaf. We say that $f$ has _full support_ ,
if each simple summand of
${{}^{\mathfrak{p}}}{\mathcal{H}}^{i}(f_{*}{\mathbb{Q}}_{X})$ has support $B$.

Next, we introduce the main characters of this paper — perverse filtrations.
The sequence of perverse truncation functors yield morphisms

${{}^{\mathfrak{p}}}\tau_{\leq k+\dim B}f_{*}{\mathbb{Q}}_{X}\to
f_{*}{\mathbb{Q}}_{X}$

which induces both the _global_ and the _local_ perverse filtrations. More
precisely, the global perverse filtration on the total cohomology of $X$ is
given by

$P_{k}H^{d}(X,{\mathbb{Q}}):=\mathrm{Im}\left\\{H^{d}(B,{{}^{\mathfrak{p}}}\tau_{\leq
k+\dim B}f_{*}{\mathbb{Q}}_{X})\to H^{d}(X,{\mathbb{Q}})\right\\},$

and, for a closed point $b\in B$, the local perverse filtration on the
cohomology of the closed fiber $X_{b}$ is given by

$P_{k}H^{d}(X_{b},{\mathbb{Q}}):={\mathcal{H}}^{d}({{}^{\mathfrak{p}}}\tau_{\leq
k+\dim B}f_{*}{\mathbb{Q}}_{X})_{b}.$

_A priori_ , the local perverse filtration for $X_{b}$ depends on the total
family $f:X\to B$.

We say that a class $\alpha\in H^{d}(X,{\mathbb{Q}})$ has _perversity_ $k$ if

$\alpha\in P_{k}H^{d}(X,{\mathbb{Q}}).$

###  1.4. Dualizable abelian fibrations

Dualizable abelian fibrations are extensions of abelian schemes involving
singular fibers, for which most properties given by the Fourier transform
described in Section 1.2 may be generalized.

Let $\pi:M\to B$ be an abelian fibration of relative dimension $g$, _i.e._
$g=\dim M-\dim B$. We say that $\pi:M\to B$ is a dualizable abelian fibration,
if it has a dual abelian fibration $\pi^{\vee}:M^{\vee}\to B$ (see Section
1.3.1) satisfying the following.

  1. (a1)

(Poincaré) There is a Poincaré complex ${{\mathcal{P}}}$ which admits an
inverse ${{\mathcal{P}}}^{-1}$, _i.e._ ,

(11)
${{\mathcal{P}}}^{-1}\circ{{\mathcal{P}}}\simeq{\mathcal{O}}_{\Delta_{M^{\vee}/B}},\quad{{\mathcal{P}}}\circ{\mathcal{P}}^{-1}\simeq{\mathcal{O}}_{\Delta_{M/B}}.$

  2. (a2)

(Convolution) There is an object ${\mathcal{K}}\in
D^{b}\mathrm{Coh}(M^{\vee}\times_{B}M^{\vee}\times_{B}M^{\vee})$ of
codimension $g$ such that

(12)
${{\mathcal{P}}}\circ{\mathcal{K}}\simeq{\mathcal{O}}_{\Delta^{\mathrm{sm}}_{M/B}}\circ({{\mathcal{P}}}\boxtimes{{\mathcal{P}}})\in
D^{b}\mathrm{Coh}(M^{\vee}\times_{B}M^{\vee}\times_{B}M).$

Here $\Delta^{\mathrm{sm}}_{M/B}$ is the small diagonal of the relative triple
product $M\times_{B}M\times_{B}M$, and the notation
${\mathcal{O}}_{\Delta^{\mathrm{sm}}_{M/B}}\circ({{\mathcal{P}}}\boxtimes{{\mathcal{P}}})$
stands for composing ${\mathcal{P}}$ with
${\mathcal{O}}_{\Delta^{\mathrm{sm}}_{M/B}}$ via the first (resp. second)
factor of $M\times_{B}M\times_{B}M$.

  3. (b)

(Support) The morphism $\pi$ has full support.

Here (a1, a2) serve as the detailed version of (a) in Section 0.2. We list
some consequences of (a1, a2) in terms of Fourier–Mukai transforms. Using
(a1), we can define the Fourier–Mukai transforms
$\mathrm{FM}_{{\mathcal{P}}}:D^{b}\mathrm{Coh}(M^{\vee})\to
D^{b}\mathrm{Coh}(M)$ and its inverse

$\mathrm{FM}^{-1}_{{\mathcal{P}}}=\mathrm{FM}_{{{\mathcal{P}}}^{-1}}:D^{b}\mathrm{Coh}(M)\to
D^{b}\mathrm{Coh}(M^{\vee}).$

Next, we define a convolution product via ${\mathcal{K}}$ in (a2):

$\ast:D^{b}\mathrm{Coh}(M^{\vee})\times D^{b}\mathrm{Coh}(M^{\vee})\to
D^{b}\mathrm{Coh}(M^{\vee}),\quad F_{1}\ast F_{2}=p_{3*}(p_{1}^{*}F_{1}\otimes
p_{2}^{*}F_{2}\otimes{\mathcal{K}}).$

Then (12) implies the interaction between Fourier–Mukai transforms and the
convolution:

$\mathrm{FM}_{{\mathcal{P}}}(F_{1})\otimes\mathrm{FM}_{{\mathcal{P}}}(F_{2})\simeq\mathrm{FM}_{{\mathcal{P}}}(F_{1}\ast
F_{2}),\quad F_{1},~{}F_{2}\in D^{b}\mathrm{Coh}(M^{\vee}).$

We also note that dualizable abelian fibrations extend abelian schemes.

######  Proposition 1.2.

An abelian scheme $\pi:A\to B$ is dualizable in the sense above.

###### Proof.

The abelian schemes $\pi:A\to B$ and $\pi^{\vee}:A^{\vee}\to B$ form dual
abelian fibrations with ${\mathcal{P}}$ given by the normalized Poincaré line
bundle ${\mathcal{L}}$. The properties concerning the convolution were proven
in [41]. In particular, restricting over point $b\in B$ with fiber
$A^{\vee}_{b}=\mathrm{Pic}^{0}(A_{b})$, the convolution kernel is the
structure sheaf of the locus

$\\{(l_{1},l_{2},l_{3})\in\mathrm{Pic}^{0}(A_{b})^{\times 3}\,|\,l_{1}\otimes
l_{2}\simeq l_{3}\\}.$

This is clearly a codimension $g$ locus over $B$. Finally, we note that each
simple summand of $R^{i}\pi_{*}{\mathbb{Q}}_{A}$ is a local system with
support $B$, which guarantees (b). ∎

Another class of dualizable fibrations is given by compactified Jacobian
fibrations, which we will discuss systematically in Section 3. Arinkin’s
theory of the Poincaré sheaf [5, 6] plays a crucial role. We will also treat
twisted compactified Jacobian fibrations in Section 4.

##  2\. Multiplicativity and perversity

###  2.1. Overview

In this section, we apply Fourier transforms to study the perverse filtrations
for dualizable abelian fibrations. After the necessary preparations, we state
our main result in Theorem 2.4 which is an extension of the Beauville
decomposition and a motivic enhancement of the multiplicativity and the
“Perverse $\supset$ Chern” statements of Theorem 0.1. Along the way, we note
in Corollary 2.5 that our approach actually confirms the motivic decomposition
conjecture of Corti–Hanamura [15] for a dualizable abelian fibration
satisfying the Fourier vanishing. The proofs are given in Section 2.5.

###  2.2. Relative correspondences

As our strategy involves passing from coherent derived categories to cycle
classes, we find it convenient to use the language of _relative Chow motives_
developed by Corti–Hanamura [15]. This notion has its advantage in the various
realizations: Chow groups, mixed Hodge modules, constructible sheaves, and
particularly, global cohomology equipped with mixed Hodge structures. Some of
these realizations are relevant to our results while others might draw
potential interests. We begin with a quick review of the theory.

####  2.2.1. Relative Chow motives

We fix a nonsingular base variety $B$. All Chow groups $\mathrm{CH}_{*}(-)$
are taken with ${\mathbb{Q}}$-coefficients. Let $f_{1}:X_{1}\to
B,f_{2}:X_{2}\to B$ be two proper morphisms with $X_{1},X_{2}$ nonsingular. If
$X_{2}$ is equidimensional,333If $X_{2}$ is not equidimensional, we decompose
$X_{2}=\sqcup_{\alpha}X_{2,\alpha}$ and define the group of correspondences
accordingly. we define the group of degree $k$ relative correspondences
between $X_{1}$ and $X_{2}$ to be

$\mathrm{Corr}^{k}_{B}(X_{1},X_{2}):=\mathrm{CH}_{\dim
X_{2}-k}(X_{1}\times_{B}X_{2}).$

Compositions of correspondences are similar to the compositions of
Fourier–Mukai kernels in (10). For $f_{3}:X_{3}\to B$ proper with $X_{3}$
nonsingular, and

$\Gamma\in\mathrm{Corr}_{B}^{k}(X_{1},X_{2}),\quad\Gamma^{\prime}\in\mathrm{Corr}_{B}^{k^{\prime}}(X_{2},X_{3}),$

we set

(13)
$\Gamma^{\prime}\circ\Gamma:=p_{13*}\delta^{!}(\Gamma\times\Gamma^{\prime})\in\mathrm{Corr}_{B}^{k+k^{\prime}}(X_{1},X_{3}),$

where $p_{13}$ is the projection to the first and the third factors and

$\delta^{!}:\mathrm{CH}_{*}((X_{1}\times_{B}X_{2})\times(X_{2}\times_{B}X_{3}))\to\mathrm{CH}_{*-\dim
X_{2}}(X_{1}\times_{B}X_{2}\times_{B}X_{3})$

is the refined Gysin pullback with respect to the regular embedding
$\Delta_{X_{2}}:X_{2}\hookrightarrow X_{2}\times X_{2}$. For $X$ nonsingular
and proper over $B$, a self-correspondence
$\mathfrak{p}\in\mathrm{Corr}_{B}^{0}(X,X)$ is a _projector_ if
$\mathfrak{p}\circ\mathfrak{p}=\mathfrak{p}$.

The category of relative Chow motives over $B$, denoted $\mathrm{CHM}(B)$,
consists of triples $(X,\mathfrak{p},m)$ where $X$ is nonsingular and proper
over $B$, $\mathfrak{p}\in\mathrm{Corr}^{0}_{B}(X,X)$ is a projector, and $m$
is an integer taking care of Tate twists. Typical examples are given by the
motive of $X$, _i.e._ ,

$h(X):=(X,[\Delta_{X/B}],0).$

Morphisms between $(X_{1},\mathfrak{p},m)$ and $(X_{2},\mathfrak{q},n)$ are
defined by

$\mathrm{Hom}_{\mathrm{CHM}(B)}((X_{1},\mathfrak{p},m),(X_{2},\mathfrak{q},n)):=\mathfrak{q}\circ\mathrm{Corr}^{n-m}_{B}(X_{1},X_{2})\circ\mathfrak{p}.$

The category $\mathrm{CHM}(B)$ is additive and pseudo-abelian by construction.
Note that however, it admits no tensor product as relative products of
nonsingular $B$-schemes are often singular.

####  2.2.2. Homological realizations

In this paper we are mostly concerned with homological realizations of
$\mathrm{CHM}(B)$.444We refer to [23] for the Chow realization. A key
observation in [15] is that if $f_{1}:X_{1}\to B$ and $f_{2}:X_{2}\to B$ are
two morphisms with $f_{1}$ proper and $X_{2}$ nonsingular, then there is a
natural isomophism

(14)
$\phi:\operatorname{Hom}_{D^{b}_{c}(B)}(f_{1*}{\mathbb{Q}}_{X_{1}}[i],f_{2*}{\mathbb{Q}}_{X_{2}}[j])\xrightarrow{~{}~{}\simeq~{}~{}}H^{\mathrm{BM}}_{2\dim
X_{2}+i-j}(X_{1}\times_{B}X_{2},{\mathbb{Q}}).$

The isomorphism $\phi$ is compatible with compositions on both sides in the
following sense. For $i=1,2,3$, let $f_{i}:X_{i}\to B$ be morphisms with
$f_{1},f_{2}$ proper and $X_{2},X_{3}$ nonsingular. Given two morphisms in
$D^{b}_{c}(B)$:

$u:f_{1*}{\mathbb{Q}}_{X_{1}}[i]\to f_{2*}{\mathbb{Q}}_{X_{2}}[j],\quad
v:f_{2*}{\mathbb{Q}}_{X_{2}}[j]\to f_{3*}{\mathbb{Q}}_{X_{3}}[k],$

we have

(15) $\phi(v\circ u)=\phi(v)\circ\phi(u)\in H^{\mathrm{BM}}_{2\dim
X_{3}+i-k}(X_{1}\times_{B}X_{3},{\mathbb{Q}})$

where the definition of compositions in Borel–Moore homology is identical to
(13); see [15, Lemmas 2.21 and 2.23].

Consequently, by the cycle class map

$\mathrm{cl}:\mathrm{CH}_{*}(-)\to H_{2*}^{\mathrm{BM}}(-,{\mathbb{Q}})$

and by showing that $D^{b}_{c}(B)$ is pseudo-abelian [15, Lemma 2.24],
Corti–Hanamura obtained a well-defined realization functor

(16) $\mathrm{CHM}(B)\to D^{b}_{c}(B)$

sending $(f:X\to B,\mathfrak{p},m)$ to
$\mathfrak{p}_{*}(f_{*}{\mathbb{Q}}_{X}[2m])$. The functor further specializes
by taking global cohomology $H^{*}(-)$.

####  2.2.3. Multiplicative structure

We now discuss the multiplicative structure of the motive $h(X)=(f:X\to
B,[\Delta_{X/B}],0)$ and its realizations. With no tensor product in
$\mathrm{CHM}(B)$, we define manually a binary operation of motives as
follows. For $i=1,2,3$, let $f_{i}:X_{i}\to B$ be proper morphisms with
$X_{i}$ nonsingular. If $X_{3}$ is equidimensional, we define the group of
degree $k$ binary relative correspondences

$\mathrm{Corr}_{B}^{k}(X_{1},X_{2};X_{3}):=\mathrm{CH}_{\dim
X_{3}-k}(X_{1}\times_{B}X_{2}\times_{B}X_{3}).$

For $i=1,2,3$, let $g_{i}:Y_{i}\to B$ be three more proper morphisms with
$Y_{i}$ nonsingular. Given four correspondences

$\Gamma_{1}\in\mathrm{Corr}_{B}^{k_{1}}(Y_{1},X_{1}),\quad\Gamma_{2}\in\mathrm{Corr}_{B}^{k_{2}}(Y_{2},X_{2}),\quad\Gamma_{3}\in\mathrm{Corr}_{B}^{k_{3}}(X_{3},Y_{3}),\quad\Gamma\in\mathrm{Corr}_{B}^{k}(X_{1},X_{2};X_{3}),$

there is the composition

$\Gamma_{3}\circ\Gamma\circ(\Gamma_{1}\times\Gamma_{2})\in\mathrm{Corr}_{B}^{k+k_{1}+k_{2}+k_{3}}(Y_{1},Y_{2};Y_{3}).$

Here the notation stands for composing $\Gamma_{1},\Gamma_{2}$ with $\Gamma$
via the $X_{1}$ and $X_{2}$ factors, and then with $\Gamma_{3}$ via the
$X_{3}$ factor, which makes sense since $X_{1},X_{2},X_{3}$ are all
nonsingular. A binary morphism of motives

(17)
$(X_{1},\mathfrak{p},m)\times(X_{2},\mathfrak{q},n)\to(X_{3},\mathfrak{r},p)$

is then given by a binary relative correspondence

$\mathfrak{r}\circ\Gamma\circ(\mathfrak{p}\times\mathfrak{q}),\quad\Gamma\in\mathrm{Corr}_{B}^{p-m-n}(X_{1},X_{2};X_{3}).$

To see the relation of (17) with the realization functor (16), we observe that
by (14) composed with the Künneth formula, there is a natural isomorphism

$\Phi:\operatorname{Hom}_{D^{b}_{c}(B)}(f_{1*}{\mathbb{Q}}_{X_{1}}[i]\otimes
f_{2*}{\mathbb{Q}}_{X_{2}}[j],f_{3*}{\mathbb{Q}}_{X_{3}}[k])\xrightarrow{~{}~{}\sim~{}~{}}H^{\mathrm{BM}}_{2\dim
X_{3}+i+j-k}(X_{1}\times_{B}X_{2}\times_{B}X_{3},{\mathbb{Q}}).$

The compatibility of $\Phi$ with compositions on both sides is summarized in
the following lemma. We include a short proof as we could not find it in the
literature.

######  Lemma 2.1.

For $i=1,2,3$, let $f_{i}:X_{i}\to B$ be as above and let $g_{i}:Y_{i}\to B$
be morphisms with $g_{1},g_{2}$ proper and $Y_{3}$ nonsingular. Given four
morphisms in $D^{b}_{c}(B)$:

$\displaystyle u:g_{1*}{\mathbb{Q}}_{Y_{1}}[i^{\prime}]\to
f_{1*}{\mathbb{Q}}_{X_{1}}[i],\quad
v:g_{2*}{\mathbb{Q}}_{Y_{2}}[j^{\prime}]\to
f_{2*}{\mathbb{Q}}_{X_{2}}[j],\quad w:f_{3*}{\mathbb{Q}}_{X_{3}}[k]\to
g_{3*}{\mathbb{Q}}_{Y_{3}}[k^{\prime}],$ $\displaystyle
T:f_{1*}{\mathbb{Q}}_{X_{1}}[i]\otimes f_{2*}{\mathbb{Q}}_{X_{2}}[j]\to
f_{3*}{\mathbb{Q}}_{X_{3}}[k],$

we have

$\Phi(w\circ T\circ(u\otimes
v))=\phi(w)\circ\Phi(T)\circ(\phi(u)\times\phi(v))\in H^{\mathrm{BM}}_{2\dim
Y_{3}+i^{\prime}+j^{\prime}-k^{\prime}}(Y_{1}\times_{B}Y_{2}\times_{B}Y_{3},{\mathbb{Q}}).$

###### Proof.

We decompose the lemma into three statements:

(18) $\displaystyle\Phi(T\circ u)=\Phi(T)\circ\phi(u),$
$\displaystyle\Phi(T\circ v)=\Phi(T)\circ\phi(v),$ (19)
$\displaystyle\Phi(w\circ T)=\phi(w)\circ\Phi(T).$

The statement (19) follows from (15) by viewing $X_{1}\times_{B}X_{2}\to B$ as
a whole and by Künneth. Indeed, the assumptions of (15) are satisfied since
$Y_{3}$ is nonsingular. For symmetry reasons it remains to prove (18). Note
that (15) does not apply directly as $X_{2}\times_{B}X_{3}$ may be singular.

The strategy is to go through the actual proof of (15) in [15, Section 3] and
reduce to the technical Lemma 3.1 therein. We first remark that $u$
corresponds via adjunction and proper base change (since $g_{1}$ is proper) to
a morphism in $D^{b}_{c}(X_{1})$:

$u^{\prime}:p_{*}{\mathbb{Q}}_{Y_{1}\times_{B}X_{1}}[i^{\prime}]\to{\mathbb{Q}}_{X_{1}}[i],$

where $p:Y_{1}\times_{B}X_{1}\to X_{1}$ is the second projection. Also by
adjunction $T$ corresponds to a morphism in $D^{b}_{c}(X_{1})$:

(20) ${\mathbb{Q}}_{X_{1}}[i]\to
f_{1}^{!}R\mathcal{H}\mathrm{om}(f_{2*}{\mathbb{Q}}_{X_{2}}[j],f_{3*}{\mathbb{Q}}_{X_{3}}[k]).$

We have natural isomorphisms

(21) $\displaystyle
f_{1}^{!}R\mathcal{H}\mathrm{om}(f_{2*}{\mathbb{Q}}_{X_{2}}[j],f_{3*}{\mathbb{Q}}_{X_{3}}[k])$
$\displaystyle\simeq{}$ $\displaystyle
R\mathcal{H}\mathrm{om}\left(f_{1}^{*}f_{2*}{\mathbb{Q}}_{X_{2}}[j],f_{1}^{!}f_{3*}{\mathbb{Q}}_{X_{3}}[k]\right)$
$\displaystyle\textrm{($f_{2}$ is proper)}\simeq{}$ $\displaystyle
R\mathcal{H}\mathrm{om}\left(p^{12}_{1*}{\mathbb{Q}}_{X_{1}\times_{B}X_{2}}[j],p^{13}_{1*}p^{13!}_{3}{\mathbb{Q}}_{X_{3}}[k]\right)$
$\displaystyle\textrm{($X_{3}$ is nonsingular)}\simeq{}$ $\displaystyle
R\mathcal{H}\mathrm{om}\left(p^{12}_{1*}{\mathbb{Q}}_{X_{1}\times_{B}X_{2}}[j],p^{13}_{1*}\omega_{X_{1}\times_{B}X_{3}}[k-2\dim
X_{3}]\right)$ $\displaystyle\textrm{($X_{1}$ is nonsingular)}\simeq{}$
$\displaystyle
R\mathcal{H}\mathrm{om}\left(p^{12}_{1*}{\mathbb{Q}}_{X_{1}\times_{B}X_{2}}[j],p^{13}_{1*}p^{13!}_{1}{\mathbb{Q}}_{X_{1}}[k+2\dim
X_{1}-2\dim X_{3}]\right)$ $\displaystyle\textrm{($p^{12}_{1}$ is
proper)}\simeq{}$ $\displaystyle
p^{12}_{1*}R\mathcal{H}\mathrm{om}\left({\mathbb{Q}}_{X_{1}\times_{B}X_{2}}[j],p^{12!}_{1}p^{13}_{1*}p^{13!}_{1}{\mathbb{Q}}_{X_{1}}[k+2\dim
X_{1}-2\dim X_{3}]\right)$ $\displaystyle\simeq{}$ $\displaystyle
p^{12}_{1*}\left(p^{12!}_{1}p^{13}_{1*}\right)p^{13!}_{1}{\mathbb{Q}}_{X_{1}}[k-j+2\dim
X_{1}-2\dim X_{3}]$ $\displaystyle\simeq{}$ $\displaystyle
p^{12}_{1*}\left(p^{123}_{12*}p^{123!}_{13}\right)p^{13!}_{1}{\mathbb{Q}}_{X_{1}}[k-j+2\dim
X_{1}-2\dim X_{3}]$ $\displaystyle\simeq{}$ $\displaystyle
p^{123}_{1*}p^{123!}_{1}{\mathbb{Q}}_{X_{1}}[k-j+2\dim X_{1}-2\dim X_{3}],$

where the $p^{-}_{-}$ stand for the various projections from relative products
of $X_{i}$. We write

$T^{\prime}:{\mathbb{Q}}_{X_{1}}[i]\to
q_{*}q^{!}{\mathbb{Q}}_{X_{1}}[k-j+2\dim X_{1}-2\dim X_{3}]$

for the morphism in $D^{b}_{c}(X_{1})$ resulted from (20) and (21), with
$q=p^{123}_{1}:X_{1}\times_{B}X_{2}\times_{B}X_{3}\to X_{1}$ the projection to
the first factor.

Finally, we obtain (18) by applying [15, Lemma 3.1] to $u^{\prime}$ and
$T^{\prime}$ together with an identical diagram chase as in [15, Proof of
Lemma 2.23]. ∎

In particular, the binary morphism (17) specializes to a well-defined morphism
in $D^{b}_{c}(B)$:

$\mathfrak{p}_{*}(f_{1*}{\mathbb{Q}}_{X_{1}}[2m])\otimes\mathfrak{q}_{*}(f_{2*}{\mathbb{Q}}_{X_{2}}[2n])\to\mathfrak{r}_{*}(f_{3*}{\mathbb{Q}}_{X_{3}}[2p]).$

One primary example of binary morphisms of motives is the cup-product

$\cup:h(X)\times h(X)\to h(X)$

corresponding to the small relative diagonal
$[\Delta^{\mathrm{sm}}_{X/B}]\in\mathrm{Corr}^{0}_{B}(X,X;X)$. It specializes
to the sheaf-theoretic cup-product in $D^{b}_{c}(B)$,

$\cup:f_{*}{\mathbb{Q}}_{X}\otimes f_{*}{\mathbb{Q}}_{X}\to
f_{*}{\mathbb{Q}}_{X},$

and further to the cup-product

$\cup:H^{d}(X_{b},{\mathbb{Q}})\times H^{d^{\prime}}(X_{b},{\mathbb{Q}})\to
H^{d+d^{\prime}}(X_{b},{\mathbb{Q}})$

by restricting to the closed point $b\in B$.555In the Chow realization, the
cup-product specializes to the intersection product on $\mathrm{CH}^{*}(X)$.

###  2.3. Chern characters

As we will be working throughout with coherent sheaves on singular varieties,
we need a systematic definition of Chern characters taking values in Chow
groups. For convenience we use the Riemann–Roch theorem of
Baum–Fulton–MacPherson in [21, Chapter 18].

Let $K_{*}(-)$ (resp. $K^{*}(-)$) denote the Grothendieck group of coherent
sheaves (resp. vector bundles). By [21, Theorem 18.3], for any finite type
scheme $X$ there is a homomorphism

$\tau:K_{*}(X)\to\mathrm{CH}_{*}(X)$

satisfying the following properties.

  1. (i)

For $f:X\to Y$ proper and $F\in K_{*}(X)$, we have
$f_{*}\tau(F)=\tau(f_{*}F)$.

  2. (ii)

For $E\in K^{*}(X),F\in K_{*}(X)$, we have $\tau(E\otimes
F)={\mathrm{ch}}(E)\cap\tau(F)$.

  3. (iii)

If $i:X\hookrightarrow Y$ is a closed embedding with $Y$ nonsingular, then for
$F\in K_{*}(X)$,

(22)
$\tau(F)={\mathrm{ch}}_{X}^{Y}(F_{\bullet})\cap{\mathrm{td}}(Y)\cap[Y]={\mathrm{td}}(i^{*}T_{Y})\cap{\mathrm{ch}}_{X}^{Y}(F_{\bullet})\cap[Y].$

Here ${\mathrm{ch}}_{X}^{Y}(-)$ is the localized Chern character in [21,
Section 18.1] and $F_{\bullet}$ is any locally free resolution of $i_{*}F$ on
$Y$.

  4. (iv)

If $f:X\to Y$ is an l.c.i. morphism and both $X$ and $Y$ admit closed
embeddings in nonsingular varieties666The assumption on the embeddings was
recently removed by [2]., then for $F\in K_{*}(Y)$,

$\tau(f^{*}F)={\mathrm{td}}(T_{f})\cap f^{*}\tau(F).$

Here $T_{f}\in K^{*}(X)$ is the virtual tangent bundle of $f$.

Intuitively, one can think of $\tau$ as a covariant (for proper morphisms),
Todd-twisted Chern character for coherent sheaves on arbitrary varieties.

###  2.4. Statement of the main theorem

Let $\pi:M\to B$ be a dualizable abelian fibration of relative dimension $g$
in the sense of Section 1.4, and let $\pi^{\vee}:M^{\vee}\to B$ be its dual
abelian fibration. We consider the Poincaré complex ${\mathcal{P}}\in
D^{b}(M^{\vee}\times_{B}M)$, and define

(23)
$\mathfrak{F}:={\mathrm{td}}(-T_{M^{\vee}\times_{B}M})\cap\tau({\mathcal{P}})\in\mathrm{CH}_{*}(M^{\vee}\times_{B}M),$

where $T_{M^{\vee}\times_{B}M}$ is the virtual tangent bundle of
$M^{\vee}\times_{B}M$ (since $M^{\vee}\times_{B}M$ is l.c.i.). By (22) applied
to the closed embedding $i:M^{\vee}\times_{B}M\hookrightarrow M^{\vee}\times
M$, we also have

(24)
$\mathfrak{F}={\mathrm{td}}((\pi^{\vee}\times_{B}\pi)^{*}T_{B})\cap{\mathrm{ch}}_{M^{\vee}\times_{B}M}^{M^{\vee}\times
M}({\mathcal{P}}_{\bullet})\cap[M^{\vee}\times M],$

where $\pi^{\vee}\times_{B}\pi:M^{\vee}\times_{B}M\to B$ is the natural
morphism. On the other hand, for the inverse of ${\mathcal{P}}$ we set

(25) $\displaystyle\mathfrak{F}^{-1}$
$\displaystyle:={\mathrm{td}}(-(\pi\times_{B}\pi^{\vee})^{*}T_{B})\cap\tau({\mathcal{P}}^{-1})$
$\displaystyle\phantom{:}={\mathrm{td}}(T_{M\times_{B}M^{\vee}})\cap{\mathrm{ch}}_{M\times_{B}M^{\vee}}^{M\times
M^{\vee}}({\mathcal{P}}^{-1}_{\bullet})\cap[M\times
M^{\vee}]\in\mathrm{CH}_{*}(M\times_{B}M^{\vee}).$

We view both $\mathfrak{F}$ and $\mathfrak{F}^{-1}$ as mixed-degree relative
correspondences and call them the Chow-theoretic _Fourier transform_ and its
inverse.

######  Lemma 2.2.

We have

$\mathfrak{F}^{-1}\circ\mathfrak{F}=[\Delta_{M^{\vee}/B}]\in\mathrm{Corr}^{0}_{B}(M^{\vee},M^{\vee}),\quad\mathfrak{F}\circ\mathfrak{F}^{-1}=[\Delta_{M/B}]\in\mathrm{Corr}_{B}^{0}(M,M).$

###### Proof.

We take
${{\mathcal{P}}}^{-1}\circ{{\mathcal{P}}}\simeq{\mathcal{O}}_{\Delta_{M^{\vee}/B}}$
from (11) and apply $\tau$ on both sides. On one hand we have

(26)
$\tau({\mathcal{O}}_{\Delta_{M^{\vee}/B}})=\Delta_{M^{\vee}/B*}\tau({\mathcal{O}}_{M^{\vee}})=\Delta_{M^{\vee}/B*}\left({\mathrm{td}}(T_{M^{\vee}})\cap[M^{\vee}]\right)={\mathrm{td}}(q_{1}^{\vee*}T_{M^{\vee}})\cap[\Delta_{M^{\vee}/B}]$

where $q^{\vee}_{1}:M^{\vee}\times_{B}M^{\vee}\to M^{\vee}$ is the first
projection. On the other hand, we compute

(27) $\displaystyle\tau({\mathcal{P}}^{-1}\circ{\mathcal{P}})$
$\displaystyle=p_{13*}\tau(\delta^{*}({\mathcal{P}}\boxtimes{\mathcal{P}}^{-1}))$
$\displaystyle=p_{13*}\left({\mathrm{td}}(-p_{2}^{*}T_{M})\cap\delta^{!}\tau({\mathcal{P}}\boxtimes{\mathcal{P}}^{-1})\right)$
$\displaystyle=p_{13*}\left({\mathrm{td}}(-p_{2}^{*}T_{M})\cap\delta^{!}(\tau({\mathcal{P}})\times\tau({\mathcal{P}}^{-1}))\right).$

Here we have taken the notation from Sections 1.3.2 and 2.2.1. The second
equality uses the flatness of $\pi^{\vee}:M^{\vee}\to B$ so that the base-
changed embedding

$M^{\vee}\times_{B}M\times_{B}M^{\vee}\hookrightarrow(M^{\vee}\times_{B}M)\times(M\times_{B}M^{\vee})$

is regular (hence l.c.i.) with normal bundle identified with $p_{2}^{*}T_{M}$,
and pulling back along this embedding coincides with the Gysin pullback
$\delta^{!}$. The third equality uses [21, Example 18.3.1].

By capping both (26) and (27) with
${\mathrm{td}}(-q_{1}^{\vee*}T_{M^{\vee}})$, we find

$\displaystyle[\Delta_{M^{\vee}/B}]$
$\displaystyle={\mathrm{td}}(-q_{1}^{\vee*}T_{M^{\vee}})\cap
p_{13*}\left({\mathrm{td}}(-p_{2}^{*}T_{M})\cap\delta^{!}(\tau({\mathcal{P}})\times\tau({\mathcal{P}}^{-1}))\right)$
$\displaystyle=p_{13*}\left({\mathrm{td}}(-p_{1}^{*}T_{M^{\vee}})\cap{\mathrm{td}}(-p_{2}^{*}T_{M})\cap\delta^{!}(\tau({\mathcal{P}})\times\tau({\mathcal{P}}^{-1}))\right)$
$\displaystyle=p_{13*}\delta^{!}\left(\left({\mathrm{td}}(-T_{M^{\vee}\times_{B}M})\cap\tau({\mathcal{P}})\right)\times\left({\mathrm{td}}(-(\pi\times_{B}\pi^{\vee})^{*}T_{B})\cap\tau({\mathcal{P}}^{-1})\right)\right)$
$\displaystyle=p_{13*}\delta^{!}({\mathfrak{F}}\times{\mathfrak{F}}^{-1})={\mathfrak{F}}^{-1}\circ{\mathfrak{F}},$

which proves the first statement of the lemma. The second statement is almost
identical. ∎

######  Remark 2.3.

The seemingly asymmetric assignments for $\mathfrak{F}$ and
$\mathfrak{F}^{-1}$ are deliberate. For one thing $\mathfrak{F}$ restricts to
the Fourier transform in [16] over the open $U\subset B$ where $\pi:M_{U}\to
U$ is an abelian scheme,

$\mathfrak{F}|_{U}={\mathrm{ch}}({\mathcal{L}})\cap[M^{\vee}_{U}\times_{U}M_{U}]=\exp(c_{1}({\mathcal{L}}))\cap[M^{\vee}_{U}\times_{U}M_{U}],$

with no Todd contribution whatsoever. Another reason is the compatibility with
tautological classes; see Section 5.2.

We decompose both ${\mathfrak{F}}$ and ${\mathfrak{F}}^{-1}$ into

$\displaystyle\mathfrak{F}=\sum_{i}\mathfrak{F}_{i},\quad\mathfrak{F}_{i}\in\mathrm{Corr}_{B}^{i-g}(M^{\vee},M),$
$\displaystyle\mathfrak{F}^{-1}=\sum_{i}\mathfrak{F}^{-1}_{i},\quad\mathfrak{F}^{-1}_{i}\in\mathrm{Corr}_{B}^{i-g}(M,M^{\vee}).$

Now we can formulate the Fourier vanishing mentioned in Section 0.2 in terms
of correspondences. We say that the dualizable abelian fibration $\pi:M\to B$
of relative dimension $g$ satisfies the Fourier vanishing (FV), if

(FV)
${\mathfrak{F}}^{-1}_{i}\circ{\mathfrak{F}}_{j}=0\in\mathrm{Corr}_{B}^{i+j-2g}(M^{\vee},M^{\vee}),\quad
i+j<2g.$

The main theorem of this section is a motivic version of Theorem 0.1.

######  Theorem 2.4.

Let $\pi:M\to B$ be a dualizable abelian fibration of relative dimension $g$
which satisfies (FV).

  1. (i)

(Decomposition) For each $k$, the pair of self-correspondences

$\mathfrak{p}_{k}:=\sum_{i\leq
k}{\mathfrak{F}}_{i}\circ{\mathfrak{F}}^{-1}_{2g-i}\in\mathrm{Corr}^{0}_{B}(M,M),\quad\mathfrak{q}_{k+1}:=\sum_{i\geq
k+1}{\mathfrak{F}}_{i}\circ{\mathfrak{F}}^{-1}_{2g-i}\in\mathrm{Corr}^{0}_{B}(M,M)$

are orthogonal projectors, and induce a decomposition of motives

(28) $h(M)=P_{k}h(M)\oplus Q_{k+1}h(M)\in\mathrm{CHM}(B)$

with
$P_{k}h(M)=(M,{\mathfrak{p}}_{k},0),\,Q_{k+1}h(M)=(M,{\mathfrak{q}}_{k+1},0)$.

  2. (ii)

(Realization) For each $k$, the homological realization of $P_{k}h(M)$
together with the inclusion $P_{k}h(M)\to h(M)$ is the natural morphism

${{}^{\mathfrak{p}}}\tau_{\leq k+\dim
B}\pi_{*}{\mathbb{Q}}_{M}\to\pi_{*}{\mathbb{Q}}_{M}.$

Similarly, the homological realization of $h(M)\to Q_{k+1}h(M)$ is the natural
morphism

$\pi_{*}{\mathbb{Q}}_{M}\to{{}^{\mathfrak{p}}}\tau_{\geq k+1+\dim
B}\pi_{*}{\mathbb{Q}}_{M}.$

  3. (iii)

(Multiplicativity) For each pair of $k,l$, the cup-product

$\cup:h(M)\times h(M)\to h(M)$

restricts to zero on

$\cup:P_{k}h(M)\times P_{l}h(M)\to Q_{k+l+1}h(M).$

  4. (iv)

(Perverse $\supset$ Chern) For each $k$, the morphism
${\mathfrak{F}}_{k}:h(M^{\vee})(g-k)\to h(M)$ restricts to zero on

${\mathfrak{F}}_{k}:h(M^{\vee})(g-k)\to Q_{k+1}h(M).$

In view of the discussion in Section 2.2.3, Theorem 2.4(iii) specializes to
the sheaf-theoretic multiplicativity

$\cup:{{}^{\mathfrak{p}}}\tau_{\leq k+\dim
B}\pi_{*}{\mathbb{Q}}_{M}\otimes{{}^{\mathfrak{p}}}\tau_{\leq l+\dim
B}\pi_{*}{\mathbb{Q}}_{M}\to{{}^{\mathfrak{p}}}\tau_{\leq k+l+\dim
B}\pi_{*}{\mathbb{Q}}_{M},$

which further specializes to Theorem 0.1(i). Theorem 2.4(iv) specializes
immediately to Theorem 0.1(ii) by taking global cohomology. Assuming Theorem
0.2 (which will be proven in Section 3) and Theorem 3.1 (the intrinsity of the
perverse filtration (6), Theorem 2.4(iii) also specializes to Theorem 0.7
concerning the cohomology of a single curve with planar singularities.

######  Corollary 2.5.

Let $\pi:M\to B$ be a dualizable abelian fibration of relative dimension $g$
which satisfies (FV). There exists a decomposition of motives

$h(M)=\bigoplus_{i=0}^{2g}h_{i}(M)\in\mathrm{CHM}(B),$

whose homological specialization recovers the decomposition theorem for
$\pi:M\to B$.

###  2.5. Proofs

We now prove the four statements of Theorem 2.4. Afterwards, we note that
Corollary 2.5 follows immediately.

####  2.5.1. Decomposition

By Lemma 2.2, we have

$[\Delta_{M/B}]={\mathfrak{F}}\circ{\mathfrak{F}}^{-1}=\sum_{i=0}^{2g}{\mathfrak{F}}_{i}\circ{\mathfrak{F}}^{-1}_{2g-i}={\mathfrak{p}}_{k}+{\mathfrak{q}}_{k+1}\in\mathrm{Corr}_{B}^{0}(M,M),$

where the second equality holds for degree reasons. Applying the condition
(FV), we find

$\displaystyle{\mathfrak{p}}_{k}=[\Delta_{M/B}]\circ{\mathfrak{p}}_{k}=\left(\sum_{i=0}^{2g}{\mathfrak{F}}_{i}\circ{\mathfrak{F}}^{-1}_{2g-i}\right)\circ\left(\sum_{j\leq
k}{\mathfrak{F}}_{j}\circ{\mathfrak{F}}^{-1}_{2g-j}\right)$
$\displaystyle=\left(\sum_{i\leq
k}{\mathfrak{F}}_{i}\circ{\mathfrak{F}}^{-1}_{2g-i}\right)\circ\left(\sum_{j\leq
k}{\mathfrak{F}}_{j}\circ{\mathfrak{F}}^{-1}_{2g-j}\right)={\mathfrak{p}}_{k}\circ{\mathfrak{p}}_{k},$
$\displaystyle{\mathfrak{q}}_{k+1}=[\Delta_{M/B}]-{\mathfrak{p}}_{k}=([\Delta_{M/B}]-{\mathfrak{p}}_{k})\circ([\Delta_{M/B}]-{\mathfrak{p}}_{k})={\mathfrak{q}}_{k+1}\circ{\mathfrak{q}}_{k+1},$
$\displaystyle{\mathfrak{p}}_{k}\circ{\mathfrak{q}}_{k+1}={\mathfrak{p}}_{k}\circ([\Delta_{M/B}]-{\mathfrak{p}}_{k})=0,$
$\displaystyle{\mathfrak{q}}_{k+1}\circ{\mathfrak{p}}_{k}=([\Delta_{M/B}]-{\mathfrak{p}}_{k})\circ{\mathfrak{p}}_{k}=0.$

This shows that ${\mathfrak{p}}_{k}$ and ${\mathfrak{q}}_{k+1}$ are orthogonal
projectors adding up to $[\Delta_{M/B}]$, which proves the motivic
decomposition (28).

####  2.5.2. Realization

The realization statement is a consequence of the support condition (b) of
Section 1.4. We first remark that by restricting to the open subset $U\subset
B$ where $\pi_{U}:M_{U}\to U$ is an abelian scheme, the self-correspondences

$\overline{{\mathfrak{p}}}_{k}|_{U}:={\mathfrak{F}}_{k}|_{U}\circ{\mathfrak{F}}^{-1}_{2g-k}|_{U}\in\mathrm{Corr}^{0}_{U}(M_{U},M_{U})$

are projectors giving rise to the motivic decomposition in [16]:

(29) $h(M_{U})=\bigoplus_{i=0}^{2g}h_{i}(M_{U})\in\mathrm{CHM}(U),\quad
h_{i}(M_{U})=(M_{U},\overline{{\mathfrak{p}}}_{i}|_{U},0).$

The above formula for the projectors is written down explicitly in [31,
Section 2]. Moreover, it was shown in [16, Remarks after Corollary 3.2] that
the homological realization of (29) yields a canonical decomposition into
shifted local systems

$\pi_{U*}{\mathbb{Q}}_{M_{U}}=\bigoplus_{i=0}^{2g}R^{i}\pi_{U*}{\mathbb{Q}}_{M_{U}}[-i].$

In particular, for the realization of
$P_{k}h(M)|_{U}=(M_{U},{\mathfrak{p}}_{k}|_{U},0)$ we have

${\mathfrak{p}}_{k}|_{U*}(\pi_{U*}{\mathbb{Q}}_{M_{U}})=\bigoplus_{i=0}^{k}R^{i}\pi_{U*}{\mathbb{Q}}_{M_{U}}[-i]$

with
${\mathfrak{p}}_{k}|_{U*}(\pi_{U*}{\mathbb{Q}}_{M_{U}})\to\pi_{U*}{\mathbb{Q}}_{M_{U}}$
given by the natural inclusion.

Consequently, for the realization of the inclusion $P_{k}h(M)\to h(M)$, _i.e._
,

(30) ${\mathfrak{p}}_{k*}(\pi_{*}{\mathbb{Q}}_{M})\to\pi_{*}{\mathbb{Q}}_{M},$

we know that the restriction to $U$ of the perverse cohomology

(31) ${{}^{\mathfrak{p}}}{\mathcal{H}}^{i+\dim
B}({\mathfrak{p}}_{k*}(\pi_{*}{\mathbb{Q}}_{M}))$

is zero for $i>k$, and the restriction to $U$ of the induced morphism

(32) ${{}^{\mathfrak{p}}}{\mathcal{H}}^{i+\dim
B}(-):{{}^{\mathfrak{p}}}{\mathcal{H}}^{i+\dim
B}({\mathfrak{p}}_{k*}(\pi_{*}{\mathbb{Q}}_{M}))\to{{}^{\mathfrak{p}}}{\mathcal{H}}^{i+\dim
B}(\pi_{*}{\mathbb{Q}}_{M})$

is an isomorphism for $i\leq k$. The full support assumption then implies that
(31) and (32) must stay so over the entire base $B$. Therefore we have

${\mathfrak{p}}_{k*}(\pi_{*}{\mathbb{Q}}_{M})\simeq{{}^{\mathfrak{p}}}\tau_{\leq
k+\dim B}\pi_{*}{\mathbb{Q}}_{M}$

with (30) given by the natural morphism ${{}^{\mathfrak{p}}}\tau_{\leq k+\dim
B}\pi_{*}{\mathbb{Q}}_{M}\to\pi_{*}{\mathbb{Q}}_{M}$. The realization
statement for $Q_{k+1}h(M)$ follows from its $P_{k}h(M)$ counterpart together
with the motivic decomposition (28).

####  2.5.3. Multiplicativity

The motivic multiplicativity is proven via the convolution product. Recall the
convolution kernel ${\mathcal{K}}\in
D^{b}\mathrm{Coh}(M^{\vee}\times_{B}M^{\vee}\times_{B}M^{\vee})$ from the
condition (a2) of Section 1.4. We define the Chow-theoretic _convolution_
(also known as the _Pontryagin product_) to be

(33)
${\mathfrak{C}}:={\mathrm{td}}(-q^{\vee*}_{12}T_{M^{\vee}\times_{B}M^{\vee}})\cap\tau({\mathcal{K}})\in\mathrm{CH}_{*}(M^{\vee}\times_{B}M^{\vee}\times_{B}M^{\vee})$

where $q^{\vee}_{12}:M^{\vee}\times_{B}M^{\vee}\times_{B}M^{\vee}\to
M^{\vee}\times_{B}M^{\vee}$ is the projection to the first two factors and
$T_{M^{\vee}\times_{B}M^{\vee}}$ is the virtual tangent bundle of
$M^{\vee}\times_{B}M^{\vee}$ (since $M^{\vee}\times_{B}M^{\vee}$ is l.c.i.).
After Section 2.2.3, we view ${\mathfrak{C}}$ as a mixed-degree binary
relative correspondence.

######  Lemma 2.6.

We have

${\mathfrak{F}}\circ{\mathfrak{C}}\circ({\mathfrak{F}}^{-1}\times{\mathfrak{F}}^{-1})=[\Delta^{\mathrm{sm}}_{M/B}]\in\mathrm{Corr}_{B}^{0}(M,M;M).$

###### Proof.

The proof is similar to that of Lemma 2.2. We take the identity
${{\mathcal{P}}}\circ{\mathcal{K}}\simeq{\mathcal{O}}_{\Delta^{\mathrm{sm}}_{M/B}}\circ({{\mathcal{P}}}\boxtimes{{\mathcal{P}}})$
from (12) and rewrite it as

${{\mathcal{P}}}\circ{\mathcal{K}}\circ({{\mathcal{P}}}^{-1}\boxtimes{{\mathcal{P}}}^{-1})\simeq{\mathcal{O}}_{\Delta^{\mathrm{sm}}_{M/B}}\in
D^{b}\mathrm{Coh}(M\times_{B}M\times_{B}M).$

Then we apply $\tau$ on both sides. On one hand we have

(34)
$\tau({\mathcal{O}}_{\Delta^{\mathrm{sm}}_{M/B}})=\Delta^{\mathrm{sm}}_{M/B*}\tau({\mathcal{O}}_{M})=\Delta^{\mathrm{sm}}_{M/B*}\left({\mathrm{td}}(T_{M})\cap[M]\right)={\mathrm{td}}(q_{3}^{*}T_{M})\cap[\Delta^{\mathrm{sm}}_{M/B}]$

where $q_{3}:M\times_{B}M\times_{B}M\to M$ is the third projection. On the
other hand, we first compute $\tau({\mathcal{K}}\circ{\mathcal{P}}^{-1})$
where the composition is on the first factor of
$M^{\vee}\times_{B}M^{\vee}\times_{B}M^{\vee}$. We have

$\displaystyle\tau({\mathcal{K}}\circ{\mathcal{P}}^{-1})$
$\displaystyle=p_{134*}\tau(\delta^{*}({\mathcal{P}}^{-1}\boxtimes{\mathcal{K}}))$
$\displaystyle=p_{134*}\left({\mathrm{td}}(-p_{2}^{*}T_{M^{\vee}})\cap\delta^{!}\tau({\mathcal{P}}^{-1}\boxtimes{\mathcal{K}})\right)$
$\displaystyle=p_{134*}\left({\mathrm{td}}(-p_{2}^{*}T_{M^{\vee}})\cap\delta^{!}(\tau({\mathcal{P}}^{-1})\times\tau({\mathcal{K}}))\right)$
$\displaystyle=p_{134*}\delta^{!}\left(\left({\mathrm{td}}(-(\pi\times_{B}\pi^{\vee})^{*}T_{B})\cap\tau({\mathcal{P}}^{-1})\right)\times\left({\mathrm{td}}(-q_{1}^{\vee*}T_{\pi^{\vee}})\cap\tau({\mathcal{K}})\right)\right)$
$\displaystyle=({\mathrm{td}}(-q_{1}^{\vee*}T_{\pi^{\vee}})\cap\tau({\mathcal{K}}))\circ{\mathfrak{F}}^{-1},$

where $q_{i}^{\vee}:M^{\vee}\times_{B}M^{\vee}\times_{B}M^{\vee}\to M^{\vee}$
is the $i$-th projection. Similarly, we have

$\tau({\mathcal{K}}\circ({\mathcal{P}}^{-1}\boxtimes{\mathcal{P}}^{-1}))=({\mathrm{td}}(-q_{1}^{\vee*}T_{\pi^{\vee}}-q_{2}^{\vee*}T_{\pi^{\vee}})\cap\tau({\mathcal{K}}))\circ({\mathfrak{F}}^{-1}\times{\mathfrak{F}}^{-1}).$

Finally, comparing with (34) we find

$\displaystyle[\Delta^{\mathrm{sm}}_{M/B}]$
$\displaystyle={\mathrm{td}}(-q_{3}^{*}T_{M})\cap\tau({\mathcal{O}}_{\Delta^{\mathrm{sm}}_{M/B}})$
$\displaystyle={\mathrm{td}}(-q_{3}^{*}T_{M})\cap\tau({{\mathcal{P}}}\circ{\mathcal{K}}\circ({{\mathcal{P}}}^{-1}\boxtimes{{\mathcal{P}}}^{-1}))$
$\displaystyle={\mathrm{td}}(-q_{3}^{*}T_{M})\cap
p_{124*}\tau(\delta^{*}(({\mathcal{K}}\circ({{\mathcal{P}}}^{-1}\boxtimes{{\mathcal{P}}}^{-1}))\boxtimes{\mathcal{P}}))$
$\displaystyle={\mathrm{td}}(-q_{3}^{*}T_{M})\cap
p_{124*}\left({\mathrm{td}}(-p_{3}^{*}T_{M^{\vee}})\cap\delta^{!}\left(\tau({\mathcal{K}}\circ({\mathcal{P}}^{-1}\boxtimes{\mathcal{P}}^{-1}))\times\tau({\mathcal{P}})\right)\right)$
$\displaystyle={\mathrm{td}}(-q_{3}^{*}T_{M})\cap
p_{124*}\left({\mathrm{td}}(-p_{3}^{*}T_{M^{\vee}})\phantom{{\mathfrak{F}}^{-1}}\right.$
$\displaystyle\quad\quad\quad\cap\left.\delta^{!}\left(\left(({\mathrm{td}}(-q_{1}^{\vee*}T_{\pi^{\vee}}-q_{2}^{\vee*}T_{\pi^{\vee}})\cap\tau({\mathcal{K}}))\circ({\mathfrak{F}}^{-1}\times{\mathfrak{F}}^{-1})\right)\times\tau({\mathcal{P}})\right)\right)$
$\displaystyle=p_{124*}\delta^{!}\left(\left(({\mathrm{td}}(-q_{12}^{\vee*}T_{M^{\vee}\times_{B}M^{\vee}})\cap\tau({\mathcal{K}}))\circ({\mathfrak{F}}^{-1}\times{\mathfrak{F}}^{-1})\right)\times\left({\mathrm{td}}(-T_{M^{\vee}\times_{B}M})\cap\tau({\mathcal{P}})\right)\right)$
$\displaystyle=p_{124*}\delta^{!}\left(({\mathfrak{C}}\circ({\mathfrak{F}}^{-1}\times{\mathfrak{F}}^{-1}))\times{\mathfrak{F}}\right)={\mathfrak{F}}\circ{\mathfrak{C}}\circ({\mathfrak{F}}^{-1}\times{\mathfrak{F}}^{-1}),$

which proves the lemma. ∎

The key input from the condition (a2) of Section 1.4 is a degree estimate

(35) ${\mathfrak{C}}\in\mathrm{Corr}^{\geq-g}_{B}(M^{\vee},M^{\vee};M^{\vee})$

since the convolution kernel ${\mathcal{K}}\in
D^{b}\mathrm{Coh}(M^{\vee}\times_{B}M^{\vee}\times_{B}M^{\vee})$ is of
codimension $g$. Now the restriction of the cup-product $\cup:h(M)\times
h(M)\to h(M)$ to

$\cup:P_{k}h(M)\times P_{l}h(M)\to Q_{k+l+1}h(M)$

is given by the composition

(36)
${\mathfrak{q}}_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{M/B}]\circ({\mathfrak{p}}_{k}\times{\mathfrak{p}}_{l})\in\mathrm{Corr}^{0}_{B}(M,M;M).$

By Lemma 2.6 we have

(37)
${\mathfrak{q}}_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{M/B}]\circ({\mathfrak{p}}_{k}\times{\mathfrak{p}}_{l})={\mathfrak{q}}_{k+l+1}\circ{\mathfrak{F}}\circ{\mathfrak{C}}\circ\left(({\mathfrak{F}}^{-1}\circ{\mathfrak{p}}_{k})\times({\mathfrak{F}}^{-1}\circ{\mathfrak{p}}_{l})\right).$

Expanding the right-hand side of (37) and applying (FV), we find

(38) $\displaystyle\left(\sum_{i_{3}\geq
k+l+1}{\mathfrak{F}}_{i_{3}}\circ{\mathfrak{F}}^{-1}_{2g-i_{3}}\right)\circ\left(\sum_{j_{3}}{\mathfrak{F}}_{j_{3}}\right)\circ{\mathfrak{C}}$
$\displaystyle\quad\circ\left(\left(\left(\sum_{j_{1}}{\mathfrak{F}}^{-1}_{j_{1}}\right)\circ\left(\sum_{i_{1}\leq
k}{\mathfrak{F}}_{i_{1}}\circ{\mathfrak{F}}_{2g-i_{1}}\right)\right)\times\left(\left(\sum_{j_{2}}{\mathfrak{F}}^{-1}_{j_{2}}\right)\circ\left(\sum_{i_{2}\leq
l}{\mathfrak{F}}_{i_{2}}\circ{\mathfrak{F}}_{2g-i_{2}}\right)\right)\right)$
$\displaystyle{}={}$ $\displaystyle\left(\sum_{i_{3}\geq
k+l+1}{\mathfrak{F}}_{i_{3}}\circ{\mathfrak{F}}^{-1}_{2g-i_{3}}\right)\circ\left(\sum_{j_{3}\geq
k+l+1}{\mathfrak{F}}_{j_{3}}\right)\circ{\mathfrak{C}}$
$\displaystyle\quad\circ\left(\left(\left(\sum_{j_{1}\geq
2g-k}{\mathfrak{F}}^{-1}_{j_{1}}\right)\circ\left(\sum_{i_{1}\leq
k}{\mathfrak{F}}_{i_{1}}\circ{\mathfrak{F}}_{2g-i_{1}}\right)\right)\times\left(\left(\sum_{j_{2}\geq
2g-l}{\mathfrak{F}}^{-1}_{j_{2}}\right)\circ\left(\sum_{i_{2}\leq
l}{\mathfrak{F}}_{i_{2}}\circ{\mathfrak{F}}_{2g-i_{2}}\right)\right)\right)$
$\displaystyle{}={}$
$\displaystyle{\mathfrak{q}}_{k+l+1}\circ\left(\sum_{j_{3}\geq
k+l+1}{\mathfrak{F}}_{j_{3}}\right)\circ{\mathfrak{C}}\circ\left(\left(\left(\sum_{j_{1}\geq
2g-k}{\mathfrak{F}}^{-1}_{j_{1}}\right)\circ{\mathfrak{p}}_{k}\right)\times\left(\left(\sum_{j_{2}\geq
2g-l}{\mathfrak{F}}^{-1}_{j_{2}}\right)\circ{\mathfrak{p}}_{l}\right)\right).$

We observe that

$\displaystyle\sum_{j_{1}\geq
2g-k}{\mathfrak{F}}^{-1}_{j_{1}}\in\mathrm{Corr}_{B}^{\geq
g-k}(M,M^{\vee}),\quad\sum_{j_{2}\geq
2g-l}{\mathfrak{F}}^{-1}_{j_{2}}\in\mathrm{Corr}_{B}^{\geq g-l}(M,M^{\vee}),$
$\displaystyle\sum_{j_{3}\geq
k+l+1}{\mathfrak{F}}_{j_{3}}\in\mathrm{Corr}_{B}^{\geq k+l+1-g}(M^{\vee},M).$

This, together with the degree estimate (35), yields for (38) a total degree
of

$\geq-g+(g-k)+(g-l)+(k+l+1-g)=1.$

In other words, we find
${\mathfrak{q}}_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{M/B}]\circ({\mathfrak{p}}_{k}\times{\mathfrak{p}}_{l})$
to lie strictly in $\mathrm{Corr}^{\geq 1}_{B}(M,M;M)$ which, compared with
(36), implies the vanishing

${\mathfrak{q}}_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{M/B}]\circ({\mathfrak{p}}_{k}\times{\mathfrak{p}}_{l})=0.$

This proves the multiplicativity statement.

####  2.5.4. Perverse $\supset$ Chern

The final statement amounts to showing that

${\mathfrak{q}}_{k+1}\circ{\mathfrak{F}}_{k}=0\in\mathrm{Corr}_{B}^{k-g}(M^{\vee},M),$

which follows immediately from (FV) since

${\mathfrak{q}}_{k+1}\circ{\mathfrak{F}}_{k}=\left(\sum_{i\geq
k+1}{\mathfrak{F}}_{i}\circ{\mathfrak{F}}^{-1}_{2g-i}\right)\circ{\mathfrak{F}}_{k}=\sum_{i\geq
k+1}{\mathfrak{F}}_{i}\circ\left({\mathfrak{F}}^{-1}_{2g-i}\circ{\mathfrak{F}}_{k}\right)=0.$

The proof of Theorem 2.4 is now complete. ∎

####  2.5.5. Motivic decomposition

Finally, we note that our construction yields a motivic decomposition of
$\pi:M\to B$ which proves Corollary 2.5.

One subtlety of the sequence of motives $P_{k}h(M)$ is that we do not know if
$P_{k}h(M)$ is a summand of $P_{l}h(M)$ for $k<l$. While the condition (FV)
immediately implies

(39) ${\mathfrak{p}}_{l}\circ{\mathfrak{p}}_{k}={\mathfrak{p}}_{k},\quad k<l,$

it is unclear if
${\mathfrak{p}}_{k}\circ{\mathfrak{p}}_{l}={\mathfrak{p}}_{k}$. However, we
can fix this issue by introducing a minor modification

$\widetilde{{\mathfrak{p}}}_{k}:={\mathfrak{p}}_{k}\circ\cdots\circ{\mathfrak{p}}_{2g}\in\mathrm{Corr}^{0}_{B}(M,M),\quad\overline{{\mathfrak{p}}}_{k}:=\widetilde{{\mathfrak{p}}}_{k}-\widetilde{{\mathfrak{p}}}_{k-1}\in\mathrm{Corr}^{0}_{B}(M,M).$

Now it is easy to verify that both $\widetilde{{\mathfrak{p}}}_{k}$ and
$\overline{{\mathfrak{p}}}_{k}$ are projectors, and

$\widetilde{{\mathfrak{p}}}_{k}\circ\widetilde{{\mathfrak{p}}}_{l}=\widetilde{{\mathfrak{p}}}_{l}\circ\widetilde{{\mathfrak{p}}}_{k}=\widetilde{{\mathfrak{p}}}_{k},\quad
k<l.$

For instance, we have by (39)

$\widetilde{{\mathfrak{p}}}_{k}\circ\widetilde{{\mathfrak{p}}}_{l}={\mathfrak{p}}_{k}\circ\cdots\circ{\mathfrak{p}}_{2g}\circ{\mathfrak{p}}_{l}\circ\cdots\circ{\mathfrak{p}}_{2g}={\mathfrak{p}}_{k}\circ\cdots\circ{\mathfrak{p}}_{2g}=\widetilde{{\mathfrak{p}}}_{k},\quad
k\leq l.$

The resulting motives
$\widetilde{P}_{k}h(M)=(M,\widetilde{{\mathfrak{p}}}_{k},0)$ and
$h_{k}(M)=(M,\overline{{\mathfrak{p}}}_{k},0)$ satisfy

$\widetilde{P}_{k}h(M)=\widetilde{P}_{k-1}h(M)\oplus
h_{k}(M)\in\mathrm{CHM}(B).$

By induction, we also find that the $\overline{{\mathfrak{p}}}_{k}$ are
mutually orthogonal projectors giving rise to a motivic decomposition

(40) $h(M)=\bigoplus_{i=0}^{2g}h_{i}(M)\in\mathrm{CHM}(B).$

The support argument of Section 2.5.2 then shows that the homological
realization of (40) provides a decomposition

$\pi_{*}{\mathbb{Q}}_{M}=\bigoplus_{i=0}^{2g}{{}^{\mathfrak{p}}}{\mathcal{H}}^{i+\dim
B}(\pi_{*}{\mathbb{Q}}_{M})[-i-\dim B]\in D^{b}_{c}(B).$

This finishes the proof of Corollary 2.5. ∎

##  3\. Compactified Jacobians

###  3.1. Overview

In this section, we treat the geometry of compactified Jacobian fibrations;
our main purpose is to prove Theorem 0.2.

Throughout this section, we let $C\to B$ be a flat family of integral
projective curves of arithmetic genus $g$ with planar singularities over an
irreducible base $B$. We further assume that $C\to B$ admits a section. We
denote by $\pi:\overline{J}_{C}\to B$ the compactified Jacobian fibration, and
assume that the total space $\overline{J}_{C}$ is nonsingular. In particular,
$B$ is nonsingular and $\pi:\overline{J}_{C}\to B$ is flat with integral
fibers of dimension $g$.

###  3.2. Compactified Jacobians

Let $J_{C}\subset\overline{J}_{C}$ be the Jacobian fibration parameterizing
line bundles. It is an open dense subset of $\overline{J}_{C}$, which forms a
nonsingular commutative group scheme over $B$.

The Fourier–Mukai theory of compactified Jacobian fibrations was developed by
Arinkin [5, 6]. More precisely, Arinkin constructed in [5] the Poincaré line
bundle ${\mathcal{P}}$ over the product $J_{C}\times_{B}\overline{J}_{C}$
extending the standard normalized Poincaré line bundle for Jacobians
associated with nonsingular curves. Then in [6] Arinkin further extended this
line bundle ${\mathcal{P}}$ to a Cohen–Macaulay coherent sheaf (which we still
denote by ${\mathcal{P}}$ for notational convenience) on
$\overline{J}_{C}\times_{B}\overline{J}_{C}$, and proved that ${\mathcal{P}}$
induces a derived auto-equivalence

$\mathrm{FM}_{{\mathcal{P}}}:D^{b}\mathrm{Coh}(\overline{J}_{C})\xrightarrow{~{}~{}\simeq~{}~{}}D^{b}\mathrm{Coh}(\overline{J}_{C}).$

The inverse of this transform is induced by

${\mathcal{P}}^{-1}:={\mathcal{P}}^{\vee}\otimes
p_{2}^{*}\omega_{\pi}[g],\quad{\mathcal{P}}^{\vee}:={{\mathcal{H}}}om({\mathcal{P}},{\mathcal{O}}_{\overline{J}_{C}\times_{B}\overline{J}_{C}}).$

Here $\omega_{\pi}$ is the relative canonical bundle with respect to
$\pi:\overline{J}_{C}\to B$, and
$p_{2}:\overline{J}_{C}\times_{B}\overline{J}_{C}\to B$ is the natural
morphism.

Furthermore, the support theorem of Ngô [42] and the Severi inequality [35,
Lemma 4.1] ensure that the decomposition theorem associated with
$\pi:\overline{J}_{C}\to B$ has full support. Therefore, in order to prove
Theorem 0.2, it remains to verify:

  1. (i)

the convolution kernel

(41) ${\mathcal{K}}\in
D^{b}\mathrm{Coh}(\overline{J}_{C}\times_{B}\overline{J}_{C}\times_{B}\overline{J}_{C})$

is of codimension $g$;

  2. (ii)

the condition (FV) holds for $\pi:\overline{J}_{C}\to B$.

We will discuss (i), which is a result of Arinkin, in Section 3.3, and will
discuss (ii) in Section 3.5.

We conclude this section by recalling the _canonical_ perverse filtration
constructed in [38] for the compactified Jacobian $\overline{J}_{C_{0}}$
associated with _any_ integral projective curve $C_{0}$ with planar
singularities; this is the filtration used in Theorem 0.7. For such a $C_{0}$,
we include it in a family of curves $C\to B$ over an irreducible $B$ such that
the associated compactified Jacobian fibration $\pi:\overline{J}_{C}\to B$ is
nonsingular and $\overline{J}_{C_{0}}\subset\overline{J}_{C}$ is the closed
fiber over $0\in B$. Then the construction of Section 1.3.3 yields a perverse
filtration

$P_{0}H^{*}(\overline{J}_{C_{0}},{\mathbb{Q}})\subset
P_{1}H^{*}(\overline{J}_{C_{0}},{\mathbb{Q}})\subset\cdots\subset
H^{*}(\overline{J}_{C_{0}},{\mathbb{Q}})$

via the morphism $\pi:\overline{J}_{C}\to B$.

######  Theorem 3.1 (Maulik–Yun [38, Theorem 1.1]).

The perverse filtration defined above is independent of the choice of the
family $C\to B$ of curves.

###### Proof.

In [38, Theorem 1.1], the family $C\to B$ is chosen with certain assumptions.
However, as pointed out in [38, Proposition 2.15], the smoothness of
$\overline{J}_{C}$ suffices. ∎

In particular, Theorem 0.7 follows from Theorem 2.4(iii) and Theorem 0.2,
where we include $C_{0}$ in a family $\pi:C\to B$ as a closed fiber.

###  3.3. Arinkin’s dimension bound and the convolution kernel

We first give an expression of the convolution kernel (41). Denote by
$\overline{J}_{C}^{n+1}$ the $(n+1)$-th relative product. It carries natural
projections

$\pi_{n+1}:\overline{J}_{C}^{n+1}\to B,\quad
p_{i}:\overline{J}_{C}^{n+1}\to\overline{J}_{C},\quad
p_{i,j}:\overline{J}_{C}^{n+1}\to\overline{J}^{2}_{C},\quad\cdots.$

We consider the object

${\mathcal{K}}_{n}:=p_{1,\dots,n,*}\left(p_{1,n+1}^{*}{\mathcal{P}}\otimes
p_{2,n+1}^{*}{\mathcal{P}}\otimes\cdots\otimes
p_{n,n+1}^{*}{\mathcal{P}}\right)\otimes\pi_{n}^{*}\omega_{\pi}[g]\in
D^{b}\mathrm{Coh}(\overline{J}_{C}^{n}).$

Here the boundedness of ${\mathcal{K}}_{n}$ is a consequence of the flatness
of ${\mathcal{P}}$ with respect to both factors [6, Theorem A and Lemma 6.1].
Let $\nu:\overline{J}_{C}\to\overline{J}_{C}$ be the involution sending a
sheaf to its dual, so that we can write

${\mathcal{P}}^{\vee}=(\mathrm{id}_{\overline{J}_{C}}\times_{B}\nu)^{*}{\mathcal{P}}.$

Then by definition, we may express the convolution kernel (41) as

${\mathcal{K}}=\left(\mathrm{id}_{\overline{J}_{C}\times_{B}\overline{J}_{C}}\times_{B}\nu\right)^{*}{\mathcal{K}}_{3}.$

Therefore, the condition (i) of Section 3.2 is a special case of the following
result of Arinkin [6, Section 7.1].

######  Proposition 3.2 (Arinkin).

For any $n$, we have

$\mathrm{codim}_{\overline{J}^{n}_{C}}(\mathrm{supp}({\mathcal{K}}_{n}))\geq
g.$

Note that ${\mathcal{K}}_{n}$ is a classical object for low values of $n$.
When $n=1$ the proposition was a step towards establishing [5, Theorem A]
concerning the Fourier–Mukai transform of the structure sheaf. When $n=2$, the
proposition was used in [6] to show that $\mathrm{FM}_{\mathcal{P}}$ is a
derived equivalence. The same argument works for the general case, which we
review as follows.

###### Proof of Proposition 3.2.

We first fix some notation. For an integral projective curve $C_{b}$, we
denote by $g_{a}(C_{b})$ its arithmetic genus, and $\widetilde{C}_{b}$ its
normalization. We denote by $\delta(C_{b})\in{\mathbb{Z}}_{\geq 0}$ the
dimension of the maximal affine group of $\mathrm{Pic}(C_{b})$. Clearly, when
$C_{b}$ has planar singularities, we have

$\delta(C_{b})=g_{a}(C_{b})-g_{a}(\widetilde{C}_{b}).$

This yields a constructible function

$\delta:B\to{\mathbb{Z}}_{\geq 0},\quad b\mapsto\delta(C_{b})$

associated with $C\to B$. For an irreducible subvariety $Z\subset B$, we
define $\delta(Z)$ to be the value of $\delta(b)$ at a general point $b\in Z$.

Step 1. Recall that the smoothness of $\overline{J}_{C}$ yields the Severi
inequality [35, Lemma 4.1]

$\mathrm{codim}_{B}(Z)\geq\delta_{Z}$

for any irreducible $Z\subset B$. Therefore, it suffices to show that for any
curve $C_{b}$ in the family $C\to B$, we have

$\mathrm{codim}_{\overline{J}_{C_{b}}}\left(\mathrm{supp}({\mathcal{K}}_{n})\cap\overline{J}_{C_{b}}\right)\geq
g_{a}(\widetilde{C}_{b}).$

Step 2. Now we focus on a fixed curve $C_{b}$. For a point
$F\in\overline{J}_{C_{b}}$, we denote by ${\mathcal{P}}_{F}$ the restriction
of ${\mathcal{P}}$ to $\overline{J}_{C_{b}}$. If we further restrict this
sheaf to the open subset $J_{C_{b}}\subset\overline{J}_{C_{b}}$ parameterizing
line bundles on $C_{b}$, it is a line bundle.

If $(F_{1},F_{2},\cdots,F_{n})\in\overline{J}^{n}_{C_{b}}$ lies in the support
of ${\mathcal{K}}_{n}$, then there is a nontrivial cohomology

(42)
$H^{*}(\overline{J}_{C_{b}},{\mathcal{P}}_{F_{1}}\otimes{\mathcal{P}}_{F_{2}}\otimes\cdots\otimes{\mathcal{P}}_{F_{n}})\neq
0$

By the same argument as in [6, Proposition 7.5], (42) ensures that

(43)
$\bigotimes_{i=1}^{n}({\mathcal{P}}_{F_{i}}|_{J_{C_{b}}})\simeq{\mathcal{O}}_{J_{C_{b}}}.$

Step 3. Finally, we claim that (43) expresses a codimension
$g_{a}(\widetilde{C}_{b})$ condition for

$(F_{1},F_{2},\cdots,F_{n})\in\overline{J}^{n}_{C_{b}}.$

This was explained in [6, Corollary 7.3 and Proposition 7.4]. Indeed, pulling
back the isomorphism (43) via the Abel–Jacobi map
$C_{b}\hookrightarrow\overline{J}_{C_{b}}$, we obtain that

(44)
$\bigotimes_{i=1}^{n}({F_{i}}|_{C^{\mathrm{reg}}_{b}})\simeq{\mathcal{O}}_{C^{\mathrm{reg}}_{b}}.$

Then the same argument as in [6, Proposition 7.4] applied to the action
morphism

$\mu\times\mathrm{id}_{\overline{J}_{C_{b}}}:J_{C_{b}}\times\overline{J}^{n}_{C_{b}}\to\overline{J}^{n}_{C_{b}}$

shows that (44) is a codimension $g_{a}(\widetilde{C}_{b})$ condition. This
proves Proposition 3.2. ∎

We have thus verified (i) of Section 3.2.

###  3.4. Remarks on the Fourier vanishing for abelian schemes

We first discuss two proofs of (FV) for abelian schemes.

As in Section 1.2, we let $\pi:A\to B$ be an abelian scheme of relative
dimension $g$ with $\pi^{\vee}:A^{\vee}\to B$ its dual. Then by Lemma 2.2, for
degree reasons we have for any $k\neq 2g$,

(45)
$\sum_{i+j=k}{\mathfrak{F}}^{-1}_{i}\circ{\mathfrak{F}}_{j}=0\in\mathrm{Corr}_{B}^{k-2g}(A^{\vee},A^{\vee}).$

The idea of [7, 16] is that, if we apply the pullback associated with the
multiplication by $N$ map on the first factor

$[N]:A^{\vee}\times_{B}A^{\vee}\to A^{\vee}\times_{B}A^{\vee},$

we may scale each individual term of (45) by a different constant $N^{j}$:

(46) $\sum_{i+j=k}N^{j}{\mathfrak{F}}^{-1}_{i}\circ{\mathfrak{F}}_{j}=0.$

Since this holds for every $N$, we obtain the desired vanishing

${\mathfrak{F}}^{-1}_{i}\circ{\mathfrak{F}}_{j}=0,\quad i+j\neq 2g.$

However, this argument does not work for abelian fibrations with singular
fibers, since it relies on the facts that there is a multiplication by $N$ map
on the total space $A^{\vee}$ and the Poincaré complex is a line bundle. Both
fail when there are singular fibers.

We now present another proof of the weaker version (FV) whose idea can be
generalized. For convenience, we focus on the Jacobian fibration $\pi:J_{C}\to
B$ associated with a family of nonsingular projective curves $C\to B$ of genus
$g$. Recall that we have the normalized Poincaré line bundle ${\mathcal{L}}$.
The discussion in Section 3.3 actually yields the dimension bound

$\mathrm{codim}_{J^{2}_{C}}\left(\mathrm{supp}({\mathcal{L}}^{-1}\circ{\mathcal{L}}^{\otimes
N})\right)\geq g$

for any $N\in{\mathbb{Z}}$. Therefore, if we take the Chern character of
${\mathcal{L}}^{-1}\circ{\mathcal{L}}^{\otimes N}$, we obtain the equation
(46) only for $i+j<2g$. This yields (FV) for $\pi:J_{C}\to B$.

The second proof above relies on the following two ingredients:

  1. (i)

the Arinkin-type dimension bound of Section 3.3, and

  2. (ii)

the $K$-theoretic operations $(-)^{\otimes N}$ which help us to scale the
Chern character of different degrees distinctly.

In the next section, we will generalized them to treat
$\pi:\overline{J}_{C}\to B$ where the second ingredient is replaced by the
Adams operations from $K$-theory.

###  3.5. Adams operations and Fourier vanishing

We now verify the condition (FV) for $\pi:\overline{J}_{C}\to B$, which proves
(ii) of Section 3.2. Consequently, this completes the proofs of Theorems 0.2,
0.3, and 0.7.

####  3.5.1. Dimension bound

We consider the closed embedding

(47)
$i:\overline{J}_{C}\times_{B}\overline{J}_{C}\hookrightarrow\overline{J}_{C}\times\overline{J}_{C}.$

For any $N\in{\mathbb{Z}}_{>0}$, the object

$(i_{*}{\mathcal{P}})^{\otimes N}\in
D^{b}\mathrm{Coh}(\overline{J}_{C}\times\overline{J}_{C})$

is bounded, and is exact off
$\overline{J}_{C}\times_{B}\overline{J}_{C}\subset\overline{J}_{C}\times\overline{J}_{C}$.
Then we consider

$\widetilde{{\mathcal{K}}}(N):={\mathcal{P}}^{-1}\circ(i_{*}{\mathcal{P}})^{\otimes
N}\in D^{b}\mathrm{Coh}(\overline{J}_{C}\times\overline{J}_{C}).$

By definition (and by the flatness of ${\mathcal{P}}$ with respect to both
factors), it is constructed by pulling back the two objects
$(i_{*}{\mathcal{P}})^{\otimes N},{\mathcal{P}}^{-1}$ to the triple product

$\overline{J}_{C}\times\overline{J}_{C}\times_{B}\overline{J}_{C},$

taking their tensor product, and pushing forward to the first and the third
factors.

######  Proposition 3.3.

For any $N$, the object $\widetilde{{\mathcal{K}}}(N)\in
D^{b}\mathrm{Coh}(\overline{J}_{C}\times\overline{J}_{C})$ is supported on a
codimension $g$ subset of $\overline{J}_{C}\times_{B}\overline{J}_{C}$.

###### Proof.

The proof relies on Arinkin’s dimension estimate which goes through the
following steps. It is essentially parallel to the proof of Proposition 3.2.

Step 1. We first show that the set-theoretic support of
$\widetilde{{\mathcal{K}}}(N)$ is contained in
$\overline{J}_{C}\times_{B}\overline{J}_{C}$. By definition and the projection
formula, the object $\widetilde{{\mathcal{K}}}(N)$ is obtained via the
pushforward of an object supported on

$\overline{J}_{C}\times_{B}\overline{J}_{C}\times_{B}\overline{J}_{C}\hookrightarrow\overline{J}_{C}\times\overline{J}_{C}\times_{B}\overline{J}_{C}$

Therefore, it can be written as the pushforward through the following chain of
maps

$\overline{J}_{C}\times_{B}\overline{J}_{C}\times_{B}\overline{J}_{C}\rightarrow\overline{J}_{C}\times_{B}\overline{J}_{C}\hookrightarrow\overline{J}_{C}\times\overline{J}_{C}$

where the first map is the projection to the first and the third factors, and
the second map is the natural inclusion $i$. This proves the claim.

Step 2. Now any closed point in the support of $\widetilde{K}(N)$ can be
represented by a pair of sheaves on $C_{b}\subset C$:

(48)
$(F_{1},F_{2})\in\mathrm{supp}(\widetilde{K}(N))\cap\overline{J}_{C_{b}}.$

As in Step 2 of the proof of Proposition 3.2, it suffices to show that

(49)
$\mathrm{codim}_{\overline{J}_{C_{b}}}\left(\mathrm{supp}(\widetilde{{\mathcal{K}}}(N))\cap\overline{J}_{C_{b}}\right)\geq
g_{a}(\widetilde{C}_{b})$

where $\widetilde{C}_{b}$ is the normalization of $C_{b}$.

Step 3. For a pair $(F_{1},F_{2})$ as in (48), by base change we obtain a
nontrivial cohomology

$H^{*}\left(\overline{J}_{C},(\iota_{*}{\mathcal{P}}_{F_{1}})^{\otimes
N}\otimes\iota_{*}{\mathcal{P}}^{\vee}_{F_{2}}\right)\neq 0.$

Here $\iota:\overline{J}_{C_{b}}\hookrightarrow\overline{J}_{C}$ is the closed
embedding of a closed fiber and the tensor product takes place on the ambient
variety $\overline{J}_{C}$. Equivalently, we may write this nontrivial
cohomology on the fiber $\overline{J}_{C_{b}}$ as

(50)
$H^{*}\left(\overline{J}_{C_{b}},(\iota^{*}\iota_{*}{\mathcal{P}}_{F_{1}})^{\otimes
N}\otimes{\mathcal{P}}^{\vee}_{F_{2}}\right)\neq 0.$

Since $\overline{J}_{C}$ is nonsingular, the complex
$\iota^{*}\iota_{*}{\mathcal{P}}_{F_{1}}$ is _perfect_ on the possibly
singular fiber $\overline{J}_{C_{b}}$. The following claim describes all the
cohomology sheaves of this complex.

Claim. Each cohomology sheaf
${\mathcal{H}}^{k}(\iota^{*}\iota_{*}{\mathcal{P}}_{F_{1}})\in\mathrm{Coh}(\overline{J}_{C_{b}})$
is a direct sum of finite copies of ${\mathcal{P}}_{F_{1}}$ on
$\overline{J}_{C_{b}}$.

###### Proof of the claim.

We first note the decomposition

$\iota^{*}\iota_{*}{\mathcal{O}}_{\overline{J}_{C_{b}}}\simeq\bigoplus_{k}{\mathcal{H}}^{k}(\iota^{*}\iota_{*}{\mathcal{O}}_{\overline{J}_{C_{b}}})[-k]$

where each term in the right-hand side is a free
${\mathcal{O}}_{\overline{J}_{C_{b}}}$-module. This can be seen from base
change and the corresponding statement for the embedding
$\\{b\\}\hookrightarrow B$.

Next we relate the two objects777The two objects (51) may not be isomorphic in
$D^{b}\mathrm{Coh}(\overline{J}_{C_{b}})$, since the second may not be perfect
while the first is always perfect.

(51)
$\iota^{*}\iota_{*}{\mathcal{P}}_{F_{1}},\quad\iota^{*}\iota_{*}{\mathcal{O}}_{\overline{J}_{C_{b}}}\otimes{\mathcal{P}}_{F_{1}}\in
D^{b}\mathrm{Coh}(\overline{J}_{C_{b}}).$

By the projection formula, we see that they are isomorphic after pushing
forward to $\overline{J}_{C}$,

$\iota_{*}(\iota^{*}\iota_{*}{\mathcal{P}}_{F_{1}})\simeq\iota_{*}\left(\iota^{*}\iota_{*}{\mathcal{O}}_{\overline{J}_{C_{b}}}\otimes{\mathcal{P}}_{F_{1}}\right).$

Therefore, the cohomology sheaves of the two objects (51), which are
${\mathcal{O}}_{\overline{J}_{C_{b}}}$-modules themselves, are isomorphic as
${\mathcal{O}}_{\overline{J}_{C}}$-modules. This forces them to be isomorphic
as ${\mathcal{O}}_{\overline{J}_{C_{b}}}$-modules:

${\mathcal{H}}^{k}(\iota^{*}\iota_{*}{\mathcal{P}}_{F_{1}})\simeq{\mathcal{H}}^{k}\left(\iota^{*}\iota_{*}{\mathcal{O}}_{\overline{J}_{C_{b}}}\otimes{\mathcal{P}}_{F_{1}}\right)\in\mathrm{Coh}(\overline{J}_{C_{b}}).$

The claim then follows from the decomposition of
$\iota^{*}\iota_{*}{\mathcal{O}}_{\overline{J}_{C_{b}}}$. ∎

Step 4. By the claim of Step 3, we know that the object
$\iota^{*}\iota_{*}{\mathcal{P}}_{F_{1}}$ admits an increasing filtration
induced by the standard truncation functors, whose graded pieces are direct
sums of finite copies of ${\mathcal{P}}_{F_{1}}$. Hence the cohomology (50)
admits a filtration whose graded pieces are direct sums of finite copies of

(52) $H^{*}\left(\overline{J}_{C_{b}},{\mathcal{P}}_{F_{1}}^{\otimes
N}\otimes{\mathcal{P}}^{-1}_{F_{2}}\right).$

In particular, we obtain from (50) that there exists a nontrivial cohomology
of the type (52).

By the same argument as in [6, Section 7.1] or Steps 2 and 3 in the proof of
Proposition 3.2, the non-vanishing cohomology (52) expresses a codimension
$g_{a}(\widetilde{C}_{b})$ condition for the pair
$(F_{1},F_{2})\in\overline{J}^{2}_{C_{b}}$: the constraint is given by

$({\mathcal{P}}_{F_{1}}|_{J_{C_{b}}})^{\otimes
N}\otimes({\mathcal{P}}_{F_{2}}|_{J_{C_{b}}})^{\vee}\simeq{\mathcal{O}}_{J_{C_{b}}},$

which further yields

$(F_{1}\big{|}_{C^{\mathrm{reg}}_{b}})^{\otimes
N}\otimes(F_{2}\big{|}_{C^{\mathrm{reg}}_{b}})^{\vee}\simeq{\mathcal{O}}_{C^{\mathrm{reg}}_{b}}.$

This is a codimension $g_{a}(\widetilde{C}_{b})$ condition, and the proof of
(49) is complete. ∎

For our purpose we also consider variants of $\widetilde{{\mathcal{K}}}(N)$,
replacing the (derived) tensor product partially by the (derived) exterior
product:

$\widetilde{{\mathcal{K}}}^{\lambda}(N_{1},N_{2},\dots,N_{k}):={\mathcal{P}}^{-1}\circ\bigotimes_{i=1}^{k}\left(\wedge^{N_{i}}(i_{*}{\mathcal{P}})\right)\in
D^{b}\mathrm{Coh}(\overline{J}_{C}\times\overline{J}_{C}).$

Since $\wedge^{N_{i}}(i_{*}{\mathcal{P}})$ is an isotypic component of the
natural $\mathfrak{S}_{n}$-action on $(i_{*}{\mathcal{P}})^{\otimes N_{i}}$,
the object above is a direct summand of
$\widetilde{{\mathcal{K}}}(\sum_{i}N_{i})$. This gives the following.

######  Corollary 3.4.

For any $k$-tuple $(N_{1},N_{2},\dots,N_{k})$, the object
$\widetilde{{\mathcal{K}}}^{\lambda}(N_{1},N_{2},\dots,N_{k})$ is supported on
a codimension $g$ subset of $\overline{J}_{C}\times_{B}\overline{J}_{C}$.

####  3.5.2. Adams operations

With the Arinkin-type dimension bound in hand, we proceed to $K$-theoretic
operations following Gillet–Soulé [22].

Let $i:X\hookrightarrow Y$ be a closed embedding of finite type schemes. We
denote by $K_{X}(Y)$ the Grothendieck group of bounded complexes of locally
free sheaves on $Y$ which are exact off $X$. In [22, Section 4], a sequence of
_Adams operations_

$\psi^{N}:K_{X}(Y)\to K_{X}(Y),\quad N\geq 1$

were constructed using the $\lambda$-ring structure on $K_{X}(Y)$ given by the
(derived) exterior product, together with the induction formula

(53)
$\psi^{N}-\psi^{N-1}\otimes\lambda^{1}+\cdots+(-1)^{N-1}\psi^{1}\otimes\lambda^{N-1}+(-1)^{N}N\lambda^{N}=0.$

The compatibility of the Adams operations with the localized Chern character
${\mathrm{ch}}_{X}^{Y}(-)$ in [21, Section 18.1] is described in _e.g._ [32,
Theorem 3.1]. Namely for $F_{\bullet}\in K_{X}(Y)$, we have

(54)
${\mathrm{ch}}_{X,k}^{Y}(\psi^{N}(F_{\bullet}))=N^{k}{\mathrm{ch}}_{X,k}^{Y}(F_{\bullet})\in\mathrm{CH}^{k}(X\to
Y).$

Here $\mathrm{CH}^{k}(X\to Y)$ is the degree $k$ bivariant Chow group of
$i:X\hookrightarrow Y$. From (54) we see how localized Chern characters in
different degrees are scaled differently by the Adams operations.

We now return to the closed embedding (47) and fix it for the rest of this
section.888Alternatively, one could use the Adams operations $\psi_{N}$ in [2,
26] which are canonical and independent of the closed embedding. Since
$\overline{J}_{C}\times\overline{J}_{C}$ is nonsingular, capping with
${\mathcal{O}}_{\overline{J}_{C}\times\overline{J}_{C}}\in
K_{*}(\overline{J}_{C}\times\overline{J}_{C})$ induces a canonical isomorphism

(55)
$\cap{\mathcal{O}}_{\overline{J}_{C}\times\overline{J}_{C}}:K_{\overline{J}_{C}\times_{B}\overline{J}_{C}}(\overline{J}_{C}\times\overline{J}_{C})\xrightarrow{~{}~{}\simeq~{}~{}}K_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C})$

whose inverse is the map that sends $F\in
K_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C})$ to any locally free
resolution $F_{\bullet}$ of $i_{*}F$ on
$\overline{J}_{C}\times\overline{J}_{C}$; see [22, Lemma 1.9]. Similarly, by
[21, Propositions 17.3.1 and 17.4.2] there are canonical isomorphisms

(56)
$\cap[\overline{J}_{C}\times\overline{J}_{C}]:\mathrm{CH}^{k}(\overline{J}_{C}\times_{B}\overline{J}_{C}\to\overline{J}_{C}\times\overline{J}_{C})\xrightarrow{~{}~{}\simeq~{}~{}}\mathrm{CH}_{2\dim\overline{J}_{C}-k}(\overline{J}_{C}\times_{B}\overline{J}_{C}).$

Under (55) and (56) the Adams operations (with respect to the closed embedding
(47)) take the form

$\psi^{N}:K_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C})\to
K_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C}),\quad N\geq 1$

and the localized Chern character
${\mathrm{ch}}_{\overline{J}_{C}\times_{B}\overline{J}_{C}}^{\overline{J}_{C}\times\overline{J}_{C}}(-)$
yields a morphism (with simplified notation)

$\widetilde{{\mathrm{ch}}}:K_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C})\to\mathrm{CH}_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C}),$

such that for $F\in K_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C})$ we have

(57)
$\widetilde{{\mathrm{ch}}}_{k}(\psi^{N}(F))=N^{k}\widetilde{{\mathrm{ch}}}_{k}(F)\in\mathrm{CH}_{2\dim\overline{J}_{C}-k}(\overline{J}_{C}\times_{B}\overline{J}_{C}).$

Moreover, Corollary 3.4 together with the expression (53) immediately implies
the following.

######  Corollary 3.5.

For any $N$, the class ${\mathcal{P}}^{-1}\circ\psi^{N}({\mathcal{P}})\in
K_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C})$ is supported on a
codimension $g$ subset of $\overline{J}_{C}\times_{B}\overline{J}_{C}$.

####  3.5.3. Proof of (FV) for $\pi:\overline{J}_{C}\to B$

We apply $\tau(-)$ to the $K$-theory class

${\mathcal{P}}^{-1}\circ\psi^{N}({\mathcal{P}})\in
K_{*}(\overline{J}_{C}\times_{B}\overline{J}_{C}).$

Following the calculations in (27), we find

(58)
$\tau({\mathcal{P}}^{-1}\circ\psi^{N}({\mathcal{P}}))=p_{13*}\left({\mathrm{td}}(-p_{2}^{*}T_{\overline{J}_{C}})\cap\delta^{!}(\tau(\psi^{N}({\mathcal{P}}))\times\tau({\mathcal{P}}^{-1}))\right).$

Here we have taken the notation from Sections 1.3.2 and 2.2.1. Then we cap
both sides of (58) with
${\mathrm{td}}(-q_{1}^{*}T_{\overline{J}_{C}}-\pi_{2}^{*}T_{B})$, where
$q_{i}:\overline{J}_{C}\times_{B}\overline{J}_{C}\to\overline{J}_{C}$ is the
$i$-th projection and $\pi_{2}:\overline{J}_{C}\times_{B}\overline{J}_{C}\to
B$ is the natural map. We have by (22)

(59)
$\displaystyle{\mathrm{td}}(-q_{1}^{*}T_{\overline{J}_{C}}-\pi_{2}^{*}T_{B})\cap\tau({\mathcal{P}}^{-1}\circ\psi^{N}({\mathcal{P}}))$
$\displaystyle={}$
$\displaystyle{\mathrm{td}}(-q_{1}^{*}T_{\overline{J}_{C}}-\pi_{2}^{*}T_{B})\cap
p_{13*}\left({\mathrm{td}}(-p_{2}^{*}T_{\overline{J}_{C}})\cap\delta^{!}(\tau(\psi^{N}({\mathcal{P}}))\times\tau({\mathcal{P}}^{-1}))\right)$
$\displaystyle={}$ $\displaystyle
p_{13*}\delta^{!}\left(\left({\mathrm{td}}(-q_{1}^{*}T_{\overline{J}_{C}}-q_{2}^{*}T_{\overline{J}_{C}})\cap\tau(\psi^{N}({\mathcal{P}}))\right)\times({\mathrm{td}}(-\pi_{2}^{*}T_{B})\cap\tau({\mathcal{P}}^{-1}))\right)$
$\displaystyle={}$
$\displaystyle{\mathfrak{F}}^{-1}\circ\widetilde{{\mathrm{ch}}}(\psi^{N}({\mathcal{P}})).$

The dimension bound of Corollary 3.5 now implies that both ends of (59) vanish
in codimensions $<g$. In other words, we have for $k<2g$,

$\sum_{i+j=k}{\mathfrak{F}}^{-1}_{i}\circ\widetilde{{\mathrm{ch}}}_{j+\dim
B}(\psi^{N}({\mathcal{P}}))=\sum_{i+j=k}N^{j+\dim
B}{\mathfrak{F}}^{-1}_{i}\circ\widetilde{{\mathrm{ch}}}_{j+\dim
B}({\mathcal{P}})=0\in\mathrm{Corr}_{B}^{k-2g}(\overline{J}_{C},\overline{J}_{C}).$

Note that the first equality uses the scaling (57). Since the second equality
holds for every $N$, we obtain the vanishing of each individual term

(60) ${\mathfrak{F}}^{-1}_{i}\circ\widetilde{{\mathrm{ch}}}_{j+\dim
B}({\mathcal{P}})=0,\quad i+j<2g.$

Finally, we notice that by (24) we have

${\mathfrak{F}}={\mathrm{td}}(\pi_{2}^{*}T_{B})\cap\widetilde{{\mathrm{ch}}}({\mathcal{P}}).$

Since the Todd class ${\mathrm{td}}(\pi_{2}^{*}T_{B})$ is pulled back from the
base $B$, we conclude from (60) that

$\displaystyle{\mathfrak{F}}^{-1}_{i}\circ{\mathfrak{F}}_{j}$
$\displaystyle={\mathfrak{F}}^{-1}_{i}\circ\left(\sum_{j^{\prime}+j^{\prime\prime}=j}{\mathrm{td}}_{j^{\prime}}(\pi_{2}^{*}T_{B})\cap\widetilde{{\mathrm{ch}}}_{j^{\prime\prime}+\dim
B}({\mathcal{P}})\right)$
$\displaystyle=\sum_{j^{\prime}+j^{\prime\prime}=j}{\mathrm{td}}_{j^{\prime}}(\pi_{2}^{*}T_{B})\cap({\mathfrak{F}}^{-1}_{i}\circ\widetilde{{\mathrm{ch}}}_{j^{\prime\prime}+\dim
B}({\mathcal{P}}))=0$

for $i+j<2g$. This completes the proof of (FV) for $\pi:\overline{J}_{C}\to
B$. ∎

###  3.6. Variants

Recall from [5, 6] that Arinkin’s normalized Poincaré sheaf ${\mathcal{P}}$ on
$\overline{J}_{C}\times_{B}\overline{J}_{C}$ is constructed from the
normalized universal family ${\mathcal{F}}$ on $C\times_{B}\overline{J}_{C}$
of degree $0$ rank $1$ torsion-free sheaves. Here the normalization of
${\mathcal{F}}$ is characterized by the trivializations along the sections

$B\subset C,\quad 0_{J}\subset\overline{J}_{C}.$

In this section, we write

${\mathcal{P}}={\mathcal{P}}({\mathcal{F}},{\mathcal{F}})$

to indicate the dependence of the normalized Poincaré sheaf on the universal
sheaf ${\mathcal{F}}$, where the two factors correspond to the two factors in
Arinkin’s formula [6, (1.1)] respectively; see also the formula after (2.2) in
[1].

We have shown that this normalized Poincaré sheaf ${\mathcal{P}}$ yields a
motivic lifting of the perverse filtration associated with
$\pi:\overline{J}_{C}\to B$. Now we discuss variants of the construction
above, which will be used in Section 4.

Using the section of $C\to B$, we can identify the degree $d$ compactified
Jacobian $\overline{J}^{d}_{C}$ with the (degree $0$) compactified Jacobian
$\overline{J}_{C}$. A universal family of rank $1$ degree $d$ torsion-free
sheaves on $C\times_{B}\overline{J}^{d}_{C}=C\times_{B}\overline{J}_{C}$ is
given by

${\mathcal{F}}\otimes p_{C}^{*}O_{C}(dB)\otimes p_{J}^{*}{\mathcal{L}},$

where $p_{C},p_{J}$ are the natural projections from
$C\times_{B}\overline{J}_{C}$, and ${\mathcal{L}}$ is a line bundle on
$\overline{J}_{C}$. Now for two integers $d,e$, we consider the twisted
Poincaré sheaf

${\mathcal{P}}^{\prime}:={\mathcal{P}}({\mathcal{F}}\otimes
p_{C}^{*}O_{C}(eB)\otimes p_{J}^{*}{\mathcal{L}}_{e},\,{\mathcal{F}}\otimes
p_{C}^{*}O_{C}(dB)\otimes
p_{J}^{*}{\mathcal{L}}_{d})\in\mathrm{Coh}(\overline{J}_{C}\times_{B}\overline{J}_{C}),$

where we impose the condition that ${\mathcal{L}}_{d},{\mathcal{L}}_{e}$ are
line bundles on $\overline{J}_{C}$ whose restrictions to a nonsingular fiber
${J}_{C_{b}}$ lie in $\mathrm{Pic}^{0}({J}_{C_{b}})$. By the proof of [1,
Proposition 2.1(a)], we have

${\mathcal{P}}^{\prime}\simeq{\mathcal{P}}\otimes\left({\mathcal{L}}_{e}^{\otimes
d}\boxtimes{\mathcal{L}}_{d}^{\otimes
e}\right),\quad{\mathcal{P}}^{\prime-1}\simeq{\mathcal{P}}^{-1}\otimes\left({\mathcal{L}}_{d}^{\vee\otimes
e}\boxtimes{\mathcal{L}}_{e}^{\vee\otimes d}\right).$

Parallel to the construction of the Fourier theory using ${\mathcal{P}}$, we
can also use ${\mathcal{P}}^{\prime}$ to define the Fourier transforms
$\mathfrak{F}^{\prime},\mathfrak{F}^{\prime-1}$, the projectors
$\mathfrak{p}^{\prime}_{k},\mathfrak{q}^{\prime}_{k+1}$, and the motives
$P^{\prime}_{k}h(\overline{J}_{C})$ with orthogonal complements
$Q^{\prime}_{k+1}h(\overline{J}_{C})$. These operators and submotives _a
priori_ depend on the choices of $d,e,{\mathcal{L}}_{d},{\mathcal{L}}_{e}$.

The following proposition shows that, changing the degree, or twisting the
normalized Poincaré sheaf by fiberwise homologically trivial line bundles does
not influence the realization.

######  Proposition 3.6.

For ${\mathcal{P}}^{\prime}$ as above, the homological realization of
$P^{\prime}_{k}h(\overline{J}_{C})$ together with the inclusion
$P^{\prime}_{k}h(\overline{J}_{C})\to h(\overline{J}_{C})$ is also given by
the natural morphism

${{}^{\mathfrak{p}}}\tau_{\leq k+\dim
B}\pi_{*}{\mathbb{Q}}_{\overline{J}_{C}}\to\pi_{*}{\mathbb{Q}}_{\overline{J}_{C}}.$

###### Proof.

By the support argument of Section 2.5.2, it suffices to work with the
Jacobian fibration $\pi_{U}:J_{U}\to U$ over an (arbitrary) open subset
$U\subset B$ parameterizing nonsingular curves, and show that the composition

(61)
${\mathfrak{p}}^{\prime}_{k}|_{U*}(\pi_{U*}{\mathbb{Q}}_{J_{U}})\rightarrow\pi_{U*}{\mathbb{Q}}_{J_{U}}\rightarrow{\mathfrak{p}}_{k}|_{U*}(\pi_{U*}{\mathbb{Q}}_{J_{U}})$

is an isomorphism.

Note that both sides of (61) are direct sums of (shifted) local systems. In
order to show that (61) is an isomorphism, it suffices to check that it
induces an isomorphism on the stalks over each $b\in U$. Then the desired
statement follows clearly: since ${\mathcal{L}}_{d},{\mathcal{L}}_{e}$ lie in
$\mathrm{Pic}^{0}(J_{C_{b}})$ for $b\in U$, the restrictions of
${\mathfrak{p}}^{\prime}_{k}$ and ${\mathfrak{p}}_{k}$ over $b\in U$ are
cohomologically identical. ∎

##  4\. Twisted compactified Jacobians

###  4.1. Overview and setup

The purpose of the this section is to extend the Fourier theory for
compactified Jacobians established in Section 3 to a family of genus $g$
curves without a section. This situation occurs in the study of Hitchin
systems and the Le Potier moduli spaces of $1$-dimensional sheaves on
${\mathbb{P}}^{2}$. We will discuss applications in Section 5.

Throughout this section, we let $C\to B$ be a flat family of integral
projective curves of arithmetic genus $g$ with planar singularities over an
irreducible base $B$. We assume that the total space $C$ is nonsingular (hence
$B$ is nonsingular), and there is a multisection

$D\subset C\to B$

of degree $r$ which is finite and flat over $B$. For any integer $d$, the
degree $d$ compactified Jacobian exists as a scheme étale locally over the
base $B$ since there is always a section of $C\to B$ étale locally. Globally,
the degree $d$ compactified Jacobian $\overline{J}^{d}_{C}$ exists only as an
algebraic space. For convenience, throughout we assume that for any degree
$d$, the compactified Jacobian $\overline{J}^{d}_{C}$ is a quasi-projective
variety. This is satisfied when $C\to B$ is given by a linear system of a
nonsingular surface; in this case, the compactified Jacobian arises as the
moduli space of certain semistable torsion sheaves on the surface, so quasi-
projectivity follows from the general construction of moduli spaces of
semistable sheaves. We further assume that $\overline{J}^{d}_{C}$ is
nonsingular.

We denote the natural projection map by

$\pi_{d}:\overline{J}^{d}_{C}\to B$

to indicate its dependence on $d$.

###  4.2. Universal families and gerbes

For arbitrary degree $d$, two issues arise if we want to establish a Fourier
theory for $\overline{J}^{d}_{C}$ governing the perverse filtration.

  1. (i)

There may not exist a universal sheaf on $C\times_{B}\overline{J}^{d}_{C}$.

  2. (ii)

Even if a universal sheaf exists, it is not unique. Different choices of a
universal sheaf influence the Poincaré sheaf and the Fourier transform. In
particular, for an arbitrary choice of a universal sheaf, the induced motivic
filtration may not recover the perverse filtration via homological
realization.999This is already the case for the Jacobian fibration associated
with a family of nonsingular curves.

When $C\to B$ admits a section, a standard solution to (ii) is that we can
force the universal family to be trivialized along the section, which
eliminates the ambiguity given by a line bundle pulled back from the moduli
space $\overline{J}^{d}_{C}$. Since now we only have a multisection, we first
introduce the notion of _trivialization along a multisection_.

For any $B$-scheme $T$, we consider a flat family of rank $1$ torsion-free
sheaves of degree $d$ on the curves parameterized by $T$:

${\mathcal{F}}^{d}_{T}\rightsquigarrow C\times_{B}T,$

and we define

(62)
${\mathcal{R}}^{d}_{T}:=\mathrm{det}\left(p_{T*}({\mathcal{F}}^{d}_{T}|_{D\times_{B}T})\right)\in\mathrm{Pic}(T).$

Note that ${\mathcal{F}}^{d}_{T}|_{D\times_{B}T}$ is Tor-finite over $T$ so
that the pushforward to $T$ is a perfect complex and we can take its
determinant; see [52, Tag 08IS]. We say that this family over $T$ is
trivialized along the multisection $D\subset C$, if there is a specified
isomorphism

${\mathcal{R}}^{d}_{T}\simeq{\mathcal{O}}_{T}\in\mathrm{Pic}(T).$

However, we note that unlike the case with a section, in general we can not
find a universal sheaf on $C\times_{B}\overline{J}^{d}_{C}$ which is
trivialized along the multisection $D$, even if a universal sheaf
${\mathcal{F}}^{d}$ exists. This is because globally on $\overline{J}^{d}_{C}$
there may not exist a line bundle which is an $r$-th root of the line bundle
${\mathcal{R}}^{d}$. Nevertheless, we can always solve both issues (i, ii)
above by passing to a $\mu_{r}$-gerbe over $\overline{J}^{d}_{C}$ as follows.

The idea is to modify the moduli functor defining the compactified Jacobian.
Let $\overline{{\mathcal{J}}}^{d}_{C}$ be the functor sending any $B$-scheme
$T$ to a groupoid given by the data

${\mathcal{F}}^{d}_{T}\rightsquigarrow C\times_{B}T$

satisfying the same conditions as for the stack of the degree $d$ compactified
Jacobian, with an extra assumption that ${\mathcal{F}}^{d}_{T}$ is trivialized
along the multisection $D$.

######  Proposition 4.1.

The functor $\overline{{\mathcal{J}}}^{d}_{C}$ is represented by a
Deligne–Mumford stack which is a $\mu_{r}$-gerbe over $\overline{J}^{d}_{C}$.

###### Proof.

Let $\overline{\mathfrak{J}}^{d}_{C}$ denote the stack of degree $d$
compactified Jacobian; it is a ${\mathbb{G}}_{m}$-gerbe over
$\overline{J}^{d}_{C}$. The line bundles (62) defines a morphism

$\overline{\mathfrak{J}}^{d}_{C}\to B{\mathbb{G}}_{m}.$

Imposing the trivialization condition along $D$ is equivalent to taking the
fiber product of this morphism with the structure map $\mathrm{pt}\to
B{\mathbb{G}}_{m}$:

$\overline{{\mathcal{J}}}^{d}_{C}=\overline{\mathfrak{J}}^{d}_{C}\times_{B{\mathbb{G}}_{m}}\mathrm{pt}.$

This is representable by a Deligne–Mumford stack. Finally it is a
$\mu_{r}$-gerbe over $\overline{J}^{d}_{C}$ since the perfect complex
${\mathcal{R}}^{d}_{\overline{\mathfrak{J}}^{d}_{C}}$ has rank $r$ (which can
be seen directly from nonsingular fibers). ∎

Consequently, we have constructed for any $d$ a nonsingular Deligne–Mumford
stack $\overline{{\mathcal{J}}}^{d}_{C}$ realized as a $\mu_{r}$-gerbe over
$\overline{J}^{d}_{C}$, together with the universal family ${\mathcal{F}}^{d}$
of rank $1$ degree $d$ torsion-free sheaves on
$C\times_{B}\overline{{\mathcal{J}}}^{d}_{C}$, trivialized along the
multisection $D$:

$\mathrm{det}\left(p_{J*}({\mathcal{F}}^{d}|_{D\times_{B}\overline{{\mathcal{J}}}_{C}^{d}})\right)\simeq{\mathcal{O}}_{\overline{{\mathcal{J}}}^{d}_{C}}\in\mathrm{Pic}(\overline{{\mathcal{J}}}^{d}_{C}).$

To set up the Fourier–Mukai transforms, we first define the isotypic category
$\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})_{(k)}$ (resp.
$D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})_{(k)}$) to be the full
subcategory of $\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})$ (resp.
$D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})$) consisting of objects
for which the action of $\mu_{r}$ on fibers is given by the character
$\lambda\mapsto\lambda^{k}$ of $\mu_{r}$. Then we consider two integers $d,e$.
As in Section 3.6, we plug ${\mathcal{F}}^{d},{\mathcal{F}}^{e}$ in Arinkin’s
formula [6, (1.1)] and obtain a Poincaré sheaf

${\mathcal{P}}_{e,d}:={\mathcal{P}}({\mathcal{F}}^{e},{\mathcal{F}}^{d})\in\mathrm{Coh}(\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{{\mathcal{J}}}^{d}_{C})_{(d,e)}.$

We define

${\mathcal{P}}_{e,d}^{-1}:={\mathcal{H}}\mathrm{om}_{\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{{\mathcal{J}}}^{d}_{C}}({\mathcal{P}}_{e,d},\,p_{2}^{*}\omega_{\pi_{d}})[g]$

where $\omega_{\pi_{d}}$ is the relative canonical bundle with respect to
$\pi_{d}:\overline{{\mathcal{J}}}^{d}_{C}\to B$, and we view
${\mathcal{P}}_{e,d}^{-1}$ as in
$D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C}\times_{B}\overline{{\mathcal{J}}}^{e}_{C})_{(-e,-d)}$.
Because ${\mathcal{P}}_{e,d}$ lies in the isotypic category, the Fourier–Mukai
transform

$\mathrm{FM}_{{\mathcal{P}}_{e,d}}:D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{e}_{C})\rightarrow
D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})$

is only nonzero on the following isotypic components:

$\mathrm{FM}_{{\mathcal{P}}_{e,d}}:D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{e}_{C})_{(-d)}\rightarrow
D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})_{(e)}.$

######  Proposition 4.2.

The objects ${\mathcal{P}}_{e,d},{\mathcal{P}}^{-1}_{e,d}$ are inverse to each
other as Fourier–Mukai kernels:

$\mathrm{FM}_{{\mathcal{P}}_{e,d}}:D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{e}_{C})_{(-d)}\xrightarrow{~{}~{}\simeq~{}~{}}D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})_{(e)},\quad\mathrm{FM}_{{\mathcal{P}}^{-1}_{e,d}}=\mathrm{FM}^{-1}_{{\mathcal{P}}_{e,d}}.$

###### Proof.

Recall that for ${\mathcal{P}}_{e,d}$ and ${\mathcal{P}}^{-1}_{e,d}$ we always
have the canonical adjunction morphisms

$\mathrm{id}_{D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{e}_{C})_{(-d)}}\to\mathrm{FM}_{{\mathcal{P}}^{-1}_{e,d}}\circ\mathrm{FM}_{{\mathcal{P}}_{e,d}},\quad\mathrm{FM}_{{\mathcal{P}}_{e,d}}\circ\mathrm{FM}_{{\mathcal{P}}^{-1}_{e,d}}\to\mathrm{id}_{D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})_{(e)}}.$

It suffices to show that their cones in the derived categories are $0$; this
is a property which can be checked étale locally over the base $B$. We
postpone the étale local calculations to Corollary 4.4 below. ∎

###  4.3. Étale local structures

In this section, we carry out the étale local calculation for the
Fourier–Mukai transforms given by
${\mathcal{P}}_{e,d},{\mathcal{P}}^{-1}_{e,d}$. This leads to two
consequences. First, the calculation completes the proof of Proposition 4.2.
Second, we will show as in Proposition 3.6 that the motivic decomposition
induced by the Fourier transforms $\mathfrak{F},\mathfrak{F}^{-1}$ associated
with ${\mathcal{P}}_{e,d},{\mathcal{P}}^{-1}_{e,d}$ specializes to the
(standard) perverse filtration; see Section 4.4.

Now we work étale locally over the base $B$. Let $U$ be an étale neighborhood
of the base. We may assume that $C\to U$ admits simultaneously a section and a
multisection

$U\subset C\to U,\quad D\subset C\to U.$

They are independent and do not have any nontrivial relation. Due to the
existence of the section, the relative compactified Jacobian
$\overline{J}^{d}_{C}$ are identified for any choices of $d$. As before, we
denote by $\overline{J}_{C}$ the degree $0$ compactified Jacobian, and we want
to compare ${\mathcal{F}}^{d}$ on
$C\times_{U}\overline{{\mathcal{J}}}^{d}_{C}$ (which is trivialized along the
multisection $D$) with the normalized universal sheaf ${\mathcal{F}}$ on
$C\times_{U}\overline{J}_{C}$ (which is trivialized along the section $U$).

######  Proposition 4.3.

There is a $U$-morphism

$\sigma_{d}:\overline{{\mathcal{J}}}^{d}_{C}\to\overline{J}_{C}$

satisfying the following properties.

  1. (i)

We have

${\mathcal{F}}^{d}\simeq(\mathrm{id}_{C}\times_{U}\sigma_{d})^{*}{\mathcal{F}}\otimes
p_{C}^{*}{\mathcal{O}}_{C}(dU)\otimes
p_{\mathcal{J}}^{*}{\mathcal{L}}_{d},\quad{\mathcal{L}}_{d}\in\mathrm{Pic}(\overline{{\mathcal{J}}}^{d}_{C}),$

where $p_{C},p_{\mathcal{J}}$ are the natural projections from
$C\times_{U}\overline{{\mathcal{J}}}^{d}_{C}$.

  2. (ii)

For any $b\in U$ with $C_{b}$ a nonsingular curve, the restriction of
${\mathcal{L}}_{d}$ to the fiber ${\mathcal{J}}^{d}_{C_{b}}$ has trivial first
Chern class in $H^{2}({{\mathcal{J}}}^{d}_{C_{b}},{\mathbb{Q}})$.

###### Proof.

The desired morphism $\sigma_{d}$ can be constructed using the map from the
Deligne–Mumford stack to its coarse moduli space, in view of the
identification $\overline{J}^{d}_{C}\simeq\overline{J}_{C}$. For our purpose,
we describe it in a more direct way as follows.

Note that $\overline{J}_{C}$ is a fine moduli space. Therefore, describing a
morphism from $\overline{{\mathcal{J}}}^{d}_{C}$ is equivalent to constructing
a family of rank $1$ torsion-free sheaves of degree $0$ trivialized along the
section $U\subset C$, for which we may use ${\mathcal{F}}^{d}$. More
precisely, we define

${\mathcal{G}}^{d}:={\mathcal{F}}^{d}\otimes
p_{C}^{*}{\mathcal{O}}_{C}(-dU)\otimes
p_{\mathcal{J}}^{*}\left(({\mathcal{F}}^{d}\otimes
p_{C}^{*}{\mathcal{O}}_{C}(-dU))|_{U\times_{U}\overline{{\mathcal{J}}}^{d}_{C}}\right)^{\vee}.$

Here the second factor adjusts the degree from $d$ to $0$, and the third
factor is included to trivialize the universal sheaf along the section. This
yields a morphism
$\sigma_{d}:\overline{{\mathcal{J}}}^{d}_{C}\to\overline{J}_{C}$ satisfying

$(\mathrm{id}_{C}\times_{U}\sigma_{d})^{*}{\mathcal{F}}\simeq{\mathcal{G}}^{d}.$

This already proves (i).

To prove (ii), it suffices to show that, for a fixed nonsingular curve $C_{b}$
with $s\in C_{b}$ the intersection with the section $U\subset C$, the line
bundle ${\mathcal{F}}^{d}|_{s\times{\mathcal{J}}_{C_{b}}}$ has homologically
trivial first Chern class. Assume $D\cap C_{b}$ is given by
$s_{1},\dots,s_{e}\in C_{b}$ (counted with multiplicities). Then the condition
that ${\mathcal{F}}^{d}$ is trivialized along the multisection $D$ implies

$\sum_{i=1}^{e}c_{1}\left({\mathcal{F}}^{d}|_{s_{i}\times{\mathcal{J}}_{C_{b}}}\right)=0\in
H^{2}({{\mathcal{J}}}^{d}_{C_{b}},{\mathbb{Q}}).$

This shows

$c_{1}\left({\mathcal{F}}^{d}|_{s\times{\mathcal{J}}_{C_{b}}}\right)=0\in
H^{2}({\mathcal{J}}^{d}_{C_{b}},{\mathbb{Q}})$

as desired, since the first Chern class of the left-hand side does not depend
on the point in $C_{b}$. The proof of (ii) is complete. ∎

Now we consider

$\sigma_{e}\times_{U}\sigma_{d}:\overline{{\mathcal{J}}}^{e}_{C}\times_{U}\overline{{\mathcal{J}}}^{d}_{C}\to\overline{J}_{C}\times_{U}\overline{J}_{C}.$

The normalized Poincare sheaf ${\mathcal{P}}$ and its inverse
${\mathcal{P}}^{-1}$ are canonically defined on the target. The following is
an immediate consequence of Proposition 4.3 as in Section 3.6 (using the
argument of [1, Proposition 2.1(a)]). It provides an étale local description
of the Poincaré sheaf ${\mathcal{P}}_{e,d}$ and its inverse
${\mathcal{P}}^{-1}_{e,d}$.

######  Corollary 4.4.

We have

${\mathcal{P}}_{e,d}\simeq(\sigma_{e}\times_{U}\sigma_{d})^{*}{\mathcal{P}}\otimes\left({\mathcal{L}}_{e}^{\otimes
d}\boxtimes{\mathcal{L}}_{d}^{\otimes
e}\right),\quad{\mathcal{P}}^{-1}_{e,d}\simeq(\sigma_{d}\times_{U}\sigma_{e})^{*}{\mathcal{P}}^{-1}\otimes\left({\mathcal{L}}_{d}^{\vee\otimes
e}\boxtimes{\mathcal{L}}_{e}^{\vee\otimes d}\right).$

Here ${\mathcal{L}}_{d},{\mathcal{L}}_{e}$ are line bundles on
$\overline{{\mathcal{J}}}^{d}_{C},\overline{{\mathcal{J}}}^{e}_{C}$
respectively. Moreover, for any $b\in U$ with $C_{b}$ nonsingular, the
restrictions of these line bundles to the fibers over $b$ have homologically
trivial first Chern classes.

###  4.4. Multiplicativity and perversity

The goal of this section is to demonstrate the following result for the degree
$d$ compactified Jacobian $\pi_{d}:\overline{J}^{d}_{C}\to B$. We state it as
a corollary since the proof essentially replicates what has been done in
Sections 2 and 3.

######  Corollary 4.5.

All four statements in Theorem 2.4, as well as Corollary 2.5, hold equally for
$\pi_{d}:\overline{J}^{d}_{C}\to B$.

Note that we chose to first work out a special case of Corollary 4.5 in the
setting of Section 3.1 for purely expository reasons, so that the Fourier
theory can be streamlined and separated from the language of stacks. The
general case requires the following three additional ingredients.

  1. (i)

Since we assume $\overline{J}^{d}_{C}$ to be a quasi-projective variety, by
[28] the Brauer map

$\mathrm{Br}(\overline{J}^{d}_{C})\to
H^{2}(\overline{J}^{d}_{C},{\mathbb{G}}_{m})_{\mathrm{tors}}$

is an isomorphism. Then [20, Theorem 3.6] implies that the $\mu_{r}$-gerbe
$\overline{{\mathcal{J}}}^{d}_{C}$ over $\overline{J}^{d}_{C}$ is a quotient
stack. This also applies to fiber products of
$\overline{{\mathcal{J}}}^{d}_{C}$ for various $d$ over the base scheme $B$.

  2. (ii)

The intersection theory of quotient stacks has been developed by a series of
papers of Edidin and Graham; we refer to [17] for a summary of the theory and
the references therein. In particular, the Riemann–Roch functor $\tau(-)$ for
quotient stacks was constructed in [19] and was shown to satisfy the same
functorial properties as in Section 2.3 except for the covariance under
arbitrary proper morphisms. The Edidin–Graham functor $\tau(-)$ is only
covariant under proper _representable_ morphisms of quotient stacks.

We also note that with ${\mathbb{Q}}$-coefficients, the Edidin–Graham Chow
groups of a Deligne–Mumford quotient stack are canonically isomorphic (via
pushforward) to the Chow groups of the coarse moduli space; see [18, 54].

  3. (iii)

The $K$-theory of quotient stacks has been studied by Anderson, Gonzales, and
Payne [4, 2] in the style of Fulton and Edidin–Graham. For our purposes we
shall need the construction of Adams operations and the scaling (54) for
closed embeddings of (possibly singular) quotient stacks, both of which were
established in [2, Section 4].

We are now ready to run the entire argument of Sections 2 and 3 for
$\pi_{d}:\overline{J}^{d}_{C}\to B$.

Step 1. Choosing an arbitrary integer $e$, we consider the Poincaré sheaf
${\mathcal{P}}_{e,d}$ and its inverse ${\mathcal{P}}_{e,d}^{-1}$. We define
the Fourier transforms

${\mathfrak{F}}_{e,d}\in\mathrm{CH}_{*}(\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{{\mathcal{J}}}^{d}_{C})=\mathrm{CH}_{*}(\overline{J}^{e}_{C}\times_{B}\overline{J}^{d}_{C}),\quad{\mathfrak{F}}_{e,d}^{-1}\in\mathrm{CH}_{*}(\overline{{\mathcal{J}}}^{d}_{C}\times_{B}\overline{{\mathcal{J}}}^{e}_{C})=\mathrm{CH}_{*}(\overline{J}^{d}_{C}\times_{B}\overline{J}^{e}_{C})$

following the recipe of (23) and (25) and using the Edidin-Graham functor
$\tau(-)$.

To prove the identities parallel to Lemma 2.2, we first note that the
Fourier–Mukai kernel ${\mathcal{E}}$ of
$\mathrm{id}_{D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{e}_{C})_{(-d)}}$ is
given by a line bundle on a $\mu_{r}\times\mu_{r}$-gerbe over the relative
diagonal
$\overline{J}^{e}_{C}\subset\overline{J}^{e}_{C}\times_{B}\overline{J}^{e}_{C}$.
The line bundle, which corresponds to the character
$(\lambda,\lambda^{\prime})\mapsto\lambda^{d}\lambda^{\prime-d}$ of
$\mu_{r}\times\mu_{r}$, has its $r$-th tensor power the trivial bundle. Hence
by the functorialities of $\tau(-)$, we find that $\tau({\mathcal{E}})$ (when
pushed forward to $\overline{J}^{e}_{C}\times\overline{J}^{e}_{C}$) is as
before given by

$\mathrm{td}(q_{1}^{e*}T_{\overline{J}^{e}_{C}})\cap[\Delta_{\overline{J}^{e}_{C}/B}]\in\mathrm{CH}_{*}(\overline{J}^{e}_{C}\times\overline{J}^{e}_{C})$

up to a factor of $1/r^{2}$, where
$q_{1}^{e}:\overline{J}^{e}_{C}\times_{B}\overline{J}^{e}_{C}\to\overline{J}^{e}_{C}$
is the first projection. To compute
$\tau({\mathcal{P}}_{e,d}^{-1}\circ{\mathcal{P}}_{e,d})$ we observe that there
is exactly one appearance of a non-representable proper pushforward in the
whole process, namely the pushforward to the first and the third factors. On
the other hand, since the object

$p_{12}^{*}{\mathcal{P}}_{e,d}\otimes p_{23}^{*}{\mathcal{P}}^{-1}_{e,d}\in
D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{{\mathcal{J}}}^{d}_{C}\times\overline{{\mathcal{J}}}^{e}_{C})_{(d,0,-d)}$

descends to an object in
$D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{J}^{d}_{C}\times_{B}\overline{{\mathcal{J}}}^{e}_{C})$,
it suffices to apply the covariance of $\tau(-)$ under the proper
_representable_ morphism

$p_{13}:\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{J}^{d}_{C}\times_{B}\overline{{\mathcal{J}}}^{e}_{C}\to\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{{\mathcal{J}}}^{e}_{C}.$

This yields the identity

${\mathfrak{F}}_{e,d}^{-1}\circ{\mathfrak{F}}_{e,d}=\frac{1}{r^{4}}[\Delta_{\overline{J}^{e}_{C}/B}]\in\mathrm{Corr}^{0}_{B}(\overline{J}^{e}_{C},\overline{J}^{e}_{C}),$

where the factor of $1/r^{4}$ is due to the fact that each
$\overline{{\mathcal{J}}}^{d}_{C}\to\overline{J}^{d}_{C}$ is of degree $1/r$.
The other identity is parallel.

Step 2. The next logical step is the Fourier vanishing (FV). We first remark
that the Arinkin-type dimension bounds of Sections 3.3 and 3.5 are properties
of coherent sheaves which can be checked étale locally.

To proceed with the Adams argument, we consider a sequence of $K$-theory
classes supported in codimension $g$:

(63) ${\mathcal{P}}_{e,d}^{-1}\circ\psi^{N}({\mathcal{P}}_{e,d})\in
K_{*}(\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{{\mathcal{J}}}^{e}_{C}),\quad
N\equiv 1\pmod{r},$

where $\psi^{N}(-)$ stand for the Adams operations of Anderson–Gonzales–Payne.
Here the congruence condition on $N$ guarantees that the class

$p_{12}^{*}\psi^{N}({\mathcal{P}}_{e,d})\otimes
p_{23}^{*}{\mathcal{P}}^{-1}_{e,d}\in
K_{*}(\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{{\mathcal{J}}}^{d}_{C}\times_{B}\overline{{\mathcal{J}}}^{e}_{C})_{(d,0,-d)}$

descends to a class in
$K_{*}(\overline{{\mathcal{J}}}^{e}_{C}\times_{B}\overline{J}^{d}_{C}\times_{B}\overline{{\mathcal{J}}}^{e}_{C})$
so that the Riemann–Roch calculations work in the same way as in Step 1. We
then repeat the calculations of Section 3.5.3 using the infinite sequence
(63), and deduce the Fourier vanishing

(FV)
$\quad\quad({\mathfrak{F}}_{e,d}^{-1})_{i}\circ({\mathfrak{F}}_{e,d})_{j}=0,\quad
i+j<2g.$

Consequently, we obtain pairs of orthogonal projectors
${\mathfrak{p}}_{k}^{e,d},{\mathfrak{q}}_{k+1}^{e,d}$ (depending on $e$) and
the corresponding motives
$P_{k}^{e}h(\overline{J}_{C}^{d}),Q_{k+1}^{e}h(\overline{J}_{C}^{d})$. We also
deduce the Perverse $\supset$ Chern statement of Theorem 2.4 for
$\pi_{d}:\overline{J}^{d}_{C}\to B$, as well as the decomposition part of
Corollary 2.5:

(64)
$h(\overline{J}_{C}^{d})=\bigoplus_{i=0}^{2g}h_{i}^{e}(\overline{J}_{C}^{d})\in\mathrm{CHM}(B).$

Step 3. For the multiplicativity statement we choose two arbitrary integers
$e_{1},e_{2}$. We consider the convolution kernel

${\mathcal{K}}_{e_{1},e_{2},d}\in
D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{e_{1}}_{C}\times_{B}\overline{{\mathcal{J}}}^{e_{2}}_{C}\times_{B}\overline{{\mathcal{J}}}^{e_{1}+e_{2}}_{C})_{(d,d,-d)}$

obtained from ${\mathcal{P}}_{e_{1},d},{\mathcal{P}}_{e_{2},d}$ and
${\mathcal{P}}^{-1}_{e_{1}+e_{2},d}$ together with (12). As in Step 2 an étale
local verification shows that ${\mathcal{K}}_{e_{1},e_{2},d}$ is supported in
codimension $g$. We define the Chow class ${\mathfrak{C}}_{e_{1},e_{2},d}$
following (33).

The identity parallel to Lemma 2.6 is proven in the same way as in Step 1. The
Fourier–Mukai kernel of

$\otimes:D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})_{(e_{1})}\times
D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})_{(e_{2})}\to
D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{d}_{C})_{(e_{1}+e_{2})}$

is given by a torsion line bundle on a
$\mu_{r}\times\mu_{r}\times\mu_{r}$-gerbe over the small relative diagonal
$\overline{J}^{d}_{C}\subset\overline{J}^{d}_{C}\times_{B}\overline{J}^{d}_{C}\times_{B}\overline{J}^{d}_{C}$.
On the other hand, the computation of

$\tau\left({\mathcal{P}}_{e_{1}+e_{2},d}\circ{\mathcal{K}}_{e_{1},e_{2},d}\circ({\mathcal{P}}^{-1}_{e_{1},d}\boxtimes{\mathcal{P}}^{-1}_{e_{2},d})\right)$

only involves pushforwards via proper representable morphisms.

The rest of the proof is identical to the untwisted case in Section 2.5.3.

Step 4. Finally, we verify for $\pi_{d}:\overline{J}^{d}_{C}\to B$ the
realization statement of Theorem 2.4 together with the realization part of
Corollary 2.5. Ngô’s support theorem holds equally in the twisted case: all
perverse sheaves appearing in the decomposition theorem for
$\pi_{d}:\overline{J}^{d}_{C}\to B$ have full support. Then, as in Section
2.5.2, it suffices to check the homological realization of
$P_{k}^{e}h(\overline{J}_{C}^{d})$ over an open subset of the base $B$
supporting nonsingular curves. Note that by étale descent of perverse sheaves,
we can even replace the open subset by an _étale_ neighborhood $U$ of $B$.

We are placed in the setup of Section 4.3. By Corollary 4.4, over $U$ the
Poincaré sheaf ${\mathcal{P}}_{e,d}$ differs from the normalized
${\mathcal{P}}$ only by line bundles which are fiberwise homologically
trivial. Now a proof identical to that of Proposition 3.6 shows that the
homological realization of $P_{k}^{e}h(\overline{J}_{C}^{d})\to
h(\overline{J}_{C}^{d})$ is precisely

${{}^{\mathfrak{p}}}\tau_{\leq k+\dim
B}\pi_{d*}{\mathbb{Q}}_{\overline{J}^{d}_{C}}\to\pi_{d*}{\mathbb{Q}}_{\overline{J}^{d}_{C}}.$

The homological realizations of $Q_{k+1}^{e}h(\overline{J}_{C}^{d})$ and of
(64) are parallel. This finishes the proof of Corollary 4.5. ∎

###  4.5. Abel–Jacobi maps

In addition, we treat the Abel–Jacobi map which recovers the universal family
of $1$-dimensional sheaves from the Poincaré sheaf. We follow essentially [1,
Proposition 2.1(b)].

Recall the Abel–Jacobi map to the degree $1$ compactified Jacobian associated
with $C\to B$:

$\mathrm{AJ}:C\rightarrow\overline{J}^{1}_{C},\quad(x\in
C_{b})\mapsto\mathfrak{m}^{\vee}_{x/C_{b}}.$

Pulling back along the $\mu_{r}$-gerbe
$\overline{{\mathcal{J}}}^{1}_{C}\to\overline{J}^{1}_{C}$, we obtain the
Abel–Jacobi map between Deligne–Mumford stacks

$\mathrm{AJ}:{\mathcal{C}}\to\overline{{\mathcal{J}}}^{1}_{C}.$

Here ${\mathcal{C}}$ is a $\mu_{r}$-gerbe over $C$ with the structure map
$\sigma_{\mathcal{C}}:{\mathcal{C}}\to C$.

For any integer $d$, we consider the universal family ${\mathcal{F}}^{d}$
normalized along the multisection $D$ (see Section 4.2), and the Poincaré
sheaf ${\mathcal{P}}_{1,d}$ on
$\overline{{\mathcal{J}}}_{C}^{1}\times_{B}\overline{{\mathcal{J}}}_{C}^{d}$.
The Abel–Jacobi map above induces

$\mathrm{AJ}\times_{B}\mathrm{id}_{\overline{{\mathcal{J}}}^{d}_{C}}:{\mathcal{C}}\times_{B}\overline{{\mathcal{J}}}^{d}_{C}\to\overline{{\mathcal{J}}}_{C}^{1}\times_{B}\overline{{\mathcal{J}}}_{C}^{d}.$

######  Proposition 4.6 (_c.f._ [1, Proposition 2.1(b)]).

We have

$(\mathrm{AJ}\times_{B}\mathrm{id}_{\overline{{\mathcal{J}}}^{d}_{C}})^{*}{\mathcal{P}}_{1,d}\simeq(\sigma_{\mathcal{C}}\times_{B}\mathrm{id}_{\overline{{\mathcal{J}}}^{d}_{C}})^{*}{\mathcal{F}}^{d}\otimes
p_{\mathcal{C}}^{*}{\mathcal{N}}\otimes{\mathcal{B}}.$

Here ${\mathcal{N}}$ is a line bundle on ${\mathcal{C}}$ given by a
${\mathbb{Q}}$-divisor proportional to $D$, and ${\mathcal{B}}$ is the
pullback of a line bundle on the base $B$.

###### Proof.

We consider

$\mathrm{id}_{C}\times_{B}\mathrm{AJ}:C\times_{B}{\mathcal{C}}\to
C\times_{B}\overline{{\mathcal{J}}}^{1}_{C}.$

By the definition of the Abel–Jacobi map, we may write

(65)
$(\mathrm{id}_{C}\times_{B}\mathrm{AJ})^{*}{\mathcal{F}}^{1}\simeq(\mathrm{id}_{C}\times_{B}\sigma_{\mathcal{C}})^{*}{\mathcal{I}}^{\vee}_{\Delta}\otimes
p^{*}_{\mathcal{C}}{\mathcal{N}}^{\prime}$

where ${\mathcal{N}}^{\prime}$ is a line bundle on ${\mathcal{C}}$, and
${\mathcal{I}}_{\Delta}$ is the ideal sheaf of the relative diagonal on
$C\times_{B}C$. The calculation in the proof of [1, Proposition 2.1(b)] shows
that the difference between

$(\mathrm{AJ}\times_{B}\mathrm{id}_{\overline{{\mathcal{J}}}^{d}_{C}})^{*}{\mathcal{P}}_{1,d},\quad(\sigma_{\mathcal{C}}\times_{B}\mathrm{id}_{\overline{{\mathcal{J}}}^{d}_{C}})^{*}{\mathcal{F}}^{d}$

is given by $d$-th power of the line bundle ${\mathcal{N}}^{\prime}$ of (65);
see (2.9) and the last equation of the proof of [1, Proposition 2.1(b)].

It suffices to show that, modulo a line bundle pulled back from $B$,
${\mathcal{N}}^{\prime}$ is given by a divisor proportional to $D$. We
calculate ${\mathcal{N}}^{\prime}$ via (65). By the condition that
${\mathcal{F}}^{1}$ is trivialized along the multisection, the right-hand side
of (65) is trivialized along $D\subset C$ (of the first factor). The desired
statement then follows from a direct calculation

$\mathrm{det}\left(p_{C*}({\mathcal{I}}^{\vee}_{\Delta}\big{|}_{D\times_{B}C})\right)\simeq{\mathcal{O}}_{C}(D)\otimes{\mathcal{B}}$

with ${\mathcal{B}}$ a line bundle pulled back from the base $B$. This
completes the proof. ∎

######  Remark 4.7.

In the case when there is a (non-canonical) universal family $F^{d}$ of rank
$1$ degree $d$ torsion-free sheaves on $C\times_{B}\overline{J}^{d}_{C}$, as
in Proposition 4.3 there is a $B$-morphism

$\widetilde{\sigma}_{d}:\overline{{\mathcal{J}}}^{d}_{C}\to\overline{J}^{d}_{C}$

so that