# Algebraic cycles and Hitchin systems

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

The purpose of this paper is to study motivic aspects of the Hitchin system
for $\mathrm{GL}_{n}$. Our results include the following. (a) We prove the
motivic decomposition conjecture of Corti–Hanamura for the Hitchin system; in
particular, the decomposition theorem associated with the Hitchin system is
induced by algebraic cycles. This yields an unconditional construction of the
motivic perverse filtration for the Hitchin system, which lifts the
cohomological/sheaf-theoretic perverse filtration. (b) We prove that the
inverse of the relative Hard Lefschetz symmetry is induced by a relative
algebraic correspondence, confirming the relative Lefschetz standard
conjecture for the Hitchin system. (c) We show a strong perversity bound for
the normalized Chern classes of a universal bundle with respect to the motivic
perverse filtration; this specializes to the sheaf-theoretic result obtained
earlier by Maulik–Shen. (d) We prove a $\chi$-independence result for the
relative Chow motives associated with Hitchin systems.

Our methods combine Fourier transforms for compactified Jacobian fibrations
associated with integral locally planar curves, nearby and vanishing cycle
techniques, and a Springer-theoretic interpretation of parabolic Hitchin
moduli spaces.

###### Contents

  1. 0 Introduction
  2. 1 Moduli of Higgs bundles
  3. 2 Projectors for compactified Jacobians
  4. 3 Projectors and Springer theory
  5. 4 Sheaf-theoretic operations
  6. 5 Questions and conjectures

##  0\. Introduction

The topology of the Hitchin system has been intensively studied over decades.
In [45], Ngô proved the fundamental lemma of the Langlands program by reducing
it to the study of the decomposition theorem [7] for the Hitchin system; the
topological mirror symmetry conjecture of Hausel–Thaddeus [26] was proven by
comparing the decomposition theorem for Hitchin systems associated with the
Langlands dual groups $\mathrm{SL}_{n}$ and $\mathrm{PGL}_{n}$ [23, 35]; the
decomposition theorem for the Hitchin system further yields a perverse
filtration on the singular cohomology of the Hitchin moduli space, whose
interaction with the non-abelian Hodge theory is the main theme of the P=W
conjecture [9], recently proven [37, 24, 38]; the decomposition theorem for
the Hitchin system also appears naturally in enumerative geometry,
particularly in the study of BPS invariants [15, 36, 30, 17].

Most of these developments relied heavily on topological and sheaf-theoretic
approaches. On the other hand, hints from several recent results [22, 33, 29,
38] suggest that there should be a motivic theory hidden behind the
cohomological structures mentioned above. The purpose of this work is to study
motivic aspects of the decomposition theorem and the perverse filtration for
the Hitchin system. The upshot is that the geometry of the Hitchin system
provides us powerful tools to construct algebraic cycles on the moduli of
Higgs bundles, lifting many cohomological statements to the motivic level.

###  0.1. Hitchin systems

We work over the complex numbers ${\mathbb{C}}$. Let $\Sigma$ be a nonsingular
projective curve of genus $g\geq 2$. Let $M_{n,d}$ be the moduli space of
stable Higgs bundles with coprime rank $n$ and degree $d$ on $\Sigma$; it
carries the structure of an integrable system, known as the _Hitchin system_ :

(1) $f:M_{n,d}\to B,$

which is a proper and surjective morphism to an affine space $B$. We use
$R_{f}$ to denote the relative dimension of (1). By the decomposition theorem
of Beilinson, Bernstein, Deligne, and Gabber [7], the derived pushforward
$Rf_{*}{\mathbb{Q}}_{M_{n,d}}\in D_{c}^{b}(B)$ admits a decomposition into
(shifted) semisimple perverse sheaves:

(2) $Rf_{*}{\mathbb{Q}}_{M_{n,d}}[\dim
M_{n,d}-R_{f}]\simeq\bigoplus_{i=0}^{2R_{f}}{{}^{\mathfrak{p}}}{\mathcal{H}}^{i}\left(Rf_{*}{\mathbb{Q}}_{M_{n,d}}[\dim
M_{n,d}-R_{f}]\right)\left[-i\right]\in D^{b}_{c}(B)$

where the semisimple summands on the right-hand side satisfy the Lefschetz
symmetry

(3)
$\sigma^{i}:{{}^{\mathfrak{p}}}{\mathcal{H}}^{R_{f}-i}\left(Rf_{*}{\mathbb{Q}}_{M_{n,d}}[\dim
M_{n,d}-R_{f}]\right)\xrightarrow{~{}~{}\simeq~{}~{}}{{}^{\mathfrak{p}}}{\mathcal{H}}^{R_{f}+i}\left(Rf_{*}{\mathbb{Q}}_{M_{n,d}}[\dim
M_{n,d}-R_{f}]\right)$

induced by cup-product with $i$-th power of a relative ample class $\sigma$.
The decomposition (2) is an important invariant for the Hitchin system (1) and
has played a crucial role in all the stories mentioned above. For example,
recent resolutions of the P=W conjecture show that the (cohomological)
perverse filtration

(4) $P_{0}H^{*}(M_{n,d},{\mathbb{Q}})\subset
P_{1}H^{*}(M_{n,d},{\mathbb{Q}})\subset\cdots\subset
H^{*}(M_{n,d},{\mathbb{Q}})$

induced by (2) matches the double-indexed weight filtration associated with
the corresponding character variety via non-abelian Hodge theory:

$P_{k}H^{*}(M_{n,d},{\mathbb{Q}})=W_{2k}H^{*}(M_{\mathrm{char}},{\mathbb{Q}}).$

Furthermore, the Lefschetz symmetry (3) matches the _curious Hard Lefschetz_
for the character variety [25, 40].

Our first result is that both the decomposition (2) and the filtration (4) are
motivic in a strong sense. In particular, this confirms the _motivic
decomposition conjecture_ of Corti–Hanamura [16] for the Hitchin system.

######  Theorem 0.1 (Motivic decomposition).

There exists a decomposition of the relative diagonal into orthogonal
projectors:

$[\Delta_{M_{n,d}/B}]=\sum_{i=0}^{2R_{f}}\mathfrak{r}_{i},\quad\quad\mathfrak{r}_{i}\circ\mathfrak{r}_{i}=\mathfrak{r}_{i},\quad\mathfrak{r}_{i}\circ\mathfrak{r}_{j}=0,\quad
i\neq j,$

which induces a sheaf-theoretic decomposition (2). Here $[\Delta_{M_{n,d}/B}]$
and $\mathfrak{r}_{i}$ lie in the Chow group of the relative product
$M_{n,d}\times_{B}M_{n,d}$, and the composition is taken as relative
correspondences.

Our method also yields immediately the relative Lefschetz standard conjecture
for the Hitchin system, which shows the algebraicity of the inverse of (3).

######  Theorem 0.2 (Relative Lefschetz).

For any relative ample class $\sigma$ and $i>0$, the inverse of the Lefschetz
symmetry (3) is induced by an algebraic cycle $\mathfrak{Z}_{\sigma,i}$ on
$M_{n,d}\times_{B}M_{n,d}$:

$\mathfrak{Z}_{\sigma,i}:{{}^{\mathfrak{p}}}{\mathcal{H}}^{R_{f}+i}\left(Rf_{*}{\mathbb{Q}}_{M_{n,d}}[\dim
M_{n,d}-R_{f}]\right)\xrightarrow{~{}~{}\simeq~{}~{}}{{}^{\mathfrak{p}}}{\mathcal{H}}^{R_{f}-i}\left(Rf_{*}{\mathbb{Q}}_{M_{n,d}}[\dim
M_{n,d}-R_{f}]\right).$

Under the language of [16], we obtain from Theorem 0.1 a decomposition in the
category of relative Chow motives over $B$:

(5) $h\left(M_{n,d}\right)=\bigoplus_{i}h_{i}(M_{n,d})\in\mathrm{CHM}(B)$

with

$h(M_{n,d})=(M_{n,d},[\Delta_{M_{n,d}/B}],0),\quad
h_{i}(M_{n,d})=(M_{n,d},\mathfrak{r}_{i},0)$

whose homological realization under the Corti–Hanamura functor

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

recovers (2). As a consequence, we obtain a motivic perverse filtration

(6) $P_{\bullet}h(M_{n,d})\subset h(M_{n,d});$

see Section 2.1. This motivic perverse filtration specializes to both the
cohomological perverse filtration (4) and a perverse filtration on the Chow
groups of $M_{n,d}$; the latter has not yet been much studied.

###  0.2. Motivic strong perversity of Chern classes

Let ${\mathcal{U}}$ be a universal Higgs bundle over $\Sigma\times M_{n,d}$.
After normalization, it induces a canonical Chern character

$\widetilde{\mathrm{ch}}({\mathcal{U}}):=\sum_{k}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\in\mathrm{CH}^{*}(\Sigma\times
M_{n,d}),\quad\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\in\mathrm{CH}^{k}(\Sigma\times
M_{n,d})$

which is independent on the choice of the universal bundle ${\mathcal{U}}$. It
was proven in [37] that the cohomology class
$\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})$ has strong perversity $k$, _i.e._
, its action on the direct image complex

$\widetilde{\mathrm{ch}}_{k}({\mathcal{U}}):Rf_{*}{\mathbb{Q}}_{\Sigma\times
M_{n,d}}\to Rf_{*}{\mathbb{Q}}_{\Sigma\times M_{n,d}}[2k]$

via cup-product satisfies that

(7) $\widetilde{\mathrm{ch}}_{k}({\mathcal{U}}):{{}^{\mathbf{p}}\tau_{\leq
i}Rf_{*}{\mathbb{Q}}_{\Sigma\times
M_{n,d}}}\to\left({{}^{\mathbf{p}}\tau_{\leq
i+k}}Rf_{*}{\mathbb{Q}}_{\Sigma\times M_{n,d}}\right)[2k].$

Here, we also use $f$ to denote the product morphism $\Sigma\times
M_{n,d}\rightarrow\Sigma\times B$ and consider the corresponding perverse
filtration. Specializing to global cohomology, the equation (7) yields

$\displaystyle\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\cup-:H^{*}(\Sigma\times
M_{n,d},{\mathbb{Q}})$ $\displaystyle\to H^{*+2k}(\Sigma\times
M_{n,d},{\mathbb{Q}}),$ $\displaystyle P_{i}H^{*}(\Sigma\times
M_{n,d},{\mathbb{Q}})$ $\displaystyle\mapsto P_{i+k}H^{*+2k}(\Sigma\times
M_{n,d},{\mathbb{Q}}),$

which further implies the P=W conjecture; we refer to [37, Section 2] for
details on the reduction.

In this paper, we show that the sheaf-theoretic strong perversity (7) for
Chern classes can be lifted motivically using the motivic perverse filtration
(6).

######  Theorem 0.3 (Motivic strong perversity of Chern classes).

We have for any $i,k\in{\mathbb{Z}}$:

$\widetilde{\mathrm{ch}}_{k}({\mathcal{U}}):P_{i}h(\Sigma\times M_{n,d})\to
P_{i+k}h(\Sigma\times M_{n,d})(k).$

Here the action is given by cup-product and $(k)$ stands for the $k$-th Tate
twist.

###  0.3. Motivic $\chi$-independence

Now we consider two degrees $d,d^{\prime}$ which are both coprime to $n$. The
Hitchin systems associated with $M_{n,d},M_{n,d^{\prime}}$ share the same
base,

$f_{d}:M_{n,d}\to B,\quad f_{d^{\prime}}:M_{n,d^{\prime}}\to B.$

Here we use $f_{\bullet}$ to indicate the dependence on the degree.

The following result was predicted by the Hausel–Thaddeus conjecture on the
topological mirror symmetry [26], and was first proven by
Groechenig–Wyss–Ziegler [23] for perverse sheaves via point counting over
finite fields and Maulik–Shen [36] for Hodge modules using support theorem and
the vanishing cycle functor:

(8) $Rf_{d*}{\mathbb{Q}}_{M_{n,d}}\simeq
Rf_{d^{\prime}*}{\mathbb{Q}}_{M_{n,d^{\prime}}}.$

This equation is an instance of the _$\chi$ -independence_ phenomenon: the
moduli space of Higgs bundles can be viewed as the moduli space of
$1$-dimensional stable sheaves properly supported on the surface $T^{*}C$ in
the curve class $n[C]$, and (8) implies that the decomposition theorem is
independent on the choice of the Euler characteristic
$\chi\left(=\mathrm{deg}+(1-g)n\right)$ of the sheaves. This phenomenon has
tight connections to enumerative geometry [46], and has been generalized to
singular moduli spaces using cohomological Donaldson–Thomas theory [36, 30].
Recently, Hoskins and Pepin Lehalleur [29] proved the $\chi$-independence of
the Chow motives of the moduli of Higgs bundles:

(9) $h(M_{n,d})=h(M_{n,d^{\prime}})\in\mathrm{CHM}(\mathrm{pt}).$

We prove a motivic $\chi$-independence result for the relative Chow motives
over $B$, which enhances both the sheaf-theoretic result (8) and the motivic
result (9).

######  Theorem 0.4 (Motivic $\chi$-independence).

Assume $(n,d)=(n,d^{\prime})=1$, we have an isomorphism of relative Chow
motives:

(10) $h(M_{n,d})\simeq h(M_{n,d^{\prime}})\in\mathrm{CHM}(B)$

preserving the motivic perverse filtrations

$P_{k}h(M_{n,d})\simeq P_{k}h(M_{n,d^{\prime}})\in\mathrm{CHM}(B).$

######  Remark 0.5.

  1. (a)

To the best of our knowledge, the isomorphism of the relative Chow motives
(10) (even without the motivic perverse filtrations) was not known before.
After pushing to a point, this recovers the isomorphism (9) of Hoskins and
Pepin Lehalleur.

  2. (b)

By Proposition 2.4, we obtain that $f_{d}:M_{n,d}\to B$ and
$f_{d^{\prime}}:M_{n,d^{\prime}}\to B$ have isomorphic motivic decompositions;
_i.e._ , there exists motivic decompositions

$h(M_{n,d})=\bigoplus_{i}h_{i}(M_{n,d}),\quad
h(M_{M_{n,d^{\prime}}})=\bigoplus_{i}h_{i}(M_{n,d^{\prime}})$

satisfying that

$h_{i}(M_{n,d})\simeq h_{i}(M_{n,d^{\prime}})\in\mathrm{CHM}(B).$

###  0.4. Background and relations to other work

In this section, we discuss briefly some background and relations to some
previous work.

####  0.4.1. Motivic decomposition conjecture

Let $f:X\to B$ be a proper morphism of nonsingular varieties. The motivic
decomposition conjecture of Corti–Hanamura [16] predicts that in general the
sheaf-theoretic decomposition

$Rf_{*}{\mathbb{Q}}_{X}[\dim
X-R_{f}]\simeq\bigoplus_{i=0}^{2R_{f}}{{}^{\mathfrak{p}}}{\mathcal{H}}^{i}\left(Rf_{*}{\mathbb{Q}}_{X}[\dim
X-R_{f}]\right)[-i],\quad R_{f}:=\dim X\times_{B}X-\dim X$

admits a _motivic_ lifting. More precisely, as in the statement of Theorem
0.1, they conjectured that there is a decomposition of the relative diagonal
cycle into orthogonal projectors

$[\Delta_{X/B}]=\sum_{i=0}^{2R_{f}}\mathfrak{r}_{i},\quad\quad\mathfrak{r}_{i}\circ\mathfrak{r}_{i}=\mathfrak{r}_{i},\quad\mathfrak{r}_{i}\circ\mathfrak{r}_{j}=0,\quad
i\neq j,$

which induces a sheaf-theoretic decomposition as above. In general, the
conjectural existence of the motivic decomposition is wide open. We note two
major challenges in constructing such a decomposition. First, in the special
case when $X$ is a nonsingular projective variety and $B$ is a point, the
motivic decomposition conjecture is equivalent to the existence of a
Chow–Künneth decomposition for $X$. This was conjectured by Murre [44], and
has only been verified in very limited cases. Second, even if we are able to
find orthogonal projectors over a Zariski dense open subset $U\subset B$
(_e.g._ the locus where $f$ is smooth), it is unclear if these projectors over
$U$ can be extended to orthogonal projectors over $B$. In particular, the
cycles given by the Zariski closures of the orthogonal projectors over $U$ are
not orthogonal projectors in general, and understanding the behaviour of the
“boundary” is crucial.

Recently there have been some attempts to apply the theory of motivic
intermediate extension of Wildeshaus to construct motivic decompositions for
certain proper morphisms with _Tate fibers_ ; we refer to [13] and references
therein for more details.

####  0.4.2. Derived equivalences

For a certain class of examples, one approach to the motivic decomposition
conjecture is to use symmetries of the derived category of coherent sheaves.
It was first found by Beauville [5] and Deninger–Murre [18] over 35 years ago
that descending Mukai’s Fourier transform [43] from the derived category of
coherent sheaves to algebraic cycles yields a motivic decomposition for an
abelian scheme $f:A\to B$. Later, the Fourier transform was further used by
Künnerman [31] to construct a Lefschetz decomposition. In this case the
morphism $f$ is smooth and each summand appeared in the decomposition theorem
is a (shifted) semisimple local system.

The case of abelian fibrations with singular fibers is more complicated, and
the perverse $t$-structure comes into play. On the derived category side,
Arinkin [3] extended the Fourier–Mukai transform to compactified Jacobian
fibrations associated with a family of projective integral locally planar
curves. Arinkin’s work was motivated by an attempt to understanding the
classical limit of the geometric Langlands correspondence for Hitchin systems
[3, Section 4.5]. Recently, motivated by the P=W conjecture for Hitchin moduli
spaces [9, 12, 37, 24], a theory of Fourier transform for relative Chow
motives was established in [38] using the Arinkin sheaf. As a consequence of
the Fourier theory and Ngô’s support theorem [45], it was obtained in [38]
that the motivic decomposition conjecture holds for compactified Jacobians
fibrations associated with projective integral locally planar curves. We note
that, for the singular fibers, this approach matches the complexity of the
Arinkin sheaf with the complexity of intermediate extensions of the local
systems.

####  0.4.3. Hitchin systems

For the Hitchin system (1), the techniques described above imply the motivic
decomposition conjecture for the Hitchin system over a Zariski open subset
$B^{\mathrm{ell}}\subset B$, called _the elliptic locus_ , formed by integral
spectral curves. This open subset is exactly the locus over which the Hitchin
fibers are integral. However, to the best of our knowledge, it is not yet
known how to extend Arinkin’s derived equivalence [3] from the elliptic locus
to the total Hitchin base $B$. Even worse, the full support theorem _fails_
for (1); the results of [10] suggest that the supports of the decomposition
theorem associated with (1) are very complicated and there is no complete
description by far. In view of the approach of [1], the lack of the control of
the supports provides also the difficulty in proving the relative Lefschetz
standard conjecture. The main point of this paper is to give a proof of these
conjectures for the Hitchin system (1) in the absence of the aforementioned
geometric ingredients. As a byproduct, we show that Corti–Hanamura’s motivic
decomposition conjecture is preserved under specialization; see Corollary 4.2.

###  0.5. Acknowledgements

J.S. gratefully acknowledges the hospitality of both the math department of
MIT and the Isaac Newton Institute at Cambridge during his stay in the spring
of 2024 where part of this work was completed.

D.M. was supported by a Simons Investigator Grant. J.S. was supported by the
NSF grant DMS-2301474, a Sloan Research Fellowship, and a Simons Fellowship
during his visit at the Isaac Newton Institute.

##  1\. Moduli of Higgs bundles

In this section, we review the geometry of the moduli of Higgs bundles.
Throughout, we fix a nonsingular projective curve $\Sigma$ of genus $g\geq 2$,
and fix two integers $n,d$ with $(n,d)=1$.

###  1.1. Moduli spaces

We denote by $M_{n,d}$ the moduli space of stable Higgs bundles

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

where the stability condition is with respect to the slope
$\mu({\mathcal{E}},\theta)=\mathrm{deg}({\mathcal{E}})/\mathrm{rk}({\mathcal{E}})$.
The moduli space admits a proper morphism to an affine space

(11) $f:M_{n,d}\to
B:=\bigoplus_{i=1}^{n}H^{0}\left(\Sigma,\omega_{\Sigma}^{\otimes i}\right).$

This map is given by calculating the characteristic polynomial of the Higgs
field $\theta$:

$f({\mathcal{E}},\theta)=(\mathrm{tr}(\theta),\mathrm{tr}(\wedge^{2}\theta),\cdots,\mathrm{det}(\theta))\in
B,$

and is now known as the _Hitchin system_.111Later, when we need to consider
the moduli spaces $M_{n,d}$ with different $d$, we will write $f_{d}$ to
indicate the dependence of the Hitchin system on the degree. The moduli space
$M_{n,d}$ is a holomorphic symplectic variety and the Hitchin system $f$ is a
Lagrangian fibration [27, 28].

By the Beauville–Narasimhan–Ramanan correspondence [6], the Hitchin base $B$
can be viewed as the parameter space of spectral curves; these are properly
supported curves in the open surface $\mathrm{Tot}_{\Sigma}(\omega_{\Sigma})$
that are degree $n$ covers of the $0$-section $\Sigma$. We define the elliptic
locus $B^{\mathrm{ell}}\subset B$ to be the open subset formed by integral
spectral curves. The results of [38, Theorem 0.3 and Section 4] thus establish
the motivic decomposition for the restricted Hitchin system

$f^{\mathrm{ell}}:M^{\mathrm{ell}}_{n,d}\to B^{\mathrm{ell}}.$

As we commented in Section 0.4.2, at the level of algebraic cycles it is not
easy to extend orthogonal projectors from $B^{\mathrm{ell}}$ to $B$. A
plausible approach is to extend first Arinkin’s Fourier–Mukai transform from
$M^{\mathrm{ell}}_{n,d}$ to the total space $M_{n,d}$, and then extend the
Fourier theory of relative Chow motives of [38] as well. Even if we are able
to achieve this, a potential difficulty is the complexity of the supports (see
Remark 1.1 below) which makes it hard to calculate the homological realization
of the projectors.

Instead, our approach in this paper is to first construct orthogonal
projectors for certain parabolic Higgs moduli space which we review in the
following sections; then we show that these projectors induce the desired
orthogonal projectors of the original Hitchin system. This path of reduction
using parabolic Higgs moduli spaces was applied in proving the P=W conjecture:
it was first used in the proof of Hausel–Mellit–Minets–Schiffmann [24], and
was later also used in the proof of Maulik–Shen–Yin [38, Section 5.4].

######  Remark 1.1.

We can also consider the moduli space of stable $\mathfrak{D}$-twisted Higgs
bundles

$({\mathcal{E}},\theta),\quad\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\omega_{\Sigma}(\mathfrak{D})$

as in [14] with $\mathfrak{D}\geq 0$ an effective divisor. The case of
$\mathfrak{D}=0$ recovers the Hitchin system (11) we considered above. We note
that the decomposition theorem for the Hitchin system are very different when
$\mathfrak{D}=0$ and $\mathfrak{D}>0$ respectively. For $\mathfrak{D}>0$,
Chaudouard–Laumon [14] showed that every simple summand that appeared in the
decomposition theorem has full support $B$. This full-support property has
been generalized to singular moduli spaces without coprime assumption on $n$
and $d$ [36] and has many applications [35, 22, 29, 37]. However, when $D=0$,
the behaviour of the supports for the Hitchin system is very different. In
[10], the authors classified all the supports over an open subset of the
Hitchin base formed by _reduced_ spectral curves. However it is not well
understood what are all the possible supports outside this open subset. A
question raised by Mauri–Migliorini [39, Question 1.5] asked if the full
support property holds for the singular moduli space of rank $n$ and degree
$d=0$. In Section 4.3.5, we will discuss briefly how to extend the results of
this paper to the $\mathfrak{D}$-twisted case.

###  1.2. Parabolic Higgs bundles

In this section, we review the parabolic Higgs moduli space (_c.f._ [24,
Section 8.4]).

We fix a point $p\in\Sigma$. Denote by ${M}^{\mathrm{par}}$ the moduli space
of stable parabolic Higgs bundles $({\mathcal{E}},\theta,F_{p})$, where
$({\mathcal{E}},\theta)$ is a meromorphic Higgs bundle

$({\mathcal{E}},\theta),\quad\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\omega_{\Sigma}(p),\quad\mathrm{rk}({\mathcal{E}})=n,\quad\mathrm{deg}({\mathcal{E}})=d,$

$F_{p}$ is a complete flag of ${\mathcal{E}}$ over the point $p$ such that the
residue $\mathrm{res}_{p}(\theta)$ of the Higgs field over the point $p$
preserves the flag $F_{p}$. The stability condition here requires that any
parabolic sub-Higgs bundle has a smaller slope.

We record the following lemma which is a direct consequence of the
Riemann–Roch formula.

######  Lemma 1.2.

We have

$H^{0}(\Sigma,\omega_{\Sigma})=H^{0}(\Sigma,\omega_{\Sigma}(p)).$

Note that, since $\mathrm{res}_{p}(\theta)$ preserves a complete flag, its
eigenvalues naturally lie in an affine space

${\mathbb{A}}^{n}:=\left\\{(\lambda_{1},\cdots,\lambda_{n})|~{}\lambda_{i}\in{\mathbb{A}}^{1}\right\\},$

and Lemma 1.2 ensures that the sum of the eigenvalues is always $0$; in other
words, the image of $\chi_{p}$ lies in a codimension $1$ hyperplane

${\mathcal{A}}=\left\\{(\lambda_{1},\cdots,\lambda_{n})|\sum_{i}\lambda_{i}=0\right\\}\subset{\mathbb{A}}^{n}.$

As a result, the moduli space admits a natural morphism

(12) $\chi_{p}:{M}^{\mathrm{par}}\to{\mathcal{A}}\simeq{\mathbb{A}}^{n-1}$

which is in fact smooth. Each closed fiber is a moduli of stable parabolic
Higgs bundles with specified eigenvalues at $p$; this is a nonsingular
symplectic variety.

For our purposes, we take a general line
$T(\simeq{\mathbb{A}}^{1})\subset{\mathcal{A}}$ passing through the origin and
restrict the family (12) over $T$:

$\chi_{p}:M^{\mathrm{par}}_{T}\to T.$

This family connects two types of Hitchin moduli spaces which we now describe:

  1. (a)

Let $\eta\in T$ be a general point which represents $n$ _distinct_ eigenvalues
satisfying the condition that no proper subset of the eigenvalues has sum $0$.
We denote by $W_{\eta}$ the linear system parameterizing spectral curves with
fixed eigenvalues $\eta$ over $p\in\Sigma$. Let $C\to W_{\eta}$ be the
universal spectral curve. By the generality of $\eta$ and Lemma 1.2, every
curve in the family $C\to W_{\eta}$ is integral. The fiber
$M^{\mathrm{par}}_{\eta}:=\chi_{p}^{-1}(\eta)$ fits into the following
Cartesian diagram:

${M^{\mathrm{par}}_{\eta}}$${\overline{J}^{e}_{C}}$${\widetilde{W}_{\eta}}$${W_{\eta}.}$

Here the right vertical arrow is the (twisted) compactified Jacobian fibration
of degree $e$, $\widetilde{W}_{\eta}$ can be viewed as the parameter space of
spectral curves lying in $W_{\eta}$ with a marking over $p\in\Sigma$, and the
bottom horizontal arrow is the projection to $W_{\eta}$ which is a natural
$\mathfrak{S}_{n}$-quotient.222Here $e$ is an integer determined by the degree
$d$ of the Higgs bundles and is coprime to $n$. The Hitchin fibration
associated with $M^{\mathrm{par}}_{\eta}$ is the composition of a vertical and
a horizontal maps in the diagram above

$h_{\eta}:M^{\mathrm{par}}_{\eta}\to W_{\eta}.$

Let $T^{\circ}\subset T$ be the Zariski open subset such that the spectral
curves in $W_{\eta}$ are integral for any $\eta\in T^{\circ}$. We thus obtain
a morphism

$h^{\circ}:M^{\mathrm{par},\circ}\to W^{\circ}$

satisfying that $M^{\mathrm{par,\circ}}$ is a compactified Jacobian fibration
over a base $\widetilde{W}^{\circ}$, and $W^{\circ}$ is the quotient of
$\widetilde{W}^{\circ}$ by the permutation group $\mathfrak{S}_{n}$.

  2. (b)

The central fiber $M^{\mathrm{par}}_{0}:=\chi_{p}^{-1}(0)$ together with its
associated Hitchin system

$h_{0}:M^{\mathrm{par}}_{0}\to W_{0}$

is a classical object, known as the moduli of stable strongly parabolic Higgs
bundles. The variety $M^{\mathrm{par}}_{0}$ parameterizes triples
$({\mathcal{E}},\theta,F_{p})$ with similar condition as for stable parabolic
Higgs bundle, but the main change is that the residue of the Higgs field
preserves the flag in a stronger sense:

$\mathrm{res}_{p}(\theta):F_{p,j}\to F_{p,{j+1}}.$

In other words, $M_{0}^{\mathrm{par}}$ is the moduli of stable parabolic Higgs
bundles with nilpotent residue.

In conclusion, by specializing over $T$, we are able to connect the geometry
of a compactified Jacobian fibration to the Hitchin system associated with the
moduli of strongly parabolic Higgs bundles:

$h^{\circ}:M^{\mathrm{par},\circ}\to W^{\circ}\quad\rightsquigarrow\quad
h_{0}:M^{\mathrm{par}}_{0}\to W_{0}.$

We conclude this section by further connecting the latter to the more
classical Hitchin system $f:M_{n,d}\to B$ via a correspondence following [24,
Section 8.6].

###  1.3. Correspondences

We consider the closed subvariety parameterizing parabolic Higgs bundles with
$0$-residue at $p$,

(13) $\widetilde{M}_{0}=\\{\mathrm{res}_{p}(\theta)=0\\}\subset
M_{0}^{\mathrm{par}}=\\{\mathrm{res}_{p}(\theta)\textrm{ is nilpotent}\\}.$

In particular, every parabolic Higgs bundle in $\widetilde{M}_{0}$ has no pole
at $p\in\Sigma$. Hence there is a well-defined forgetful map:

(14) $\widetilde{M}_{0}\to
M_{n,d},\quad({\mathcal{E}},\theta,F_{p})\mapsto({\mathcal{E}},\theta)$

where every closed fiber is isomorphic to a flag variety $\mathrm{Fl}_{n}$
parameterizing all possible choices of flags.

We summarize the discussion above in the diagram

(15)
${\widetilde{M}_{0}}$${M_{0}^{\mathrm{par}}}$${M_{n,d}}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$

where $\iota$ is the closed embedding (13) and $\pi$ is the smooth projection
(14). All the three moduli spaces in (15) map to a common base $W_{0}$ with
all the induced natural diagrams commuting: the map from
$M_{0}^{\mathrm{par}}$ is the parabolic Hitchin map $h_{0}$, the map from
$\widetilde{M}_{0}$ is the induced map, and the map from $M_{n,d}$ is the
Hitchin map $f:M_{n,d}\to B$ composed with the natural inclusion of the linear
subspace $B\subset W_{0}$.

Our purpose is to construct certain algebraic cycles on
$M_{n,d}\times_{B}M_{n,d}$ which serve as orthogonal projectors. The diagram
(15) allows us to construct cycles for $M_{n,d}$ from $M_{0}^{\mathrm{par}}$,
_i.e._ , we have

$\Gamma:\mathrm{CH}_{*}({M^{\mathrm{par}}_{0}\times_{W_{0}}M^{\mathrm{par}}_{0}})\to\mathrm{CH}_{*}(M_{n,d}\times_{B}M_{n,d})$

given by the formula

$\Gamma(\alpha):={{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ\alpha\circ[\widetilde{M}_{0}].$

Throughout, the Chow groups are taken with rational coefficients, and here we
view $[\widetilde{M}_{0}]$ to be a correspondence from $M_{n,d}$ to
$M^{\mathrm{par}}_{0}$ via (15) and ${{}^{\mathfrak{t}}[\widetilde{M}_{0}]}$
to be its transpose. The change of indices of the Chow groups through $\Gamma$
is given by the dimension formula:

$\mathrm{codim}(\iota)=\mathrm{rel.dim}(\pi)=\binom{n}{2};$

in particular, $\Gamma$ sends $\mathrm{CH}_{\dim
M_{0}^{\mathrm{par}}}({M^{\mathrm{par}}_{0}\times_{W_{0}}M^{\mathrm{par}}_{0}})$
to $\mathrm{CH}_{\dim M_{n,d}}\left(M_{n,d}\times_{B}M_{n,d}\right)$.

The following lemma gives a sufficient condition for $\Gamma(\alpha)$ to
vanish.

######  Lemma 1.3.

Suppose we are given a flat, surjective morphism $p:Z\to W_{0}$ with $Z$
irreducible and nonsingular and that there exists a factorization of $\alpha$

$\alpha:=\alpha_{2}\circ\alpha_{1}\in\mathrm{CH}_{\dim
M_{0}^{\mathrm{par}}}({M^{\mathrm{par}}_{0}\times_{W_{0}}M^{\mathrm{par}}_{0}})$

such that either

$\alpha_{1}\in\mathrm{CH}_{>\dim(M_{0}^{\mathrm{par}}\times_{W_{0}}Z)-\binom{n}{2}}(M_{0}^{\mathrm{par}}\times_{W_{0}}Z)\textup{
or
}\alpha_{2}\in\mathrm{CH}_{>\dim(M_{0}^{\mathrm{par}}\times_{W_{0}}Z)-\binom{n}{2}}(Z\times_{W_{0}}M_{0}^{\mathrm{par}}).$

Then we have

$\Gamma(\alpha)=0\in\mathrm{CH}_{\dim
M_{n,d}}(M_{n,d}\times_{W_{0}}M_{n,d})=\mathrm{CH}_{\dim
M_{n,d}}(M_{n,d}\times_{B}M_{n,d}).$

###### Proof.

We prove the vanishing under the first hypothesis; the argument under the
second hypothesis is identical. If we consider the composition
$\alpha_{1}\circ[\widetilde{M}_{0}]\in\mathrm{CH}(M_{n,d}\times_{W_{0}}Z)$,
the assumption on $\alpha_{1}$ implies that this lives in degree strictly
larger than

$\dim(M_{0}^{\mathrm{par}}\times_{W_{0}}Z)-2\binom{n}{2}.$

The assumption on $p$ implies that this bound equals

$\dim M_{0}^{\mathrm{par}}+\dim Z-\dim W_{0}-2\binom{n}{2}=\dim M_{n,d}+\dim
Z-\dim W_{0}$

which is the dimension of $M_{n,d}\times_{W_{0}}Z$. As a result, we have the
vanishing of $\alpha_{1}\circ[\widetilde{M}_{0}]$ so $\Gamma(\alpha)$ vanishes
as well. ∎

We conclude this section by giving an overview of the rest of the paper. Our
strategy is to first construct projectors for compactified Jacobian fibrations
and show that these algebraic cycles satisfy the desired properties as in
Theorems 0.3 and 0.4 using the techniques of [38]. We will discuss this in
Section 2, which solves the problems for $h_{\eta}:M^{\mathrm{par}}_{\eta}\to
W_{\eta}$ of Section 1.2(a). Then applying the specialization and the
correspondence $\Gamma$, we obtain algebraic cycles for $M_{n,d}$; we further
apply a Springer theory argument to show that these cycles form orthogonal
projectors as desired. This is treated in Section 3. Finally, in Section 4, we
show that the projectors obtained via $\Gamma$ have the correct homological
realization under the Corti–Hanamura functor.

##  2\. Projectors for compactified Jacobians

In this section, we first introduce the notion of motivic perverse
filtrations. Then we treat the case of compactified Jacobians associated with
locally planar curves, where our main tool is the Fourier theory developed in
[38]. For our purpose, we need a slight extension of the results of [38] in
Section 2.3. Then we apply these results to the parabolic Hitchin system of
Section 1.2(a), which further allows us to prove analogues of Theorems 0.1,
0.3, and 0.4 for $h_{0}:M^{\mathrm{par}}_{0}\to W_{0}$.

###  2.1. Motivic perverse filtrations

In this section, we define a _motivic perverse filtration_ , which lifts its
cohomological counterpart to the theory of relative Chow motives of
Corti–Hanamura.

The theory of relative Chow motives [16] is built to be compatible with the
decomposition theorem, adapts well to non-proper bases, and admits natural
Chow/homological realizations [21]. We refer to [38, Section 2.2] for a brief
review of the theory.

We work over a nonsingular base variety $B$. Recall that the group of degree
$k$ relative correspondences between two nonsingular proper $B$-schemes $X,Y$
is333Here we assume $Y$ irreducible. For reducible $Y$, we decompose
$Y=\sqcup_{\alpha}Y_{\alpha}$ and define the group of correspondences
accordingly.

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

Compositions of relative correspondences are defined via refined intersection
theory. The category of relative Chow motives $\mathrm{CHM}(B)$ consists of
objects triples $(X,\mathfrak{p},m)$ where $X$ is a nonsingular proper
$B$-scheme, $\mathfrak{p}\in\mathrm{Corr}^{0}_{B}(X,X)$ is a projector, and
$m\in\mathbb{Z}$. For example, the motive of $X$ is

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

where $\Delta_{X/B}$ is the relative diagonal. Morphisms between two motives
$M=(X,\mathfrak{p},m)$, $N=(Y,\mathfrak{q},n)$ are

$\operatorname{Hom}_{\mathrm{CHM}(B)}(M,N):=\mathfrak{q}\circ\mathrm{Corr}^{n-m}_{B}(X,Y)\circ\mathfrak{p}.$

Note that in [38, Section 2.3.3], we also introduced relative binary
correspondences

$\mathrm{Corr}_{B}^{k}(X,Y;Z):=\mathrm{CH}_{\dim
Z-k}(X\times_{B}Y\times_{B}Z)$

where $X,Y,Z$ are nonsingular proper $B$-schemes. Binary morphisms from
$M=(X,{\mathfrak{p}},m)$, $N=(Y,{\mathfrak{q}},n)$ to $P=(Z,{\mathfrak{r}},p)$
are given by

${\mathfrak{r}}\circ\mathrm{Corr}_{B}^{p-m-n}(X,Y;Z)\circ({\mathfrak{p}}\times{\mathfrak{q}}).$

For example, the motivic cup-product

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

is induced by the class of the relative small diagonal

$[\Delta^{\mathrm{sm}}_{X/B}]\in\mathrm{Corr}_{B}^{0}(X,X;X);$

see [38, Section 2.3.3] for more details. Finally, the degree $k$ Chow group
of $M=(X,\mathfrak{p},m)$ is

$\mathrm{CH}^{k}(M):=\operatorname{Hom}_{\mathrm{CHM}(B)}(h(B/B),M(k))$

where $M(k):=(X,\mathfrak{p},m+k)$ is the $k$-th Tate twist of $M$.

Recall the Corti–Hanamura realization functor

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

which sends the motive $h(X)$ for $f:X\to B$ to the direct image complex
$Rf_{*}{\mathbb{Q}}_{X}$. The latter admits a natural (sheaf-theoretic)
perverse filtration induced by the perverse truncation functor.

Let $f:X\to B$ be a proper morphism with $X$ irreducible and nonsingular, and
let $R_{f}$ be the defect of semismallness $\dim X\times_{B}X-\dim X$.

######  Definition 2.1.

We say that a sequence of motives

$P_{k}h(X)=(X,\mathfrak{p}_{k},0),\quad 0\leq k\leq 2R_{f},$

which are all summands of $h(X)$, form a _motivic perverse filtration_ for
$f$, if

  1. (a)

(Termination) $P_{2R_{f}}h(X)=h(X)$;

  2. (b)

(Realization) Under the Corti–Hanamura functor (16) the natural inclusion

$P_{\bullet}h(X)\hookrightarrow h(X)$

specializes to the natural inclusion induced by the perverse truncation
functor

${}^{\mathfrak{p}}\tau_{\leq\bullet+(\dim
X-R_{f})}Rf_{*}{\mathbb{Q}}_{X}\hookrightarrow Rf_{*}{\mathbb{Q}}_{X};$

  3. (c)

(Semi-orthogonality) for any $k$ we have the relation

$\mathfrak{p}_{k+1}\circ\mathfrak{p}_{k}=\mathfrak{p}_{k}.$

We note that (c) together with an easy induction implies

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

Indeed, we have

$\mathfrak{p}_{l}\circ\mathfrak{p}_{k}=\mathfrak{p}_{l}\circ\mathfrak{p}_{k+1}\circ\mathfrak{p}_{k}=\mathfrak{p}_{l}\circ\mathfrak{p}_{l-1}\circ\cdots\circ\mathfrak{p}_{k+1}\circ\mathfrak{p}_{k}=\mathfrak{p}_{l-1}\circ\cdots\circ\mathfrak{p}_{k+1}\circ\mathfrak{p}_{k}=\mathfrak{p}_{k}.$

On the other hand, this does not mean that $P_{k}h(X)$ is a summand of
$P_{l}h(X)$ if $k<l$. So if we define directly

$\mathfrak{r}_{k}:=\mathfrak{p}_{k}-\mathfrak{p}_{k-1},$

they are only semi-orthogonal:

${\mathfrak{r}}_{l}\circ{\mathfrak{r}}_{k}=0,\quad k<l,$

and we lose control of the other half of the orthogonality

${\mathfrak{r}}_{l}\circ{\mathfrak{r}}_{k}\stackrel{{\scriptstyle?}}{{=}}0,\quad
k\geq l.$

In particular, the $\mathfrak{r}_{k}$ are not necessarily projectors. However,
the following proposition in [38] shows that the existence of a motivic
perverse filtration is equivalent to the existence of a motivic decomposition.

######  Proposition 2.2.

[Motivic perverse filtration $=$ motivic decomposition] A motivic perverse
filtration induces a motivic decomposition

$h(X)=\bigoplus_{i=0}^{2R_{f}}h_{i}(X)\in\mathrm{CHM}(B).$

###### Proof.

The proof was given in the [38, Section 2.5.5]; we include it here for
completeness. Assume that the motives $P_{k}h(X)=(X,\mathfrak{p}_{k},0)$
induce a motivic perverse filtration. We consider the modification of
projectors:

$\widetilde{{\mathfrak{p}}}_{k}:={\mathfrak{p}}_{k}\circ\cdots\circ{\mathfrak{p}}_{2R_{f}},\quad\overline{{\mathfrak{p}}}_{k}:=\widetilde{{\mathfrak{p}}}_{k}-\widetilde{{\mathfrak{p}}}_{k-1}.$

Using (17), it is straightforward to verify that both
$\widetilde{{\mathfrak{p}}}_{k}$ and $\overline{{\mathfrak{p}}}_{k}$ are
projectors satisfying the relations

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

In particular, we have

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

with

$\widetilde{P}_{k}h(X):=(X,\widetilde{{\mathfrak{p}}}_{k},0),\quad
h_{k}(X):=(X,\overline{\mathfrak{p}}_{k},0).$

We see from (b) that both motives $\widetilde{P}_{k}h(X)$ and $P_{k}h(X)$ have
the same realization under (16), which is the one given by the perverse
truncation functor. In particular, the projector $\overline{\mathfrak{p}}_{k}$
realizes taking the corresponding perverse cohomology of the complex
$Rf_{*}{\mathbb{Q}}_{X}$. This finishes the proof of the proposition. ∎

In the following, we further extend the equivalence between motivic perverse
filtrations and motivic decompositions of Proposition 2.2 to isomorphism
classes.

Assume that

$f:X\to B,\quad f^{\prime}:X^{\prime}\to B$

admit motivic perverse filtrations

(18) $P_{k}h(X)=(X,\mathfrak{p}_{k},0),\quad
P_{k}h(X^{\prime})=(X^{\prime},\mathfrak{p}^{\prime}_{k},0),\quad 0\leq k\leq
2R_{f}=2R_{f^{\prime}}.$

######  Definition 2.3.

We say that an isomorphism of relative Chow motives

$\mathfrak{C}:h(X)\xrightarrow{~{}\simeq~{}}h(X^{\prime}),\quad\mathfrak{C}^{-1}:h(X^{\prime})\xrightarrow{~{}\simeq~{}}h(X)$

with

$\mathfrak{C}\in\mathrm{Corr}_{B}^{0}(X,X^{\prime}),\quad\mathfrak{C}^{-1}\in\mathrm{Corr}_{B}^{0}(X^{\prime},X)$

preserves the motivic perverse filtrations (18), if for any $k$ we have

(19)
$\mathfrak{q}^{\prime}_{k+1}\circ\mathfrak{C}\circ\mathfrak{p}_{k}=0,\quad\mathfrak{q}_{k+1}\circ\mathfrak{C}^{-1}\circ\mathfrak{p}^{\prime}_{k}=0.$

Here

$\mathfrak{q}_{k+1}:=[\Delta_{X/B}]-\mathfrak{p}_{k},\quad\mathfrak{q}^{\prime}_{k+1}:=[\Delta_{X^{\prime}/B}]-\mathfrak{p}^{\prime}_{k}.$

The following proposition shows that, for two relative Chow motives, an
isomorphism preserving motivic filtrations automatically yields isomorphic
motivic decompositions as well.

######  Proposition 2.4.

With the notation as above, assume that there is an isomorphism between the
relative Chow motives $h(X)$ and $h(X^{\prime})$ preserving the motivic
perverse filtrations (18). Then the following holds:

  1. (a)

For any $k$ we have an isomorphism

$P_{k}h(X)\simeq P_{k}h(X^{\prime})\in\mathrm{CHM}(B).$

compatible with the inclusions $P_{k}h(X)\hookrightarrow h(X)$ and
$P_{k}h(X^{\prime})\hookrightarrow h(X^{\prime})$.

  2. (b)

The motivic decompositions of $h(X),h(X^{\prime})$ given by Proposition 2.2
are isomorphic, _i.e._ , for any $k$ we have an isomorphism

$h_{k}(X)\simeq h_{k}(X^{\prime})\in\mathrm{CHM}(B).$

###### Proof.

We first prove (a). Assume that the correspondences
$\mathfrak{C},\mathfrak{C}^{-1}$ induce an isomorphism between
$h(X),h(X^{\prime})$ preserving the perverse filtrations. We claim that the
morphisms between the relative Chow motives

$\mathfrak{p}^{\prime}_{k}\circ\mathfrak{C}\circ\mathfrak{p}_{k}:P_{k}h(X)\to
P_{k}h(X^{\prime}),\quad\mathfrak{p}_{k}\circ\mathfrak{C}^{-1}\circ\mathfrak{p}^{\prime}_{k}:P_{k}h(X^{\prime})\to
P_{k}h(X)$

are inverse to each other. In fact, we have

$\displaystyle\mathfrak{p}_{k}=\mathfrak{p}_{k}\circ\mathfrak{C}^{-1}\circ\mathfrak{C}\circ\mathfrak{p}_{k}$
$\displaystyle=\mathfrak{p}_{k}\circ\mathfrak{C}^{-1}\circ(\mathfrak{p}^{\prime}_{k}+\mathfrak{q}^{\prime}_{k+1})\circ\mathfrak{C}\circ\mathfrak{p}_{k}$
$\displaystyle=\mathfrak{p}_{k}\circ\mathfrak{C}^{-1}\circ\mathfrak{p}^{\prime}_{k}\circ\mathfrak{C}\circ\mathfrak{p}_{k}$

where we used
$\mathfrak{q}^{\prime}_{k+1}\circ\mathfrak{C}\circ\mathfrak{p}_{k}=0$ in the
last equation. This proves

$\mathfrak{p}_{k}\circ\mathfrak{C}^{-1}\circ\mathfrak{p}^{\prime}_{k}\circ\mathfrak{C}\circ\mathfrak{p}_{k}=\mathfrak{p}_{k};$

the other identity is parallel. To show that this isomorphism is compatible
with $\mathfrak{C}$ and the inclusions into $h(X)$ and $h(X^{\prime})$, it
suffices to show that

$\mathfrak{C}\circ\mathfrak{p}_{k}=\mathfrak{p}^{\prime}_{k}\circ(\mathfrak{p}^{\prime}_{k}\circ\mathfrak{C}\circ\mathfrak{p}_{k})$

which again follows via the condition that $\mathfrak{C}$ preserves the
motivic filtration. We have completed the proof of (a).

Now we prove (b); we follow the notation in the proof of Proposition 2.2. As
in (a), it suffices to show that

(20)
$\left(\widetilde{\mathfrak{p}}_{k}-\widetilde{\mathfrak{p}}_{k-1}\right)\circ\mathfrak{C}^{-1}\circ\left(\widetilde{\mathfrak{p}}^{\prime}_{k}-\widetilde{\mathfrak{p}}^{\prime}_{k-1}\right)\circ\mathfrak{C}\circ\left(\widetilde{\mathfrak{p}}_{k}-\widetilde{\mathfrak{p}}_{k-1}\right)=\widetilde{\mathfrak{p}}_{k}-\widetilde{\mathfrak{p}}_{k-1};$

the other identity follows from a parallel argument. Expanding the left-hand
side of (20), we obtain 8 terms, each of which is of the form

$\pm~{}\widetilde{\mathfrak{p}}_{i_{1}}\circ\mathfrak{C}^{-1}\circ\widetilde{\mathfrak{p}}^{\prime}_{i_{2}}\circ\mathfrak{C}\circ\widetilde{\mathfrak{p}}_{i_{3}},\quad
i_{1},i_{2},i_{3}\in\\{k-1,k\\}.$

We claim that

(21)
$\widetilde{\mathfrak{p}}_{i_{1}}\circ\mathfrak{C}^{-1}\circ\widetilde{\mathfrak{p}}^{\prime}_{i_{2}}\circ\mathfrak{C}\circ\widetilde{\mathfrak{p}}_{i_{3}}=\widetilde{\mathfrak{p}}_{\mathrm{min}\\{i_{1},i_{2},i_{3}\\}}.$

Since at least two of the three indices $i_{1},i_{2},i_{3}$ are equal, without
loss of generality we treat here the case $i_{2}=i_{3}$. By (17) and the first
equation of (19), we have

$\mathfrak{C}\circ\mathfrak{p}_{k}=\mathfrak{p}_{l}^{\prime}\circ\mathfrak{C}\circ\mathfrak{p}_{k},\quad
k\leq l,$

which further yields

$\displaystyle\widetilde{\mathfrak{p}}^{\prime}_{i_{2}}\circ\mathfrak{C}\circ\widetilde{\mathfrak{p}}_{i_{2}}$
$\displaystyle={\mathfrak{p}}^{\prime}_{i_{2}}\circ\cdots\circ{\mathfrak{p}}^{\prime}_{2R_{f}}\circ{\mathfrak{C}}\circ{\mathfrak{p}}_{i_{2}}\circ\cdots\circ{\mathfrak{p}}_{2R_{f}}$
$\displaystyle={\mathfrak{C}}\circ{\mathfrak{p}}_{i_{2}}\circ\cdots\circ{\mathfrak{p}}_{2R_{f}}$
$\displaystyle={\mathfrak{C}}\circ\widetilde{{\mathfrak{p}}}_{i_{2}}.$

Consequently, (21) follows:

$\displaystyle\widetilde{\mathfrak{p}}_{i_{1}}\circ\mathfrak{C}^{-1}\circ\widetilde{\mathfrak{p}}^{\prime}_{i_{2}}\circ\mathfrak{C}\circ\widetilde{\mathfrak{p}}_{i_{2}}$
$\displaystyle=\widetilde{\mathfrak{p}}_{i_{1}}\circ\mathfrak{C}^{-1}\circ\mathfrak{C}\circ\widetilde{\mathfrak{p}}_{i_{2}}$
$\displaystyle=\widetilde{\mathfrak{p}}_{i_{1}}\circ\widetilde{\mathfrak{p}}_{i_{2}}$
$\displaystyle=\widetilde{{\mathfrak{p}}}_{\mathrm{min}\\{i_{1},i_{2}\\}}.$

Finally, (20) is deduced immediately from (21), which completes the proof of
(b). ∎

In many cases, it is more natural to work with a motivic perverse filtration,
rather than a motivic decomposition. This is because they carry the same
information, but the extra complexity of obtaining the orthogonal projectors
from the semi-orthogonal projectors, as in the proof of Proposition 2.2, is
formal and not essential. This in particular is the case for compactified
Jacobian fibrations [38, Section 2.5.5] as we will also review below, where
the semi-orthogonal projectors are of better form; see Theorem 2.5.

###  2.2. Compactified Jacobians

Throughout this section, we work with the general setting that $C\to B$ is a
flat family of integral projective curves of arithmetic genus $g$ with planar
singularities over an irreducible base $B$; we will specialize to the case of
spectral curves in Section 2.4. We assume that the total space $C$ is
nonsingular with a multi-section

$D\subset C\to B$

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, while
globally the degree $d$ compactified Jacobian $\overline{J}^{d}_{C}$ exists
only as an algebraic space. For convenience, we assume that for any degree
$d$, the compactified Jacobian $\overline{J}^{d}_{C}$ is a quasi-projective
variety. This is, for example, satisfied when $C\to B$ arises as a linear
system of a nonsingular surface, where the quasi-projectivity follows from the
fact that the compactified Jacobian arises as the moduli space of certain
semistable torsion sheaves on the surface. We further assume that
$\overline{J}^{d}_{C}$ is nonsingular. In order to work with the universal
family, we also need to consider the moduli stack of the degree $d$
compactified Jacobian, which is a ${\mathbb{G}}_{m}$-gerbe over
$\overline{J}^{d}_{C}$. Following [38, Section 4.2], we rigidify this stack by
imposing the condition that the universal family is _trivialized_ along the
multi-section; this gives a Deligne–Mumford stack which is a $\mu_{r}$-gerbe
over the scheme:

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

see [38, Proposition 4.1]. The rigidification condition we imposed implies
that there is a universal sheaf ${\mathcal{F}}^{d}$ over
$C\times_{B}\overline{{\mathcal{J}}}^{d}_{C}$ satisfying

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

Here $p_{J}$ is the projection to compactified Jacobian factor.

Now we consider two integers $d,e$. Arinkin’s work [3] shows that the
universal sheaves ${\mathcal{F}}^{d},{\mathcal{F}}^{e}$ induce a pair of
Fourier–Mukai kernels

(22) ${\mathcal{P}}_{e,d}\in
D^{b}\mathrm{Coh}\left(\overline{{\mathcal{J}}}^{e}_{C}\times\overline{{\mathcal{J}}}^{d}_{C}\right),\quad{\mathcal{P}}^{-1}_{e,d}\in
D^{b}\mathrm{Coh}\left(\overline{{\mathcal{J}}}^{d}_{C}\times\overline{{\mathcal{J}}}^{e}_{C}\right).$

The associated Fourier–Mukai transforms are equivalences of categories

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

where $D^{b}\mathrm{Coh}(\overline{{\mathcal{J}}}^{m}_{C})_{(k)}$ is the
isotypic category consisting of objects for which the action of $\mu_{r}$ on
the fibers is given by the character $\lambda\mapsto\lambda^{k}$ of $\mu_{r}$.
The objects (22) induce (Chow-theoretic) Fourier transforms

$\displaystyle\mathfrak{F}_{e,d}$
$\displaystyle:={\mathrm{td}}\left(-T_{\overline{{\mathcal{J}}}_{C}^{e}\times_{B}\overline{{\mathcal{J}}}^{d}_{C}}\right)\cap\tau({\mathcal{P}}_{e,d})\in\mathrm{CH}_{*}(\overline{{\mathcal{J}}}_{C}^{e}\times_{B}\overline{{\mathcal{J}}}^{d}_{C})=\mathrm{CH}_{*}(\overline{J}_{C}^{e}\times_{B}\overline{J}^{d}_{C}),$
$\displaystyle\mathfrak{F}_{e,d}^{-1}$
$\displaystyle:={\mathrm{td}}(-(\pi_{d}\times_{B}\pi_{e})^{*}T_{B})\cap\tau({\mathcal{P}}_{e,d}^{-1})\in\mathrm{CH}_{*}(\overline{{\mathcal{J}}}_{C}^{d}\times_{B}\overline{{\mathcal{J}}}^{e}_{C})=\mathrm{CH}_{*}(\overline{J}_{C}^{d}\times_{B}\overline{J}^{e}_{C}).$

Here $\pi_{\bullet}$ is the natural projection to the base $B$,
$T_{\overline{{\mathcal{J}}}_{C}^{e}\times_{B}\overline{{\mathcal{J}}}^{d}_{C}}$
is the virtual tangent bundle, $\tau(-)$ is the tau-class used in the
Riemann–Roch theorem of Baum–Fulton–MacPherson [4], and we identify the
(rational) Chow groups of the coarse moduli spaces and the gerbes; we refer to
[38, Section 4] for more details.

The correspondences $\mathfrak{F}_{e,d},\mathfrak{F}^{-1}_{e,d}$ are of mixed
degrees, and we have decompositions

$\displaystyle\mathfrak{F}_{e,d}=\sum_{i\geq
0}(\mathfrak{F}_{e,d})_{i},\quad({\mathfrak{F}}_{e,d})_{i}\in\mathrm{Corr}_{B}^{i-g}(\overline{J}_{C}^{e},\overline{J}_{C}^{d})$
$\displaystyle\mathfrak{F}_{e,d}^{-1}=\sum_{i\geq
0}(\mathfrak{F}^{-1}_{e,d})_{i},\quad({\mathfrak{F}}^{-1}_{e,d})_{i}\in\mathrm{Corr}_{B}^{i-g}(\overline{J}_{C}^{d},\overline{J}_{C}^{e}).$

For a fixed $d$, in order to define a motivic perverse filtration associated
with $\pi_{d}:\overline{J}^{d}_{C}\to B$ using the Fourier transform, we need
to choose a degree $e$, whose associated compactified Jacobian fibration will
serve as the “dual” side; see Step 1 of [38, Section 4.4]. In this paper, we
choose this degree uniformly to be $1$; this choice will be important for
applying the Abel–Jacobi map $C\to\overline{J}^{1}_{C}$ to build a connection
between the Fourier transform and the normalized Chern classes of a universal
bundle for a Hitchin-type moduli space. We define

(23) $\mathfrak{p}_{k}:=\sum_{i\leq
k}({\mathfrak{F}}_{1,d})_{i}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-i}\in\mathrm{Corr}_{B}^{0}(\overline{J}_{C}^{d},\overline{J}^{d}_{C}).$

The following theorem proven in [38] confirmed the motivic decomposition
conjecture for the compactified Jacobian fibration associated with $C\to B$,
with the projectors constructed above.

######  Theorem 2.5 ([38, Corollary 4.5(i)]).

For every $0\leq k\leq 2g$, the self-correspondence $\mathfrak{p}_{k}$ is a
projector for $\pi_{d}:\overline{J}^{d}_{C}\to B$, and the motives

$P_{k}h(\overline{J}^{d}_{C}):=(\overline{J}^{d}_{C},\mathfrak{p}_{k},0)\in\mathrm{CHM}(B),\quad
0\leq k\leq 2g$

form a motivic perverse filtration. Moreover, every submotive
$P_{k}h(\overline{J}^{d}_{C})$ admits a canonical orthogonal complement
$Q_{k+1}h(\overline{J}^{d}_{C})$:

$h(\overline{J}^{d}_{C})=P_{k}h(\overline{J}^{d}_{C})\oplus
Q_{k+1}h(\overline{J}^{d}_{C}),\quad
Q_{k+1}h(\overline{J}^{d}_{C})=(\overline{J}^{d}_{C},\mathfrak{q}_{k+1},0)$

with

$\mathfrak{q}_{k+1}:=[\Delta_{\overline{J}^{d}_{C}/B}]-\mathfrak{p}_{k}=\sum_{i\geq
k+1}({\mathfrak{F}}_{1,d})_{i}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-i}.$

The orthogonal complements $Q_{\bullet}h(\overline{J}^{d}_{C})$ allow us to
directly formulate the condition that a correspondence to
$h(\overline{J}^{d}_{C})$ has image lying in a certain piece of the motivic
perverse filtration; we refer to the statements in Theorem 2.6 below as
examples.

Now we consider two natural operations on the relative Chow motive
$h(\overline{J}^{d}_{C})$. The first is induced by the correspondence
$(\mathfrak{F}_{1,d})_{k}$:

$(\mathfrak{F}_{1,d})_{k}:h(\overline{J}^{1}_{C})(g-k)\to
h(\overline{J}^{d}_{C}).$

The second is the multiplicative structure given by the cup-product:

$\cup:h(\overline{J}^{d}_{C})\times h(\overline{J}^{d}_{C})\to
h(\overline{J}^{d}_{C}).$

The following theorem shows the compatibility between the motivic perverse
filtration of Theorem 2.5 and the two operations above.

######  Theorem 2.6.

The following hold for $\pi_{d}:\overline{J}^{d}_{C}\to B$.

  1. (a)

We have a morphism

$(\mathfrak{F}_{1,d})_{k}:h(\overline{J}^{1}_{C})(g-k)\to
P_{k}h(\overline{J}^{d}_{C}).$

More precisely, the following composition is zero:

$h(\overline{J}^{1}_{C})(g-k)\xrightarrow{(\mathfrak{F}_{1,d})_{k}}h(\overline{J}^{d}_{C})\to
Q_{k+1}h(\overline{J}^{d}_{C}).$

  2. (b)

The motivic perverse filtration is multiplicative:

$\cup:P_{k}h(\overline{J}^{d}_{C})\times P_{l}h(\overline{J}^{d}_{C})\to
P_{k+l}h(\overline{J}_{C}^{d}).$

More precisely, the following composition is zero:

$P_{k}h(\overline{J}^{d}_{C})\times
P_{l}h(\overline{J}^{d}_{C})\xrightarrow{~{}\cup~{}}h(\overline{J}^{d}_{C})\to
Q_{k+l+1}h(\overline{J}_{C}^{d}).$

Statement (a) was proven in [38, Corollary 4.5(iv)] by setting $e=1$. However,
(b) does not follow directly from the multiplicativity result [38, Corollary
4.5(iii)] (since we cannot make $e_{1}=e_{2}=e_{1}+e_{2}=1$); we postpone the
proof to Section 2.3. The following result is an immediate consequence of the
multiplicativity.

######  Corollary 2.7.

Assume that $\alpha\in\mathrm{CH}^{l}(\overline{J}^{d}_{C})$ lies in the
$k$-th piece of the perverse filtration, _i.e._ ,

$\alpha\in\mathrm{CH}^{l}(P_{k}h(\overline{J}^{d}_{C})).$

Then cupping with $\alpha$ interacts with the motivic perverse filtration as
follows:

$\alpha:P_{i}h(\overline{J}^{d}_{C})\to P_{i+k}h(\overline{J}^{d}_{C})(l).$

###### Proof.

It suffices to prove the relation

$\mathfrak{q}_{i+k+1}\circ\Delta_{*}\alpha\circ\mathfrak{p}_{i}=0$

on the relative product $\overline{J}^{d}_{C}\times_{B}\overline{J}^{d}_{C}$,
where
$\Delta:\overline{J}^{d}_{C}\hookrightarrow\overline{J}^{d}_{C}\times_{B}\overline{J}^{d}_{C}$
is the diagonal embedding. Theorem 2.6(b) reads

$\mathfrak{q}_{i+k+1}\circ[\Delta^{\mathrm{sm}}_{\overline{J}_{C}^{d}/B}]\circ\left(\mathfrak{p}_{k}\times\mathfrak{p}_{i}\right)=0$

which is a relation on the relative triple product. Therefore, we have

$\displaystyle 0$
$\displaystyle=p_{23*}\left(p_{1}^{*}\alpha\cap\left(\mathfrak{q}_{i+k+1}\circ[\Delta^{\mathrm{sm}}_{\overline{J}_{C}^{d}/B}]\circ\left(\mathfrak{p}_{k}\times\mathfrak{p}_{i}\right)\right)\right)$
$\displaystyle=\mathfrak{q}_{i+k+1}\circ
p_{23*}\left(p_{1}^{*}\alpha\cap\left([\Delta^{\mathrm{sm}}_{\overline{J}_{C}^{d}/B}]\circ\left(\mathfrak{p}_{k}\times\mathfrak{p}_{i}\right)\right)\right)$
$\displaystyle=\mathfrak{q}_{i+k+1}\circ\Delta_{*}{\mathfrak{p}_{k}(\alpha)}\circ\mathfrak{p}_{i}$
$\displaystyle=\mathfrak{q}_{i+k+1}\circ\Delta_{*}{\alpha}\circ\mathfrak{p}_{i}.$

Here $p_{ij},p_{i}$ are the natural projections from the relative triple
product
$\overline{J}^{d}_{C}\times_{B}\overline{J}^{d}_{C}\times_{B}\overline{J}^{d}_{C}$
to the corresponding factors. This completes the proof. ∎

Next, we discuss the motivic $\chi$-independence phenomenon for compactified
Jacobian fibrations. For two integers $d,d^{\prime}$, we consider the
correspondence

$\mathfrak{C}_{d,d^{\prime}}:=\sum_{i=0}^{2g}({\mathfrak{F}}_{1,d^{\prime}})_{i}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-i}\in\mathrm{Corr}_{B}^{0}(\overline{J}_{C}^{d},\overline{J}^{d^{\prime}}_{C}).$

Recall that $r$ is the degree of the multisection $D\subset C$ we fixed at the
beginning.

######  Theorem 2.8.

Assume $d,d^{\prime}$ are two integers coprime to $r$. Then
$\mathfrak{C}_{d,d^{\prime}}$ induces an isomorphism

$\mathfrak{C}_{d,d^{\prime}}:h(\overline{J}^{d}_{C})\xrightarrow{\simeq}h(\overline{J}^{d^{\prime}}_{C})$

preserving the motivic perverse filtrations

$\mathfrak{C}_{d,d^{\prime}}:P_{k}h(\overline{J}^{d}_{C})\xrightarrow{\simeq}P_{k}h(\overline{J}^{d^{\prime}}_{C}).$

Moreover, the inverse of $\mathfrak{C}_{d,d^{\prime}}$ is given by
$\mathfrak{C}_{d^{\prime},d}$,

(24)
$\mathfrak{C}_{d^{\prime},d}\circ\mathfrak{C}_{d,d^{\prime}}=[\Delta_{\overline{J}^{d}_{C}/B}],\quad\mathfrak{C}_{d,d^{\prime}}\circ\mathfrak{C}_{d^{\prime},d}=[\Delta_{\overline{J}^{d^{\prime}}_{C}/B}].$

Both Theorem 2.6(b) and Theorem 2.8 are proven in the next section, which rely
on a generalized version of the Fourier vanishing established in [38].

###  2.3. Proofs of Theorem 2.6(b) and Theorem 2.8

In [38, Section 4.4], we establish a series of relations between the Fourier
components, known as Fourier vanishing (FV). For two integers $d,e$, this
vanishing states that

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

In this section, we prove a generalized Fourier vanishing given as follows:

######  Proposition 2.9.

Let $d,d^{\prime},e,e^{\prime}$ be integers. Then if $e$ is coprime to $r$, we
have

(FV1)
$({\mathfrak{F}}_{e^{\prime},d}^{-1})_{i}\circ({\mathfrak{F}}_{e,d})_{j}=0,\quad
i+j<2g.$

If $d$ is coprime to $r$, we have

(FV2)
$({\mathfrak{F}}_{e,d})_{i}\circ({\mathfrak{F}}_{e,d^{\prime}}^{-1})_{j}=0,\quad
i+j<2g.$

###### Proof.

The proof of (FV1) follows closely that of (FV) in [38, Proof of Corollary
4.5(i)], where we used the Adams operations $\psi^{N}(-)$ of [2, Section 4] to
scale the Poincaré sheaf ${\mathcal{P}}_{e,d}$ in $K$-theory. By applying the
tau-class $\tau(-)$ to the $K$-theory classes supported in codimension $g$:

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

for infinitely many $N$, we were able to obtain (FV) for each pair of $i,j$
satisfying $i+j<2g$. The crucial point here is that the functor $\tau(-)$ of
[20] is only covariant under proper _representable_ morphisms of quotient
stacks. By choosing

$N\equiv 1\pmod{r}$

we made sure that the class

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

descends to a class in
$K_{*}(\overline{\mathcal{J}}_{C}^{e}\times_{B}\overline{J}_{C}^{d}\times_{B}\overline{\mathcal{J}}_{C}^{e})$
in order to apply $\tau(-)$ to the proper representable morphism

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

Here $p_{ij}$ are the natural projections to the corresponding factors.

Now, under the assumptions of (FV1), we consider the $K$-theory classes
supported in codimension $g$:

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

Since $(e,r)=1$, we have $ae\equiv 1\pmod{r}$ for some integer $a$. Then by
taking

$N=aN^{\prime},\quad N^{\prime}\equiv e^{\prime}\pmod{r},$

we get an infinite sequence of classes

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

which descend to
$K_{*}(\overline{\mathcal{J}}_{C}^{e}\times_{B}\overline{J}_{C}^{d}\times_{B}\overline{\mathcal{J}}_{C}^{e^{\prime}})$.
The argument of [38, Proof of Corollary 4.5(i)] then applies _verbatim_ to
yield (FV1).

For the proof of (FV2), we can either repeat the whole argument by switching
the composition order of the Fourier transform and its inverse, or simply
observe that the transpose of

$({\mathfrak{F}}_{e,d})_{i}\circ({\mathfrak{F}}_{e,d^{\prime}}^{-1})_{j}$

is precisely

(24)
$({\mathfrak{F}}^{-1}_{d^{\prime},e})_{j}\circ({\mathfrak{F}}_{d,e})_{i}.$

This is an immediate consequence of the symmetry in Arinkin’s formula [3,
(1.1)] for the Poincaré sheaf; see also [38, Footnote 12]. Therefore, (FV2)
follows from (FV1) and (24) by replacing $e$ with $d$, $e^{\prime}$ with
$d^{\prime}$, $d$ with $e$, and by switching $i$ and $j$. ∎

We are ready to prove Theorems 2.6(b) and 2.8. We begin with the latter.

###### Proof of Theorem 2.8.

We first prove the identities (24). We expand

$\displaystyle[\Delta_{\overline{J}^{d}_{C}/B}]$
$\displaystyle={\mathfrak{F}}_{1,d}\circ{\mathfrak{F}}^{-1}_{1,d^{\prime}}\circ{\mathfrak{F}}_{1,d^{\prime}}\circ{\mathfrak{F}}^{-1}_{1,d}$
(25)
$\displaystyle=\sum_{i,j,l,m}({\mathfrak{F}}_{1,d})_{i}\circ({\mathfrak{F}}^{-1}_{1,d^{\prime}})_{j}\circ({\mathfrak{F}}_{1,d^{\prime}})_{l}\circ({\mathfrak{F}}^{-1}_{1,d})_{m}.$

For degree reasons the terms in (25) vanish unless $i+j+l+m=4g$. Applying the
Fourier vanishing (FV2) with $d,d^{\prime}$ coprime to $r$, we find that the
nonvanishing terms in (25) must also satisfy $i+j\geq 2g$, $l+m\geq 2g$. Hence
both inequalities are actual equalities and we have

$[\Delta_{\overline{J}^{d}_{C}/B}]=\left(\sum_{i=0}^{2g}({\mathfrak{F}}_{1,d})_{i}\circ({\mathfrak{F}}^{-1}_{1,d^{\prime}})_{2g-i}\right)\circ\left(\sum_{l=0}^{2g}({\mathfrak{F}}_{1,d^{\prime}})_{l}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-l}\right)={\mathfrak{C}}_{d^{\prime},d}\circ{\mathfrak{C}}_{d,d^{\prime}},$

which proves the first identity in (24). The second identity follows by
symmetry.

It remains to verify that
${\mathfrak{C}}_{d,d^{\prime}},{\mathfrak{C}}_{d^{\prime},d}$ preserve the
motivic perverse filtrations. This amounts to showing the relations (19) where
${\mathfrak{p}}_{k},{\mathfrak{q}}_{k+1}$ (resp.
${\mathfrak{p}}^{\prime}_{k},{\mathfrak{q}}^{\prime}_{k+1}$) denote the
projectors for $\overline{J}^{d}_{C}$ (resp. $\overline{J}^{d^{\prime}}_{C}$).
By symmetry it suffices to prove the first identity:

${\mathfrak{q}}^{\prime}_{k+1}\circ\mathfrak{C}_{d,d^{\prime}}\circ{\mathfrak{p}}_{k}=0.$

The left-hand side can be expressed as

(26) $\left(\sum_{i\geq
k+1}({\mathfrak{F}}_{1,d^{\prime}})_{i}\circ({\mathfrak{F}}^{-1}_{1,d^{\prime}})_{2g-i}\right)\circ\left(\sum_{j=0}^{2g}({\mathfrak{F}}_{1,d^{\prime}})_{j}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-j}\right)\circ\left(\sum_{m\leq
k}({\mathfrak{F}}_{1,d})_{m}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-m}\right).$

This time applying (FV1) with $e=1$ to (26), we see that the nonvanishing
terms must satisfy

$2g-i+j\geq 2g,\quad 2g-j+m\geq 2g,\quad i\geq k+1,\quad m\leq k,$

or, equivalently,

$k\geq m\geq j\geq i\geq k+1.$

This is absurd, and hence (26) vanishes. ∎

The proof of Theorem 2.6(b) is slightly more complicated but follows a similar
pattern. For two integers $e_{1},e_{2}$, recall from [38, Proof of Corollary
4.5(iii)] the Chow-theoretic convolution kernel444The convolution kernel was
denoted ${\mathfrak{C}}_{e_{1},e_{2},d}$ in [38].

${\mathfrak{K}}_{e_{1},e_{2},d}\in\mathrm{Corr}_{B}^{\geq-g}(\overline{J}_{C}^{e_{1}},\overline{J}_{C}^{e_{2}};\overline{J}_{C}^{e_{1}+e_{2}}),$

which satisfies

(27)
${\mathfrak{F}}_{e_{1}+e_{2},d}\circ{\mathfrak{K}}_{e_{1},e_{2},d}\circ\left({\mathfrak{F}}^{-1}_{e_{1},d}\times{\mathfrak{F}}^{-1}_{e_{2},d}\right)=[\Delta^{\mathrm{sm}}_{\overline{J}^{d}_{C}/B}].$

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

We need to show the vanishing

(28)
${\mathfrak{q}}_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{\overline{J}^{d}_{C}/B}]\circ({\mathfrak{p}}_{k}\times{\mathfrak{p}}_{l})=0\in\mathrm{Corr}_{B}^{0}(\overline{J}^{d}_{C},\overline{J}^{d}_{C};\overline{J}^{d}_{C}).$

By (27) we have

$\displaystyle{\mathfrak{q}}_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{\overline{J}^{d}_{C}/B}]\circ({\mathfrak{p}}_{k}\times{\mathfrak{p}}_{l})$
$\displaystyle={\mathfrak{q}}_{k+l+1}\circ{\mathfrak{F}}_{e_{1}+e_{2},d}\circ{\mathfrak{K}}_{e_{1},e_{2},d}\circ\left(({\mathfrak{F}}^{-1}_{e_{1},d}\circ{\mathfrak{p}}_{k})\times({\mathfrak{F}}^{-1}_{e_{2},d}\circ{\mathfrak{p}}_{l})\right)$
(29) $\displaystyle=\left(\sum_{i_{3}\geq
k+l+1}({\mathfrak{F}}_{1,d})_{i_{3}}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-i_{3}}\right)\circ\left(\sum_{j_{3}}({\mathfrak{F}}_{e_{1}+e_{2},d})_{j_{3}}\right)\circ{\mathfrak{K}}_{e_{1},e_{2},d}$
$\displaystyle\quad\quad\circ\left(\left(\left(\sum_{j_{1}}({\mathfrak{F}}^{-1}_{e_{1},d})_{j_{1}}\right)\circ\left(\sum_{i_{1}\leq
k}({\mathfrak{F}}_{1,d})_{i_{1}}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-i_{1}}\right)\right)\right.$
$\displaystyle\quad\quad\quad\times\left.\left(\left(\sum_{j_{2}}({\mathfrak{F}}^{-1}_{e_{2},d})_{j_{2}}\right)\circ\left(\sum_{i_{2}\leq
l}({\mathfrak{F}}_{1,d})_{i_{2}}\circ({\mathfrak{F}}^{-1}_{1,d})_{2g-i_{2}}\right)\right)\right).$

So far the choices of $e_{1},e_{2}$ have been arbitrary. But if we choose
$e_{1},e_{2}$ such that $e_{1}+e_{2}$ is coprime to $r$, then by applying
(FV1) with $e=1$ or $e=e_{1}+e_{2}$ we see that the nonvanishing terms in (29)
must satisfy

$j_{1}+i_{1}\geq 2g,\quad j_{2}+i_{2}\geq 2g,\quad 2g-i_{3}+j_{3}\geq 2g.$

This implies

$j_{1}\geq 2g-k,\quad j_{2}\geq 2g-l,\quad j_{3}\geq k+l+1.$

Therefore we have

(30)
${\mathfrak{q}}_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{\overline{J}^{d}_{C}/B}]\circ({\mathfrak{p}}_{k}\times{\mathfrak{p}}_{l})={\mathfrak{q}}_{k+l+1}\circ\left(\sum_{j_{3}\geq
k+l+1}({\mathfrak{F}}_{e_{1}+e_{2},d})_{j_{3}}\right)\circ{\mathfrak{K}}_{e_{1},e_{2},d}\\\
\circ\left(\left(\left(\sum_{j_{1}\geq
2g-k}({\mathfrak{F}}^{-1}_{e_{1},d})_{j_{1}}\right)\circ{\mathfrak{p}}_{k}\right)\times\left(\left(\sum_{j_{2}\geq
2g-l}({\mathfrak{F}}^{-1}_{e_{2},d})_{j_{2}}\right)\circ{\mathfrak{p}}_{l}\right)\right)$

Comparing degrees, we find that the right-hand side of (30) lies in

$\mathrm{Corr}_{B}^{\geq-g+(k+l+1-g)+(g-k)+(g-l)}(\overline{J}^{d}_{C},\overline{J}^{d}_{C};\overline{J}^{d}_{C})=\mathrm{Corr}_{B}^{\geq
1}(\overline{J}^{d}_{C},\overline{J}^{d}_{C};\overline{J}^{d}_{C}),$

while the left-hand side lies in
$\mathrm{Corr}^{0}_{B}(\overline{J}^{d}_{C},\overline{J}^{d}_{C};\overline{J}^{d}_{C})$.
This shows the desired vanishing (28). ∎

###  2.4. Parabolic Higgs bundles: general residue

Theorems 2.5, 2.6, and 2.8 in Section 2.2 are analogues of our main theorems
in the general setting of compactified Jacobians. For our purpose, we
specialize to the case where $C\to B$ arises as a family of spectral curves.
The main results of this section are Theorems 2.11, 2.12, and 2.13 which treat
the case for the Hitchin system associated with parabolic Higgs bundles with
general residue.

For the curve $\Sigma$, we consider nonsingular surface

$S:=\mathrm{Tot}_{\Sigma}(\omega_{\Sigma}(p)).$

Now we work with a family of curves $C\to B$ as in Section 2.2, imposing the
extra condition that they lie in the linear system $|n\Sigma|$ in the surface
$S$; in particular, every curve $C_{b}\subset S$ is a degree $n$ finite cover
over the zero-section $\Sigma$. We have a natural evaluation map

$\mathrm{ev}:C\to S\times B.$

The composition of $\mathrm{ev}$ with the projection $S\times B\to\Sigma\times
B$ gives the spectral cover together with a (degree $n$) multisection:

$\pi:C\to\Sigma\times B,\quad D:=\pi^{-1}(p\times B).$

For each curve $C_{b}$ with a degree $n$ spectral cover
$\pi_{b}:C_{b}\to\Sigma$, the multisection over this fiber is
${D}_{b}=\pi_{b}^{-1}(p)$.

Next, we introduce the normalized Chern character of Section 0.2. Let
$M^{\mathrm{mero}}$ be the moduli space of stable meromorphic Higgs bundles on
the curve $\Sigma$, which parameterizes slope stable (meromorphic) Higgs
bundles:

$({\mathcal{E}},\theta),\quad\theta:{\mathcal{E}}\to{\mathcal{E}}\otimes\omega_{\Sigma}(p),\quad\mathrm{rk}({\mathcal{E}})=n,\quad\mathrm{deg}({\mathcal{E}})=d.$

Here we call them meromorphic Higgs bundles since they can be viewed as Higgs
bundles which are allowed to have a simple pole at $p$. The variety
$M^{\mathrm{mero}}$ is nonsingular, and the coprime condition $(n,d)=1$
ensures the existence of a universal bundle ${\mathcal{U}}$ on $\Sigma\times
M^{\mathrm{mero}}$. The reason we work first with this moduli space is that
all the moduli spaces in Section 1.2, including the original moduli space
$M_{n,d}$, admit natural morphisms to $M^{\mathrm{mero}}$, and the pullback of
the universal bundle gives a universal bundle on each moduli space. So the
normalized Chern character

$\widetilde{\mathrm{ch}}({\mathcal{U}})\in\mathrm{CH}^{*}\left(\Sigma\times
M^{\mathrm{mero}}\right),$

yields the normalized Chern character for each moduli space of Section 1.2.
This strategy was taken in [38, Section 5.4].

In cohomology, the normalization is given by a class

$\delta:=\pi^{*}_{\Sigma}\delta_{\Sigma}+\pi^{*}_{M}\delta_{M}\in
H^{2}(\Sigma\times M^{\mathrm{mero}},{\mathbb{Q}}),$

with $\pi_{\bullet}$ the natural projections, satisfying that

$c_{1}({\mathcal{U}})+\delta\in H^{1}(\Sigma,{\mathbb{Q}})\otimes
H^{1}(M^{\mathrm{mero}},{\mathbb{Q}})\subset H^{2}(\Sigma\times
M^{\mathrm{mero}},{\mathbb{Q}});$

this determines $\delta$ completely. Then the normalized Chern character is
defined to be

$\widetilde{\mathrm{ch}}({\mathcal{U}}):=\mathrm{ch}({\mathcal{U}})\cup\mathrm{exp}\left(\delta\right)\in
H^{*}(\Sigma\times M^{\mathrm{mero}},{\mathbb{Q}}),$

which is canonical and independent of the choice of ${\mathcal{U}}$. We refer
to [12, Section 0.3] for more details.

For our purpose, we need to lift $\delta$ (associated with a universal bundle
${\mathcal{U}}$) canonically to a divisor

(31)
$\delta:=\pi_{\Sigma}^{*}D_{\Sigma}+\pi_{M}^{*}D_{M}\in\mathrm{CH}^{1}(\Sigma\times
M^{\mathrm{mero}}).$

We first determine $D_{M}$ using the marked point $p\in\Sigma$:

$D_{M}:=c_{1}({\mathcal{U}})|_{p\times
M^{\mathrm{mero}}}\in\mathrm{CH}^{1}(M^{\mathrm{mero}}).$

Now we determine $D_{\Sigma}$, which is eventually written as a linear
combination of the point class of $p$ and the canonical divisor $K_{\Sigma}$
on the curve $\Sigma$. Pick a nonsingular spectral curve
$C_{b}\subset\mathrm{Tot}_{\Sigma}(\omega_{\Sigma}(p))$; a Higgs bundle in
$M^{\mathrm{mero}}$ associated with the spectral curve $C_{b}$ can be
represented by a line bundle ${\mathcal{L}}_{C_{b}}$ on $C_{b}$ of degree
$l(n,d)\in{\mathbb{Z}}$, where $l(n,d)$ is a number only depending on $n,d$
and is coprime to $n$. In particular, we have

$c_{1}({\mathcal{L}}_{C_{b}})-\frac{l(n,d)}{n}\cdot{D}_{b}=c_{1}({\mathcal{L}}_{C_{b}})-\frac{l(n,d)}{n}\cdot\pi_{b}^{*}[p]=0\in
H^{2}(C_{b},{\mathbb{Q}}).$

Finally, there is a unique constant $\lambda_{0}\in\frac{1}{n}{\mathbb{Z}}$
such that

$c_{1}({\mathcal{U}})-\frac{l(n,d)}{n}\cdot\pi_{b}^{*}[p]+\lambda_{0}\cdot
K_{\Sigma}=0\in H^{2}(\Sigma,0).$

We set

$D_{\Sigma}:=-\frac{l(n,d)}{n}\cdot[p]+\lambda_{0}\cdot K_{\Sigma},$

in (31), which defines the normalized Chern character

(32)
$\widetilde{\mathrm{ch}}({\mathcal{U}}):=\mathrm{ch}({\mathcal{U}})\cup\mathrm{exp}(\delta)\in\mathrm{CH}^{*}(\Sigma\times
M^{\mathrm{mero}}).$

We denote by $\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})$ its degree $k$ part:

$\widetilde{\mathrm{ch}}({\mathcal{U}})=\sum_{k}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}}),\quad\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\in\mathrm{CH}^{k}(\Sigma\times
M^{\mathrm{mero}}).$

By definition, the normalized Chern character (32) does not depend on the
choice of the universal bundle ${\mathcal{U}}$, and its image under the cycle
class map recovers the normalized Chern character in cohomology.

Next, we give an explicit description of the normalized Chern character for
the moduli space $M_{\eta}^{\mathrm{par}}$ for a general $\eta\in T$ as in
Section 1.2(a); it admits a compactified Jacobian fibration over
$\widetilde{W}_{\eta}$. For notational convenience, we denote by
$C\to\widetilde{W}_{\eta}$ the family of spectral curves. Therefore we have

$M_{\eta}^{\mathrm{par}}=\overline{J}^{e}_{C},\quad e=l(n,d).$

We consider the diagram

(33)
${C}$${\overline{J}^{1}_{C}}$${\Sigma\times\widetilde{W}_{\eta}}$$\scriptstyle{\mathrm{AJ}}$$\scriptstyle{\pi}$

where the horizontal map is the Abel–Jacobi map

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

The diagram (33) yields a relative correspondence over $\widetilde{W}_{\eta}$:

$[C]\in\mathrm{CH}_{\dim C}(\Sigma\times\overline{J}^{1}_{C}).$

######  Proposition 2.10 ([38]).

Let ${\mathcal{U}}$ be a universal bundle over
$\Sigma\times\overline{J}^{e}_{C}=\Sigma\times M^{\mathrm{par}}_{\eta}$. Then
we have

$\widetilde{\mathrm{ch}}({\mathcal{U}})=\mathfrak{F}_{1,d}\circ[C]\in\mathrm{CH}^{*}(\Sigma\times\overline{J}^{d}_{C}).$

###### Proof.

Applying the Grothendieck–Riemann–Roch (GRR) formula to [38, Proposition 4.7]
as in the proof of [38, Proposition 5.1(a)], we express the Chern character of
the universal $1$-dimensional torsion free sheaf over the universal spectral
curve $C$ in terms of $\mathfrak{F}_{1,d}$. Then the result follows from
applying further the GRR formula along
$\pi:C\to\Sigma\times\widetilde{W}_{\eta}$, expressing the Chern character of
a universal bundle over $\Sigma$ in terms of the Chern character of a
universal torsion free sheaf over $C$ [38, Section 5].

Note that the term $-\frac{l(n,d)}{n}\cdot[p]$ in the normalization comes from
the line bundle ${\mathcal{N}}$ of [38, Proposition 4.7], and the second term
$\lambda_{0}\cdot K_{C}$ arise from the Todd contribution of the GRR
calculation along $\pi:C\to\Sigma\times\widetilde{W}_{\eta}$. ∎

The morphism $h_{\eta}:M^{\mathrm{par}}_{\eta}\to W_{\eta}$ can be written as
a composition

$M^{\mathrm{par}}_{\eta}=\overline{J}^{e}_{C}\to\widetilde{W}_{\eta}\to
W_{\eta}$

where the first map is the compactified Jacobian fibration associated with
$C\to\widetilde{W}$, and the second is a finite map given by
$\mathfrak{S}_{n}$-quotient.

######  Theorem 2.11.

The morphism $h_{\eta}:M^{\mathrm{par}}_{\eta}\to W_{\eta}$ admits a motivic
perverse filtration, where the projectors are induced by those for the
compactified Jacobian fibration $\overline{J}^{e}_{C}\to\widetilde{W}_{\eta}$
given in Theorem 2.5.

###### Proof.

By Theorem 2.5, we have the semi-orthogonal projectors for the map
$\overline{J}^{e}_{C}\to\widetilde{W}_{\eta}$,

$\mathfrak{p}_{k}\in\mathrm{CH}_{\dim\overline{J}^{e}_{C}}(\overline{J}_{C}^{e}\times_{\widetilde{W}_{\eta}}\overline{J}^{e}_{C}),\quad
0\leq k\leq 2g.$

Then under the natural morphism

$\mathrm{CH}_{\dim\overline{J}^{e}_{C}}(\overline{J}_{C}^{e}\times_{\widetilde{W}_{\eta}}\overline{J}^{e}_{C})\to\mathrm{CH}_{\dim\overline{J}^{e}_{C}}(\overline{J}_{C}^{e}\times_{{W}_{\eta}}\overline{J}^{e}_{C})$

induced by $\widetilde{W}_{\eta}\to W_{\eta}$, we obtain semi-orthogonal
projectors for the latter:

$\mathfrak{p}_{k}\in\mathrm{CH}_{\dim\overline{J}^{e}_{C}}(\overline{J}_{C}^{e}\times_{{W}_{\eta}}\overline{J}^{e}_{C}),\quad
0\leq k\leq 2g.$

Finally, we need to check that these projectors have the correct homological
realization, which follows from the fact that the pushforward along the finite
map

$D^{b}_{c}(\widetilde{W}_{\eta})\to D_{c}^{b}(W_{\eta})$

is $t$-exact with respect to the perverse $t$-structure. ∎

The following theorem determines the perversity of the normalized Chern
character.

######  Theorem 2.12.

Taking cup-product with the normalized Chern character satisfies

$\widetilde{\mathrm{ch}}_{k}({\mathcal{U}}):P_{i}h(\Sigma\times\overline{J}^{e}_{C})\to
P_{i+k}h(\Sigma\times\overline{J}^{e}_{C})(k)\in\mathrm{CHM}(\Sigma\times{W}_{\eta}).$

Here the motivic perverse filtration for
$\Sigma\times\overline{J}^{e}_{C}\to\Sigma\times W_{\eta}$ is given by the
pullback of the motivic perverse filtration for
$h_{\eta}:\overline{J}^{e}_{C}\to W_{\eta}$.

###### Proof.

As in the proof of Theorem 2.11, it suffices to prove the corresponding
statement in the category $\mathrm{CHM}(\Sigma\times\widetilde{W}_{\eta})$.
Then pushing forward along $\Sigma\times\widetilde{W}_{\eta}\to\Sigma\times
W_{\eta}$ yields the desired statement over $W_{\eta}$.

Recall the perverse filtration on $\mathrm{CH}_{*}(\overline{J}^{e}_{C})$
induced by the motivic perverse filtration associated with the compactified
Jacobian fibration $\overline{J}^{e}_{C}\to\widetilde{W}_{\eta}$. We first
note that

$\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\in
P_{k}\mathrm{CH}^{k}(\Sigma\times\overline{J}^{e}_{C}).$

This is a direct consequence of Proposition 2.10:

$\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})=(\mathfrak{F}_{1,d})_{k}\circ[C]$

combined with Theorem 2.6(a). Then applying Corollary 2.7 to

$\Sigma\times\overline{J}^{e}_{C}\to\Sigma\times\widetilde{W}_{\eta}$

completes the proof of the theorem. ∎

The next theorem concerning the $\chi$-independence phenomenon is immediately
deduced from Theorem 2.8 by pushing forward along $\widetilde{W}_{\eta}\to
W_{\eta}$.

######  Theorem 2.13.

Assume $e,e^{\prime}$ are two integers coprime to $n$. Then
$\mathfrak{C}_{e,e^{\prime}}$ from Theorem 2.8 induces an isomorphism

$\mathfrak{C}_{e,e^{\prime}}:h(\overline{J}^{e}_{C})\xrightarrow{\simeq}h(\overline{J}^{e^{\prime}}_{C})\in\mathrm{CHM}(W_{\eta})$

preserving the motivic perverse filtrations

$\mathfrak{C}_{e,e^{\prime}}:P_{k}h(\overline{J}^{e}_{C})\xrightarrow{\simeq}P_{k}h(\overline{J}^{e^{\prime}}_{C}).$

Moreover, the inverse of $\mathfrak{C}_{e,e^{\prime}}$ is given by
$\mathfrak{C}_{e^{\prime},e}$,

$\mathfrak{C}_{e^{\prime},e}\circ\mathfrak{C}_{e,e^{\prime}}=[\Delta_{\overline{J}^{e}_{C}/W_{\eta}}],\quad\mathfrak{C}_{e,e^{\prime}}\circ\mathfrak{C}_{e^{\prime},e}=[\Delta_{\overline{J}^{e^{\prime}}_{C}/W_{\eta}}].$

###  2.5. Parabolic Higgs bundles: nilpotent residue

In the previous section, we have established the desired results for

$h_{\eta}:M^{\mathrm{par}}_{\eta}\to W_{\eta}$

associated with parabolic Higgs bundles with general residue. The proofs in
fact give the results for the family

$h^{\circ}:M^{\mathrm{par},\circ}\to W^{\circ}$

over $T^{\circ}$ introduced in Section 1.2(a), which puts together the
fibrations $h_{\eta}$ for $\eta\in T^{\circ}$. By applying the specialization
map

$\eta\in T^{\circ}\quad\rightsquigarrow\quad 0\in T$

for Chow groups, all the results concerning relations of algebraic cycles hold
for

$h_{0}:M^{\mathrm{par}}_{0}\to W_{0}.$

######  Theorem 2.14.

The morphism $h_{0}:M^{\mathrm{par}}_{0}\to W_{0}$ admits a motivic perverse
filtration, where the projectors are induced by specializing the projectors of
$h_{\eta}:M^{\mathrm{par}}_{\eta}\to\widetilde{W}_{\eta}$ with $\eta\in
T^{\circ}$ given in Theorem 2.11.

As we discussed above, the projectors we obtain from specialization
automatically satisfy the conditions (a, c) in the definition of motivic
perverse filtration. It remains to prove (b), _i.e._ , the homological
realization recovers the perverse truncation functor. This will be proven in
Section 4.1.

######  Theorem 2.15.

Let ${\mathcal{U}}$ be a universal bundle over $\Sigma\times
M^{\mathrm{par}}_{0}$. Then we have

$\widetilde{\mathrm{ch}}_{k}({\mathcal{U}}):P_{i}h(\Sigma\times
M^{\mathrm{par}}_{0})\to P_{i+k}h(\Sigma\times
M^{\mathrm{par}}_{0})(k)\in\mathrm{CHM}(\Sigma\times{W}_{0}).$

###### Proof.

Since the statement can be expressed in terms of a relation of algebraic
cycles:

$\mathfrak{q}_{i+k+1}\circ\Delta_{*}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\circ\mathfrak{p}_{i}=0$

with $\Delta:M^{\mathrm{par}}_{0}\hookrightarrow
M^{\mathrm{par}}_{0}\times_{W_{0}}M^{\mathrm{par}}_{0}$ the relative diagonal,
it follows from specializing the corresponding relation for
$h_{\eta}:M^{\mathrm{par}}_{\eta}\to W_{\eta}$ established in Theorem 2.12. ∎

Similarly, by a straightforward specialization argument, we obtain from
Theorem 2.13 the following theorem concerning the $\chi$-independence.

######  Theorem 2.16.

Assume $d,d^{\prime}$ are two integers coprime to $n$. Let
$M^{\mathrm{par}}_{0},{M^{\mathrm{par}}_{0}}^{\prime}$ be the parabolic moduli
spaces associated with the degrees $d,d^{\prime}$ respectively. Then there is
an isomorphism

$h(M^{\mathrm{par}}_{0})\simeq h({M^{\mathrm{par}}_{0}}^{\prime})$

preserving the motivic perverse filtrations

$P_{k}h(M^{\mathrm{par}}_{0})\simeq P_{k}h({M^{\mathrm{par}}_{0}}^{\prime}).$

######  Remark 2.17.

We conclude this section by noting that the motivic perverse filtration we
introduced in Theorem 2.14 is independent of the choice of the $1$-dimensional
family $T$. Indeed, we consider the Hitchin system associated with the total
space of the moduli of stable parabolic Higgs bundles

$M^{\mathrm{par}}\to W$

in Section 1.2. The Fourier transform can be defined over a Zariski open
subset $W^{\circ}$ of the base $W$ formed by integral spectral curves, and the
projectors $\mathfrak{p}_{k}$ we obtained in Theorem 2.15 is the
specialization of the projectors over $W^{\circ}$. Here we choose to work with
a $1$-dimensional family $T$ for convenience (_e.g._ in the sheaf-theoretic
argument of Section 4).

##  3\. Projectors and Springer theory

In this section, we study the correspondences given by (15) connecting
$M^{\mathrm{par}}_{0}$ and $M_{n,d}$. Our main purpose is to prove Proposition
3.5, which yields a candidate for the motivic perverse filtration (41) for
$f:M_{n,d}\to B$. For the approach, we reduce the Higgs case to a _local
model_ where the correspondence (15) is understood via classical Springer
theory. This lifts several cohomological results of [24, Section 8]
motivically. Springer theoretic interpretations of parabolic Hitchin moduli
spaces have previously used in [19] in the study of Kac polynomials.

###  3.1. Springer theory

For our purpose, we consider the case $G=\mathrm{PGL}_{n}$. We denote by $B,T$
the Borel subgroup given by upper triangular matrices and the maximal torus
given by diagonal matrices respectively. We use
$\mathfrak{g},\mathfrak{b},\mathfrak{t}$ to denote the corresponding Lie
algebra. The Weyl group is the permutation group $\mathfrak{S}_{n}$ which acts
on $\mathfrak{t}$ naturally. Let

$\mathfrak{c}:=\mathfrak{g}\mathbin{/\mkern-6.0mu/}G\simeq\mathfrak{t}\mathbin{/\mkern-6.0mu/}\mathfrak{S}_{n}$

be the categorical quotient, and we have a natural morphism

$\chi:\mathfrak{g}\to\mathfrak{c}$

which can be viewed as calculating the characteristic polynomial of a matrix.
We consider the Grothendieck–Springer resolution over the base $\mathfrak{c}$:

${\widetilde{\mathfrak{g}}}$${\mathfrak{g}}$${\mathfrak{c}.}$$\scriptstyle{q}$$\scriptstyle{\chi}$

The fibers over $0\in\mathfrak{c}$ recovers the symplectic resolution of the
nilpotent cone

$q_{0}:\widetilde{{\mathcal{N}}}_{0}\to{\mathcal{N}}_{0}$

by the total cotangent bundle of the flag variety
$\widetilde{{\mathcal{N}}}=T^{*}\mathrm{Fl}$, while the fibers over a general
$\eta\in\mathfrak{c}$ (_e.g._ the corresponding matrix has distinct
eigenvalues) is a natural quotient of the Weyl group:

$q_{\eta}:\widetilde{{\mathcal{N}}}_{\eta}\to{{\mathcal{N}}}_{\eta}=\widetilde{{\mathcal{N}}_{\eta}}/\mathfrak{S}_{n}.$

From this description, we see that any element $w\in\mathfrak{S}_{n}$ induces
a self-correspondence of $\widetilde{{\mathcal{N}}}_{\eta}$. As we will
illustrate in the next section, the local analog of the correspondence (15) is
(the $G$-equivariant version of) the following:

(34)
${\mathrm{Fl}}$${\widetilde{{\mathcal{N}}}_{0}=T^{*}\mathrm{Fl}}$${\mathrm{pt}}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$

where $\iota$ is the closed embedding of the $0$-section. The following lemma
collects a few standard results in Springer theory, which concerns the self-
correspondences

(35)
$[\mathrm{Fl}]\circ{{}^{\mathfrak{t}}}[\mathrm{Fl}]\in\mathrm{CH}_{*}(\widetilde{{\mathcal{N}}_{0}}\times\widetilde{{\mathcal{N}}_{0}})=\mathrm{Corr}^{*}(\widetilde{{\mathcal{N}}_{0}},\widetilde{{\mathcal{N}}_{0}})$

and

(36)
${{}^{\mathfrak{t}}}[\mathrm{Fl}]\circ[\mathrm{Fl}]\in\mathrm{CH}_{*}(\mathrm{pt})=\mathrm{Corr}^{*}(\mathrm{pt},\mathrm{pt}).$

######  Lemma 3.1.

  1. (a)

The self-correspondence (35) of $\widetilde{{\mathcal{N}}}_{0}$ is the
specialization of the self-correspondence of
$\widetilde{{\mathcal{N}}}_{\eta}$ induced by the longest element $w_{0}$ in
the Weyl group $\mathfrak{S}_{n}$.

  2. (b)

The self-correspondence of (36) of a point is

$\epsilon n!\cdot\mathrm{id}\in\mathrm{Corr}^{0}(\mathrm{pt},\mathrm{pt}).$

Here $\epsilon=\pm 1$ is a sign dependent on $n$.

  3. (c)

There is a natural Weyl group $\mathfrak{S}_{n}$ action on the cohomology
$H^{*}(\widetilde{{\mathcal{N}}_{0}},{\mathbb{Q}})$; the anti-invariant part
is $1$-dimensional concentrated in the top degree cohomology:

$H^{*}(\widetilde{{\mathcal{N}}_{0}},{\mathbb{Q}})^{\mathrm{sgn}}={\mathbb{Q}}\left[2\binom{n}{2}\right].$

Here the shift is given by the dimension $\dim\mathrm{Fl}=\binom{n}{2}$.
Furthermore, the cohomological correspondences:

$[\mathrm{Fl}]:H^{*}(\widetilde{N}_{0},{\mathbb{Q}})\to
H^{*-\binom{n}{2}}(\mathrm{pt},{\mathbb{Q}})={\mathbb{Q}},\quad{{}^{\mathfrak{t}}}[\mathrm{Fl}]:H^{*-\binom{n}{2}}(\mathrm{pt},{\mathbb{Q}})={\mathbb{Q}}\to
H^{*}(\widetilde{N}_{0},{\mathbb{Q}})$

are projecting to the anti-invariant part, and the natural inclusion from the
anti-invariant part respectively.

######  Remark 3.2.

The correspondences $[\mathrm{Fl}],{{}^{\mathfrak{t}}}[\mathrm{Fl}]$ are
naturally $G$-equivariant and relative over ${\mathcal{N}}_{0}$. All the
statements above concerning the correspondences hold $G$-equivariantly and
relatively. For example, the statement of (c) actually is obtained from a
sheaf-theoretic result: the Weyl group $\mathfrak{S}_{n}$ acts on the derived
push-forward

$Rq_{0*}{\mathbb{Q}}_{\widetilde{{\mathcal{N}}_{0}}}\in
D^{b}_{c}({\mathcal{N}}_{0})$

induced by the $\mathfrak{S}_{n}$-structure of
$q_{\eta}:\widetilde{{\mathcal{N}}}_{\eta}\to{\mathcal{N}}_{\eta}$. The
correspondences

$[\mathrm{Fl}]:Rq_{0*}{\mathbb{Q}}_{\widetilde{{\mathcal{N}}}_{0}}\to{\mathbb{Q}}_{0}\left[-2\binom{n}{2}\right],\quad{{}^{\mathfrak{t}}}[\mathrm{Fl}]:{\mathbb{Q}}_{0}\left[-2\binom{n}{2}\right]\to
Rq_{0*}{\mathbb{Q}}_{\widetilde{{\mathcal{N}}}_{0}}\in
D^{b}_{c}({\mathcal{N}}_{0})$

are projecting to the anti-invariant part and the natural inclusion from the
anti-invariant part respectively. Then the cohomological statement of Lemma
3.1(c) is obtained by further pushing forward the sheaf-theoretic statements
to a point.

As a consequence of Lemma 3.1(a), if we take a self-correspondence of
${\mathcal{N}}_{\eta}$, pull it back along

$q_{\eta}:\widetilde{{\mathcal{N}}}_{\eta}\times\widetilde{{\mathcal{N}}}_{\eta}\to{\mathcal{N}}_{\eta}\times{\mathcal{N}}_{\eta},$

and specialize it to a self-correspondence of $\widetilde{{\mathcal{N}}}_{0}$,
then this resulting self-correspondence commute with (35). In the next
section, we will use a global version of this result to analyze the
correspondence

(37)
$[\widetilde{M}_{0}]\circ{{}^{\mathfrak{t}}}[\widetilde{M}_{0}]\in\mathrm{Corr}_{W_{0}}^{*}(M_{0}^{\mathrm{par}},M_{0}^{\mathrm{par}}).$

###  3.2. Hitchin moduli spaces

Recall the moduli space $M^{\mathrm{mero}}$ of stable meromorphic Higgs
bundles of Section 2.4. It admits a natural evaluation map at the point
$p\in\Sigma$:

(38) $\mathrm{ev}_{p}:M^{\mathrm{mero}}\to\mathfrak{g}/G$

as we introduce below.

We denote by $M^{\mathrm{ab}}$ the _abelian_ Hitchin moduli space

$M^{\mathrm{ab}}=\mathrm{Pic}^{0}(\Sigma)\times
H^{0}(\Sigma,\omega_{\Sigma}(p))$

which parameterize rank $1$ Higgs bundles. There is a natural action of
$M^{\mathrm{ab}}$ on $M^{\mathrm{mero}}$ by

$({\mathcal{L}},\sigma)\cdot({\mathcal{E}},\theta)=({\mathcal{L}}\otimes{\mathcal{E}},\sigma+\theta),\quad({\mathcal{L}},\sigma)\in
M^{\mathrm{ab}},\quad({\mathcal{E}},\theta)\in M^{\mathrm{mero}}$

where the sum $\sigma+\theta$ is defined in the sense that we view both
$\sigma,\theta$ as morphisms

${\mathcal{L}}\otimes{\mathcal{E}}\to{\mathcal{L}}\otimes{\mathcal{E}}\otimes\omega_{\Sigma}(p).$

The quotient space

$\widehat{M}^{\mathrm{mero}}:=M^{\mathrm{mero}}/M^{\mathrm{ab}}$

is a nonsingular Deligne–Mumford stack, known as the moduli space of stable
meromorphic $\mathrm{PGL}_{n}$-Higgs bundles, which admits a natural
evaluation map at the point $p$. The evaluation map (38) for the original
moduli space the composition of the quotient map and the
$\mathrm{PGL}_{n}$-evaluation map:

$M^{\mathrm{mero}}\twoheadrightarrow\widehat{M}^{\mathrm{mero}}\xrightarrow{\mathrm{ev}_{p}}\mathfrak{g}/G.$

######  Proposition 3.3.

The morphism (38) is smooth and surjective.

###### Proof.

The quotient map
$M^{\mathrm{mero}}\twoheadrightarrow\widehat{M}^{\mathrm{mero}}$ is smooth.
Therefore, the smoothness of the composition follows from the smoothness of
the second evaluation map which has been proven in [35, Propositin 4.1] via
deformation theory. ∎

######  Remark 3.4.

We do not work directly with the evaluation map associated with
$\mathrm{GL}_{n}$ for the following reasons. First, Lemma 1.2 implies that the
image of $\mathrm{ev}_{p}$ for a $\mathrm{GL}_{n}$-Higgs bundle has trace $0$;
in particular the trace lies naturally in

$\mathfrak{g}=\\{\mathrm{tr}=0\\}\subset\mathfrak{gl}_{n}.$

Second, when we pass from the moduli stack of $\mathrm{GL}_{n}$-stable Higgs
bundle to the corresponding moduli space, we need to rigidify the automorphism
group ${\mathbb{C}}^{*}\subset\mathrm{GL}_{n}$. Working with the evaluation
map associated with $G=\mathrm{PGL}_{n}$ is more convenient which resolves
both issues.

Now we may use the evaluation map to pull back natural morphisms in Springer
theory. As a first example, the pullback of (38) via the ($G$-equivariant)
Groethendieck–Springer resolution recovers the parabolic moduli space

${M^{\mathrm{par}}}$${\widetilde{\mathfrak{g}}/G}$${M^{\mathrm{mero}}}$${\mathfrak{g}/G.}$$\scriptstyle{\mathrm{ev}_{p}}$$\scriptstyle{q}$$\scriptstyle{\mathrm{ev}_{p}}$

This further yields the specialization

$M^{\mathrm{par}}_{\eta}\xrightarrow{\mathrm{ev}_{p}}\widetilde{{\mathcal{N}}}_{\eta}/G\quad\rightsquigarrow\quad
M^{\mathrm{par}}_{0}\xrightarrow{\mathrm{ev}_{p}}\widetilde{{\mathcal{N}}}_{0}/G.$

Finally, the pullback along $\mathrm{ev}_{p}$ of the $G$-equivariant version
of the diagram (34)

(39)
${\mathrm{Fl}/G}$${\widetilde{{\mathcal{N}}}_{0}/G}$${\mathrm{pt}/G}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$

recovers the diagram (15). Since all the maps $\mathrm{ev}_{p}$ are obtained
via base change from (38), they are all smooth and surjective by Proposition
3.3.

Recall the $\mathfrak{S}_{n}$-quotient map

$M^{\mathrm{par}}_{\eta}\to\overline{J}^{e}_{C}$

of Section 1.2(a) for general $\eta\in T$.

######  Proposition 3.5.

Assume that the self-correspondence

$\mathfrak{Z}_{\eta}\in\mathrm{Corr}_{W_{\eta}}^{*}(M^{\mathrm{par}}_{\eta},M^{\mathrm{par}}_{\eta})$

is pulled back from
$\overline{J}^{e}_{C}\times_{W_{\eta}}\overline{J}^{e}_{C}$. Let

$\mathfrak{Z}_{0}\in\mathrm{Corr}^{*}_{W_{0}}(M^{\mathrm{par}}_{0},M^{\mathrm{par}}_{0})$

be the specialization of $\mathfrak{Z}_{\eta}$. Then (37) commutes with
$\mathfrak{Z}_{0}$, _i.e._ , we have

$\left([\widetilde{M}_{0}]\circ{{}^{\mathfrak{t}}}[\widetilde{M}_{0}]\right)\circ\mathfrak{Z}_{0}=\mathfrak{Z}_{0}\circ\left([\widetilde{M}_{0}]\circ{{}^{\mathfrak{t}}}[\widetilde{M}_{0}]\right)\in\mathrm{Corr}^{*}_{W_{0}}(M^{\mathrm{par}}_{0},M^{\mathrm{par}}_{0}).$

###### Proof.

The correspondence (15) is induced by the pullback of the correspondence (39)
via the evaluation maps, we obtain that

(40)
$\mathrm{ev}^{*}\left([\mathrm{Fl}/G]\circ{{}^{\mathfrak{t}}}[\mathrm{Fl}/G]\right)=[\widetilde{M}_{0}]\circ{{}^{\mathfrak{t}}}[\widetilde{M}_{0}].$

Here the evaluation map is given by

$\mathrm{ev}:M_{0}^{\mathrm{par}}\times_{W_{0}}M_{0}^{\mathrm{par}}\to\widetilde{{\mathcal{N}}}_{0}/G\times\widetilde{{\mathcal{N}}}_{0}/G.$

The $G$-equivariant version of Lemma 3.1(a) then implies that the
correspondence (40) is the specialization of a self-correspondence of
$M^{\mathrm{par}}_{\eta}$ induced by $w_{0}\in\mathfrak{S}_{n}$. The latter
commutes with the pullback of any self-correspondence of
$\overline{J}_{C}^{e}=M^{\mathrm{par}}_{\eta}/\mathfrak{S}_{n}$, which proves
the proposition. ∎

Recall the morphism $\Gamma$ from Section 1.3. From now on, we use

$\mathfrak{p}^{\mathrm{par}}_{k}\in\mathrm{Corr}_{W_{0}}^{0}(M_{0}^{\mathrm{par}},M^{\mathrm{par}}_{0}),\quad
k=0,1,\cdots,2(\dim M_{0}^{\mathrm{par}}-\dim W_{0})$

to denote the projectors associated with the motivic perverse filtration of
Theorem 2.14.

######  Proposition 3.6.

The following statements hold.

  1. (a)

We have

$\Gamma([\Delta_{M_{0}^{\mathrm{par}}/W_{0}}])=\epsilon
n!\cdot[\Delta_{M_{n,d}/B}].$

Here $\epsilon=\pm 1$ is as in Lemma 3.1(b).

  2. (b)

We have the vanishing

$\Gamma(\mathfrak{p}_{k}^{\mathrm{par}})=0,\quad k<\binom{n}{2}.$

  3. (c)

We have the stabilization

$\Gamma(\mathfrak{p}_{k}^{\mathrm{par}})=[\Delta_{M_{n,d}/B}],\quad k\geq
2(\dim M_{0}^{\mathrm{par}}-\dim W_{0})-\binom{n}{2}.$

  4. (d)

We have the semi-orthogonality relation

$\Gamma(\mathfrak{p}^{\mathrm{par}}_{l})\circ\Gamma(\mathfrak{p}^{\mathrm{par}}_{k})=\epsilon
n!\cdot\Gamma(\mathfrak{p}^{\mathrm{par}}_{k}),\quad k<l.$

###### Proof.

(a) follows from pulling back the local statement of Lemma 3.1(b). (b) and (c)
follow from Lemma 1.3 and the construction of
$\mathfrak{p}^{\mathrm{par}}_{k}$ in terms of the specialization of algebraic
cycles of the form (23). (d) is a consequence of part (a) and Proposition 3.5:

$\displaystyle\Gamma(\mathfrak{p}^{\mathrm{par}}_{l})\circ\Gamma(\mathfrak{p}^{\mathrm{par}}_{k})$
$\displaystyle={{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ\mathfrak{p}^{\mathrm{par}}_{l}\circ\left([\widetilde{M}_{0}]\circ{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\right)\circ\mathfrak{p}^{\mathrm{par}}_{k}\circ[\widetilde{M}_{0}]$
$\displaystyle=\left({{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ[\widetilde{M}_{0}]\right)\circ{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ\left(\mathfrak{p}^{\mathrm{par}}_{l}\circ\mathfrak{p}^{\mathrm{par}}_{k}\right)\circ[\widetilde{M}_{0}]$
$\displaystyle=\epsilon
n!\cdot{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ\mathfrak{p}^{\mathrm{par}}_{k}\circ[\widetilde{M}_{0}]$
$\displaystyle=\epsilon n!\cdot\Gamma(\mathfrak{p}^{\mathrm{par}}_{k}).$

Here we have used that every $\mathfrak{p}_{k}^{\mathrm{par}}$ satisfies the
assumption of $\mathfrak{Z}_{0}$ in Proposition 3.5, and we applied (a) to
deduce the third identity. ∎

By a direct dimension calculation, we have

$(\dim M_{0}^{\mathrm{par}}-\dim W_{0})-\binom{n}{2}=R_{f}.$

Therefore, after rescaling and relabelling as

$\mathfrak{p}_{k}:=\frac{\epsilon}{n!}\cdot\Gamma\left(\mathfrak{p}^{\mathrm{par}}_{k-\binom{n}{2}}\right),$

we obtain the projectors

(41) $\mathfrak{p}_{k}\in\mathrm{Corr}_{B}^{0}(M_{n,d},M_{n,d}),\quad
k=0,1,\cdots,2R_{f}.$

In Section 4, we will prove that (41) forms a motivic perverse filtration; as
for Theorem 2.14, we only need to show that the homological realization of
$\mathfrak{p}_{k}$ recovers the perverse truncation functor.

##  4\. Sheaf-theoretic operations

The purpose of this section is to prove a general specialization result for
the decomposition theorem (Proposition 4.1). Then we apply it to the Hitchin
systems we studied in the previous sections and complete the proofs of all the
main theorems.

###  4.1. Specializations

In this section, we finish the proof of Theorem 2.14 by addressing the
compatibility of the Chow and sheaf-theoretic specialization maps, and their
relationship with the perverse truncation.

Our setup is the following: let $T$ be a nonsingular curve and let $\pi:B\to
T$ be a smooth morphism. Consider two proper morphisms $f:X\to B$, $g:Y\to B$
such that the compositions $\pi\circ f:X\to T$, $\pi\circ g:Y\to T$ are
smooth:

${X}$${Y}$${B}$${T}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\pi}$

We think of $f:X\to B$, $g:Y\to B$ as two families of proper morphisms over
$T$.

Let $\eta\in T$ (resp. $0\in T$) be a general (resp. the special) point.
Suppose we are given a correspondence

${\mathfrak{Z}}_{\eta}\in\mathrm{Corr}^{*}_{B_{\eta}}(X_{\eta},Y_{\eta})=\mathrm{CH}_{*}(X_{\eta}\times_{B_{\eta}}Y_{\eta}).$

Then by the specialization map for Chow groups via

$X_{\eta}\to B_{\eta}\quad\rightsquigarrow\quad X_{0}\to B_{0},$

we obtain a correspondence

${\mathfrak{Z}}_{0}\in\mathrm{Corr}^{*}_{B_{\eta}}(X_{\eta},Y_{\eta})=\mathrm{CH}_{*}(X_{0}\times_{B_{0}}Y_{0}).$

Also recall from [16, Lemma 2.21] that the cycle class
$\mathrm{cl}({\mathfrak{Z}}_{\eta})\in
H^{\mathrm{BM}}_{2*}(X_{\eta}\times_{B_{\eta}}Y_{\eta},{\mathbb{Q}})$ induces
a morphism in $D^{b}_{c}(B_{\eta})$:

(42) $\mathrm{cl}({\mathfrak{Z}}_{\eta}):Rf_{\eta*}{\mathbb{Q}}_{X_{\eta}}\to
Rg_{\eta*}{\mathbb{Q}}_{Y_{\eta}}[*],$

where we ignore the precise shifting index. Similarly, we have a morphism in
$D^{b}_{c}(B_{0})$:

(43) $\mathrm{cl}({\mathfrak{Z}}_{0}):Rf_{0*}{\mathbb{Q}}_{X_{0}}\to
Rg_{0*}{\mathbb{Q}}_{Y_{0}}[*].$

######  Proposition 4.1.

Let ${\mathfrak{Z}}_{\eta},{\mathfrak{Z}}_{0}$ be as above. For
$k\in{\mathbb{Z}}$, if

(44)
${{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\mathrm{cl}({\mathfrak{Z}}_{\eta})):{{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(Rf_{\eta*}{\mathbb{Q}}_{X_{\eta}})\to{{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(Rg_{\eta*}{\mathbb{Q}}_{Y_{\eta}}[*])$

is an isomorphism (resp. zero), then the same holds for

(45)
${{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\mathrm{cl}({\mathfrak{Z}}_{0})):{{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(Rf_{0*}{\mathbb{Q}}_{X_{0}})\to{{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(Rg_{0*}{\mathbb{Q}}_{Y_{0}}[*]).$

###### Proof.

Let $\overline{{\mathfrak{Z}}}\in\mathrm{CH}_{*}(X\times_{B}Y)$ be the Zariski
closure of (the spread-out of)
${\mathfrak{Z}}_{\eta}\in\mathrm{CH}_{*}(X_{\eta}\times_{B_{\eta}}Y_{\eta})$.
The cycle class $\mathrm{cl}(\overline{{\mathfrak{Z}}})\in
H^{\mathrm{BM}}_{2*}(X\times_{B}Y,{\mathbb{Q}})$ induces a morphism in
$D^{b}_{c}(B)$:

(46) $\mathrm{cl}(\overline{{\mathfrak{Z}}}):Rf_{*}\mathbb{Q}_{X}\to
Rg_{*}\mathbb{Q}_{Y}[*].$

Then by construction and proper base change, the morphism
$\mathrm{cl}({\mathfrak{Z}}_{0})$ in (43) is obtained by applying $i^{*}$ to
(46) for the inclusion $i:B_{0}\hookrightarrow B$:

$\mathrm{cl}({\mathfrak{Z}}_{0})=i^{*}\mathrm{cl}(\overline{{\mathfrak{Z}}}):Rf_{0*}{\mathbb{Q}}_{X_{0}}\to
Rg_{0*}{\mathbb{Q}}_{Y_{0}}[*].$

We give an alternative construction of (43) via nearby and vanishing cycles
following [11, 8]. Recall the distinguished triangle

(47)
$i^{*}[-1]\to\psi_{B}[-1]\xrightarrow{\mathrm{can}}\phi_{B}\xrightarrow{+1}$

where $\psi_{B},\phi_{B}:D^{b}_{c}(B)\to D^{b}_{c}(B_{0})$ are the nearby and
vanishing cycle functors respectively. Since $X$ is smooth over $T$, we have
$\phi_{X}\mathbb{Q}_{X}\simeq 0\in D^{b}_{c}(X_{0})$ and by proper base change
for $\phi$,

$\phi_{B}Rf_{*}\mathbb{Q}_{X}\simeq Rf_{0*}\phi_{X}\mathbb{Q}_{X}\simeq 0\in
D^{b}_{c}(B_{0}).$

Similarly, we have

$\phi_{B}Rg_{*}\mathbb{Q}_{Y}\simeq Rg_{0*}\phi_{Y}\mathbb{Q}_{Y}\simeq 0\in
D^{b}_{c}(B_{0}).$

We now apply the distinguished triangle (47) to the morphism
$\mathrm{cl}(\overline{{\mathfrak{Z}}})$ in (46), which yields a commutative
diagram

(48)
${Rf_{0*}{\mathbb{Q}}_{X_{0}}}$${Rg_{0*}{\mathbb{Q}}_{Y_{0}}[*]}$${\psi_{B}Rf_{*}{\mathbb{Q}}_{X}}$${\psi_{B}Rg_{*}{\mathbb{Q}}_{Y}[*].}$$\scriptstyle{\mathrm{cl}({\mathfrak{Z}}_{0})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\psi_{B}\mathrm{cl}(\overline{{\mathfrak{Z}}})}$

Here both columns are isomorphisms since the vanishing cycles are trivial, and
the crucial point is that the bottom morphism
$\psi_{B}\mathrm{cl}(\overline{{\mathfrak{Z}}})$ is fully determined by the
morphism $\mathrm{cl}({\mathfrak{Z}}_{\eta})$ in (42).

Finally, we recall the fact that $\psi_{B}[-1]$ is perverse $t$-exact, and
hence commutes with taking perverse cohomology. Therefore, if (44) is an
isomorphism (resp. zero), then

${{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\psi_{B}\mathrm{cl}(\overline{{\mathfrak{Z}}})):{{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\psi_{B}Rf_{*}{\mathbb{Q}}_{X})\to{{}^{\mathfrak{p}}{\mathcal{H}}}^{k}(\psi_{B}Rg_{*}{\mathbb{Q}}_{Y}[*])$

is also an isomorphism (resp. zero). By the diagram (48) the same holds for
(45). ∎

A useful consequence of Proposition 4.1 is that the validity of the
Corti–Hanamura motivic decomposition conjecture is preserved under
specialization. This also completes the proof of Theorem 2.14.

######  Corollary 4.2.

Let $X\xrightarrow{f}B\xrightarrow{\pi}T$ be as above, with $f$ proper and
$\pi,\pi\circ f$ smooth. If the motivic decomposition conjecture holds for
$X_{\eta}\to B_{\eta}$, then the specializations of the projectors provide a
motivic decomposition for $X_{0}\to B_{0}$.

###### Proof.

The (semi-)orthogonal projectors of $X_{\eta}\to B_{\eta}$ specialize
immediately to (semi-)orthogonal projectors of $X_{0}\to B_{0}$, while the
relative diagonal class $[\Delta_{X_{\eta}/B_{\eta}}]$ specializes to
$[\Delta_{X_{0}/B_{0}}]$. Now Proposition 4.1 guarantees that the homological
realization of the specialized projectors recovers the decomposition theorem
for $X_{0}\to B_{0}$. ∎

###  4.2. Correspondences

For our purpose, we connect the decomposition theorem associated with the
Hitchin maps:

$h_{0}:M^{\mathrm{par}}\to W_{0},\quad f:M_{n,d}\to B\subset W_{0}.$

The diagram (15) yields sheaf-theoretic correspondences

(49) $[\widetilde{M}_{0}]:Rh_{0*}{\mathbb{Q}}_{M^{\mathrm{par}}_{0}}\to
Rf_{*}{\mathbb{Q}}_{M_{n,d}}\left[-2\binom{n}{2}\right],\quad{{}^{\mathfrak{t}}}[\widetilde{M}_{0}]:Rf_{*}{\mathbb{Q}}_{M_{n,d}}\left[-2\binom{n}{2}\right]\to
Rh_{0*}{\mathbb{Q}}_{M^{\mathrm{par}}_{0}}$

which take places in the derived category $D^{b}_{c}(W_{0})$.

The following lemma is a global version of Lemma 3.1(c).

######  Lemma 4.3.

There is a natural Weyl group $\mathfrak{S}_{n}$action on the object

$Rh_{0*}{\mathbb{Q}}_{M^{\mathrm{par}}_{0}}\in D^{b}_{c}(W_{0});$

the anti-invariant part is

$\left(Rh_{0*}{\mathbb{Q}}_{M^{\mathrm{par}}_{0}}\right)^{\mathrm{sgn}}=Rf_{*}{\mathbb{Q}}_{M_{n,d}}\left[-2\binom{n}{2}\right]\in
D^{b}_{c}(W_{0}).$

Furthermore, under this isomorphism the correspondences (49) are projecting to
the anti-invariant part and the natural inclusion from the anti-invariant part
respectively.

###### Proof.

By Proposition 3.3, the diagram (15) is the pullback of local case (34) along
smooth morphisms $\mathrm{ev}_{p}$. Therefore, Lemma 4.3 follow from pulling
back the $G$-equivariant version of Lemma 3.1(c) (see Remark 3.2). ∎

###  4.3. Proofs of Theorems 0.1, 0.2, 0.3, 0.4

In this section, we complete the proofs of all the main theorems.

####  4.3.1. Proof of Theorem 0.1

We start with the proof of Theorem 0.1. By Proposition 2.2, it suffices to
show that the correspondences we constructed in (41) form a motivic perverse
filtration. By Proposition 3.6, it remains to check that the homological
realization

$\mathfrak{p}_{k}Rf_{*}{\mathbb{Q}}_{M_{n,d}}\rightarrow
Rf_{*}{\mathbb{Q}}_{M_{n,d}}$

recovers the perverse truncation

${}^{\mathfrak{p}}\tau_{\leq
k+\dim{B}}Rf_{*}{\mathbb{Q}}_{M_{n,d}}\hookrightarrow
Rf_{*}{\mathbb{Q}}_{M_{n,d}}.$

We know that this statement holds for $h_{0}:M^{\mathrm{par}}_{0}\to W_{0}$
and $\mathfrak{p}_{k}^{\mathrm{par}}$ by Theorem 2.14 (proven in Section 4.1).
Now, we consider the correspondence of $\mathfrak{p}_{k}$ given by the
composition:

$\mathfrak{p}_{k}:Rf_{*}{\mathbb{Q}}_{M_{n,d}}\xrightarrow{[\widetilde{M}_{0}]}Rh_{0*}{\mathbb{Q}}_{M_{0}^{\mathrm{par}}}\left[2\binom{n}{2}\right]\xrightarrow{\mathfrak{p}^{\mathrm{par}}_{k}}Rh_{0*}{\mathbb{Q}}_{M_{0}^{\mathrm{par}}}\left[2\binom{n}{2}\right]\xrightarrow{{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}}Rf_{*}{\mathbb{Q}}_{M_{n,d}}.$

By Lemma 4.3, we have a direct sum decomposition induced by the
$\mathfrak{S}_{n}$-action:

(50)
$Rh_{0*}{\mathbb{Q}}_{M_{0}^{\mathrm{par}}}\left[2\binom{n}{2}\right]=(Rh_{0*}{\mathbb{Q}}_{M_{0}^{\mathrm{par}}})^{\mathrm{sgn}}\left[2\binom{n}{2}\right]\oplus\mathrm{others},$

and applying $\mathfrak{p}_{k}$ to $Rf_{*}{\mathbb{Q}}_{M_{n,d}}$ is induced
by applying $\mathfrak{p}_{k}^{\mathrm{par}}$ on the first factor above. In
particular, it coincides with the perverse truncation. This completes the
proof. ∎

####  4.3.2.  Proof of Theorem 0.2

We first note the following result of [1, Proposition 6.4].

Fact 1. The compactified Jacobian fibration

$\overline{J}^{e}_{C}\to W_{\eta}$

of Section 1.2(a) satisfies the relative Lefschetz standard conjecture.

More precisely, this is due to the resolutions of the Lefschetz standard
conjecture for abelian varieties [32, 31] and the fact that the decomposition
theorem associated with a compactified Jacobian fibration associated with
integral locally planar curves has _full support_ by Ngô support theorem [45].

Our strategy is to reduce the case of $f:M_{n,d}\to B$ to
$\overline{J}^{e}_{C}\to W_{\eta}$. Let $\sigma\in
H^{2}(M_{n,d},{\mathbb{Q}})$ be a relative ample class. By [34], we can write
$\sigma$ in terms of tautological classes as follows. We first recall that we
have a canonical isomorphism

(51) $H^{2}(M_{n,d},{\mathbb{Q}})=H^{2}(\widehat{M}_{n,d},{\mathbb{Q}})\oplus
H^{2}(J_{\Sigma},{\mathbb{Q}})$

where $\widehat{M}_{n,d}$ is the $\mathrm{PGL}_{n}$-Higgs moduli space and
$J_{\Sigma}$ is the Jacobian variety of the curve $\Sigma$; see [9, Section
2.4]. The first summand on the right-hand side of (51) spans a $1$-dimensional
subspace, generated by the relative ample tautological class given by the
Künneth component of normalized second Chern character
$\widetilde{\mathrm{ch}}_{2}({\mathcal{U}})$ with respect to the point class
$[\mathrm{pt}]\in H^{2}(\Sigma,{\mathbb{Q}})$ on the curve. This class can be
lifted to the (${\mathbb{Q}}$-)divisor class

$c_{2}([\Sigma]):=\pi_{M*}\left(\pi_{\Sigma}^{*}[\Sigma]\cup\widetilde{\mathrm{ch}}_{2}({\mathcal{U}})\right)\in\mathrm{Pic}(M_{n,d})_{\mathbb{Q}}.$

Therefore, under the decomposition (51), we may present the relative class
$\eta$ as

(52) $\sigma=c_{2}([\Sigma])+\theta$

where $\theta$ is induced by an ample divisor on $J_{\Sigma}$. The right-hand
side is the pullback of a divisor on $M^{\mathrm{mero}}$ with the same
tautological presentation (52). As in Section 2.4, pulling back this class
yields relative ample classes on each variety $M^{\mathrm{par}}_{\eta}$ with
$\eta\in T$.

In conclusion, we obtain the following fact.

Fact 2. There exists a divisor
$\sigma_{0}^{\mathrm{par}}\in\mathrm{Pic}(M_{0}^{\mathrm{par}})_{\mathbb{Q}}$
satisfying

(53) $\pi^{*}\sigma=\iota^{*}\sigma_{0}^{\mathrm{par}}$

where $\iota,\pi$ are defined in the diagram (15). Moreover, this divisor
$\sigma_{0}^{\mathrm{par}}$ is obtained as the specialization of a relative
ample divisor

$\sigma_{\eta}^{\mathrm{par}}\in\mathrm{Pic}(M^{\mathrm{par}}_{\eta})_{{\mathbb{Q}}}$

for general $\eta$.

Note that in general the cup-product with respect to an algebraic cycle $E$ is
induced by a correspondence given by the $\Delta_{*}E$, where $\Delta$ is the
(relative) diagonal. Now we consider the correspondences

$\Delta_{*}{\sigma^{i}}\in\mathrm{Corr}^{*}_{B}(M_{n,d},M_{n,d}),\quad\Delta_{*}{(\sigma^{\mathrm{par}}_{0})^{i}}\in\mathrm{Corr}_{W_{0}}^{*}(M_{0}^{\mathrm{par}},M_{0}^{\mathrm{par}}).$

Fact 3. We have

$\Gamma\left(\Delta_{*}{(\sigma^{\mathrm{par}}_{0})^{i}}\right)=\epsilon
n!\cdot\Delta_{*}{\sigma^{i}}\in\mathrm{Corr}^{*}_{B}(M_{n,d},M_{n,d}).$

In particular, after the homological realization we obtain the commutative
diagram

${Rf_{*}{\mathbb{Q}}_{M_{n,d}}}$${Rf_{*}{\mathbb{Q}}_{M_{n,d}}[2i]}$${Rh_{0*}{\mathbb{Q}}_{M^{\mathrm{par}}_{0}}\left[2\binom{n}{2}\right]}$${Rh_{0*}{\mathbb{Q}}_{M^{\mathrm{par}}_{0}}\left[2\binom{n}{2}+2i\right].}$$\scriptstyle{\epsilon
n!\cdot(\cup\sigma^{i})}$$\scriptstyle{[\widetilde{M}_{0}]}$$\scriptstyle{\cup(\sigma_{0}^{\mathrm{par}})^{i}}$$\scriptstyle{{{}^{\mathfrak{t}}}[\widetilde{M}_{0}]}$

###### Proof of Fact 3.

It suffices to show the following relation of correspondences

(54)
${{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ\Delta_{*}(\sigma_{0}^{\mathrm{par}})^{i}=\Delta_{*}\sigma^{i}\circ{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\in\mathrm{Corr}^{*}_{W_{0}}(M_{0}^{\mathrm{par}},M_{n,d}).$

Then by Proposition 3.6(a), we have

$\Gamma(\Delta_{*}(\sigma_{0}^{\mathrm{par}})^{i})={{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ\Delta_{*}(\sigma_{0}^{\mathrm{par}})^{i}\circ[\widetilde{M}_{0}]=\Delta_{*}\sigma^{i}\circ{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ[\widetilde{M}_{0}]=\epsilon
n!\cdot\Delta_{*}\sigma^{i},$

which proves Fact 3. To see the relation (54), we expand the left-hand side

$\displaystyle{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\circ\Delta_{*}(\sigma_{0}^{\mathrm{par}})^{i}$
$\displaystyle=p_{13*}\delta^{!}\left(\Delta_{*}(\sigma_{0}^{\mathrm{par}})^{i}\times{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\right)$
$\displaystyle=p_{13*}\delta^{!}\left(\left(q_{2}^{*}(\sigma_{0}^{\mathrm{par}})^{i}\cap[\Delta_{M_{0}^{\mathrm{par}}/W_{0}}]\right)\times{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\right)$
$\displaystyle=p_{13*}\delta^{!}\left([\Delta_{M_{0}^{\mathrm{par}}/W_{0}}]\times\left(r_{1}^{*}(\sigma_{0}^{\mathrm{par}})^{i}\cap{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\right)\right)$
$\displaystyle=r_{1}^{*}(\sigma_{0}^{\mathrm{par}})^{i}\cap{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}={{}^{\mathfrak{t}}(\iota^{*}(\sigma_{0}^{\mathrm{par}})^{i}\cap[\widetilde{M}_{0}])}.$

Here $\delta^{!}$ is the refined Gysin pullback with respect to the regular
(absolute diagonal) embedding $\Delta:M_{0}^{\mathrm{par}}\to
M_{0}^{\mathrm{par}}\times M_{0}^{\mathrm{par}}$, and
$p_{13}:M_{0}^{\mathrm{par}}\times_{W_{0}}M_{0}^{\mathrm{par}}\times_{W_{0}}M_{n,d}\to
M_{0}^{\mathrm{par}}\times_{W_{0}}M_{n,d}$,
$q_{2}:M_{0}^{\mathrm{par}}\times_{W_{0}}M_{0}^{\mathrm{par}}\to
M_{0}^{\mathrm{par}}$, and $r_{1}:M_{0}^{\mathrm{par}}\times_{W_{0}}M_{n,d}\to
M_{0}^{\mathrm{par}}$ are the natural projections. Similarly, we compute the
right-hand side of (54):

$\displaystyle\Delta_{*}\sigma^{i}\circ{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}$
$\displaystyle=p_{13*}\delta^{!}\left({{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\times\Delta_{*}\sigma^{i}\right)$
$\displaystyle=p_{13*}\delta^{!}\left({{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\times\left(r_{1}^{*}\sigma^{i}\cap[\Delta_{M_{n,d}/B}]\right)\right)$
$\displaystyle=p_{13*}\delta^{!}\left(\left(q_{2}^{*}\sigma^{i}\cap{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}\right)\times[\Delta_{M_{n,d}/B}]\right)$
$\displaystyle=q_{2}^{*}\sigma^{i}\cap{{}^{\mathfrak{t}}[\widetilde{M}_{0}]}={{}^{\mathfrak{t}}(\pi^{*}\sigma^{i}\cap[\widetilde{M}_{0}])}.$

This time $\delta^{!}$ is the refined Gysin pullback with respect to
$\Delta:M_{n,d}\to M_{n,d}\times M_{n,d}$, and $p_{13},q_{2},r_{1}$ are the
corresponding natural projections. Hence (54) follows from (53). ∎

Now we construct the cycle $\mathfrak{Z}_{\sigma,i}$ for the relative ample
class $\sigma$ on $M_{n,d}$. We start with the cycle given by Fact 1 for
$\overline{J}^{e}_{C}$. Its pullback to $M^{\mathrm{par}}_{\eta}$ yields a
correspondence

$\mathfrak{Z}_{\sigma^{\mathrm{par}}_{\eta},i}\in\mathrm{Corr}_{W_{\eta}}^{*}(M^{\mathrm{par}}_{\eta},M^{\mathrm{par}}_{\eta})$

which gives the inverse of the Lefschetz symmetry induced by
$(\sigma^{\mathrm{par}}_{\eta})^{i}$ for general $\eta$. Moreover, since the
cycle is pulled back from a self-correspondence of
$\overline{J}^{e}_{C}=M^{\mathrm{par}}_{\eta}/\mathfrak{S}_{n}$, its induced
isomorphism respects the isotypic decomposition with respect to the
$\mathfrak{S_{n}}$-action.

By Proposition 4.1, its specialization

(55)
$\mathfrak{Z}_{\sigma^{\mathrm{par}}_{0},i}\in\mathrm{Corr}_{W_{0}}^{*}(M^{\mathrm{par}}_{0},M^{\mathrm{par}}_{0})$

gives the inverse of the Lefschetz symmetry induced by
$(\sigma^{\mathrm{par}}_{0})^{i}$ and respects the decomposition given by the
$\mathfrak{S}_{n}$-structure. In particular, the correspondence (55) induces
an isomorphism on the perverse cohomology sheaves of the first summand of
(50).

Finally, we set

$\mathfrak{Z}_{\sigma,i}:=\Gamma\left(\mathfrak{Z}_{\sigma^{\mathrm{par}}_{0},i}\right)\in\mathrm{Corr}_{B}^{*}(M_{n,d},M_{n,d}).$

As in the proof of Theorem 0.1 in Section 4.3.1, the diagram of Fact 3
together with Lemma 4.3 implies that the sheaf-theoretic operator
$\cup\sigma^{i}$ coincides with the restriction of the operator
$\cup(\sigma^{\mathrm{par}}_{0})^{i}$ to the anti-invariant component with
respect to the $\mathfrak{S}_{n}$-equivariant structure. Since the inverse
operator (55) of $\cup(\sigma^{\mathrm{par}}_{0})^{i}$ also respects the
decomposition (50), we conclude that the homological realization of
$\mathfrak{Z}_{\sigma,i}$ gives the inverse of $\cup\sigma^{i}$ as desired. ∎

####  4.3.3. Proof of Theorem 0.3

We consider the diagram

(56) ${\Sigma\times\widetilde{M}_{0}}$${\Sigma\times
M_{0}^{\mathrm{par}}}$${\Sigma\times
M_{n,d}}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$

given by the base change of (15). For convenience, we use the same notation
for this diagram as for (15); _e.g._ we still use $[\widetilde{M}_{0}]$ to
denote the correspondence from $\Sigma\times M_{n,d}$ to $\Sigma\times
M_{0}^{\mathrm{par}}$.

By the construction of the normalized Chern character, we have

$\pi^{*}\widetilde{\mathrm{ch}}({\mathcal{U}})=\iota^{*}\widetilde{\mathrm{ch}}({\mathcal{U}}).$

Here the normalized Chern character on the left-hand side lies over
$\Sigma\times M_{n,d}$ and the normalized Chern character on the right-hand
side lies over $\Sigma\times M_{0}^{\mathrm{par}}$; their match after pulling
back to $\Sigma\times\widetilde{M}_{0}$ follows from the fact that both
normalized Chern characters are pulled back from $\Sigma\times
M^{\mathrm{mero}}$. By the same argument as for Fact 3 in Section 4.3.2, we
have

(57)
$\Gamma\left(\Delta_{*}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\right)=\epsilon
n!\cdot\Delta_{*}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\in\mathrm{Corr}^{*}_{\Sigma\times
B}(\Sigma\times M_{n,d},\Sigma\times M_{n,d})$

where the two sides concern the normalized Chern characters on $\Sigma\times
M^{\mathrm{par}}_{0}$ and $\Sigma\times M_{n,d}$ respectively. Recall the
projectors $\mathfrak{p}_{k}$ from (15) and we use the same notation to denote
the pullback of these projectors to the varieties in (56). We define

$\mathfrak{q}_{k+1}:=\Delta_{\Sigma\times M_{n,d}/\Sigma\times
B}-\mathfrak{p}_{k}=\frac{\epsilon}{n!}\cdot\Gamma\left(\Delta_{\Sigma\times
M_{0}^{\mathrm{par}}/\Sigma\times
B}-\mathfrak{p}^{\mathrm{par}}_{k-\binom{n}{2}}\right),$

where the second identity is a consequence of Proposition 3.6(a).

For our purpose, we want to prove the relation:

$\mathfrak{q}_{i+k+1}\circ\Delta_{*}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\circ\mathfrak{p}_{i}=0;$

using (57) this can be rewritten as

(58)
$\Gamma\left(\mathfrak{q}^{\mathrm{par}}_{i+k+1-\binom{n}{2}}\right)\circ\Gamma\left(\Delta_{*}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\right)\circ\Gamma\left(\mathfrak{p}^{\mathrm{par}}_{i-\binom{n}{2}}\right)=0.$

Since all the correspondences

$\mathfrak{q}^{\mathrm{par}}_{i+k+1-\binom{n}{2}},\quad\Delta_{*}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}}),\quad\mathfrak{p}^{\mathrm{par}}_{i-\binom{n}{2}}\in\mathrm{Corr}^{*}_{\Sigma\times
W_{0}}(\Sigma\times M_{0}^{\mathrm{par}},\Sigma\times M_{0}^{\mathrm{par}})$

are given by specializations of cycles pulled back from correspondences for
$\overline{J}^{e}_{C}$, by Proposition 3.5 the relation (58) follows from the
relation

$\mathfrak{q}^{\mathrm{par}}_{i+k+1-\binom{n}{2}}\circ\Delta_{*}\widetilde{\mathrm{ch}}_{k}({\mathcal{U}})\circ\mathfrak{p}^{\mathrm{par}}_{i-\binom{n}{2}}=0\in\mathrm{Corr}^{*}_{\Sigma\times
W_{0}}(\Sigma\times M_{0}^{\mathrm{par}},\Sigma\times M_{0}^{\mathrm{par}}).$

This is given by Theorem 2.15. The proof is completed. ∎

####  4.3.4. Proof of Theorem 0.4

We denote by

$\mathfrak{C}\in\mathrm{Corr}_{W_{0}}^{0}(M^{\mathrm{par}}_{0},{M^{\mathrm{par}}_{0}}^{\prime}),\quad\mathfrak{C}^{\prime}\in\mathrm{Corr}_{W_{0}}^{0}({M_{0}^{\mathrm{par}}}^{\prime},M_{0}^{\mathrm{par}})$

the correspondences constructed in Theorem 2.16 inducing an isomorphism

$h(M_{0}^{\mathrm{par}})\simeq h({M_{0}^{\mathrm{par}}}^{\prime}).$

Then an identical argument as above (using Proposition 3.5) implies that

$\overline{\mathfrak{C}}:=\frac{\epsilon}{n!}\cdot\mathfrak{C}\in\mathrm{Corr}_{B}^{0}(M_{n,d},M_{n,d^{\prime}}),\quad\overline{\mathfrak{C}}^{\prime}:=\frac{\epsilon}{n!}\cdot\mathfrak{C}^{\prime}\in\mathrm{Corr}_{B}^{0}(M_{n,d^{\prime}},M_{n,d})$

induces an isomorphism as desired. ∎

####  4.3.5. Remarks on the twisted case

As in Remark 1.1, one can also consider the $\mathfrak{D}$-twisted case with
$\mathfrak{D}>0$. Here we discuss briefly the possible strategies to prove the
main theorems in the $\mathfrak{D}$-twisted case.

The decomposition theorem associated with $f:M^{\mathfrak{D}}_{n,d}\to
B^{\mathfrak{D}}$ has full support; see Remark 1.1 and [14]. In this case,
Theorem 0.2 follows directly from [1, Proposition 6.4] as Fact 1 above, and
the argument of Section 4.3.3 is not needed.

On the other hand, due to the failure of the $\mathfrak{D}$-twisted version of
Lemma 1.2, for general $\eta\in T$ we can only guarantee that all the spectral
curves are reduced, but there may exist non-integral ones. Therefore, the
extension of Arinkin’s Fourier–Mukai transform over reduced locally planar
curves [41, 42] may be needed to construct the projectors for the motivic
decomposition.

##  5\. Questions and conjectures

In this section, we discuss some conjectures and open questions concerning
algebraic cycles for the Hitchin system.

###  5.1. Multiplicativity

A central question raised by de Cataldo, Hausel, and Migliorini in accompany
with their discovery of the P=W conjecture (see [9, Introduction]) is the
_multiplicativity_ of the perverse filtration:

(59) $\cup:P_{k}H^{*}(M_{n,d},{\mathbb{Q}})\times
P_{l}H^{*}(M_{n,d},{\mathbb{Q}})\rightarrow
P_{k+l}H^{*}(M_{n,d},{\mathbb{Q}}).$

This phenomenon now is a consequence of the resolutions of the P=W conjecture,
whose proofs rely heavily on the tautological generation result of Markman
[34].

In general, the multiplicativity holds for Leray filtrations associated with
proper morphisms, but fails for the perverse filtration. The main reason is
that the (sheaf-theoretic) cup-product does not interact nicely with the
perverse truncation functor in general.

For the Hitchin system, we expect that the sheaf-theoretic enhancement of (59)
holds.

######  Conjecture 5.1.

The cup-product interacts with the perverse truncation functor as follows

$\cup:{{}^{\mathfrak{p}}}\tau_{\leq
k}Rf_{*}{\mathbb{Q}}_{M_{n,d}}\times{{}^{\mathfrak{p}}}\tau_{\leq
l}Rf_{*}{\mathbb{Q}}_{M_{n,d}}\to{{}^{\mathfrak{p}}}\tau_{\leq k+l-(\dim
M_{n,d}-R_{f})}Rf_{*}{\mathbb{Q}}_{M_{n,d}}.$

By the proof of Theorem 0.1, the perverse filtration and its sheaf-theoretic
enhancement are governed by the algebraic cycles
$\mathfrak{p}_{k},\mathfrak{q}_{k}$. We propose the following conjecture which
further strengthens Conjecture 5.1; this also provides an explanation of the
multiplicativity phenomenon in terms of _tautological relations_ concerning
$\mathfrak{p}_{k},\mathfrak{q}_{k}$.

######  Conjecture 5.2.

The motivic perverse filtration (5) is multiplicative with respect to cup-
product, _i.e._ , the following relations hold

${\mathfrak{q}}_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{M_{n,d}/B}]\circ({\mathfrak{p}}_{k}\times{\mathfrak{p}}_{l})=0\in\mathrm{CH}_{*}(M_{n,d}\times_{B}M_{n,d}\times_{B}M_{n,d}).$

Here $\Delta^{\mathrm{sm}}_{M_{n,d}/B}$ is the relative small diagonal in the
relative triple product.

By Theorem 2.6(b), Conjecture 5.2 holds for the parabolic Hitchin system
$h_{\eta}:M^{\mathrm{par}}_{\eta}\to W_{\eta}$ for general $\eta$ or the
elliptic part of the Hitchin system
$f^{\mathrm{ell}}:M^{\mathrm{ell}}_{n,d}\to B^{\mathrm{ell}}$. From the
perspective of this paper, the main difficulty in approaching Conjecture 5.2
is to understand the small diagonal cycle for $M_{n,d}$ using parabolic moduli
spaces.

###  5.2. Refined motivic decomposition

Support theorems yield rich structures for the decomposition theorem
associated with the Hitchin system. For example, by studying the decomposition
theorem over the locus $B^{\mathrm{red}}\subset B$ of the Hitchin base formed
by reduced spectral curves, de Cataldo, Heinloth, and Migliorini showed in
[10] that every partition of $n$ contributes a support of
$Rf_{*}{\mathbb{Q}}_{M_{n,d}}$. Therefore, following the philosophy of
Corti–Hanamura [16], we expect to have the motives which lift the contribution
of the support indexed by each partition.

######  Question 5.3.

Can we find algebraic cycles which yield a refinement of the motivic
decomposition (5) based on supports of the decomposition theorem for
$f:M_{n,d}\to B$?

The conjectural refinement of Question 5.3 is mysterious for the following
reasons. As we commented in Remark 1.1, if we consider the decomposition
theorem for the $\mathfrak{D}$-twisted Hitchin system, the support theorem
behaves very differently for $\mathfrak{D}=0$ and $\mathfrak{D}>0$. In
particular, the decomposition theorem has full support when $\mathfrak{D}>0$.
Therefore, the conjectural motives of Question 5.3, which lift the
contribution of smaller supports of the decomposition theorem, should only
exist for $\mathfrak{D}=0$.

###  5.3. Singular moduli spaces

When the degree $d$ is not coprime to the rank $n$, the Hitchin moduli space
$M_{n,d}$ is singular. Many properties concerning the decomposition theorem
and the perverse filtration in the coprime setting are expected to be
generalized to the non-coprime setting. We expect that there should exist
algebraic cycles which induce these sheaf-theoretic structures.

A challenging question in this direction is to construct relative _BPS
motives_ for all $M_{n,d}$ without the coprime assumption. More precisely,
from the perspective of cohomological Donaldson–Thomas theory the role played
by the constant sheaf ${\mathbb{Q}}_{n,d}$ should be replaced by a mysterious
perverse sheaf, called the BPS sheaf

$\Phi_{\mathrm{BPS},d}\in\mathrm{Perv}(M_{n,d}),$

when $M_{n,d}$ is possibly singular. This perverse sheaf recovers the
(shifted) constant sheaf ${\mathbb{Q}}_{M_{n,d}}[\dim M_{n,d}]$ when
$(n,d)=1$, but is in general more complicated than the intersection cohomology
complex $\mathrm{IC}_{M_{n,d}}$; see [46] or [36, Section 0.4]. The sheaf-
theoretic $\chi$-independence result (8) is generalized by Kinjo–Koseki [30]
using the BPS perverse sheaf:

(60) $Rf_{d*}{\Phi_{\mathrm{BPS},d}}\simeq
Rf_{d^{\prime}*}\Phi_{\mathrm{BPS},d^{\prime}}.$

######  Question 5.4.

Can we lift the sheaf-theoretic $\chi$-independence identity (60) to relative
Chow motives over the Hitchin base $B$?

When $(n,d)\neq 1$, we have

$\Phi_{\mathrm{BPS},d}=\mathrm{IC}_{M_{n,d}}\oplus\varphi_{d}$

where $\varphi_{d}$ is a perverse sheaf supported on the singular locus of
$M_{n,d}$. It would be interesting to see how the extra summand $\varphi_{d}$
is detected by algebraic cycles. The BPS motive which lifts
$Rf_{d*}{\Phi_{\mathrm{BPS},d}}$ may be viewed as the motivic Gopakumar–Vafa
invariant associated with the local Calabi–Yau 3-fold
$X:=\mathrm{Tot}_{\Sigma}(\omega_{\Sigma}\oplus{\mathcal{O}}_{\Sigma})$.

## References

  * [1] G. Ancona, M. Cavicchi, R. Laterveer, and G. Saccà, Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations, arXiv:2304.00978.
  * [2] D. Anderson, R. Gonzales, and S. Payne, Equivariant Grothendieck–Riemann–Roch and localization in operational $K$-theory, with an appendix by Gabriele Vezzosi, Algebra Number Theory 15 (2021), no. 2, 341–385.
  * [3] D. Arinkin, Autoduality of compactified Jacobians for curves with plane singularities, J. Algebraic Geom. 22 (2013), no. 2, 363–388.
  * [4] P. Baum, W. Fulton, and R. MacPherson, Riemann–Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. (1975), no. 45, 101–145.
  * [5] A. Beauville, Sur l’anneau de Chow d’une variété abélienne, Math. Ann. 273 (1986), no. 4, 647–651.
  * [6] A. Beauville, M. S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169–179.
  * [7] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Soc. Math. France, Paris, 1982.
  * [8] M. A. de Cataldo, Perverse Leray filtration and specialisation with applications to the Hitchin morphism, Math. Proc. Cambridge Philos. Soc. 172 (2022), no. 2, 443–487.
  * [9] M. A. de Cataldo, T. Hausel, and L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case $A_{1}$, Ann. of Math. (2) 175 (2012), no. 3, 1329–1407.
  * [10] M.A. de Cataldo, J. Heinloth, and L. Migliorini, A support theorem for the Hitchin fibration: the case of $\mathrm{GL}_{n}$ and $K_{C}$, J. Reine Angew. Math. 780 (2021), 41–77.
  * [11] M. A. de Cataldo and D. Maulik, The perverse filtration for the Hitchin fibration is locally constant, Pure Appl. Math. Q. 16 (2020), no. 5, 1441–1464.
  * [12] M. A. de Cataldo, D. Maulik, and J. Shen, Hitchin fibrations, abelian surfaces, and the P=W conjecture, J. Amer. Math. Soc. 35 (2022), no. 3, 911–953.
  * [13] M. Cavicchi, F. Déglise, and J. Nagel, _Motivic decompositions of families with Tate fibers: smooth and singular cases,_ Int. Math. Res. Not. IMRN (2023), no. 16, 14239–14289.
  * [14] P.-H. Chaudouard and G. Laumon, Un théorème du support pour la fibration de Hitchin, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 2, 711–727.
  * [15] W. Y. Chuang, D. E. Diaconescu, and G. Pan, BPS states and the P=W conjecture, Moduli spaces, 132–150, London Math. Soc. Lecture Note Ser., 411, Cambridge Univ. Press, Cambridge, 2014.
  * [16] A. Corti and M. Hanamura, Motivic decomposition and intersection Chow groups. I, Duke Math. J. 103 (2000), no. 3, 459–522.
  * [17] B. Davison, L. Hennecart, and S. Schlegel Mejia. BPS Lie algebras for totally negative $2$-Calabi–Yau categories and nonabelian Hodge theory for stacks, arXiv:2212.07668.
  * [18] C. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991), 201–219.
  * [19] G. Dobrovolska, V. Ginzburg, and R. Travkin, Moduli spaces, indecomposable objects and potentials over a finite field, arXiv:1612.01733.
  * [20] D. Edidin and W. Graham, Riemann–Roch for equivariant Chow groups, Duke Math. J. 102 (2000), no. 3, 567–594.
  * [21] B. B. Gordon, M. Hanamura, and J. P. Murre, Relative Chow–Künneth projectors for modular varieties, J. Reine Angew. Math. 558 (2003), 1–14.
  * [22] M. Groechenig and S. Shen, Complex $K$-theory of moduli spaces of Higgs bundles, arXiv:2212.10695.
  * [23] M. Groechenig, D. Wyss, and P. Ziegler, Mirror symmetry for moduli spaces of Higgs bundles via $p$-adic integration, Invent. Math. 221 (2020), no.2, 505–596.
  * [24] T. Hausel, A. Mellit, A. Minets, and O. Schiffmann, P=W via $\mathcal{H}_{2}$, arXiv:2209.05429v1.
  * [25] T. Hausel and F. Rodriguez-Villegas, Mixed Hodge polynomials of character varieties, with an appendix by Nicholas M. Katz, Invent. Math. 174 (2008), no. 3, 555–624.
  * [26] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), no. 1, 197–229.
  * [27] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.
  * [28] N. J. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), no. 1, 91–114.
  * [29] V. Hoskins and S. Pepin Lehalleur, Motivic mirror symmetry and $\chi$-independence for Higgs bundles in arbitrary characteristic, arXiv:2205.15393.
  * [30] T. Kinjo and N. Koseki, Cohomological $\chi$-independence for Higgs bundles and Gopakumar–Vafa invariants, J. Eur. Math. Soc. (JEMS), to appear.
  * [31] K. Künnemann, A Lefschetz decomposition for Chow motives of abelian schemes, Invent. Math. 113 (1993), no. 1, 85–102.
  * [32] D. I. Lieberman, Numerical and homological equivalence of algebraic cycles on Hodge manifolds, Amer. J. Math. 90 (1968), 366–374.
  * [33] F. Loeser and D. Wyss, Motivic integration on the Hitchin fibration, Algebr. Geom. 8 (2021), no. 2, 196–230.
  * [34] E. Markman, Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces, J. Reine Angew. Math. 544 (2002), 61–82.