#  Hodge-to-singular correspondence for reduced curves

Mirko Mauri and Luca Migliorini

###### Abstract.

We study the summands of the decomposition theorem for the Hitchin system for
$\operatorname{GL}_{n}$, in arbitrary degree, over the locus of reduced
spectral curves. A key ingredient is a new correspondence between these
summands and the topology of hypertoric quiver varieties. In contrast to the
case of meromorphic Higgs fields, the intersection cohomology groups of moduli
spaces of regular Higgs bundles depend on the degree. We describe this
dependence.

Let $C$ be a compact Riemann surface of genus $g$ with canonical bundle
$\omega_{C}$. Let $d$ and $n$ be integers with $n>1$.

The _Dolbeault moduli space_ ${M}(n,d)$ is the coarse moduli space which
parametrises S-equivalence classes of semistable Higgs bundles on $C$ of rank
$n$ and degree $d$, i.e. polystable pairs $(\mathcal{E},\phi)$ consisting of a
vector bundle $\mathcal{E}$ of rank $n$ and degree $d$ and a section $\phi\in
H^{0}(C,\mathcal{E}nd(\mathcal{E})\otimes\omega_{C})$, called _Higgs field_ ;
see [67]. The Dolbeault moduli space is equipped with a projective fibration
called _Hitchin fibration_

(1) $\displaystyle\chi(n,d)\colon{M}(n,d)$ $\displaystyle\to
A_{n}\coloneqq\bigoplus^{n}_{i=1}H^{0}(C,\omega_{C}^{\otimes i}),$
$\displaystyle(\mathcal{E},\phi)$
$\displaystyle\mapsto\operatorname{char}(\phi)=(\mathrm{tr}(\phi),\mathrm{tr}(\Lambda^{2}\phi),\ldots,\mathrm{tr}(\Lambda^{i}\phi),\ldots,\det(\phi)),$

which assigns to $(\mathcal{E},\phi)$ the characteristic polynomial
$\operatorname{char}(\phi)$ of the Higgs field $\phi$.

The cohomology of ${M}(n,d)$ has been extensively studied in the literature,
especially under the assumption that $n$ and $d$ are coprime, namely when
${M}(n,d)$ is smooth; see for instance [40, 33, 36, 35, 38, 54, 31, 16, 54,
19]. Recently studies about the intersection cohomology
$I\\!H^{*}({M}(n,d),\mathbb{Q})$ of singular Dolbeault moduli spaces have
started to emerge; see for instance [26, 27, 48, 50, 51, 49, 14, 13, 43,
64].111The lists of works are not intended to be exhaustive at all. We limit
to list a series of selected papers that inspired the current article. From
this viewpoint, the decomposition theorem for the Hitchin map is a key tool to
investigate the intersection cohomology of ${M}(n,d)$: it allows to decompose
$I\\!H^{*}({M}(n,d),\mathbb{Q})$ into building blocks, which are cohomology of
some perverse sheaves on $A_{n}$.

In this article, we establish a correspondence which we find surprising
between the summands of the decomposition theorem for the Hitchin map of
${M}(n,d)$, and the singularities of ${M}(n,0)$; see Theorem 1.1. We call it
the _Hodge-to-singular correspondence_. This correspondence offers a new
perspective on the cohomology of Dolbeault moduli spaces, even in the smooth
case. What is surprising is that this relation is not realised via a direct
geometric correspondence between ${M}(n,d)$ and ${M}(n,0)$, like a deformation
or an alteration, but, as we explain below, by comparing the Hodge theory of
${M}(n,d)$ and the singularity theory of ${M}(n,0)$ on the common base of the
Hitchin fibration.

The Hodge-to-singular correspondence gives a formula to express the dependence
of the intersection cohomology of $M(n,d)$ on the degree $d$. This should be
compared with the degree-independence of spaces of meromorphic Higgs bundle
[48, Thm 0.1, 0.4]. It also extends the main theorem of [17] to arbitrary
degree, and provide a conceptual and elegant explanation for the occurence of
the cographic matroid in [17, §6].

Our correspondence suggests a unifying approach to the study of smooth and
singular Dolbeault moduli spaces that may be useful to tackle the topological
mirror symmetry conjecture [35, 51] and the P=W conjecture [16, 18]
simultaneously in both settings. It also provides new evidence for Davison’s
conjectures in [13, Conj. 5.6, Rem. 5.10].

In a different context, similar ideas have already been successfully employed
to compute the Hodge numbers of the exceptional O’Grady 10 example of compact
hyperkähler manifolds, out of the Hodge numbers of the Hilbert scheme of five
points on a K3 surface; see [22]. Indeed, both spaces can be interpreted as
special moduli spaces of sheaves on K3 surfaces corresponding to different
degrees $d$ (or rather Euler characteristics), and the computation of their
Betti numbers can be reduced to determining how their cohomology depends on
$d$. In fact, the Hodge-to-singular correspondence, formulated here for
Dolbeault moduli spaces, can be stated also for Mukai systems on moduli space
of sheaves on K3 or abelian surfaces; see Section 10.3.

##  1\. Main results

###  1.1. Hodge-to-singular correspondence

In order to state our main results, we need to recall some features of the
singularities of ${M}(n,d)$. A singular point of ${M}(n,d)$ corresponds to a
strictly polystable Higgs bundle

$(E_{1},\phi_{1})^{\oplus m_{1}}\oplus\ldots\oplus(E_{r},\phi_{r})^{\oplus
m_{r}},$

where $(\mathcal{E}_{i},\phi_{i})$ are distinct stable Higgs bundles of slope
$\deg E_{i}/\operatorname{rank}E_{i}=d/n$. In particular, the $r$-uples of
positive integers $\underline{m}=\\{m_{i}\\}$ and
$\underline{n}=\\{n_{i}\\}\coloneqq\\{\deg E_{i}\\}$ satisfy the following
relations

(2) $n=\sum^{r}_{i=1}m_{i}n_{i}\qquad n_{i}\cdot\frac{d}{n}=\deg
E_{i}\in\mathbb{Z}.$

For $\underline{m},\underline{n}\in\mathbb{Z}^{r}_{>0}$ satisfying (2), define
$M^{\circ}_{\underline{m},\underline{n}}(d)$ the locus of polystable Higgs
bundles in ${M}(n,d)$ of multirank $\underline{n}$ and multiplicity
$\underline{m}$, and call $M_{\underline{m},\underline{n}}(d)$ its closure in
${M}(n,d)$. The loci $M^{\circ}_{\underline{m},\underline{n}}(d)$ are the
strata of a complex Whitney stratification of ${M}(n,d)$

(3)
${M}(n,d)=\bigsqcup_{\underline{m},\underline{n}}M^{\circ}_{\underline{m},\underline{n}}(d),$

and an analytic normal slice $W_{\underline{m},\underline{n}}$ through
$M^{\circ}_{\underline{m},\underline{n}}(d)$ is isomorphic to a Nakajima
quiver variety; see [50, Proof of Prop. 2.10]. The Whitney stratification (3)
induces a filtration of the base $A_{n}$ of the Hitchin fibration by closed
subsets
$S_{\underline{m},\underline{n}}\coloneqq\chi(n,d)(M_{\underline{m},\underline{n}}(0))$,
and a stratification

(4) $A_{n}=\bigsqcup S^{\circ}_{\underline{m},\underline{n}}\qquad\text{ with
}S^{\circ}_{\underline{m},\underline{n}}\coloneqq
S_{\underline{m},\underline{n}}\setminus\bigcup_{\dim
S_{\underline{m}^{\prime},\underline{n}^{\prime}}<\dim
S_{\underline{m},\underline{n}}}S_{\underline{m}^{\prime},\underline{n}^{\prime}}.$

Let
$A^{\mathrm{red}}_{n}\coloneqq\bigsqcup_{\underline{m}=(1,\ldots,1)}S^{\circ}_{\underline{m},\underline{n}}\subset
A_{n}$ be the open subset of reduced characteristic polynomials. Define

$M(n,d)^{\mathrm{red}}\coloneqq\chi(n,d)^{-1}(A^{\mathrm{red}}_{n})\text{ and
}M_{\underline{m},\underline{n}}(d)^{\mathrm{red}}\coloneqq
M_{\underline{m},\underline{n}}(d)\cap M(n,d)^{\mathrm{red}}.$

When $\underline{m}=\\{1,\ldots,1\\}$, we abbreviate the notation by omitting
the subscript $\underline{m}$, e.g. $S_{\underline{n}}\coloneqq
S_{\underline{m},\underline{n}}$. Remarkably, $W_{\underline{n}}$ admits
diffeomorphic symplectic resolutions $\widetilde{W}_{\underline{n}}$, whose
top cohomology $H^{\dim
W_{\underline{n}}}(\widetilde{W}_{\underline{n}},\mathbb{Q})$ is a
representation of the fundamental group of $S^{\circ}_{\underline{n}}$; see
4.11 and Theorem 7.4.222In the following, we identify
$\widetilde{W}_{\underline{n}}$ with the hypertoric quiver variety
$Y(\Gamma^{\pm}_{\underline{n}},\theta)$. For convenience, we do not introduce
hypertoric quiver varieties here, and we postpone their definition to Section
4. Denote by $\mathcal{L}_{\underline{n}}$ the corresponding local system on
$S^{\circ}_{\underline{n}}$ (which carries a pure Hodge structure of weight
zero).

The Hodge-to-singular correspondence is a splitting of the mixed Hodge module
$R\chi(n,e)_{*}\mathbb{Q}_{M(n,e)}$ whose direct summands are controlled by
the topology of the singularities $W_{\underline{n}}$.

For an integer $d$ let $\mathcal{P}_{d}$ be the set of partitions
$\underline{n}=\\{n_{i}\\}$ of $n$ such that $n_{i}d/n\in\mathbb{Z}$ for every
$i$. Note that this set is in bijection with the partitions of $q=\gcd(n,d)$:
given a partition $\underline{q}=\\{a_{i}\\}$ of $q$, then
$\underline{q}=\\{a_{i}n/q\\}\in\mathcal{P}_{d}$.

######  Theorem 1.1 (Hodge-to-singular correspondence).

Let $\langle\bullet\rangle=[-2\bullet](-\bullet)$. Let $d$ and $e$ be
integers, with $e$ coprime to $n$. There is an isomorphism in
$D^{b}MHM_{\mathrm{alg}}(A^{\mathrm{red}}_{n})$ or in
$D^{b}(A^{\mathrm{red}}_{n})$ (ignoring the Tate shifts):

(5)
$R\chi(n,e)_{*}\mathbb{Q}_{M(n,e)}\simeq\bigoplus_{\underline{n}\in\mathcal{P}_{d}}R\chi(n,d)_{*}\operatorname{IC}(M_{\underline{n}}(d),\chi(n,d)^{*}\mathcal{L}_{\underline{n}})\langle\operatorname{codim}S_{\underline{n}}\rangle.$

Taking cohomology, we obtain a morphism of mixed Hodge structures

(6)
$H^{k}(M(n,e)^{\mathrm{red}},\mathbb{Q})\simeq\bigoplus_{\underline{n}\in\mathcal{P}_{d}}I\\!H^{k-2\operatorname{codim}S_{\underline{n}}}(M_{\underline{n}}(d)^{\mathrm{red}},\chi(n,d)^{*}\mathcal{L}_{\underline{n}})(-\operatorname{codim}S_{\underline{n}}).$

which respects the perverse filtrations333 We recall the definition of
perverse filtration in (51). carried by these vector spaces.

Note that the summand of the RHS of (5) corresponding to the trivial partition
$\underline{n}=\\{n\\}$ is
$R\chi(n,d)_{*}\operatorname{IC}(M(n,d),\mathbb{Q})$. This means that (6)
measures the difference between $H^{k}(M(n,e)^{\mathrm{red}},\mathbb{Q})$ and
$I\\!H^{k}(M(n,d)^{\mathrm{red}},\mathbb{Q})$.

Singular Dolbeault moduli space for $d=0$ were introduced to study the
topology of character varieties, to which they are homemorphic via the non-
abelian Hodge correspondence. Historically, to avoid to deal with singular
moduli spaces, the moduli problem has been slightly modified or twisted (cf.
the "bait-and-switch" in [36, §1]), and the Dolbeault moduli spaces with $d$
coprime to $n$ have received more attention in the literature. The Hodge-to-
singular correspondence suggests that the historical order, say from smooth to
singular spaces, should be reversed at least conceptually: the intersection
cohomology of singular Dolbeault moduli spaces should be considered as
"primitive" objects into which the cohomology of smooth Dolbeault moduli
spaces decomposes.

In [48, Thm 0.1], Maulik and Shen proved an analogue of Theorem 1.1 for
twisted Higgs bundles, i.e. pairs $(\mathcal{E},\phi)$ where now $\phi$ is a
global section of $\mathcal{E}nd(\mathcal{E})\otimes\mathcal{O}_{C}(D)$ for
some effective divisor $D$ with $\deg D>2g-2$, the so-called "Fano" condition.
Unfortunately, the estimates in [48, §4] are not useful in our setting, namely
$\mathcal{O}_{C}(D)\simeq\omega_{C}$ or "Calabi–Yau" condition; see [48,
§0.4], [10, §11] and 9.6. This accounts for the existence of proper supports
for the complex $R\chi(n,e)_{*}\operatorname{IC}(M(n,e),\mathbb{Q})$ in
$A_{n}$, which on the contrary do not appear under the "Fano" assumption by
[48, Thm. 0.4].

We can suggestively summarise the Hodge-to-singular correspondence in the
following table.

Hodge theory vs singularities of ${M}(n,d)$
---
$\operatorname{gcd}(n,d)=1$ |  $\operatorname{gcd}(n,d)\neq 1,d\neq 0$ | $d=0$
no singularities, more summands of dec. thm |  intermediate behaviour |  more singular strata, no proper summands of dec. thm

###  1.2. Dependence of $I\\!H^{*}({M}(n,d),\mathbb{Q})$ on degree

The Hodge-to-singular correspondence asserts that the intersection cohomology
$I\\!H^{k}(M(n,d)^{\mathrm{red}},\mathbb{Q})$ for arbitrary $d\in\mathbb{Z}$
decomposes into the direct sum of isotypic components of tensor products of
$I\\!H^{k}(M(n^{\prime},0)^{\mathrm{red}},\mathbb{Q})$ for $n^{\prime}\leq n$
under the action of a symmetric group; see 10.1.444An extension of the Hodge-
to-singular correspondence to the whole $A_{n}$ would provide a recursive
formula to compute the intersection cohomology of $M(n,d)$ for arbitrary genus
$g$ and degree $d$. See also Davison’s conjecture [13, Conj. 5.6, Rem. 5.10].
We can refine the result by showing that
$I\\!H^{k}(M(n^{\prime},0)^{\mathrm{red}},\mathbb{Q})$ is determined only by
the smooth locus of the Hitchin fibration. Remarkably, this is in contrast to
the smooth case, namely $d$ coprime to $n$, where
$R\chi(n,d)_{*}\mathbb{Q}_{M(n,d)}$ admits proper supports which cannot be
detected on the smooth locus of Hitchin fibration.

To this end, recall that any $a\in A_{n}$ can be viewed as a monic polynomial
of degree $n$ with coefficient of the degree $n-i$ term in
$H^{0}(C,\omega^{\otimes i}_{C})$, or equivalently as the curve
$C_{a}\subset\operatorname{Tot}(\omega_{C})$ of degree $n$ over $C$ cut by
this monic equation, called _spectral curve_. So an alternative definition of
$S^{\circ}_{\underline{n}}$ is the locus parametrising spectral curves
$C_{a}=\bigcup^{r}_{i=1}C_{i}$ such that the reduced irreducible component
$C_{i}$ has degree $n_{i}$ over $C$. We define also the following spaces:

  1. (i)

$S^{\times}_{\underline{n}}\subseteq S^{\circ}_{\underline{n}}$ is the locus
parametrising reducible nodal curves having smooth irreducible components of
degree $n_{i}$ over $C$;

  2. (ii)

$\pi_{\underline{n}}\colon\mathcal{C}^{\times}_{\underline{n}}\to
S^{\times}_{\underline{n}}$ is the universal spectral curve over
$S^{\times}_{\underline{n}}$;

  3. (iii)

$\mathrm{Pic}^{0}(\mathcal{C}^{\times}_{\underline{n}})\to
S^{\times}_{\underline{n}}$ is the (relative) Jacobian of the universal
spectral curve over $S^{\times}_{\underline{n}}$;

  4. (iv)

$g_{\underline{n}}:A^{\times}_{\underline{n}}\to S^{\times}_{\underline{n}}$
is the maximal abelian proper quotient of
$\mathrm{Pic}^{0}(\mathcal{C}^{\times}_{\underline{n}})$, or equivalently the
Jacobian of the normalization of $\mathcal{C}^{\times}_{\underline{n}}$;

  5. (v)

$\Lambda^{l}_{\underline{n}}\coloneqq
R^{l}(g_{\underline{n}})_{*}\mathbb{Q}_{A^{\times}_{\underline{n}}}=\Lambda^{l}R^{1}(g_{\underline{n}})_{*}\mathbb{Q}_{A^{\times}_{\underline{n}}}=\Lambda^{l}R^{1}(\pi_{\underline{n}})_{*}\mathbb{Q}_{A^{\times}_{\underline{n}}}$;
see [16, Lemma 1.3.5].

######  Definition 1.2.

The _Ngô strings_ 555The definition of Ngô strings differs from [22, Def.
A.0.4] by a shift and a Tate twist. This convention avoids to write the shifts
all the times. for the partition $\underline{n}$ of $n$ is the mixed Hodge
module supported on $S_{\underline{n}}$ given by

(7) $\mathscr{S}_{\underline{n}}\coloneqq\bigoplus^{2\dim
S_{\underline{n}}}_{l=0}\operatorname{IC}(S_{\underline{n}},\Lambda^{l}_{\underline{n}}\otimes\mathcal{L}_{\underline{n}})[-l]\langle\operatorname{codim}S_{\underline{n}}\rangle.$

Note that $\mathscr{S}_{\underline{n}}$ is independent of $d$, since
$\mathcal{L}_{\underline{n}}$ and
$\Lambda^{l}R^{1}(g_{\underline{n}})_{*}\mathbb{Q}_{A^{\times}_{\underline{n}}}$
are so.

In the coprime case $\mathrm{gcd}(e,n)=1$, the summands of the decomposition
theorem of the Hitchin fibration on the locus of reduced spectral curves have
been described in [17]. As a first application of the Hodge-to-singular
correspondence, we recover [17, Prop. 4.1, Thm 6.12].

######  Theorem 1.3 (Ngô strings in degree $e$ coprime to $n$).

Let $e$ be an integer coprime to $n$. There is an isomorphism in
$D^{b}MHM_{\mathrm{alg}}(A^{\mathrm{red}}_{n})$ or in
$D^{b}(A^{\mathrm{red}}_{n})$ (ignoring the Tate shifts):

(8)
$R\chi(n,e)_{*}\mathbb{Q}_{{M}(n,e)}|_{A^{\mathrm{red}}_{n}}\simeq\bigoplus_{\underline{n}\dashv\,n}\mathscr{S}_{\underline{n}}|_{A^{\mathrm{red}}_{n}}.$

Note that the proof of Theorem 1.3 below is not independent of [17]. What is
new is the conceptual interpretation of that result: the rather mysterious
occurrence of the cohomology of the cographic matroid in [17, Thm 6.12] can be
now interpreted in geometric terms as the local intersection cohomology of the
singularities of ${M}(n,0)$, or the top cohomology of their resolutions, in
symbols $H^{\dim
W_{\underline{n}}}(\widetilde{W}_{\underline{n}},\mathbb{Q})$. In the upcoming
paper [52] we provide a combinatorial and alternative proof of Theorems 1.3
and 1.4. In view of [17], where the locus of reduced spectral curves carries a
plethora of proper summands of the decomposition theorem, it could appear
bizarre that instead there is none if $d=0$.

######  Theorem 1.4 (Ngô strings in degree 0).

The complex $R\chi(n,0)_{*}\operatorname{IC}({M}(n,0),\mathbb{Q})$ has full
support on the reduced locus $A^{\mathrm{red}}_{n}$, i.e.

$R\chi(n,0)_{*}\operatorname{IC}({M}(n,0),\mathbb{Q})|_{A^{\mathrm{red}}_{n}}\simeq\mathscr{S}_{\\{n\\}}|_{A^{\mathrm{red}}_{n}}$

######  Question 1.5 (Full support).

Does $R\chi(n,0)_{*}\operatorname{IC}({M}(n,0),\mathbb{Q})$ have full support
on the whole Hitchin base $A_{n}$, i.e.

$R\chi(n,0)_{*}\operatorname{IC}({M}(n,0),\mathbb{Q})\simeq\mathscr{S}_{\\{n\\}}?$

As positive partial answer to 1.5, we observe that no summand of
$R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})$ is supported on
$\\{0\\}$, which generalises a result of Heinloth [38, Thm 1] to arbitrary
degree.

######  Proposition 1.6 (Vanishing of intersection form).

The intersection form on $I\\!H^{*}(M(n,d),\mathbb{Q})$ is trivial.
Equivalently the forgetful map $I\\!H_{c}^{*}(M(n,d),\mathbb{Q})\to
I\\!H^{*}(M(n,d),\mathbb{Q})$ is 0.

In Theorem 1.7 we answer positively 1.5 in rank two and illustrate its far-
reaching consequences. Note that the intersection cohomology groups of
$M(2,d)$ have been computed in [40, Thm 7.6] and [50, Thm 1.8, 1.9].

######  Theorem 1.7 (Ngô strings in rank two).

There exist isomorphisms

$\displaystyle
R\chi(2,1)_{*}\mathbb{Q}_{{M}(2,1)}\simeq\mathscr{S}_{\\{2\\}}\oplus\mathscr{S}_{\\{1,1\\}},\qquad
R\chi(2,0)_{*}\operatorname{IC}({{M}(2,0)},\mathbb{Q})\simeq\mathscr{S}_{\\{2\\}},$

in the bounded derived category $D^{b}\mathrm{MMH}(A_{n})$ of Hodge modules on
$A_{n}$. As a result, there exists a surjective map

$H^{*}({M}(2,1),\mathbb{Q})\twoheadrightarrow IH^{*}({M}(2,0),\mathbb{Q})$

strictly filtered by the perverse filtrations associated to $\chi(2,d)$.

######  Remark 1.8 (BPS sheaf).

The Hodge-to-singular correspondence should give an explicit formula (Theorem
7.5) for the so-called BPS sheaf [43]. From this viewpoint, Theorem 1.7 should
be compared with [43, Ex. 5.4] and [64]. See also [13, Rem. 5.10].

The Hodge-to-singular correspondence provides also a recursive strategy to
express the dependence of $I\\!H^{*}({M}(n,d),\mathbb{Q})$ on the degree $d$,
when $\gcd(n,d)\neq 0$ or $\neq 1$. We outline it in Section 10.1. The
explicit computation of the rank of the Ngô strings for
$R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})|_{A^{\mathrm{red}}_{n}}$
will appear in [52].

As a consequence of the Hodge-to-singular correspondence, we obtain an
immediate generalisation of [23, Thm. 3.11] and [43, Cor. 5.15, Ex. 5.18] to
arbitrary degree on the locus of reduced spectral curves.

######  Corollary 1.9.

Let $d$ $d^{\prime}$ be integers with
$\mathrm{gcd}(d,n)=\mathrm{gcd}(d^{\prime},n)$. Then we have

$R\chi(n,d)_{*}\operatorname{IC}(M(n,d),\mathbb{Q})|_{A^{\mathrm{red}}_{n}}\simeq
R\chi(n,d^{\prime})_{*}\operatorname{IC}(M(n,d^{\prime}),\mathbb{Q})|_{A^{\mathrm{red}}_{n}}.$

###  1.3. Some applications of the Hodge-to-singular correspondence

####  1.3.1. Stabilisation

In [12] Coskun and Woolf have proposed a conjecture about the stabilization of
the cohomology of moduli space of sheaves on compact complex surface. As a
final remark, we observe that $I\\!H^{*}(M(n,d),\mathbb{Q})$ does stabilise as
the rank $n$ grows, independently of the degree $d$.

######  Proposition 1.10 (Stabilisation).

The intersection Betti and Hodge numbers of $M(n,d)$ stabilize to the stable
intersection Betti and Hodge numbers of $M(n,0)$ as $n$ tends to infinity,
i.e. the limit

(9) $\lim_{n\to\infty}\dim I\\!H^{*}(M(n,d))^{p,q}$

is finite and independent of $d$.

Analogous stabilisation results for the Mukai systems on moduli space of
sheaves on K3 surfaces are discussed in Section 10.3.

####  1.3.2. Restriction to smooth fibres

Let $M$ be a projective irreducible holomorphic symplectic variety of
dimension $2r$ equipped with a Lagrangian fibration $f\colon M\to B$. In [28]
Felisetti, Shen and Yin have proved that the restriction of
$H^{*}(M,\mathbb{Q})$ to a smooth fiber of $f$ is isomorphic to
$H^{*}(\mathbb{P}^{r},\mathbb{Q})$. In general, this is false for non-compact
hyperkähler varieties, but it holds true for the Hitchin fibration for $n>1$
by [1, Thm 5], up to a copy of $H^{\bullet}(C,\mathbb{Q})$. Here we propose an
alternative proof.

######  Proposition 1.11.

The restriction of $I\\!H^{*}(M(n,d))$ to any smooth fiber $M_{a}\subset
M(n,d)$ of the Hitchin fibration $\chi(n,d)$ is given by

$\mathrm{Im}\\{I\\!H^{*}(M(n,d),\mathbb{Q})\to
H^{*}(M_{a},\mathbb{Q})\\}=H^{\bullet}(C,\mathbb{Q})\otimes{\mathbb{Q}[\alpha|_{M_{a}}]}/{(\alpha|_{M_{a}}^{\dim
M(n,d)+1})}$

where $\alpha$ is a $\chi(n,d)$-relative ample class on $M(n,d)$.

Let $\check{M}(n,d)$ be a fibre of the determinant map $M(n,d)\to M(1,nd)$,
given by $(\mathcal{E},\phi)\mapsto(\det\mathcal{E},\operatorname{tr}\phi)$.
The restriction of $I\\!H^{*}(\check{M}(n,d))$ to any smooth fiber
$\check{M}_{a}\subset\check{M}(n,d)$ of the Hitchin fibration
$\check{\chi}(n,d)$ is given by

$\mathrm{Im}\\{I\\!H^{k}(\check{M}(n,d),\mathbb{Q})\to
H^{k}(\check{M}_{a},\mathbb{Q})\\}=\begin{cases}\langle\alpha^{k/2}|_{\check{M}_{a}}\rangle&\text{
if }k\text{ is even},\\\ 0&\text{ if }k\text{ is odd},\\\ \end{cases}$

###  1.4. Strategy

In this paper we propose a geometric proof of the Hodge-to-singular
correspondence. Here we outline the main steps. It is well-known that the
Hitchin fibration is a family of compactified Jacobians of spectral curves;
see [39, 2, 63]. In Section 5, we enlarge the Hitchin fibration
$\chi(n,d)\colon M(n,d)\to A_{n}$ to a family $\pi\colon\overline{J}_{d,B}\to
B$ of compactified Jacobians over a versal deformation of spectral
curves:666Actually, to make the diagram work we should slice the Hitchin base,
restrict to the locus of nodal spectral curves, and take into consideration
the automorphisms of the curves by working over $\overline{\mathcal{M}}_{g}$.
For expository reasons, we allow this abuse here, and postpone the precise
details to Section 5.

${{M}(n,d)^{\times}}$${\overline{J}_{d,B}}$${A_{n}}$${B,}$$\scriptstyle{\chi(n,d)}$$\scriptstyle{\pi}$

Note that the relative dimension of $\pi$ is strictly smaller than $\dim B$.
This is a crucial ingredient to show that
$R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q})$ is independent of
$d$, while $R\chi(n,d)_{*}\operatorname{IC}(M(n,d),\mathbb{Q})$ does not; see
Theorem 9.5 and 9.6. In particular, we have

(10) $R\pi_{*}\operatorname{IC}(\overline{J}_{e,B},\mathbb{Q})|_{A_{n}}\simeq
R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q})|_{A_{n}}.$

If $\mathrm{gcd}(n,e)=1$, the inclusion
${M}(n,e)\hookrightarrow\overline{J}_{e,B}$ is a regular embedding of smooth
varieties, so

(11)
$\operatorname{IC}(\overline{J}_{e,B},\mathbb{Q})|_{M(n,e)}\simeq\mathbb{Q}_{\overline{J}_{e,B}}|_{M(n,e)}\simeq\mathbb{Q}_{M(n,e)},$

and the LHS of (10) is isomorphic to $R\chi(n,e)_{*}\mathbb{Q}_{M(n,e)}$. If
instead $\mathrm{gcd}(n,d)\neq 1$, the failure of (11) is measured by the
topology of the singularities of $M(n,d)$. Following [8], we identify the map
$M(n,d)\hookrightarrow\overline{J}_{d,B}$ with the inclusion of the hypertoric
quiver variety $Y(\Gamma_{\underline{n}},0)$ into the toric Lawrence variety
$X(\Gamma^{\pm}_{\underline{n}},0)$

${Y(\Gamma_{\underline{n}},0)}$${X(\Gamma_{\underline{n}}^{\pm},0)}$${\mathbb{C}^{b_{1}(\Gamma_{\underline{n}})}}$${\mathbb{C}^{s};}$$\scriptstyle{\iota_{M}}$$\scriptstyle{\chi_{\Gamma_{\underline{n}}}}$$\scriptstyle{\pi_{\Gamma_{\underline{n}}}}$$\scriptstyle{\iota_{A}}$

see Theorem 7.4 for the proof, and Section 4 for details about the notation.
Now a local version of the Hodge-to-singular correspondence for hypertoric
quiver varieties (Theorem 4.18) shapes the RHS of (10), and gives the main
result Theorem 1.1; see also Theorem 7.5. The key geometric input is the
following: $X(\Gamma_{\underline{n}}^{\pm},0)$ admits small resolutions of
singularities that restrict to semismall resolutions of
$Y(\Gamma_{\underline{n}},0)$; see Section 4.5.

###  1.5. Outline

  * •

In Section 3 we prove a version of Ngô support theorem for weakly abelian
Lagrangian fibrations on singular symplectic varieties; see Theorem 3.6. We
specialise the support theorem to the Hitchin fibration in Theorem 3.11.

  * •

In Section 4 we discuss toric Lawrence varieties and hypertoric quiver
varieties. The main result is the local Hodge-to-singular correspondence, i.e.
Theorem 4.18.

  * •

In Section 5 we explain how (and in which sense) we can embed the Dolbeault
moduli space into the universal compactified Jacobian.

  * •

In Section 7 we describe local models for this embedding; see Theorem 7.4. To
this purpose we need some auxiliary results: an explicit description of the
image of the Kodaira–Spencer map for nodal spectral curves provided in Section
6; the topological trivialization of a tubular neighbourhood of an abelian
variety in the universal compactified Jacobian in Section 8.

  * •

In Section 9 we show the degree independence of the intersection cohomology of
the universal compactified Jacobian; see Theorem 9.5.

  * •

In Section 10 we collect the proof of the main theorems stated in Section 1.

###  1.6. Notation

The intersection cohomology of a complex variety $X$ with middle perversity
and rational coefficients is denoted by $I\\!H^{*}(X,\mathbb{Q})\coloneqq
H^{*}(\operatorname{IC}(X,\mathbb{Q}))$, where
$\operatorname{IC}(X,\mathbb{Q})$ is the perverse intersection cohomology
complex of $X$ shifted by $-\dim X$. Ordinary singular cohomology with
rational coefficients is denoted by $H^{*}(X,\mathbb{Q})$. Recall that they
all carry mixed Hodge structures. Intersection complexes and perverse sheaves
on an algebraic stack are descents of perverse sheaves from a smooth atlas;
see [41, 45, 44]. The formalism of six operations works in this context too.
We use it in Section 7.

###  1.7. Acknowledgements

We would like to thank Alastair Craw, Mark de Cataldo, Jochen Heinloth, Daniel
Huybrechts, Davesh Maulik, Jesse Leo Kass, Tasuki Kinjo, Nicola Pagani,
Roberto Pagaria, Alex Perry, Giulia Saccà, Junliang Shen, Filippo Viviani for
numerous discussions, emails and helpful advice.

The first author was supported by the University of Michigan and the Max
Planck Institute for Mathematics. The second author was supported by PRIN
Project 2017 "Moduli and Lie theory".

##  2\. Stratifications

Recall that a _stratification_ of a complex variety $X$ is a finite collection
of locally closed smooth subvarieties $X_{i}\subseteq X$, called _strata_ ,
such that

  * •

$X$ is the disjoint union of $X_{i}$ with $i=1,\ldots,r$, i.e.
$X=\bigsqcup^{r}_{i=1}X_{i}$;

  * •

$X_{i}$ is contained in the closure of $X_{i+1}$.

A stratification is called a _Whitney stratification_ if all pairs of strata
satisfy the Whitney conditions A and B. We omit the precise definition of
these conditions (see for instance [30]), since in the following we will only
use the stronger property 2.1, that implies Whitney conditions A and B by
Lemma 2.2.

######  Definition 2.1.

A stratification $X=\bigsqcup_{i}X_{i}$ is analytically trivial in the normal
direction to each stratum, if for any $x\in X_{i}$ there exists a normal slice
$N_{x}$ through $X_{i}$ at $x$, and a neighbourhood of $x$ in $X$ which is
locally analytically isomorphic to $N_{x}\times T_{x}X_{i}$ at $(x,0)$.

######  Lemma 2.2.

A stratification analytically trivial in the normal direction to each stratum
is a Whitney stratification.

###### Proof.

We follow the argument in [53, §4.7]. Whitney conditions A and B are local, so
it is enough to check it for $N_{x}\times T_{x}X_{i}$ at $(x,0)$. Since
$T_{x}X_{i}$ is a smooth factor, it is enough to check it for $N_{x}$ at $x$.
Then this follows from [68, Lem. 19.3]. ∎

##  3\. Ngô theorem for Lagrangian fibrations on singular spaces

The goal of this section is to generalise the description of Ngô strings in
[22, Thm 7.0.3] to the case of weakly abelian Lagrangian fibrations on
singular symplectic varieties.

We will follow closely Maulik and Shen [48]. Let $B$ be a quasi-projective
complex variety. Let $g\colon P\to B$ be a smooth $B$-group scheme with
connected fibers, and let $f\colon X\to B$ be a projective morphism. Assume
that the group scheme $P$ acts on X via

(12) $\mathrm{act}\colon P\times_{B}X\to X.$

######  Definition 3.1 (Weak abelian fibration).

The triple $(X,P,B)$ is a weak abelian fibration of relative dimension $c$, if

  1. (1)

every fiber of the map $g$ is pure of dimension $c$, and $X$ has pure
dimension

$\dim X=c+\dim B,$

  2. (2)

the action (12) of $P$ on $X$ has affine stabilizers, and

  3. (3)

the Tate module $T_{\overline{\mathbb{Q}}_{l}}(P)$ associated with the group
scheme P is polarizable.

For a closed point $b\in B$, we define $\delta(b)$ as the dimension of the
maximal affine and connected subgroup of $P_{b}$. For any closed subvariety
$Z\subseteq B$, we denote by $\delta_{Z}$ the minimum value of the (upper
semi-continuous) function $\delta$ on $Z$.

######  Theorem 3.2.

Let $(X,P,B)$ be a weak abelian fibration of relative dimension $c$. Assume
that

  1. (1)

_( $\delta$-regularity)_ $\operatorname{codim}Z_{\delta}\geq\delta$ for any
$Z_{\delta}\coloneqq\\{b\in B\,|\,\delta(b)=\delta\\}$;

  2. (2)

$\tau_{>2c}(Rf_{*}\operatorname{IC}(X,\mathbb{Q}))=0$ for the standard
truncation functor $\tau_{>*}(-)$.

Then every support $Z$ of $Rf_{*}\operatorname{IC}(X,\mathbb{Q})$ satisfies

$\operatorname{codim}Z=\delta(Z).$

###### Proof.

[48, Theorem 1.8] says that if $f\colon X\to B$ is part of the datum of a weak
abelian fibration with the property (2), then any support $Z$ of
$Rf_{*}\operatorname{IC}(X,\mathbb{Q})$ satisfies
$\operatorname{codim}Z\leq\delta(Z)$. Together with the $\delta$-regularity,
we obtain that $\operatorname{codim}Z=\delta_{Z}$. ∎

######  Proposition 3.3 (Relative Dimension Bound).

Let $f:X\to B$ be an equidimensional morphism of complex algebraic varieties
of relative dimension $c$. Suppose that there exists a Whitney stratification
$X=\bigsqcup X_{i}$ such that for any $b\in B$ we have

(13) $\operatorname{codim}_{X}X_{i}\geq
2\operatorname{codim}_{f^{-1}(b)}(X_{i}\cap f^{-1}(b)).$

Then we have

  * •

$\tau_{>2c}(Rf_{*}\operatorname{IC}(X,\mathbb{Q}))=0$;

  * •

$R^{2c}f_{*}\operatorname{IC}(X,\mathbb{Q})=R^{2c}f_{*}\mathbb{Q}_{X}$.

###### Proof.

Via proper base change we have

$\mathcal{H}^{*}(Rf_{*}\operatorname{IC}(X,\mathbb{Q}))_{b}=H^{*}(f^{-1}(b),\operatorname{IC}(X,\mathbb{Q})|_{f^{-1}(b)})\text{
for any }b\in B.$

These cohomology groups are the limit of the Grothendieck spectral sequence

$E^{p,q}_{2}=H^{p-q}(f^{-1}(b),\mathcal{H}^{q}(\operatorname{IC}(X,\mathbb{Q}))|_{f^{-1}(b)})\Rightarrow
H^{p}(f^{-1}(b),\operatorname{IC}(X,\mathbb{Q})|_{f^{-1}(b)}).$

The closed subset $Z_{q}\subset X$ are the support of the constructible sheaf
$\mathcal{H}^{q}(\operatorname{IC}(X,\mathbb{Q}))$. In particular, we have

$E^{p,q}_{2}=0\text{ for }p-q>2\dim(Z_{q}\cap f^{-1}(b)).$

The irreducible components of $Z_{q}$ are the closure of some $X_{i}$ with
$q<\operatorname{codim}_{X}X_{i}$, or $X$ itself, if $q=0$. Indeed, the strong
support condition for intersection cohomology implies that
$\mathcal{H}^{q}(\operatorname{IC}(X,\mathbb{Q}))_{x}=0$ for any $x\in X_{i}$
and $q\geq\max\\{\operatorname{codim}_{X}X_{i},1\\}$; see [20, §2.3]. By (13),
we conclude that $E^{p,q}_{2}=0$ for

$p>q+2\dim(Z_{q}\cap
f^{-1}(b))\geq\max\\{\operatorname{codim}_{X}X_{i},1\\}+2\dim(Z_{q}\cap
f^{-1}(b)),$

i.e. for $p>2c$ and arbitrary $q$, or for $p=2c$ and $q\neq 0$. This yields

$\mathcal{H}^{>2c}(Rf_{*}\operatorname{IC}(X,\mathbb{Q}))=0,\qquad\mathcal{H}^{2c}(Rf_{*}\operatorname{IC}(X,\mathbb{Q}))=\mathcal{H}^{2c}(Rf_{*}\mathbb{Q}_{X}).$

∎

######  Proposition 3.4.

Let $X$ be a normal complex variety endowed with a holomorphic symplectic form
and a proper Lagrangian fibration $f:X\to B$. Then $X$ admits a complex
Whitney stratification

(14) $X=\bigsqcup X_{i}$

by locally closed algebraic symplectic submanifolds $X_{i}$ such that the
restriction map $f|_{X_{i}}:X_{i}\to B_{i}\coloneqq f(X_{i})$ is Lagrangian.
In particular, for any $b\in B$ we have

(15)
$\operatorname{codim}_{X}X_{i}=2\operatorname{codim}_{f^{-1}(b)}(X_{i}\cap
f^{-1}(b)).$

###### Proof.

By [42, Thm 2.3, Prop. 3.1], $X$ admits a stratification by locally closed
subsets $X_{i}$, with $i=1,\ldots,r$, such that:

  * •

$X_{i}$ are smooth and symplectic;

  * •

$X_{i+1}$ is the singular locus of $X_{i}$;

  * •

the stratification is analytically trivial in the normal direction to each
stratum, so it is a Whitney stratification by Lemma 2.2.

Applying [47, Thm 3.1] iteratively, the restrictions $f|_{X_{i}}$ are
Lagrangian.

Since both $f$ and $f|_{X_{i}}$ are Lagrangian, we obtain $\dim X-\dim
X_{i}=2\dim f^{-1}(b)-2\dim(X_{i}\cap f^{-1}(b)).$ ∎

######  Example 3.5.

The Whitney stratification (14) for the Dolbeault moduli space $X=M(n,d)$ is
the stratification $\eqref{eq:Whitneystra}$. In particular, 3.3 gives

$\tau_{>\dim
M(n,d)}(R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q}))=0,\qquad R^{\dim
M(n,d)}f_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})=R^{\dim
M(n,d)}f_{*}\mathbb{Q}_{{M}(n,d)}.$

Now let $(X,P,B)$ be a $\delta$-regular weak abelian fibration. We denote by
$I$ the set of supports $Z_{\alpha}\subseteq B$ for
$Rf_{*}\operatorname{IC}(X,\mathbb{Q})$. There exist open dense subsets
$Z^{\times}_{\alpha}$ such that the restriction $P_{\alpha}$ of the group
scheme $P$ admits a Chevalley decomposition

$0\to R_{\alpha}\to P_{\alpha}\to A_{\alpha}\to 0,$

where $R_{\alpha}\to Z^{\times}_{\alpha}$ is an affine group scheme and
$g_{\alpha}\colon A_{\alpha}\to Z^{\times}_{\alpha}$ is a proper abelian group
scheme. Let $\Lambda^{l}_{\alpha}\coloneqq
R^{l}g_{\alpha*}\mathbb{Q}_{A_{\alpha}}$.

######  Theorem 3.6 (Ngô strings from singular simplectic varieties).

Let $(X,P,B)$ be a $\delta$-regular weak abelian fibration on a normal
holomorphic symplectic variety $X$ of dimension $2c$. Then there exists an
isomorphism in $D^{b}MHM_{\text{alg}}(B)$ (resp. $D^{b}(B,\mathbb{Q})$)

(16) $Rf_{*}\operatorname{IC}(X,\mathbb{Q})\simeq\bigoplus_{\alpha\in
I}\bigoplus^{2\dim
Z_{\alpha}}_{l=0}\operatorname{IC}(Z_{\alpha},\Lambda^{l}_{\alpha}\otimes\mathscr{L}_{\alpha})[-l]\langle\operatorname{codim}(Z_{\alpha})\rangle,$

where $\mathscr{L}_{\alpha}$ are local systems on an open subset of
$Z_{\alpha}$. Moreover, we have:

  1. (i)

$\operatorname{codim}Z_{\alpha}=\delta(Z_{\alpha})$;

  2. (ii)

$R^{2c}f_{*}\mathbb{Q}_{X}$ admits the following direct summand
$\bigoplus_{\alpha\in I}i_{\alpha*}\mathscr{L}_{\alpha},$ where
$i_{\alpha}\colon Z^{\times}_{\alpha}\to B$ is the natural immersion.

###### Proof.

The constraint (i) is a combination of Theorem 3.2, Propositions 3.3 and 3.4.
In [48, Prop. 1.5] Maulik and Shen provided a generalization of Ngô freeness
[58, Prop. 7.4.10] to the singular context. This gives the special form (16)
to the decomposition theorem for $Rf_{*}\operatorname{IC}(X,\mathbb{Q})$.
Taking cohomology of (16), together with the isomorphism
$R^{2c}f_{*}\operatorname{IC}(X,\mathbb{Q})=R^{2c}f_{*}\mathbb{Q}_{X}$ in 3.3,
we obtain (ii). ∎

###  3.1. Ngô strings for Dolbeault moduli spaces

We specialize Theorem 3.6 to the Hitchin fibration $\chi(n,d)\colon M(n,d)\to
A_{n}$. We rely on the recent key result by Kinjo and Koseki [43], building on
the work of Davison [14].

######  Proposition 3.7.

Let $d$ and $e$ be integers, with $e$ coprime to $n$. The complex
$R\chi(n,d)_{*}\operatorname{IC}(M(n,d),\mathbb{Q})$ is a direct summand of
$R\chi(n,e)_{*}\mathbb{Q}_{M(n,e)}$.

###### Proof.

[14, Thm 6.6] and [43, Cor. 5.15, Ex. 5.18]. ∎

3.7 allows to extends [17, Thm 4.1] and [38, Thm 1] to Dolbeault moduli spaces
of arbitrary degree.

######  Corollary 3.8.

If $Z$ is a support of $R\chi(n,d)_{*}\operatorname{IC}(M(n,d),\mathbb{Q})$
with $Z\cap A^{\mathrm{red}}_{n}\neq\emptyset$, then $Z=S_{\underline{n}}$ for
some partition $\underline{n}$ of $n$. In particular, the generic point of any
support $Z$ of $R\chi(n,d)_{*}\operatorname{IC}(M(n,d),\mathbb{Q})$ with
$Z\cap A^{\mathrm{red}}_{n}\neq\emptyset$ lies in the locus $A^{\times}_{n}$
of nodal spectral curves.

###### Proof.

[17, Thm 4.1] and 3.7. ∎

######  Corollary 3.9 (1.6).

The intersection form on $I\\!H^{*}(M(n,d),\mathbb{Q})$ is trivial.
Equivalently the forgetful map $I\\!H_{c}^{*}(M(n,d),\mathbb{Q})\to
I\\!H^{*}(M(n,d),\mathbb{Q})$ is 0.

###### Proof.

By [37, Prop. 4] the vanishing of the intersection form is equivalent to show
that $\\{0\\}\subset A_{n}$ is not a support of
$R\chi(n,d)_{*}\operatorname{IC}(M(n,d),\mathbb{Q})$. In the coprime case,
this is proved by Heinloth in [38, Thm 1]. Together with 3.7, we obtain the
result in the non-coprime case. ∎

######  Remark 3.10.

3.9 holds for moduli space of $\mathrm{PGL}_{n}$\- and $\mathrm{SL}_{n}$-Higgs
bundles too.

######  Theorem 3.11 (Ngô strings from Dolbeault moduli spaces in arbitrary
degree).

There exists an isomorphism in $D^{b}MHM_{\text{alg}}(A^{\text{red}}_{n})$
(resp. $D^{b}(A^{\text{red}}_{n},\mathbb{Q})$)

(17)
$R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})|_{A^{\text{red}}_{n}}\simeq\bigoplus_{\underline{n}\in
I_{d}}\mathscr{S}(\mathscr{L}_{\underline{n}}(d))|_{A^{\text{red}}_{n}},$

where $I_{d}$ is a set of partition of $\underline{n}$ of $n$, and the Ngô
string $\mathscr{S}(\mathscr{L}_{\underline{n}}(d))$ is the mixed Hodge module
supported on $S_{\underline{n}}$ given by

$\mathscr{S}(\mathscr{L}_{\underline{n}}(d))\coloneqq\bigoplus^{2\dim
S_{\underline{n}}}_{l=0}\operatorname{IC}(S_{\underline{n}},\Lambda^{l}_{\underline{n}}\otimes\mathscr{L}_{\underline{n}}(d))[-l]\langle\operatorname{codim}S_{\underline{n}}\rangle,$

for certain local systems $\mathscr{L}_{\underline{n}}(d)$ supported on an
open set of $S_{\underline{n}}$.

###### Proof.

It follows from Theorem 3.6 and 3.8. ∎

The rest of the paper is devoted to the correspondence between the local
systems $\mathscr{L}_{\underline{n}}(d)$ and the singularities of
$M(n,d)^{\mathrm{red}}$.

##  4\. Hypertoric quiver varieties

The singularities of $M(n,d)^{\mathrm{red}}$ are modelled on hypertoric quiver
varieties. In this section we introduce these varieties and establish a local
version of the Hodge-to-singular correspondence (Theorem 4.18).

###  4.1. Affine toric varieties associated to a quiver

Let $Q=(V,E)$ be a connected directed graph (a _quiver_) with $r$ vertices
$V=\\{v_{1},...,v_{r}\\}$ and $s$ oriented edges $E$. Given $e\in E$, let
$s(e),t(e)\in V$ be the source and target of $e$, so that $e:s(e)\to t(e)$.
The underlying unoriented graph, denoted by $|Q|$, has first Betti number
$b_{1}\coloneqq s-r+1$.

We consider the group of all $\mathbb{Z}$-linear combinations of $V$ whose
coefficients sum to zero. We identify this group with $\mathbb{Z}^{r-1}$, by
fixing the basis $\\{v_{1}-v_{2},\ldots,v_{1}-v_{r}\\}$. We also identify
$\mathbb{Z}^{s}$ with the group of $\mathbb{Z}$-linear combinations of $E$.
The boundary map of the quiver $Q$

$A:\mathbb{Z}^{s}\to\mathbb{Z}^{r-1},\quad e\mapsto s(e)-t(e)$

induces an embedding of algebraic tori

$\mathbb{T}^{r-1}\coloneqq\operatorname{Spec}\mathbb{C}[\mathbb{Z}^{r-1}]\hookrightarrow\mathbb{T}^{s}\coloneqq\operatorname{Spec}\mathbb{C}[\mathbb{Z}^{s}].$

The torus $\mathbb{T}^{s}$ acts on the affine complex space
$\mathbb{C}^{s}=\operatorname{Spec}\mathbb{C}[z_{e}\colon e\in E]$, and so
does $\mathbb{T}^{r-1}$ by restriction. Explicitly,
$\lambda=(\lambda_{i})^{r}_{i=1}\in\mathbb{T}^{r}/\mathbb{T}=\mathbb{T}^{r-1}$
acts as $\lambda\cdot z_{e}=\lambda_{s(e)}z_{e}\lambda^{-1}_{t(e)}$.

######  Definition 4.1.

The affine toric variety $X(Q,0)$ is the affine GIT quotient of
$\mathbb{C}^{s}$ by $\mathbb{T}^{r-1}$

$X(Q,0)\coloneqq\mathbb{C}^{s}\sslash_{0}\mathbb{T}^{r-1}=\operatorname{Spec}\mathbb{C}[z_{e}:e\in
E]^{\mathbb{T}^{r-1}}.$

The cone of the toric variety $X(Q,0)$ is generated by the rows
$\mathscr{B}=\\{\beta_{1},\ldots,\beta_{s}\\}\subset\mathbb{Z}^{b_{1}}$ of a
_Gale dual_ of $A$, i.e. a $(s\times b_{1})$-matrix $B$ whose columns form a
base of
$\ker(A\colon\mathbb{Z}^{s}\to\mathbb{Z}^{r-1})\simeq\mathbb{Z}^{b_{1}}$, or
equivalently a matrix that sits in the following exact sequence:

$0\to\mathbb{Z}^{b_{1}}\xrightarrow{B}\mathbb{Z}^{s}\xrightarrow{A}\mathbb{Z}^{r-1}\to
0.$

Recall that a loop in a quiver is an edge whose head and tail coincide.

######  Lemma 4.2.

Let $Q$ be a quiver with $m$ loops, and $Q^{\prime}$ be the quiver obtained
from $Q$ by deleting all the loops. Then we have

$X(Q,0)\simeq X(Q^{\prime},0)\times\mathbb{C}^{m}.$

###### Proof.

Let $L\subset E$ be the loops of $Q$. We identify the edges $E^{\prime}$ of
$Q^{\prime}$ with $E\setminus L$. For any $l\in L$, the coordinates $z_{l}$
are $\mathbb{T}^{r-1}$-invariant by construction, so we have

$\mathbb{C}[z_{e}\colon e\in E]^{\mathbb{T}^{r-1}}\simeq\mathbb{C}[z_{e}\colon
e\in E^{\prime}]^{\mathbb{T}^{r-1}}\otimes\mathbb{C}[z_{l}\colon l\in L].$

∎

###  4.2. Semiprojective toric varieties associated to a quiver

In general, $X(Q,0)$ is a singular variety. A toric resolution of
singularities can be obtained via projective GIT as follows.

The ring of polynomials $S\coloneqq\mathbb{C}[z_{e}\colon e\in E]$ is graded
by $\mathbb{Z}^{r-1}$ in such a way that $\deg(z_{e})=Ae$. For
$\theta\in\mathbb{Z}^{r-1}$, let $S_{\theta}$ be the $\mathbb{C}$-vector space
of homogenous polynomials of degree $\theta$. For instance, $S_{0}$ is the
coordinate ring of $X(Q,0)$.

######  Definition 4.3.

The semiprojective toric variety $X(Q,\theta)$ is the projective GIT quotient
of $\mathbb{C}^{2s}$ by the torus $\mathbb{T}^{r-1}$ with respect to the
stability $\theta\in\mathbb{Z}^{r-1}$, i.e.

$X(Q,\theta)\coloneqq\mathbb{C}^{s}\sslash_{\theta}\mathbb{T}^{r-1}\coloneqq\operatorname{Proj}\big{(}\bigoplus^{\infty}_{k=0}t^{k}S_{k\theta}\big{)}.$

Consider now the coarsest complete fan $\Gamma(\mathscr{A})$ in
$\mathbb{R}^{r-1}\simeq\mathbb{Z}^{r-1}\otimes\mathbb{R}$ that refines all
complete simplicial fans in $\mathbb{R}^{r-1}$ whose rays lie in
$\mathscr{A}\coloneqq\\{Ae\colon e\in E\\}$. We say that
$\theta\in\mathbb{Z}^{r-1}$ is generic if it lies in the interior of the
maximal cones of $\Gamma(\mathscr{A})$.

######  Proposition 4.4.

_[34, Prop. 8.2]_ For generic $\theta\in\mathbb{Z}^{r-1}$, the affinization
morphism

$\pi_{X}=\pi_{X}(Q,\theta)\colon
X(Q,\theta)\to\operatorname{Spec}H^{0}(X(Q,\theta),\mathcal{O}_{X(Q,\theta)})=X(Q,0)$

is a resolution of singularities.

###  4.3. Lawrence varieties and hypertoric quiver varieties

We specialize the previous constructions to doubled quivers. Let $Q$ be a
quiver with $r$ vertices, $s$ oriented edges $E$, and first Betti number
$b_{1}$.

######  Definition 4.5.

The _double_ of $Q$, denoted by $Q^{\pm}$, is the quiver obtained from $Q$ by
replacing each edge $e$ of $Q$ with a pair of edges $e^{+}$ and $e^{-}$ having
the same endpoints as $e$ but opposite orientations.

######  Definition 4.6.

The _affine Lawrence toric variety_ associated to $Q$ is the affine GIT
quotient of $\mathbb{C}^{2s}$ by the torus $\mathbb{T}^{r-1}$

$X(Q^{\pm},0)\coloneqq\mathbb{C}^{2s}\sslash_{0}\mathbb{T}^{r-1}.$

The _Lawrence toric variety_ associated to $Q$ is the projective GIT quotient
of $\mathbb{C}^{2s}$ by the torus $\mathbb{T}^{r-1}$ with respect to a
stability $\theta\in\mathbb{Z}^{r-1}$, i.e.

$X(Q^{\pm},\theta)\coloneqq\mathbb{C}^{2s}\sslash_{\theta}\mathbb{T}^{r-1}\coloneqq\operatorname{Proj}\big{(}\bigoplus^{\infty}_{k=0}t^{k}S_{k\theta}\big{)}.$

######  Notation 4.7.

For convenience, we relabel the coordinates $z_{e^{+}}$ and $z_{e-}$ of
$\mathbb{C}^{2s}$ by $z_{e}$ and $w_{e}$, indexed by $e\in E$. Note that the
definition of $X(Q^{\pm},\theta)$ is independent of the choice of an
orientation of $Q$: a different choice corresponds to exchange some
coordinates $z_{e}$ with the corresponding $w_{e}$.

Suppose that the matrix $A=(a_{ie})$ corresponds to the boundary map for $Q$.
We consider the homogeneous ideal

(18) $\mathrm{Circ}(\mathscr{B})\coloneqq\langle\sum_{e\in
E}a_{ie}z_{e}w_{e}\colon i=2,\ldots r\rangle\subset S.$

######  Definition 4.8.

The _hypertoric quiver variety_ $Y(Q,0)$ (resp. $Y(Q,\theta)$) is the
irreducible subvariety of the affine Lawrence variety $X(Q^{\pm},0)$ (resp. of
the Lawrence variety $X(Q^{\pm},\theta)$) cut by the homogeneous ideal
$\mathrm{Circ}(\mathscr{B})$.

######  Remark 4.9.

Hypertoric quiver variety are both hypertoric varieties in the sense of [5,
32], and Nakajima quiver variety whose dimension vector has all coordinates
equals to one; see [57]. In [34], $Y(Q,0)$ is called _toric quiver variety_ ,
but note that it is only rarely toric; see [34, §10]. Moreover, Altmann and
Hille called toric quiver varieties the fibre over $0$ of the morphism
$\pi_{Y}\colon Y(Q,\theta)\to Y(Q,0)$. To avoid confusion, we will refer to
$Y(Q,0)$ and $Y(Q,\theta)$ as hypertoric quiver variety.

Since the product $z_{e}w_{e}$ is $\mathbb{T}^{r-1}$-invariant, the injective
morphism of $\mathbb{C}$-algebras

$\mathbb{C}[z_{e}w_{e}\colon e\in
E]\hookrightarrow\mathbb{C}[z_{e},w_{e}\colon e\in E]^{\mathbb{T}^{r-1}}$

induces the fibre product square

(19)
${Y(Q,0)=\operatorname{Spec}\bigg{(}\frac{\displaystyle\mathbb{C}[z_{e},w_{e}\colon
e\in
E]^{\mathbb{T}^{r-1}}}{\displaystyle\mathrm{Circ}(\mathscr{B})}\bigg{)}}$${X(Q^{\pm},0)=\operatorname{Spec}\big{(}\mathbb{C}[z_{e},w_{e}\colon
e\in
E]^{\mathbb{T}^{r-1}}\big{)}}$${\mathbb{C}^{b_{1}}=\operatorname{Spec}\bigg{(}\frac{\displaystyle\mathbb{C}[z_{e}w_{e}\colon
e\in
E]}{\displaystyle\mathrm{Circ}(\mathscr{B})}\bigg{)}}$${\mathbb{C}^{s}=\operatorname{Spec}\big{(}\mathbb{C}[z_{e}w_{e}\colon
e\in
E]\big{)}.}$$\scriptstyle{\iota_{M}}$$\scriptstyle{\chi_{Q}}$$\scriptstyle{\pi_{Q}}$$\scriptstyle{\iota_{A}}$

Hence, $Y(Q,0)$ is the inverse image under the map $\pi_{Q}$ of the linear
subspace $\mathbb{C}^{b_{1}}$.

###  4.4. Cohomology of hypertoric quiver varieties

The cohomology of hypertoric quiver varieties can be expressed in terms of the
cohomology of the cographic matroid. This permits to compare the Hodge-to-
singular correspondence with [17, Thm 6.11].

######  Definition 4.10.

The _cographic matroid_ $\mathscr{C}_{Q}$ of the unoriented graph $|Q|$ is the
matroid whose independent subsets are the subsets $I$ of unoriented edges of
$|Q|$ such that $|Q|\setminus I$ is connected. The set $\mathscr{C}_{Q}$ is
partially ordered with respect to inclusions and we denote by
$|\mathscr{C}_{Q}|$ the associated simplicial complex, i.e. the complex whose
$k$-dimensional faces are the independent subsets of cardinality $(k+1)$.

We now recall some facts about the affinization morphisms

$\pi_{X}:X(Q^{\pm},\theta)\to X(Q^{\pm},0),\qquad\pi_{Y}:Y(Q,\theta)\to
Y(Q,0).$

In particular, we observe that the top cohomology of $Y(Q,\theta)$ can be
identified with the top cohomology of $|\mathscr{C}_{Q}|$, which explains the
occurence of the cographic matroid in [17, §6] via the Hodge-to-singular
correspondence; see Sections 7 and 10.

In the following all the stability $\theta,\theta^{\prime}\in\mathbb{Z}^{r-1}$
are generic.

######  Proposition 4.11.

The following facts hold:

  1. (1)

_[34, Prop. 8.2]_ $X(Q^{\pm},\theta)$ is a smooth toric variety of dimension
$b_{1}+s$.

  2. (2)

_[34, Prop. 8.2]_ $Y(Q,\theta)$ is a smooth holomorphic symplectic variety of
dimension $2b_{1}$.

  3. (3)

_[34, Lem. 6.4]_ $\pi^{-1}_{X}(0)=\pi^{-1}_{Y}(0)\eqqcolon C(Q^{\pm},\theta)$.

  4. (4)

The irreducible components of $C(Q^{\pm},\theta)$ are Lagrangian in
$Y(Q,\theta)$. In particular, they all have dimension $b_{1}$.

  5. (5)

_[34, Rmk after Lem. 6.5]_ The spaces $C(Q^{\pm},\theta)\subset
Y(Q,\theta)\subset X(Q^{\pm},\theta)$ are deformation retracts in one another.

  6. (6)

_[32, Lem. 2.1] [34, Thm 6.3]_ The symplectic resolutions $Y(Q,\theta)$ and
$Y(Q,\theta^{\prime})$ are diffeomorphic.

  7. (7)

By (5) and (6), the cohomology

$H^{*}(C(Q^{\pm},\theta),\mathbb{Q})\simeq H^{*}(Y(Q,\theta),\mathbb{Q})\simeq
H^{*}(X(Q^{\pm},\theta),\mathbb{Q})$

is independent of $\theta$.

  8. (8)

_[6, Thm 7.3.3, 7.8.1]_ There exists an isomorphism

$H^{2b_{1}}(Y(Q,\theta),\mathbb{Q})\simeq\widetilde{H}^{2b_{1}-1}(|\mathscr{C}_{Q}|,\mathbb{Q}).$

###### Proof.

We just comment on (4). An irreducible component $F$ of $C(Q^{\pm},\theta)$ is
a fibre of $Y(Q,\theta)\to Y(Q,0)\xrightarrow{\chi_{Q}}\mathbb{C}^{b_{1}}$. By
semicontinuity, $\dim F\geq b_{1}$. On the other hand, the symplectic form
vanishes along $F$, because $F$ is toric and so $H^{0}(F,\Omega^{2}_{F})=0$.
So $F$ is isotropic, and $\dim F\leq b_{1}$. ∎

###  4.5. Decomposition theorem for hypertoric quiver varieties

The goal of this section is to describe the restriction
$\operatorname{IC}(X(Q^{\pm},0),\mathbb{Q})|_{Y(Q,0)}$; see Theorem 4.18. To
this end, we briefly recall the statement of the decomposition theorem for
semismall maps.

Let $f\colon X\to Y$ be a proper morphism of irreducible varieties. A
stratification of $f$ is a collection of finitely many locally closed subsets
$Y_{k}$ of dimension $\dim Y-k$ such that $f^{-1}(Y_{k})\to Y_{k}$ are
topologically locally trivial fibrations. A stratum $Y_{k}$ is _relevant_ if
$2\dim f^{-1}(Y_{k})-\dim(Y_{k})=\dim X$.

######  Definition 4.12 (Semismall maps).

The map $f$ is said to be _semismall_ if

$2\dim f^{-1}(Y_{k})-\dim(Y_{k})\leq\dim X\qquad\text{ for any }k\geq 0.$

Further if the stronger inequalities

$2\dim f^{-1}(Y_{k})-\dim(Y_{k})<\dim X\qquad\text{ for any }k>0$

hold, we say that $f$ is _small_.

######  Theorem 4.13 (Decomposition theorem for semismall maps).

Let $f\colon X\to Y$ be a semismall map from a smooth variety $X$. Then there
exists a canonical isomorphism in $D^{b}MHM_{\text{alg}}(Y)$ (resp.
$D^{b}(Y,\mathbb{Q})$)

$\mathrm{R}f_{*}\mathbb{Q}_{X}\simeq\bigoplus_{Y_{k}}\operatorname{IC}\big{(}\overline{Y}_{k},\mathrm{R}^{\operatorname{codim}Y_{k}}f_{*}\mathbb{Q}_{f^{-1}(Y_{k})}\big{)}\langle\operatorname{codim}Y_{k}/2\rangle,$

where the summation index runs over all the relevant strata of a
stratification of $f$.

In particular, if $f$ is small, then
$\mathrm{R}f_{*}\mathbb{Q}_{X}\simeq\operatorname{IC}(Y,\mathbb{Q})$.

Given a quiver $Q=(V,E)$, we describe complex Whitney stratifications of the
toric Lawrence variety $X(Q^{\pm},0)$ and of hypertoric quiver variety
$Y(Q,0)$.

######  Definition 4.14.

A partition $\underline{V}$ of the set $V$ is a set $\\{V_{1},\ldots,V_{r}\\}$
of disjoint subsets $V_{i}\subset V$ such that $V=\bigsqcup^{r}_{i=1}V_{i}$.
The quiver $Q_{\underline{V}}$ is obtained from $Q$ by identifying all
vertices in $V_{i}$, for any $i=1,\ldots,r$, and by deleting all loops.

######  Proposition 4.15.

The toric Lawrence variety $X(Q^{\pm},0)$ admits a complex Whitney
stratification

(20) $X(Q^{\pm},0)=\bigsqcup_{\underline{V}\dashv V}X_{\underline{V}},$

by toric-invariant strata $X_{\underline{V}}$ whose analytic normal slice is
isomorphic to $X(Q^{\pm}_{\underline{V}},0)$. This induces a complex Whitney
stratification of $Y(Q,0)$

(21) $Y(Q,0)=\bigsqcup_{\underline{V}\dashv
V}Y_{\underline{V}}\qquad\text{with }Y_{\underline{V}}\coloneqq
X_{\underline{V}}\cap Y(Q,0),$

whose normal slice is isomorphic to $Y(Q_{\underline{V}},0)$.

###### Proof.

Let $N$ be the lattice of one-parameter subgroups of the torus
$\mathbb{T}^{b_{1}+s}=N\otimes\mathbb{C}^{*}$. The affine toric variety
$X(Q^{\pm},0)$ corresponds to the cone $\sigma\subset N\otimes\mathbb{R}$
generated by the columns of a matrix $B^{\pm}$ Gale dual to the boundary map
$A^{\pm}\coloneqq(A,-A)\colon\mathbb{Z}^{2s}\to\mathbb{Z}^{r-1}$ of the quiver
$Q^{\pm}$; see Section 4.1. Let $\tau$ be a face of $\sigma$, $N_{\tau}$ be
the sublattice of $N$ spanned by $\tau\cap N$, and $N(\tau)\coloneqq
N/N_{\tau}$. The sets of rays of $\sigma$, $\tau$, and those of $\sigma$ not
in $\tau$ are denoted respectively $\sigma(1)$, $\tau(1)$ and $\tau(1)^{c}$.
The exact sequences

${0}$${\mathbb{Z}^{\tau(1)}}$${\mathbb{Z}^{\sigma(1)}}$${\mathbb{Z}^{\tau(1)^{c}}}$${0}$${0}$${N_{\tau}}$${N}$${N(\tau)}$${0}$$\scriptstyle{(B^{\pm}_{\tau})^{T}}$$\scriptstyle{(B^{\pm})^{T}}$

are dual to

${0}$${M(\tau)}$${\mathbb{Z}^{\tau(1)^{c}}}$${\mathbb{Z}^{r-1}}$${0}$${M}$${\mathbb{Z}^{\sigma(1)}}$${\mathbb{Z}^{r-1}}$${0}$${0}$${M_{\tau}}$${\mathbb{Z}^{\tau(1)}}$${\mathbb{Z}^{r_{\tau}-1}}$${0.}$$\scriptstyle{A^{\pm}|_{\tau(1)^{c}}}$$\scriptstyle{A^{\pm}}$$\scriptstyle{A^{\pm}_{\tau}}$

In particular, the resulting map $A^{\pm}_{\tau}$ is the boundary map of a
quiver $Q(\tau)$ obtained by contracting the edges corresponding to
$\tau(1)^{c}$. Up to loops, $Q(\tau)$ is isomorphic to $Q_{\underline{V}}$ for
some partition $\underline{V}$ of $V$. Therefore, $X(Q^{\pm},0)$ is covered by
toric charts of the form

(22) $X(Q(\tau),0)\times N(\tau)\otimes\mathbb{C}^{*}\simeq
X(Q^{\pm}_{\underline{V}},0)\times\text{smooth factor},$

where the isomorphism follows from Lemma 4.2. We have obtained the
stratification (20) which is Whitney by Lemma 2.2. By inspecting the
restriction of the equations (18) to each chart (22), we get the Whitney
stratification (21) too; see also [61, §2]. ∎

######  Proposition 4.16.

The map $\pi_{X}:X(Q^{\pm},\theta)\to X(Q^{\pm},0)$ is small.

###### Proof.

The fan of $X(Q^{\pm},\theta)$ is a regular triangulation of the cone
$\sigma\subset N\otimes\mathbb{R}$ of $X(Q^{\pm},0)$; see [34, §2]. Since any
regular triangulation of $\sigma$ restricts to a regular triangulation of its
face $\tau$, the restriction $\pi_{X}$ to the open set
$X(Q^{\pm}_{\underline{V}},0)\times X_{\underline{V}}$ is the resolution

$X(Q^{\pm}_{\underline{V}},[\theta])\times X_{\underline{V}}\to
X(Q^{\pm}_{\underline{V}},0)\times X_{\underline{V}}.$

Therefore, given $y\in X_{\underline{V}}\subset X(Q^{\pm},0)\times
X_{\underline{V}}$ we obtain

$2\dim\pi_{X}^{-1}(y)=2\dim
C(Q^{\pm}_{\underline{V}},[\theta])=2b_{1}(Q_{\underline{V}})<b_{1}(Q_{\underline{V}})+s=\dim
X(Q^{\pm}_{\underline{V}},0)=\dim X(Q^{\pm},0)-\dim X_{\underline{V}},$

i.e. $\pi_{X}$ is small. ∎

######  Proposition 4.17.

The map $\pi_{Y}:Y(Q,\theta)\to Y(Q,0)$ is semismall.

###### Proof.

By 4.11.(2), $Y(Q,\theta)$ is a smooth holomorphic symplectic variety. Any
symplectic resolution of singularities is semismall by [42, Lemma 2.11]. ∎

######  Theorem 4.18 (Local Hodge-to-singular correspondence).

The restriction $\operatorname{IC}(X(Q^{\pm},0),\mathbb{Q})|_{Y(Q,0)}[2b_{1}]$
is a semi-simple perverse sheaf. In particular, we have

$\operatorname{IC}(X(Q^{\pm},0),\mathbb{Q})|_{Y(Q,0)}[2b_{1}]\simeq\bigoplus_{\underline{V}\dashv
V}\operatorname{IC}(Y_{\underline{V}},\mathbb{Q})\otimes
H^{b_{1}(Q_{\underline{V}})}(Y(Q,\theta),\mathbb{Q})[2b_{1}]\langle
b_{1}(Q_{\underline{V}})\rangle.$

###### Proof.

Consider the cartesian square

${Y(Q,\theta)}$${X(Q^{\pm},\theta)}$${Y(Q,0)}$${X(Q^{\pm},0).}$$\scriptstyle{\iota_{M,\theta}}$$\scriptstyle{\pi_{Y}}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\iota_{A}}$

The map $\pi_{X}$ is small, the map $\pi_{Y}$ is semismall and the inclusion
$\iota_{M,\theta}$ is a regular embedding. Therefore, the decomposition
theorem for small maps gives $\operatorname{IC}(X(Q^{\pm},0),\mathbb{Q})\simeq
R\pi_{X,*}\mathbb{Q}_{X(Q^{\pm},\theta)}$. By proper base change, we obtain
$R\pi_{X,*}\mathbb{Q}_{X(Q^{\pm},\theta)}|_{Y(Q,0)}\simeq
R\pi_{Y,*}\mathbb{Q}_{Y(Q,\theta)}$. The statement now follows from the
decomposition theorem for semismall maps. In this case, the local systems
$\mathrm{R}^{\dim X-\dim Y_{k}}f_{*}\mathbb{Q}_{f^{-1}(Y_{k})}$ of Theorem
4.13 are trivial, supported on the relevant strata $Y_{\underline{V}}$ and
with fiber $H^{\mathrm{top}}(\pi_{Y}^{-1}(y),\mathbb{Q})\simeq
H^{b_{1}(Q_{\underline{V}})}(Y(Q,\theta),\mathbb{Q})$ for all $y\in
Y_{\underline{V}}$; see [61, §5]. ∎

##  5\. Universal compactified Jacobians

The goal of this section is to enlarge the Hitchin fibration to a universal
family of compactified Jacobians. Fix integers $g>1$, $n>1$, $d$ as above. Let
$X$ be a projective curve with ample canonical bundle $\omega_{X}$ and planar
singularities.

######  Definition 5.1.

A rank 1 torsion free sheaves $\mathcal{I}$ on $X$ is semistable with respect
to the canonical polarization $\omega_{X}$ if for any proper subcurve
$Y\subset X$ we have that

(23)
$\frac{\chi(X,\mathcal{I})}{\deg(\omega_{X})}\leq\frac{\chi(Y,\mathcal{I}_{Y})}{\deg(\omega_{X}|_{Y})},$

where $\mathcal{I}_{Y}$ is the biggest torsion free quotient of the
restriction $\mathcal{I}|_{Y}$ of $\mathcal{I}$ to the subcurve $Y$. The
compactified Jacobian $\overline{J}_{d}(X)$ is the projective coarse moduli
space of semistable rank 1 torsion free sheaves on $X$ of degree
$d+(n-n^{2})(1-g)$ with respect to the canonical polarization; see [66].

The BNR correspondence presents the Dolbeault moduli space $M(n,d)$ as the
relative compactified Jacobian of the spectral curve family; see [39, 2, 63].
It says that given a rank $1$ torsion free sheaf $\mathcal{I}$ on the spectral
curve $\pi_{a}\colon C_{a}\to C$, the sheaf
$\mathcal{E}\simeq\pi_{a,*}\mathcal{I}$ is a vector bundle of rank $n$
equipped with a Higgs field $\phi$, induced from the
$\mathcal{O}_{C_{a}}$-module structure. Viceversa, any Higgs bundle
$(\mathcal{E},\phi)$ is the pushfoward of some rank $1$ torsion free sheaf on
$C_{a}$.

######  Proposition 5.2.

[39, 2, 63, 55] The fibres of the Hitchin fibrations $\chi^{-1}(n,d)(a)$ with
$a\in A^{\text{red}}_{n}$ is isomorphic to the compactified Jacobian
$\overline{J}_{d}(C_{a})$ of the spectral curve $C_{a}$.

######  Remark 5.3.

The stability condition for rank 1 torsion free sheaves on reduced curves used
in [55, (10.5)] and [17] coincides with (23). Suppose that $X$ is a reduced
spectral curve $C_{a}=\bigcup^{r}_{i=1}C_{i}$ whose component $C_{i}$ has
degree $n_{i}$, and $n=\sum^{r}_{i=1}n_{i}$. Let $\mathcal{I}$ be a rank 1
torsion free sheaf of degree $d+(n-n^{2})(1-g)$ on $C_{a}$. By Riemann–Roch,
(23) reads as follows: for any subcurve $C_{I}=\bigcup_{i\in I}C_{i}$, with
$I\subset\\{1,\ldots,r\\}$, we have that

$\chi(Y,\mathcal{I}_{Y})\geq(d+n(1-g))\frac{\sum_{i\in I}n_{i}}{n},$

which is the stability condition [55, (10.5)].

Following Caporaso, Pandaripande and Esteves [7, 59, 25] (cf also [8, §2.3,
Fact 2.12] and [17]) we can enlarge the Hitchin fibration to a relative
compactified Jacobian of a versal deformation of $C_{a}$. To this end, let
$\overline{\mathcal{M}}_{g^{\prime}}$ be the Deligne–Mumford stack of semi-
stable curves of genus $g^{\prime}\coloneqq n^{2}(g-1)+1$, and
$A^{\times}_{n}\coloneqq\bigsqcup_{\underline{n}\dashv\,n}S^{\times}_{n}$. By
[17, Lem. 2.1] the flat universal spectral curve $\mathcal{C}^{\times}_{n}\to
A^{\times}_{n}$ induces a morphism

$f^{\times}:A^{\times}_{n}\to\overline{\mathcal{M}}_{g^{\prime}}.$

######  Proposition 5.4 (Universal compactified Jacobian).

There exists an open substack
$\mathcal{B}\subseteq\overline{\mathcal{M}}_{g^{\prime}}$ containing
$f^{\times}(A^{\times}_{n})$ and an irreducible (non necessarily smooth)
Deligne–Mumford stack $\pi:\overline{J}_{d,\mathcal{B}}\to\mathcal{B}$ that
étale locally is a relative compactified Jacobian parametrizing semistable
rank 1 torsion free sheaves of degree $d+(n-n^{2})(1-g)$ with respect to the
canonical polarization.

###### Proof.

The first paragraph of the proof of [17, Prop. 5.9] does not require the
coprimality of $n$ and $d$, and works verbatim in our context. ∎

######  Proposition 5.5.

For every partition $\underline{n}$ of $n$, let $a\in
S^{\times}_{\underline{n}}$. There exists a local multisection $A_{a}$ of
$f^{\times}:A^{\times}_{n}\to f^{\times}(A^{\times}_{n})\subseteq\mathcal{B}$
passing through $a$, i.e. $A_{a}$ is a locally closed subset of
$A^{\times}_{n}$ passing through $a$, intersecting
$S^{\times}_{\underline{n}}$ transversally, and such that the restriction
$f^{\times}|_{A_{a}}:A_{a}\to f^{\times}(A^{\times}_{n})$ is étale.
Furthermore we have a cartesian diagram

${\chi(n,d)^{-1}(A_{a})}$${\overline{J}_{d,\mathcal{B}}\times_{\mathcal{B}}A_{a}}$${\overline{J}_{d,\mathcal{B}}}$${A_{a}}$${\mathcal{B}.}$$\scriptstyle{\simeq}$$\scriptstyle{\chi(n,d)}$$\scriptstyle{\pi}$$\scriptstyle{f^{\times}|_{A_{a}}}$

###### Proof.

[17, Cor. 5.10] works verbatim. ∎

######  Remark 5.6.

The $\mathbb{G}_{m}$-action on $A_{n}$ and the translation action
$H^{0}(C,\omega_{C})\otimes A_{n}\to A_{n}$ lift to the universal spectral
curve $\mathcal{C}_{n}\to A_{n}$ and so to ${M}(n,d)$. Therefore, the group
action induces an analytic isomorphism between an open neighbourhood
$U_{a}\subseteq A_{n}$ of $A_{a}$ and a neighbourhood of the zero-section of
trivial vector bundle $p:A_{a}\times(H^{0}(C,\mathcal{O}_{C})\oplus
H^{0}(C,\omega_{C}))\to A_{a}$, such that there exists an homeomorphism

$\chi(n,d)^{-1}(U_{a})\simeq\chi(n,d)^{-1}(A_{a})\times(H^{0}(C,\mathcal{O}_{C})\oplus
H^{0}(C,\omega_{C})),$

and so

$R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})|_{U_{a}}\simeq
p^{*}(R\chi(n,d)_{*}\operatorname{IC}(\chi(n,d)^{-1}(A_{a}),\mathbb{Q})).$

##  6\. Kodaira–Spencer map for spectral curves

The differential of the map $f^{\times}\colon A_{a}\to\mathcal{B}$ in 5.5 can
be identified with the Kodaira–Spencer map; see for instance [17, §5.2]. Its
kernel is isomorphic to $H^{0}(C,\mathcal{O}_{C})\oplus H^{0}(C,\omega_{C})$;
see again [17, §5.2]. In 6.3 we study its image.

###  6.1. Kodaira–Spencer map

We start with some recollection about the deformation of nodal spectral
curves.

######  Definition 6.1.

The Kodaira–Spencer map is the map

(24) $KS_{a}\colon T_{a}A_{n}\simeq
H^{0}(C_{a},N_{C_{a}/T^{*}C_{a}})\to\mathrm{Ext}^{1}(\Omega^{1}_{C_{a}},\mathcal{O}_{C_{a}})$

obtained by applying the functor $\mathrm{Hom}(-,\mathcal{O}_{C_{a}})$ to the
relative cotangent sequence

(25) $0\to
N^{*}_{C_{a}/T^{*}C}\to\Omega^{1}_{T^{*}C}|_{C_{a}}\to\Omega^{1}_{C_{a}}\to
0.$

Applying the functor $\mathcal{H}om(-,\mathcal{O}_{C_{a}})$ to (25), we obtain
the exact sequence of coherent sheaves

$0\to T_{C_{a}}\to T_{T^{*}C}|_{C_{a}}\to
N_{C_{a}/T^{*}C}\to\mathbb{T}^{1}\coloneqq\mathcal{E}xt^{1}(\Omega^{1}_{C_{a}},\mathcal{O}_{C_{a}})\to
0.$

The first tangent sheaf $\mathbb{T}^{1}$ is a skyscraper sheaf supported at
the nodes $E(C_{a})$ of $C_{a}$ and it parametrises smoothings of the nodes.
The kernel of $N_{C_{a}/T^{*}C}\to\mathbb{T}^{1}$ is easily identified to
$N_{C_{a}/T^{*}C}\otimes\mathcal{I}_{E(C_{a})}$, where
$\mathcal{I}_{E(C_{a})}$ is the product of the maximal ideals of the nodes
$E(C_{a})$; see [65, Ex. 4.7.1]. Hence, there exists a short exact sequence

$0\to N_{C_{a}/T^{*}C}\otimes\mathcal{I}_{E(C_{a})}\to
N_{C_{a}/T^{*}C}\to\mathbb{T}^{1}\to 0$

inducing the exact sequence in cohomology

$0\to H^{0}(C_{a},N_{C_{a}/T^{*}C}\otimes\mathcal{I}_{E(C_{a})})\to
T_{a}A_{n}=H^{0}(C_{a},N_{C_{a}/T^{*}C})\to\mathbb{T}^{1}(C_{a}).$

Note that the last map $T_{a}A_{n}\to\mathbb{T}^{1}$ factors as

(26)
$T_{a}A_{a}=H^{0}(C_{a},N_{C_{a}/T^{*}C})\xrightarrow{KS_{a}}\mathrm{Ext}^{1}(\Omega^{1}_{C_{a}},\mathcal{O}_{C_{a}})\to
H^{0}(C_{a},\mathcal{E}xt^{1}(\Omega^{1}_{C_{a}},\mathcal{O}_{C_{a}}))=\mathbb{T}^{1}(C_{a}),$

where the latter map comes from the local-to-global Ext spectral sequence.

###  6.2. Image of the Kodaira–Spencer map for spectral curves

Let $\underline{n}^{\prime}$ be a partition of $n$. Then
$\chi(n,d)^{-1}(S^{\times}_{\underline{n}^{\prime}})$ admits a stratification
by type

$\chi(n,d)^{-1}(S^{\times}_{\underline{n}^{\prime}})=\bigsqcup_{\underline{n}\dashv\,n}\chi(n,d)^{-1}(S^{\times}_{\underline{n}^{\prime}})\cap
M^{\circ}_{\underline{n}}(d).$

Given the partition $\underline{n}=(n_{1},\ldots,n_{r})$, the condition
$(\mathcal{E},\phi)\in\chi(n,d)^{-1}(S^{\times}_{\underline{n}^{\prime}})\cap
M^{\circ}_{\underline{n}}(d)$ means that the Higgs bundle $(\mathcal{E},\phi)$
splits as

(27) $(\mathcal{E},\phi)=\bigoplus^{r}_{i=1}(\mathcal{E}_{i},\phi_{i}),$

where $(\mathcal{E}_{i},\phi_{i})$ are stable Higgs bundles of rank $n_{i}$
and degree $d_{i}=dn_{i}/n$. Further, the spectral curve $C_{a}$ of
$(\mathcal{E},\phi)$ is nodal and the union of the (possibly reducible)
spectral curves $C_{a_{i}}$ of $(\mathcal{E}_{i},\phi_{i})$

(28) $C_{a}=\bigcup^{r}_{i=1}C_{a_{i}}.$

A normal slice $N_{\underline{n}}\subseteq A_{a}$ to $S_{\underline{n}}$
through $a$ parametrises spectral curves obtained by smoothing the nodes
$E(\Gamma_{\underline{n}})\coloneqq\bigcup^{r}_{i,j=1}C_{a_{i}}\cap C_{a_{j}}$
of $C_{a}$.

######  Notation 6.2.

For a partition $\underline{n}$ of $n$ we will denote by
$\Gamma_{\underline{n}}$ the dual graph of any spectral curve given by a point
of $S^{\times}_{\underline{n}}$, i.e. $\Gamma_{\underline{n}}$ is the graph
with vertices $[r]\coloneqq\\{1,\ldots,r\\}$ corresponding to the irreducible
components of the curve and $n_{i}n_{j}(2g-2)$ edges between the vertices $i$,
$j$, corresponding to the intersection points of the components. By abuse of
notation we continue to denote by $E(\Gamma_{\underline{n}})$ the set of edges
of $\Gamma_{\underline{n}}$.

######  Proposition 6.3.

The image of the map

$T_{a}N_{\underline{n}}\hookrightarrow
T_{a}A_{a}\xrightarrow{df^{\times}}\mathrm{Ext}^{1}(\Omega^{1}_{C_{a}},\mathcal{O}_{C_{a}})\twoheadrightarrow\bigoplus_{e\in
E(\Gamma_{\underline{n}})}H^{0}(C_{a},\mathcal{E}xt^{1}(\Omega^{1}_{C_{a}},\mathcal{O}_{C_{a}})_{e})\simeq\operatorname{Spec}\mathbb{C}[T_{e}\colon
e\in E(\Gamma_{\underline{n}})]$

is cut by the linear equations

$\mathrm{Circ}(\mathscr{B})=\langle\sum_{e\in E}a_{ie}T_{e}\colon i=2,\ldots
r\rangle,$

where $A=(a_{ie})$ is the boundary map of a quiver underlying the graph
$\Gamma_{\underline{n}}$ (cf Section 4).

###### Proof.

Let $\nu_{\underline{n}}\colon
C_{\underline{n},a}\coloneqq\bigsqcup^{r}_{i=1}C_{a_{i}}\to C_{a}$ be the
partial normalization obtained by blowing-up $C_{a}$ at
$E(\Gamma_{\underline{n}})$. Applying the functor
$\mathcal{H}om(-,\mathcal{\omega}_{C_{a}})$ to the short exact sequence

$0\to\mathcal{O}_{C_{a}}\to\nu_{\underline{n},*}\mathcal{O}_{C_{\underline{n},a}}\to\bigoplus_{e\in
E(\Gamma_{\underline{n}})}\mathcal{O}_{e}\to 0,$

we obtain the short exact sequence

$0\to N_{C_{a}/T^{*}C}\otimes\mathcal{I}_{E(\Gamma_{\underline{n}})}\to
N_{C_{a}/T^{*}C}\to\mathbb{T}^{1}_{\underline{n}}\coloneqq\bigoplus_{e\in
E(\Gamma_{\underline{n}})}\mathcal{E}xt^{1}(\mathcal{O}_{e},\omega_{C_{a}})\to
0,$

where $\mathcal{I}_{E(\Gamma_{\underline{n}})}$ is the product of the maximal
ideals of the nodes $E(\Gamma_{\underline{n}})$, and $\omega_{C_{a}}$ is
identified with $N_{C_{a}/T^{*}C}$ by adjunction. Taking global cohomology, we
get the long exact sequence

$\displaystyle 0\to
H^{0}(C_{a},N_{C_{a}/T^{*}C}\otimes\mathcal{I}_{E(\Gamma_{\underline{n}})})\to
T_{a}A_{n}=H^{0}(C_{a},N_{C_{a}/T^{*}C})\to\mathbb{T}^{1}_{\underline{n}}(C_{a})\to$
(29) $\displaystyle\to
H^{1}(C_{a},N_{C_{a}/T^{*}C}\otimes\mathcal{I}_{E(\Gamma_{\underline{n}})})\to
H^{1}(C_{a},N_{C_{a}/T^{*}C})\to 0.$

By Serre duality we see

$\displaystyle
H^{0}(C_{a},N_{C_{a}/T^{*}C}\otimes\mathcal{I}_{E(\Gamma_{\underline{n}})})\simeq$
$\displaystyle
H^{0}(C_{a},\mathcal{H}om(\nu_{\underline{n},*}\mathcal{O}_{C_{\underline{n},a}},\omega_{C_{a}}))\simeq
H^{1}(C_{a},\nu_{\underline{n},*}\mathcal{O}_{C_{\underline{n},a}})^{\vee}$
(30)
$\displaystyle\simeq\bigoplus^{r}_{i=1}H^{1}(C_{i},\mathcal{O}_{C_{i}})^{\vee}\simeq\bigoplus^{r}_{i=1}H^{0}(C_{i},\omega_{C_{i}})\simeq
T_{a}S_{\underline{n}}.$

Combining (26), (29) and (6.2), the image of the map

$df^{\times}\colon T_{a}N_{\underline{n}}\simeq
T_{a}A_{n}/T_{a}S_{\underline{n}}\to\mathbb{T}^{1}_{\underline{n}}(C_{a})$

coincides with the kernel of
$\mathbb{T}^{1}_{\underline{n}}(C_{a})\to\ker\\{H^{1}(C_{a},N_{C_{a}/T^{*}C}\otimes\mathcal{I}_{E(\Gamma_{\underline{n}})})\to
H^{1}(C_{a},N_{C_{a}/T^{*}C})\\}$. Again via Serre duality, this map is
isomorphic to

$\mathbb{C}^{E(\Gamma_{\underline{n}})}\simeq\bigoplus_{e\in
E(\Gamma_{\underline{n}})}H^{0}(\mathcal{O}_{e})^{\vee}\to\ker\\{H^{0}(C_{a},\nu_{\underline{n},*}\mathcal{O}_{C_{\underline{n},a}})^{\vee}\to
H^{0}(C_{a},\mathcal{O}_{C_{a}})^{\vee}\\}\simeq\mathbb{C}^{r-1},$

which in turn can be identified with the boundary map of a quiver underlying
the graph $\Gamma_{\underline{n}}$. ∎

##  7\. Singularities of universal compactified Jacobians

By the BNR correspondence a Higgs bundle $(\mathcal{E},\phi)$ corresponds to a
rank 1 torsion free sheaf $\mathcal{I}$ on the spectral curve $C_{a}$, where
$\mathcal{E}$ is isomorphic to $\pi_{a,*}\mathcal{I}$ via the projection
$\pi_{a}\colon C_{a}\to C$; see [39, 2, 63]. Suppose that $C_{a}$ is nodal,
and let $\Sigma$ be the set of the nodes where $\mathcal{I}$ fails to be
locally free. Define $M^{\times}_{\underline{n}}(d)$ as the open subsets of
Higgs bundles of $M^{\circ}_{\underline{n}}(d)$ with nodal spectral curve.

######  Proposition 7.1.

Let $\underline{n}=(n_{1},\ldots,n_{r})$ be a partition of $n$. Then
$(\mathcal{E},\phi)$ lies in $M^{\times}_{\underline{n}}(d)$ if and only if
there exists a partial normalization $\nu_{\Sigma}\colon C_{\Sigma}\to C_{a}$
obtained by blowing-up the nodes $\Sigma$ and a line bundle $\mathcal{L}$ on
$C_{\Sigma}$ such that (1) $C_{\Sigma}$ has exactly $r$ connected components,
and (2) $\mathcal{E}\simeq\nu_{\Sigma,*}\mathcal{L}$.

###### Proof.

By assumption, $(\mathcal{E},\phi)$ splits as
$(\mathcal{E},\phi)=\bigoplus^{r}_{i=1}(\mathcal{E}_{i},\phi_{i})$, where
$(\mathcal{E}_{i},\phi_{i})$ are stable Higgs bundles of rank $n_{i}$ and
degree $d_{i}=dn_{i}/n$. The spectral curve $C_{a}$ of $(\mathcal{E},\phi)$ is
the union of the spectral curves $C_{a_{i}}$

$C_{a_{i}}\xhookrightarrow{j_{i}}C_{a}=\bigcup^{r}_{i=1}C_{a_{i}}\xrightarrow{\pi_{a}}C.$

By the BNR correspondence, there exist rank $1$ torsion free sheaves
$\mathcal{I}$ and $\mathcal{I}_{i}$ on $C_{a}$ and $C_{a_{i}}$ respectively
such that $\mathcal{E}=\pi_{a,*}\mathcal{I}$, $\mathcal{E}_{i}=(\pi_{a}\circ
j_{i})_{*}\mathcal{I}_{i}$, and

(31) $\mathcal{I}=\bigoplus^{r}_{i=1}j_{i,*}\mathcal{I}_{i}.$

Note that $\mathcal{I}$ fails to be locally free at the node $C_{a_{i}}\cap
C_{a_{j}}$, $i\neq j$. Therefore, the blow-up $\nu_{\Sigma}\colon
C_{\Sigma}\to C_{a}$ of $C_{a}$ at $\Sigma$ factors through the disjoint union
of $C_{a_{i}}$

(32) $\nu_{\Sigma}\colon C_{\Sigma}\to\bigsqcup^{r}_{i=1}C_{a_{i}}\to C_{a}.$

Since $C_{a}$ has only nodal singularities, there exists a line bundle
$\mathcal{L}$ on $C_{\Sigma}$ such that
$\mathcal{I}=\nu_{\Sigma,*}\mathcal{L}$; see [29, Prop. 3.4]. By (32), the
curve $C_{\Sigma}$ has at least $r$ components, and exactly $r$ by the
stability of $(\mathcal{E}_{i},\phi_{i})$. Otherwise, the decomposition (31)
could be further refined, and so (27) could be refined too, which is a
contradiction. ∎

######  Proposition 7.2.

Suppose that $(\mathcal{E},\phi)$ lies in $M^{\times}_{\underline{n}}(d)$.
After removing all the loops, the dual graph $\Gamma_{C_{a}}(\Sigma)$ of the
curve obtained by smoothing the nodes not in $\Sigma$ is the graph
$\Gamma_{\underline{n}}$.

###### Proof.

As in the proof of 7.1, we denote the spectral curves of $(\mathcal{E},\phi)$
and of its stable summands by $C_{a}$ and $C_{a_{i}}$ respectively, and we
write

$C_{a}=\bigcup^{r}_{i=1}C_{a_{i}}=\bigcup^{r^{\prime}}_{j=1}C_{j}\qquad
C_{a_{i}}=\bigcup_{j\in S_{i}}C_{j},$

where $C_{j}$ are the irreducible components of $C_{a}$, and
$[r^{\prime}]=\bigsqcup^{r}_{i=1}S_{i}$ is a partition of the set of
$r^{\prime}$ elements. Clearly if $C_{j}$ has degree $n^{\prime}_{j}$ over
$C$, the spectral curves $C_{a_{i}}$ has degree $n_{i}=\sum_{j\in
S_{i}}n^{\prime}_{j}$.

The graph $\Gamma_{C_{a}}(\Sigma)$ is obtained from the dual graph
$\Gamma_{C_{a}}$ of the spectral curve $C_{a}$ by contracting all the edges
not in $\Sigma$.777By abuse of notation, we identify the set of nodes $\Sigma$
with the corresponding set of edges in the dual graph $\Gamma_{C_{a}}$. Let
$q\colon\Gamma_{C_{a}}\to\Gamma_{C_{a}}(\Sigma)$ be the contraction map. If
some edge between $v_{j}$ and $v_{k}$ is not contained in $\Sigma$, then
$q(v_{j})=q(v_{k})$, and all edges in $\Sigma$ are sent to loops in
$\Gamma_{C_{a}}(\Sigma)$. The graph $\Gamma_{C_{a}}(\Sigma)$ has as many
vertices as connected components in
$\Gamma_{C_{a}}\setminus\Sigma\simeq\Gamma_{C_{\Sigma}}$, i.e. $r$ by 7.1. The
edges between two distinct vertices $w_{i}$ and $w_{i^{\prime}}$ in
$\Gamma_{C_{a}}(\Sigma)$ are exactly

$\displaystyle\sum_{\begin{subarray}{c}v_{j}\in q^{-1}(w_{i}),\\\ v_{k}\in
q^{-1}(w_{i^{\prime}})\end{subarray}}(\\#\text{ edges between }v_{j}\text{ and
}v_{k})$ $\displaystyle=\sum_{\begin{subarray}{c}v_{j}\in q^{-1}(w_{i}),\\\
v_{k}\in
q^{-1}(w_{i^{\prime}})\end{subarray}}n^{\prime}_{j}n^{\prime}_{k}(2g-2)$
$\displaystyle=\bigg{(}\sum_{j\in
S_{i}}n^{\prime}_{j}\bigg{)}\cdot\bigg{(}\sum_{k\in
S_{i^{\prime}}}n^{\prime}_{k}\bigg{)}(2g-2)=n_{i}n_{i^{\prime}}(2g-2).$

We conclude that the graphs $\Gamma_{C_{a}}(\Sigma)$ and
$\Gamma_{\underline{n}}$ coincide after removing all loops in
$\Gamma_{C_{a}}(\Sigma)$. ∎

######  Remark 7.3.

The dual graph of the spectral curve may depend on the choice of
$(\mathcal{E},\phi)$ in $M^{\times}_{\underline{n}}(d)$. 7.2 asserts that up
to loops the dual graph $\Gamma_{C_{a}}(\Sigma)$ does not!

The singularities of the universal compactified Jacobian
$\overline{J}_{d,\mathcal{B}}$ are affine Lawrence varieties
$X(\Gamma_{\underline{n}}^{\pm},0)$, as explained in [8] and [9]. The
corresponding hypertoric quiver variety $Y(\Gamma_{\underline{n}},0)$ provides
the local model for the singularities of ${M}(n,d)$.

######  Theorem 7.4 (Local model of $\overline{J}_{d,\mathcal{B}}$ and
${M}(n,d)$).

Let $a\in S^{\times}_{\underline{n}^{\prime}}$, and $A_{a}$ be the local
multisection of $f^{\times}:A^{\times}_{n}\to
f^{\times}(A^{\times}_{n})\subseteq\mathcal{B}$ defined in 5.5. Let
$(\mathcal{E},\phi)$ be a Higgs bundle in $M^{\times}_{\underline{n}}(d)$, and
$\mathcal{I}$ be the corresponding rank 1 torsion free sheaf on the spectral
curve $C_{a}$. Then there exists an analytic neighbourhood $V$ of an étale
chart of $\overline{J}_{d,\mathcal{B}}$ centered at $(C_{a},\mathcal{I})$ such
that the restriction to $V$ of the fibre product square in 5.5

${{M}(n,d)\supset
U}$${V\subset\overline{J}_{d,\mathcal{B}}}$${\chi(n,d)(U)}$${\pi(V),}$$\scriptstyle{\chi(n,d)}$$\scriptstyle{\pi}$$\scriptstyle{f^{\times}}$

with $U\coloneqq V\cap\chi(n,d)^{-1}(A_{a})$, is locally isomorphic to

(33) ${Y(\Gamma_{\underline{n}},0)\times\mathbb{C}^{\dim
M_{\underline{n}}-1-g}}$${X(\Gamma_{\underline{n}}^{\pm},0)\times\mathbb{C}^{c(\underline{n},g)}}$${\mathbb{C}^{b_{1}(\Gamma_{\underline{n}})}\times\mathbb{C}^{d(\underline{n},g)}}$${\mathbb{C}^{s}\times\mathbb{C}^{3n^{2}(g-1)-s},}$$\scriptstyle{(\iota_{M},l_{1})}$$\scriptstyle{(\chi_{\Gamma_{\underline{n}}},l_{2})}$$\scriptstyle{(\pi_{\Gamma_{\underline{n}}},l_{4})}$$\scriptstyle{(\iota_{A},l_{3})}$

where the maps $\chi$, $\pi$, $\iota_{M}$ and $\iota_{A}$ are defined in (19),
the maps $l_{i}$ are linear, $s\coloneqq\\#E(\Gamma_{\underline{n}})$,
$d(\underline{n},g)\coloneqq\frac{1}{2}(\dim{M}(n,d)-\dim
Y(\Gamma_{\underline{n}},0))-g-1=(n^{2}-1)(g-1)-1-b_{1}(\Gamma_{\underline{n}})$
and $c(\underline{n},g)\coloneqq\dim\overline{J}_{d,\mathcal{B}}-\dim
X(\Gamma_{\underline{n}}^{\pm},0)=4n^{2}(g-1)+1-b_{1}(\Gamma_{\underline{n}})-s$.

###### Proof.

Let $\Sigma$ be the set of the nodes where $\mathcal{I}$ fails to be locally
free, and $\Gamma_{C_{a}}(\Sigma)$ be the dual graph of the curve obtained by
smoothing the nodes not in $\Sigma$. By [8, Theorem A] there exists an
analytic neighbourhoods $V$ of $(C_{a},\mathcal{I})$ in
$\overline{J}_{d,\mathcal{B}}$ such that

$V\simeq_{\mathrm{loc}}X(\Gamma_{C_{a}}(\Sigma)^{\pm},0)\times\mathbb{C}^{c(\underline{n},\Sigma)},$

with $c(\underline{n},\Sigma)\coloneqq
4g(C_{a})-3-b_{1}(\Gamma_{C_{a}}(\Sigma))-\\#E(\Gamma_{C_{a}}(\Sigma))$. By
[9, Lemma 4.3] and 7.2 we obtain that

$V\simeq_{\mathrm{loc}}X(\Gamma_{C_{a}}(\Sigma)^{\pm},0)\times\mathbb{C}^{c(\underline{n},\Sigma)}\simeq
X(\Gamma_{\underline{n}}^{\pm},0)\times\mathbb{C}^{c(\underline{n},g)}.$

The fact that the morphism $\pi\colon
V\subset\overline{J}_{d,\mathcal{B}}\to\pi(V)$ can be identified with
$(\pi_{\Gamma_{\underline{n}}},l_{4})$ is explained in [9, §7.2]. The
differential of the map $f^{\times}$ is the Kodaira-Spencer map $KS_{a}\colon
T_{a}A_{a}\to H^{1}(C_{a},\mathbb{T}^{1}_{C_{a}})$, and so the identification
of the Hitchin fibration $\chi(n,d)$ with the product of the map
$\chi_{\Gamma_{\underline{n}}}\colon
Y(\Gamma_{\underline{n}},0)\to\mathbb{C}^{b_{1}(\Gamma_{\underline{n}})}$ with
a linear map $l_{2}$ follows from 6.3 and 4.8. ∎

######  Theorem 7.5.

Let $A_{a}\subset A_{n}$ as in 5.5. Then
$\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})|_{\chi(n,d)^{-1}(A_{a})}[\dim
M(n,d)]$ is a semi-simple perverse sheaf. In particular, we have

$\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})|_{\chi(n,d)^{-1}(A_{a})}[\dim
M(n,d)]\simeq\bigoplus_{\begin{subarray}{c}\underline{n}\in\mathcal{P}_{d}\end{subarray}}\operatorname{IC}(M_{\underline{n}}(d),\mathcal{L}^{\prime}_{\underline{n}})|_{\chi(n,d)^{-1}(A_{a})}[\dim
M(n,d)]\langle\operatorname{codim}S_{\underline{n}}\rangle,$

where
$\mathcal{L}^{\prime}_{\underline{n}}\coloneqq\mathcal{H}^{\operatorname{codim}M_{\underline{n}}(d)}(\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q}))$
is a local system on $M_{\underline{n}}(d)$ of rank
$h^{b_{1}(\Gamma_{\underline{n}})}(Y(\Gamma_{\underline{n}},0),\mathbb{Q})$.

We anticipate that the local systems $\mathcal{L}^{\prime}_{\underline{n}}$
are pullback of the line bundles $\mathcal{L}_{\underline{n}}$ in [17, Cor.
6.20]; see 7.6 and the proof of Theorem 1.3 below. In particular, their
monodromy is known and induced by the automorphism of the graph
$\Gamma_{\underline{n}}$; see 10.1.

The idea of the proof of Theorem 7.5 is to glue the local Hodge-to-singular
correspondence (Theorem 4.18) given the local models in Theorem 7.4.

###### Proof of Theorem 7.5.

By Theorem 4.18 and 7.4, the complex
$\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})|_{\chi(n,d)^{-1}(A_{a})}[\dim
M(n,d)]$ is self-dual, i.e. it coincides with its own Verdier dual. Since
$\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})$ is pure and self-
dual,

$\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})|_{\chi(n,d)^{-1}(A_{a})}[\dim
M(n,d)]=i^{*}\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})=i^{!}\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})$

is a pure perverse sheaf, thus semi-simple by [62, Prop. 1.7 and Prop. 1.9]. ∎

######  Proposition 7.6.

Define
$M^{\circ\circ}_{\underline{n}}(d)\coloneqq\chi(n,d)^{-1}(S^{\times}_{\underline{n}})\cap
M_{\underline{n}}(d)$. The local system $\mathcal{L}^{\prime}_{\underline{n}}$
on $M^{\circ\circ}_{\underline{n}}(d)$ descends to a local system
$\mathcal{L}_{\underline{n}}$ on $S^{\times}_{\underline{n}}$

$\mathcal{L}^{\prime}_{\underline{n}}=\chi(n,d)^{*}\mathcal{L}_{\underline{n}}.$

###### Proof.

Denote by $F_{b}$ the fibre of $\chi(n,d)\colon
M^{\circ\circ}_{\underline{n}}(d)\to S_{\underline{n}}$. By 8.1, the
restriction
$\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})|_{F_{b}}$ is
isomorphic to the complex

$\mathbb{Q}_{F_{b}}\otimes
I\\!H^{i}(X(\Gamma_{\underline{n}}^{\pm},0),\mathbb{Q})[-i]\simeq\mathbb{Q}_{F_{b}}\otimes
H^{i}(Y(\Gamma_{\underline{n}},0),\mathbb{Q})[-i]$

of constant sheaves with trivial differential. Hence,
$\mathcal{L}^{\prime}_{\underline{n}}|_{F_{b}}\simeq\mathbb{Q}_{F_{b}}\otimes
H^{b_{1}(\Gamma_{\underline{n}})}(Y(\Gamma_{\underline{n}},0),\mathbb{Q})$ is
a component of the complex, so a trivial local system. Therefore,
$\mathcal{L}^{\prime}_{\underline{n}}$ descends to a local system on
$S^{\times}_{\underline{n}}$. ∎

##  8\. Tubular neighbourhood of the Jacobian of the normalisation of a
spectral curve

To complete the proof of 7.6, we must trivialize a tubular neighbourhood of
the subvariety $F_{b}$ in the universal compactified Jacobian. This is a
global statement which asserts that the local trivializations in Theorem 7.4
glue along $F_{b}$.

Let $f\colon\mathcal{C}\to B$ be a flat projective versal family of semistable
curves and arithmetic genus $g^{\prime}$, and $\pi\colon\overline{J}_{d,B}\to
B$ be the relative compactified Jacobian with respect to the canonical
polarization. For any $b\in B$, we denote by $F_{b}$ the subvariety in
$\pi^{-1}(b)$ parametrising rank 1 torsion free sheaves on $C_{b}\coloneqq
f^{-1}(b)$ of the shape $\mathcal{I}=\nu_{*}M$ where $M$ is a line bundle on
the normalization $\nu\colon C^{\nu}_{b}=\bigsqcup^{r}_{i=1}C_{b,i}\to C_{b}$.

######  Proposition 8.1.

There exists a homeomorphism between a euclidean neighbourhood of $F_{b}$ in
$\overline{J}_{d,B}$ and the product

(34) $F_{b}\times X(\Gamma(C_{b})^{\pm},0)\times\mathbb{C}^{3g^{\prime}-3-s},$

where $\Gamma(C_{b})$ is the dual graph of $C_{b}$ with $s$ edges $\Sigma$.

######  Remark 8.2.

Let $B_{\Gamma}\subseteq B$ the locus of curve with the same dual graph as
$C_{b}$, and let $\mathcal{C}^{\nu}\to\mathcal{C}\times_{B}B_{\Gamma}$ the
simultaneous normalization of $C_{b}$. 8.1 says in particular that a euclidean
neighbourhood of $F_{b}$ in the relative Picard scheme
$\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})\subseteq\overline{J}_{d,B}$
is homeomorphic to $F_{b}\times\mathbb{C}^{3g^{\prime}-3-s}$, but this is not
a formal or analytic isomorphism. Otherwise
$T\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})|_{F_{b}}\simeq\mathcal{O}^{\oplus
c^{\prime}}_{F_{b}}$ with $c^{\prime}\coloneqq\dim(F_{b})+3g^{\prime}-3-s$,
and the boundary map
$H^{0}(N_{F_{b}/\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})})\to
H^{1}(TF_{b})$ of the short exact sequence

$0\to TF_{b}\to
T\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})|_{F_{b}}\to
N_{F_{b}/\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})}\to 0$

would be trivial, i.e. $\mathcal{C}^{\nu}$ is an isotrivial family, which is a
contradiction. This suggests that 8.1 is false in the analytic category.

In order to show 8.1, we need to recall the GIT description of
$\overline{J}_{d,B}$ and the deformation theory of the pair
$(C_{b},\mathcal{I})$ as in [8, 9]. The relative compactified Jacobian
$\overline{J}_{d,B}$ is a GIT-quotient of an open set $Q^{ss}$ of a relative
Quot scheme $\mathrm{Quot}(\mathcal{O}^{\oplus R})$ over the versal family
$f$, as in [8, Section 2.3].

Note that $Q^{ss}$ is smooth along the closed $\operatorname{SL}_{R}$-orbits
by [8, Lemma 6.4.(ii)] and by the smoothness of the miniversal deformation of
the pair $(C_{b},\mathcal{I})$; see for instance [8, §3].

Let $Z$ be the locus of closed orbits over
$F_{b}\subset\overline{J}_{d,B}=Q^{ss}\sslash\operatorname{SL}_{R}$. Note that
$F_{b}$ is isomorphic to the Jacobian
$\operatorname{Pic}^{d^{\prime}}(C^{\nu}_{b})$ with $d^{\prime}\coloneqq
d+(n-n^{2})(1-g)$, and $Z$ is a fiber bundle over $F_{b}$.

Now let

(35) $\mathfrak{n}^{\prime}\colon\mathcal{Q}=\mathrm{Bl}_{Z\times
0}(Q^{ss}\times\mathbb{C})\setminus Q^{ss}\times 0\to\mathbb{C}$

be the deformation of $Q^{ss}$ to the normal cone of $Z$. The central fibre is
the total space $\mathrm{Tot}(N_{Z})$ of the normal bundle $N_{Z}$ of $Z$ in
$Q^{ss}$, and the restriction of $\mathfrak{n}^{\prime}$ to $\mathbb{C}^{*}$
is the trivial fibration $Q^{ss}\times\mathbb{C}^{*}\to\mathbb{C}^{*}$. Since
$Z$ is $\operatorname{SL}_{R}$-invariant, $\mathcal{Q}$ inherits a
$\operatorname{SL}_{R}$-action preserving the fibres of
$\mathfrak{n}^{\prime}$. The quotient

$\mathfrak{n}\colon\mathcal{Q}\sslash\operatorname{SL}_{R}\to\mathbb{C}$

is a deformation of $\overline{J}_{d,\mathcal{B}}$ to
$\mathrm{Tot}(N_{Z})\sslash\operatorname{SL}_{R}$. We can then reduce the
proof of 8.1 to the following statements:

  * •

$\mathrm{Tot}(N_{Z})\sslash\operatorname{SL}_{R}$ is homeomorphic to the
product (34);

  * •

$\mathrm{Tot}(N_{Z})\sslash\operatorname{SL}_{R}$ is homeomorphic to a
euclidean neighbourhood of $F_{b}$ in $\overline{J}_{d,B}$.

We prove the statements respectively in Lemma 8.6 and Section 8.2.

######  Remark 8.3.

In general, the deformation $\mathfrak{n}$ is not the deformation of
$\overline{J}_{d,B}$ to the normal cone of $F_{b}$. Indeed, a neighbourhood
$U$ of a point in $F_{b}$ is locally analytically isomorphic to
$X(\Gamma(C_{b})^{\pm},0)\times\mathbb{C}^{c^{\prime}}$ with
$c^{\prime}\coloneqq 4g^{\prime}-3-b_{1}(\Gamma)-s$ by [8, Theorem A]. Up to a
smooth factor, the degeneration to the normal cone deforms $U$ to the normal
cone of $X(\Gamma(C_{b})^{\pm},0)$ at $0$. The latter may fail to be
isomorphic to $X(\Gamma(C_{b})^{\pm},0)$: they are isomorphic only if the
homogeneous polynomial cutting $X(\Gamma(C_{b})^{\pm},0)$ have the same
degree, equivalently if the number of edges between different vertices of
$\Gamma(C_{b})^{\pm}$ is constant. On the other hand, $\mathfrak{n}$ is
locally trivial, so it deforms $U$ to
$X(\Gamma(C_{b})^{\pm},0)\times\mathbb{C}^{c^{\prime}}$.

###  8.1. Deformation theory of rank 1 torsion free sheaves

A slice of $Q^{ss}$ normal to an $\operatorname{SL}_{R}$-orbit of the pair
$(C_{b},\mathcal{I})$ is given by a miniversal deformation
$\mathrm{Def}(C_{b},\mathcal{I})$ of the pair; see [8, §6].

The space $\mathrm{Def}(C_{b},\mathcal{I})$ is endowed with a natural action
of the torus

$\operatorname{Aut}(\mathcal{I})\simeq
H^{0}(C_{b}^{\nu},\mathcal{O}^{*}_{C_{b}^{\nu}})\simeq\bigoplus^{r}_{i=1}H^{0}(C_{b,i},\mathcal{O}^{*}_{C_{b,i}})\simeq\mathbb{T}^{r}\ni(\lambda_{i})^{r}_{i=1},$

see [8, Def. 3.4]. Since $\operatorname{Aut}(\mathcal{I})$ is reductive, the
tangent space $T\mathrm{Def}(C_{b},\mathcal{I})$ splits into the product

$T\mathrm{Def}(C_{b},\mathcal{I})_{\mathrm{inv}}\oplus
T\mathrm{Def}(C_{b},\mathcal{I})_{\mathrm{var}},$

where the invariant part $T\mathrm{Def}(C_{b},\mathcal{I})_{\mathrm{inv}}$ is
fixed by the action of $\operatorname{Aut}(\mathcal{I})$, while the variant
part $T\mathrm{Def}(C_{b},\mathcal{I})_{\mathrm{var}}$ is the unique
$\operatorname{Aut}(\mathcal{I})$-invariant complement of
$T\mathrm{Def}(C_{b},\mathcal{I})_{\mathrm{inv}}$. In fact, these subspaces
have a modular interpretation:

$T\mathrm{Def}(C_{b},\mathcal{I})_{\mathrm{inv}}\simeq
T\mathrm{Def}^{lt}(C_{b},\mathcal{I})\qquad
T\mathrm{Def}(C_{b},\mathcal{I})_{\mathrm{var}}\simeq\prod_{e\in\Sigma}T\mathrm{Def}(C_{b,e},\mathcal{I}_{e}),$

where $\mathrm{Def}^{lt}(C_{b},\mathcal{I})$ is the miniversal deformation of
locally trivial deformations of $(C_{b},\mathcal{I})$, and
$\mathrm{Def}(C_{b,e},\mathcal{I}_{e})$ is a miniversal deformation of the
localised pair $(C_{b,e},\mathcal{I}_{e})$; see [8, §3 and §5]. In particular,
it follows from [8, §5] that $\operatorname{Aut}(\mathcal{I})$ acts on
$T\mathrm{Def}(C_{b,e},\mathcal{I}_{e})\simeq\operatorname{Spec}k[z_{e},w_{e}]$
as

(36) $z_{e}\mapsto\lambda_{s(e)}z_{e}\lambda_{t(e)}^{-1}\qquad
w_{e}\mapsto\lambda_{s(e)}^{-1}w_{e}\lambda_{t(e)}.$

Now let $q:Q^{ss}\to\overline{J}_{d,B}$ be the quotient map. Choose
$\mathcal{U}$ a universal rank 1 torsion-free sheaf over
$Q^{ss}\times_{B}\mathcal{C}$, and take
$p:Q^{ss}\times_{B}\mathcal{C}\to{Q}^{ss}$ the natural projection.

######  Lemma 8.4.

Let $N_{Z}$ be the normal bundle of $Z$ in $Q^{ss}$. There exists a
$\mathbb{T}^{r}$-equivariant splitting

$N_{Z}\simeq N^{\mathbb{T}^{r}}_{Z}\oplus
R^{0}p_{*}\mathcal{E}xt^{1}_{C_{b}}(\mathcal{U}|_{Z},\mathcal{U}|_{Z}),$

where $N^{\mathbb{T}^{r}}_{Z}$ is the $\mathbb{T}^{r}$-invariant part of
$N_{Z}$.

###### Proof.

The normal space to $Z$ in $Q^{ss}$ at $z\in Z$ with
$q(z)=(C_{b},\mathcal{I})$ admits the $\mathbb{T}^{r}$-equivariant splitting

(37)
$N_{Z,z}\simeq(T_{z}\mathrm{Def}^{lt}(C_{b},\mathcal{I})/q^{*}T_{(C_{b},\mathcal{I})}F_{b})\oplus\bigoplus_{e\in\Sigma}T_{z}\mathrm{Def}(C_{b,e},\mathcal{I}_{e}),$

and $\mathbb{T}^{r}$ acts trivially on the first summand
$(T_{z}\mathrm{Def}^{lt}(C_{b},\mathcal{I})/q^{*}T_{(C_{b},\mathcal{I})}F_{b})$
and as (36) on the second summand. By [24, Lemma 7.13] and the local-to-global
Ext spectral sequence,
$\bigoplus_{e\in\Sigma}T_{z}\mathrm{Def}(C_{b,e},\mathcal{I}_{e})$ can be
identified with

$H^{0}(C_{b},\mathcal{E}xt^{1}_{C_{b}}(\mathcal{I},\mathcal{I}))\simeq\bigoplus_{e\in\Sigma}H^{0}(C_{b},\mathcal{E}xt^{1}(\mathcal{I},\mathcal{I})_{e})\simeq\mathbb{C}^{s},$

i.e. the fibre of the vector bundle
$R^{0}p_{*}\mathcal{E}xt^{1}_{C_{b}}(\mathcal{U}|_{Z},\mathcal{U}|_{Z})$. ∎

######  Lemma 8.5.

$R^{0}p_{*}\mathcal{E}xt^{1}_{C_{b}}(\mathcal{U}|_{Z},\mathcal{U}|_{Z})$ is a
trivial vector bundle on $Z$.

###### Proof.

The group $F_{b}\simeq\operatorname{Pic}^{d^{\prime}}(C_{b}^{\nu})$ acts on
$Z$. Its action lifts to the vector bundle
$R^{0}p_{*}\mathcal{E}xt^{1}_{C_{b}}(\mathcal{U}|_{Z},\mathcal{U}|_{Z})$
parallelising it, as we show below.

A point in $Z$ represents a quotient $[\mathcal{O}_{C_{b}}^{\otimes
R}\to\nu_{*}M]$, where $M$ is a line bundle on $C_{b}^{\nu}$. Note that
$\mathcal{E}xt^{1}_{C_{b}}(\nu_{*}M,\nu_{*}M)$ is independent of the choice of
a basis for $\mathcal{O}_{X}^{\otimes R}$, and its rank is independent of $M$.
Hence,
$R^{0}p_{*}\mathcal{E}xt^{1}_{C_{b}}(\mathcal{U}|_{Z},\mathcal{U}|_{Z})$ is
the pull-back of a vector bundles $A$ on $F_{b}$.

The transitive action of $\operatorname{Pic}^{d^{\prime}}(C_{b}^{\nu})$ by
translation induces identification of the fibers of $A$. Indeed, if
$\mathcal{I}\coloneqq\nu_{*}M$ and
$\mathcal{I}^{\prime}\coloneqq\nu_{*}(M\otimes L)$ with
$L\in\operatorname{Pic}^{d^{\prime}}(C_{b}^{\nu})$, then $\mathcal{I}$ and
$\mathcal{I}^{\prime}$ are locally isomorphic, and so there exists an
invertible sheaf $L^{\prime}$ on $X$ such that
$\mathcal{I}^{\prime}=\mathcal{I}\otimes L^{\prime}$ with
$\nu^{*}L^{\prime}=L$; see for instance [11, Lemma 2.2.2 or 3.1.2/3.1.3].
Hence, we have

$\mathcal{E}xt^{1}_{C_{b}}(\nu_{*}(M\otimes L),\nu_{*}(M\otimes
L))=\mathcal{E}xt^{1}_{C_{b}}(\nu_{*}M\otimes L^{\prime},\nu_{*}M\otimes
L^{\prime})=\mathcal{E}xt^{1}_{C_{b}}(\nu_{*}M,\nu_{*}M).$

This gives a global trivialization of $A$. ∎

######  Lemma 8.6.

Let $N_{F_{b}/\overline{J}^{\circ}_{d}(\Gamma)}$ be the normal bundle of
$F_{b}$ in
$\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})\subseteq\overline{J}_{d,B}$
as constructed in 8.2. Denote by $\mathrm{Tot}(N)$ the total space of a vector
bundle $N$. Then

$\mathrm{Tot}(N_{Z})\sslash\operatorname{SL}_{R}\simeq\mathrm{Tot}(N_{F_{b}/\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})})\times
X(\Gamma(C_{b})^{\pm},0).$

###### Proof.

By Lemma 8.4 we have

$\mathrm{Tot}(N_{Z})\sslash\operatorname{SL}_{R}\simeq(\mathrm{Tot}(N^{\mathbb{T}^{r}}_{Z})\times_{Z}\mathrm{Tot}(R^{0}p_{*}\mathcal{E}xt^{1}_{C_{b}}(\mathcal{U}|_{Z},\mathcal{U}|_{Z}))\sslash\mathbb{T}^{r-1}))\sslash(\operatorname{SL}_{r}/\mathbb{T}^{r-1}),$

and Lemma 8.5 gives

$\mathrm{Tot}(R^{0}p_{*}\mathcal{E}xt^{1}_{C_{b}}(\mathcal{U}|_{Z},\mathcal{U}|_{Z}))\simeq
Z\times\mathbb{C}^{s}.$

The algebraic torus $\mathbb{T}^{r-1}=\mathbb{T}^{r}/\mathbb{T}$ acts on
$\mathbb{C}^{s}$ as prescribed in (36), which coincides with the torus action
defining $X(\Gamma(C_{b})^{\pm},0)$. Therefore, we obtain

$\displaystyle\mathrm{Tot}(R^{0}p_{*}\mathcal{E}xt^{1}_{C_{b}}(\mathcal{U}|_{Z},\mathcal{U}|_{Z}))\sslash\operatorname{SL}_{r}$
$\displaystyle\simeq((Z\times\mathbb{C}^{s})\sslash\mathbb{T}^{r-1})\sslash(\operatorname{SL}_{r}/\mathbb{T}^{r-1})$
$\displaystyle\simeq Z\sslash(\operatorname{SL}_{r}/\mathbb{T}^{r-1})\times
X(\Gamma(C_{b})^{\pm},0)\simeq F_{b}\times X(\Gamma(C_{b})^{\pm},0).$

Finally, the vector bundle $N^{\mathbb{T}^{r}}_{Z}$ descends to
$N_{F_{b}/\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})}$, because
$\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})$ parametrises the locally
trivial deformations of the pair $(C_{b},\mathcal{I})$. ∎

###  8.2. Stratified Ehresmann’s theorem

We use a stratified version of Ehresmann’s theorem to prove that
$\mathfrak{n}^{-1}(0)$ is a model of a tubular neighbourhood of $F_{b}$ in
$\overline{J}_{d,B}$.

######  Lemma 8.7.

The degeneration
$\mathfrak{n}\colon\mathcal{Q}\sslash\operatorname{SL}_{R}\to\mathbb{C}$ is
locally analytically trivial along $F_{b}$, i.e. for any $x\in
F_{b}\subset\mathfrak{n}^{-1}(0)\subset\mathcal{Q}\sslash\operatorname{SL}_{R}$
there exists a trivialising neighbourhood
$U\subset\mathcal{Q}\sslash\operatorname{SL}_{R}$ of $x$, and a local analytic
isomorphism

(38) $\psi\colon U\to(\mathbb{C}^{c^{\prime}}\times
X(\Gamma(C_{b})^{\pm},0))\times\mathbb{C},$

with $c^{\prime}\coloneqq 4g^{\prime}-3-b_{1}(\Gamma)-s$, such that
$\mathfrak{n}\colon U\to\mathbb{C}$ corresponds to the third projection
$(X(\Gamma(C_{b})^{\pm},0)\times\mathbb{C}^{c^{\prime}})\times\mathbb{C}\to\mathbb{C}$.

###### Proof.

For any $z\in Z$, the degeneration $\mathfrak{n}\colon
Z\times\mathbb{C}\subseteq\mathcal{Q}\to\mathbb{C}$ admits a section
$\\{z\\}\times\mathbb{C}$. By (37), an analytic neighbourhood of
$\\{z\\}\times\mathbb{C}$ is locally isomorphic to

$(T_{z}O_{z}\times
T_{z}\mathrm{Def}^{lt}(C_{b},\mathcal{I})\times\bigoplus_{e\in\Sigma}T_{z}\mathrm{Def}(C_{b,e},\mathcal{I}_{e}))\times\mathbb{C},$

where $O_{z}$ is the $\operatorname{SL}_{R}$-orbit of $z$. By Luna’s slice
theorem, an analytic neighbourhood of $\\{[z]\\}\times\mathbb{C}$ in
$\mathcal{Q}\sslash\operatorname{SL}_{R}$ is locally isomorphic to

$\displaystyle(T_{z}\mathrm{Def}^{lt}(C_{b},\mathcal{I})\times$
$\displaystyle\bigoplus_{e\in\Sigma}T_{z}\mathrm{Def}(C_{b,e},\mathcal{I}_{e}))\sslash\mathbb{T}^{r-1}\times\mathbb{C}$
$\displaystyle\simeq(T_{(C_{b},\mathcal{I})}\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})\times
X(\Gamma(C_{b})^{\pm},0))\times\mathbb{C}\simeq(\mathbb{C}^{c^{\prime}}\times
X(\Gamma(C_{b})^{\pm},0))\times\mathbb{C}.$

∎

###### Proof of 8.1.

Let’s first show that a relatively compact euclidean neighbourhood of $F_{b}$
in $\mathfrak{n}^{-1}(0)$ is stratified diffeomorphic to a euclidean
neighbourhood of $F_{b}$ in $\mathfrak{n}^{-1}(0)$ in
$\mathfrak{n}^{-1}(t)=\overline{J}_{d,B}$, with $t\neq 0$.

By Lemma 8.7 the degeneration
$\mathfrak{n}\colon\mathcal{Q}\sslash\operatorname{SL}_{R}\to\mathbb{C}$ is
locally trivial. This means that we can cover
$F_{b}\subset\mathfrak{n}^{-1}(0)$ by trivialising open sets $U_{k}$, with $k$
in the index set $K$, as in Lemma 8.7. Take now the vector field
$v_{k}\coloneqq d\psi_{k}^{-1}(\partial_{t})$ on $U_{k}$, where $\partial_{t}$
is a non-vanishing vector field on $\mathbb{C}$ and $\psi_{k}$ is the
stratified diffeomorphism (38). Consider a partition of unity
$\\{\chi_{k}\\}_{k\in K}$ subordinate to the open cover $\\{U_{k}\\}_{k\in
K}$. Then the flow of the vector field $v\coloneqq\sum_{k\in K}\chi_{k}\cdot
v_{k}$ defines the required stratified diffeomorphism.

We conclude now by noticing that the neighbourhood of $F_{b}$ in
$\mathfrak{n}^{-1}(0)$ can be chosen homeomorphic to the product

$\mathrm{Tot}(N_{F_{b}/\operatorname{Pic}^{d^{\prime}}(\mathcal{C}^{\nu})})\times
X(\Gamma(C_{b})^{\pm},0)\underset{\text{homeo}}{\simeq}F_{b}\times\mathbb{C}^{3g^{\prime}-3-s}\times
X(\Gamma(C_{b})^{\pm},0)$

by Lemma 8.6. ∎

##  9\. Full support theorem for universal compactified Jacobians

######  Lemma 9.1 (Weak abelian fibration).

Let $f\colon\mathcal{C}\to B$ be a flat projective versal family of semistable
curves and arithmetic genus $g^{\prime}$, and $\pi^{P}\colon
P\coloneqq\operatorname{Pic}^{0}(\mathcal{C}/B)\to B$ be the relative degree 0
Picard scheme parametrising line bundles on the fibres of $f$ whose
restrictions to each irreducible components are of degree $0$. Let
$\pi\colon\overline{J}_{d,B}\to B$ be the relative compactified Jacobian with
respect to the canonical polarization. Then the triple
$(\overline{J}_{d,B},P,B)$ is a weak abelian fibration.

###### Proof.

We need to check condition (1), (2) and (3) in 3.1. The relative Picard scheme
$\pi^{P}\colon P\to B$ has fibres of pure dimension $g^{\prime}$, and
$\dim\overline{J}_{d,B}=g^{\prime}+\dim B$; see for instance Theorem 7.4.
Hence Condition (1) holds. The proof of [22, Lem. 3.4.5] gives the affineness
of the stabilisers, i.e. Condition (2). The same argument of [58, §4.12] gives
Condition (3); see also [15, Thm 3.3.1] where one can drop the assumption that
$\mathcal{C}$ is a family of curves embedded in a surface. ∎

######  Proposition 9.2 (Whitney stratification of $\overline{J}_{d,B}$).

In the hypothesis of Lemma 9.1, let $\mathcal{I}$ be a rank 1 torsion free
sheaf on $\mathcal{C}_{b}\coloneqq f^{-1}(b)$ which fails to be locally free
at the nodes $\Sigma$ of $\mathcal{C}_{b}$. Set
$\Gamma_{\mathcal{C}_{b}}(\Sigma)$ to be the dual graph of any curve obtained
from $\mathcal{C}_{b}$ by smoothing the nodes not in $\Sigma$. Then there
exists a complex Whitney stratification

$\overline{J}_{d,B}=\bigsqcup_{\Gamma}\overline{J}^{\circ}_{d}(\Gamma),$

where $\overline{J}^{\circ}_{d}(\Gamma)$ is the locus of rank 1 torsion free
sheaves in $\overline{J}_{d,B}$ whose $\Gamma_{\mathcal{C}_{b}}(\Sigma)$ is
isomorphic to $\Gamma$, and we have

(39)
$\operatorname{codim}_{\overline{J}_{d,B}}\overline{J}^{\circ}_{d}(\Gamma)\geq
2\operatorname{codim}_{\pi^{-1}(t)}(\overline{J}^{\circ}_{d}(\Gamma)\cap\pi^{-1}(t)).$

###### Proof.

The first statement is a reformulation of [8, Thm A]. For the inequality (39),
observe that if $\Gamma$ has $r$ vertices and $s$ edges, then [8, Thm A] gives

$\displaystyle\operatorname{codim}_{\overline{J}_{d,B}}\overline{J}^{\circ}_{d}(\Gamma)=\dim
X(\Gamma^{\pm},0)=2s-r+1,$
$\displaystyle\operatorname{codim}_{\pi^{-1}(t)}(\overline{J}^{\circ}_{d}(\Gamma)\cap\pi^{-1}(t))=s-r+1.$

∎

######  Lemma 9.3.

In the hypothesis of Lemma 9.1, we have

$\tau_{>2R}(R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q}))=0,\qquad
R\coloneqq\dim\overline{J}_{d,B}-\dim B=g^{\prime}.$

###### Proof.

It follows from 3.3 and 9.2. ∎

######  Lemma 9.4.

In the hypothesis of Lemma 9.1, any support $Z$ of
$R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q})$ satisfies

$\operatorname{codim}Z=\delta_{Z}$

###### Proof.

The inequality $\operatorname{codim}Z\geq\delta_{Z}$ appears for instance in
[56, Fact 2.4]. The reverse inequality $\operatorname{codim}Z\leq\delta_{Z}$
follows from Lemma 9.1, Lemma 9.3, and [48, Theorem 1.8] or Theorem 3.2. ∎

Let $\pi\colon\overline{J}_{d,B}\to B$ be the relative compactified Jacobian
with respect to the canonical polarization of a versal deformation of
semistable curves of arithmetic genus $g^{\prime}$. We define the invariant

(40) $\Psi_{g^{\prime}}\coloneqq\dim\overline{J}_{d,B}-2\dim
B=4{g^{\prime}}-3-2(3g^{\prime}-3)=3-2g^{\prime}.$

######  Theorem 9.5 (Full support for Jacobians of versal families).

Let $f\colon\mathcal{C}\to B$ be a flat projective versal family of semistable
curves and arithmetic genus $g^{\prime}$, and $U\subset B$ the open subscheme
over which the morphism $f$ is smooth. Let $\pi:\overline{J}_{d,B}\to B$ be
the relative compactified Jacobian with respect to the canonical polarization.
Then the complex $R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q})$
has full support on $B$, i.e.

(41)
$R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q})\simeq\bigoplus^{2g}_{l=0}\operatorname{IC}(B,\Lambda^{l}(R^{1}\pi_{*}\mathbb{Q})|_{U})[-l].$

In particular, for any $d,d^{\prime}\in\mathbb{Z}$ there is an isomorphism of
mixed Hodge modules

(42) $R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q})\simeq
R\pi_{*}\operatorname{IC}(\overline{J}_{d^{\prime},B},\mathbb{Q}).$

###### Proof.

The argument of [48, §3] can be adapted to work in the present setting. The
base $B$ admits a stratification by type

$B=\bigsqcup B^{\circ}_{\underline{g}^{\prime}},$

which extends (4). For any $r\geq 1$, and for any $r$-tuples of positive
integers $\underline{g}^{\prime}=(g^{\prime}_{i})$, the closed points $b\in B$
in $B^{\circ}_{\underline{g}^{\prime}}$ corresponds to the fibers of $f$ of
the form $\mathcal{C}_{b}=\bigcup_{i}\mathcal{C}_{i}$, where $\mathcal{C}_{i}$
are irreducible curves of genus $g^{\prime}_{i}=g(\mathcal{C}_{i})$.

Assume that the irreducible subvariety $Z\subseteq B$ is a support of
$R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q})$, and that the
generic point of $Z$ is contained in $B^{\circ}_{\underline{g}^{\prime}}$. As
in [48, Prop. 4.4], the inequality $\operatorname{codim}Z\geq\delta_{Z}$ can
be improved to

(43)
$\Psi_{g^{\prime}}+\operatorname{codim}Z\geq\sum^{r}_{i=1}\Psi_{g^{\prime}_{i}}+\delta_{Z}.$

Combining (43), (40) and Lemma 9.4, we obtain

$3(r-1)+2(g-\sum^{r}_{i=1}g_{i})=0.$

The only possibility is that $r=1$. Hence, $Z=B$, which implies (41) and (42).
∎

######  Remark 9.6.

The proof of Theorem 9.5 does not work for the Hitchin fibration. In this
case, the invariant $\Psi=\dim M(n,d)-2\dim A_{n}$ vanishes, and the
inequality (43) is inconclusive.

##  10\. Proof of the main theorems

###### Proof of Theorem 1.1 (Hodge-to-singular correspondence).

In view of the decomposition (17), it suffices to show that for any partition
$\underline{n}$ of $n$ the isomorphism (5) holds on an open euclidean
neighbourhood of a general $a\in S^{\times}_{\underline{n}}$ containing a
Zariski-dense open set of $S^{\times}_{\underline{n}}$. By 5.6, it is actually
enough to show the isomorphism on the subsets $A_{a}$ constructed in the proof
of 5.5.

By 5.5, proper base change gives

$R\chi(n,d)_{*}(\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q})|_{\chi(n,d)^{-1}(A_{a})})\simeq(R\pi_{*}\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q}))|_{A_{a}}.$

For any étale chart $B\to\mathcal{B}$, Theorem 9.5 says that
$R\pi_{*}\operatorname{IC}(\overline{J}_{e,B},\mathbb{Q})$ and
$R\pi_{*}\operatorname{IC}(\overline{J}_{d,B},\mathbb{Q})$ have full support,
which implies that

$(R\pi_{*}\operatorname{IC}(\overline{J}_{e,\mathcal{B}},\mathbb{Q}))|_{A_{a}}\simeq(R\pi_{*}\operatorname{IC}(\overline{J}_{d,\mathcal{B}},\mathbb{Q}))|_{A_{a}}.$

Since $\chi(n,e)^{-1}(A_{a})$ is regularly embedded in
$\overline{J}_{e,\mathcal{B}}$, we have

$\operatorname{IC}(\overline{J}_{e,\mathcal{B}},\mathbb{Q})|_{\chi(n,e)^{-1}(A_{a})}\simeq\mathbb{Q}_{\chi(n,e)^{-1}(A_{a})}$

Together with Theorem 7.5 and 7.6, we obtain Theorem 1.1. ∎

###### Proof of Theorem 1.3.

By Theorem 7.5, the descent of $\mathcal{L}^{\prime}_{\underline{n}}$ on
$S^{\times}_{\underline{n}}$ in 7.6 is a sublocal system of the sheaf
$\mathscr{L}_{\underline{n}}(e)$ in the Ngô strings of Theorem 3.11. For
dimensional reason, they must coincide by [17, Thm 6.11] and 4.11.(8). ∎

###### Proof of Theorem 1.4.

Theorem 1.1 for $d=0$ yields

(44)
$R\chi(n,e)_{*}\mathbb{Q}_{M(n,e)}|_{A^{\mathrm{red}}_{n}}\simeq\bigoplus_{\underline{n}\dashv\,n}R\chi(n,0)_{*}\operatorname{IC}(M_{\underline{n}}(0),\chi(n,0)^{*}\mathcal{L}_{\underline{n}})[2(\dim
S_{n}-c(g,n))]|_{A^{\mathrm{red}}_{n}}.$

On the open set of $S_{\underline{n}}$ over which the Hitchin map
$\chi(n,0)\colon M_{\underline{n}}(0)\to S_{\underline{n}}$ is smooth, there
are isomorphisms

$R\chi(n,0)_{*}\operatorname{IC}(M_{\underline{n}}(0),\chi(n,0)^{*}\mathcal{L}_{\underline{n}})\simeq\bigoplus^{2\dim
S_{\underline{n}}}_{l=0}\Lambda^{l}_{\underline{n}}\otimes\mathcal{L}_{\underline{n}}[-l]\simeq\mathscr{S}_{\underline{n}}.$

Therefore, by the decomposition theorem [3, 62], the string
$\mathscr{S}_{\underline{n}}$ is a direct summand of

$R\chi(n,0)_{*}\operatorname{IC}(M_{\underline{n}}(0),\chi(n,0)^{*}\mathcal{L}_{\underline{n}})$

on the whole $S_{\underline{n}}$. Now Theorem 1.3 gives

$\bigoplus_{\underline{n}\dashv\,n}\mathscr{S}_{\underline{n}}|_{A^{\mathrm{red}}_{n}}\langle\operatorname{codim}S_{\underline{n}}\rangle\simeq
R\chi(n,e)_{*}\mathbb{Q}_{{M}(n,e)}|_{A^{\mathrm{red}}_{n}}\simeq\text{ RHS of
}\eqref{eq:mainthm0}\supseteq\bigoplus_{\underline{n}\dashv\,n}\mathscr{S}_{\underline{n}}|_{A^{\mathrm{red}}_{n}}\langle\operatorname{codim}S_{\underline{n}}\rangle.$

This implies that the complexes
$R\chi(n,0)_{*}\operatorname{IC}(M_{\underline{n}}(0),\chi(n,0)^{*}\mathcal{L}_{\underline{n}})$
have full support. For $\underline{n}=\\{n\\}$, this gives that
$R\chi(n,0)_{*}\operatorname{IC}({M}(n,0),\mathbb{Q})$ has full support. ∎

###### Proof of Theorem 1.7.

By Theorem 1.3 and Theorem 1.4, it suffices to check that no summand of
$R\chi(2,1)_{*}\mathbb{Q}_{{M}(2,1)}$ and
$R\chi(2,0)_{*}\operatorname{IC}({{M}(2,0)},\mathbb{Q})$ is supported over
$A_{2}\setminus A^{\mathrm{red}}_{2}=\\{0\\}$. This is indeed the case by 1.6
(3.9). ∎

######  Remark 10.1.

Let $\underline{n}=(n_{1},\ldots,n_{r})=1^{\alpha_{1}}\cdot\ldots\cdot
n^{\alpha_{n}}$ be a partition of $n$, where $\alpha_{i}$ is the number of
elements in $n$ equal to $i$. Denote by
$\mathfrak{S}_{\underline{n}}\coloneqq\prod^{n}_{i=1}\mathfrak{S}_{\alpha_{i}}$
the subgroup of the symmetric group $\mathfrak{S}_{r}$ stabilizing
$\underline{n}$. Then the horizontal arrows of the following commutative
square

${\prod^{r}_{i=1}{M}(n_{i},d_{i})}$${M_{\underline{n}}(d)}$${(\mathcal{E}_{i},\phi_{i})^{r}_{i=1}}$${(\bigoplus^{r}_{i=1}\mathcal{E}_{i},\bigoplus^{r}_{i=1}\phi_{i})}$${\prod^{r}_{i=1}A_{n_{i}}}$${S_{\underline{n}},}$${(\mathrm{char}(\phi_{i}))^{r}_{i=1}}$${\prod^{r}_{i=1}\mathrm{char}(\phi_{i})}$$\scriptstyle{\prod^{r}_{i=1}\chi(n,d)}$$\scriptstyle{\chi(n,d)}$$\scriptstyle{\mathrm{mult}_{\underline{n}}}$

are quotient maps by the action of $\mathfrak{S}_{\underline{n}}$. By [17,
Cor. 6.20], the monodromy of the local systems $\mathcal{L}_{\underline{n}}$
is given by the restriction to the subgroup $\mathfrak{S}_{\underline{n}}$ of
the representation of $\mathfrak{S}_{r}$ induced by a primitive character of a
maximal cyclic subgroup. In particular,
$\mathrm{mult}^{*}_{\underline{n}}\mathcal{L}_{\underline{n}}$ is a trivial
local system. Hence, $\mathfrak{S}_{\underline{n}}$ acts on the sheaf
$\operatorname*{\scalerel*{\boxtimes}{\textstyle\sum}}^{r}_{i=1}R\chi(n_{i},d_{i})_{*}\operatorname{IC}(M(n_{i},d_{i}),\mathbb{Q})\otimes\mathbb{Q}^{\operatorname{rank}\mathcal{L}_{\underline{n}}}$
via the $\mathfrak{S}_{\underline{n}}$-representation
$\mathcal{L}_{\underline{n}}$. This means that the summand of the RHS of (5)
are the $\mathfrak{S}_{\underline{n}}$-invariant part of

(45)
$\operatorname*{\scalerel*{\boxtimes}{\textstyle\sum}}^{r}_{i=1}R\chi(n_{i},d_{i})_{*}\operatorname{IC}(M(n_{i},d_{i}),\mathbb{Q})\otimes\mathbb{Q}^{\operatorname{rank}\mathcal{L}_{\underline{n}}}\langle\operatorname{codim}S_{\underline{n}}\rangle.$

In particular, the summands of the RHS of (6) are
$\mathfrak{S}_{\underline{n}}$-subrepresentations of

$\bigotimes^{r}_{i=1}I\\!H^{k-2\operatorname{codim}S_{n}}(M(n_{i},d_{i})^{\mathrm{red}},\mathbb{Q}).$

###### Proof of 1.9.

The summation indices and the summands in (5) depends only on $\gcd(n,d)$. ∎

###  10.1. Dependence of $I\\!H^{*}({M}(n,d),\mathbb{Q})$ on degree

The description of the summands of the decomposition theorem for the Hitchin
fibration in the intermediate case, $\gcd(n,d)\neq 0$ or $\neq 1$, is more
subtle. It is not clear if one can get a close formula for them, but certainly
(5) offers a recursive method. Combining Theorem 1.1 and Theorem 1.4, one
obtain the isomorphism in $D^{b}MHM_{\mathrm{alg}}(A^{\mathrm{red}}_{n})$

(46) $\small
R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})\oplus\bigoplus_{\begin{subarray}{c}\underline{n}\in\mathcal{P}_{d},\,\underline{n}\neq\\{n\\}\end{subarray}}R\chi(n,d)_{*}\operatorname{IC}(M_{\underline{n}}(d),\chi(n,d)^{*}\mathcal{L}_{\underline{n}})\langle\operatorname{codim}S_{\underline{n}}\rangle\simeq\bigoplus_{\underline{n}=\\{n_{i}\\}\dashv\,n}\mathscr{S}_{\underline{n}}.$

Recursively from this identity, one can determine the Ngô strings for
$R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})$, and the subvarieties
$S_{\underline{n}}$ which are actual support of
$R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})$. We refer to [52] for
the explicit computation of the ranks of the Ngô strings. Here we limit to
illustrate the first non-trivial example $n=4$. More care is needed to
determine explicitly the monodromy of the associated local systems.

######  Example 10.2 ($n=4$).

We determine the rank of the local systems $\mathscr{L}_{\underline{n}}(d)$ in
Theorem 3.11 for $n=4$. To this end, observe that the Hodge-to-singular
correspondence gives

(47) $R\chi(2,4)_{*}\operatorname{IC}({M}(2,4),\mathbb{Q})\oplus
R\chi(2,4)_{*}\operatorname{IC}(M_{\\{2,2\\}}(2),\chi(4,2)^{*}\mathcal{L}_{\\{2,2\\}})\langle
8g-9\rangle\simeq\bigoplus_{\underline{n}=\\{n_{i}\\}\dashv\,4}\mathscr{S}_{\underline{n}}.$

Consider the commutative square

${{M}(2,1)^{2}}$${M_{\\{2,2\\}}(2)}$${A_{2}^{2}}$${S_{\\{2,2\\}}.}$$\scriptstyle{\chi(2,1)^{2}}$$\scriptstyle{\chi(4,2)}$$\scriptstyle{\mathrm{mult}_{\\{2,2\\}}}$

The map $\mathrm{mult}_{\\{2,2\\}}$ factors through the normalization
$\nu_{\\{2,2\\}}\colon\mathrm{Sym}^{2}A_{2}\to S_{\\{2,2\\}}$ of
$S_{\\{2,2\\}}$ as follows:

$\mathrm{mult}_{\\{2,2\\}}\colon
A^{2}_{2}\xrightarrow{\eta_{\\{2,2\\}}}\mathrm{Sym}^{2}A_{2}\xrightarrow{\nu_{\\{2,2\\}}}S_{\\{2,2\\}}.$

Set $\underline{n}=\\{2,2\\},\\{2,1,1\\}$ or $\\{1,1,1,1\\}$. The degree of
$\nu_{\\{2,2\\}}$ over $S_{\underline{n}}$, denoted by $d_{\underline{n}}$, is

$d_{\underline{n}}=\begin{cases}1\quad\text{ for
}\underline{n}=\\{2,2\\},\\{2,1,1\\}\text{, and}\\\ 3\quad\text{ for
}\underline{n}=\\{1,1,1,1\\}.\end{cases}$

In fact, the invariant $d_{\underline{n}}$ counts in how many ways one can sum
the elements of the partition $\underline{n}$ to get $\\{2,2\\}$. The map
$\eta_{\\{2,2\\}}$ instead is étale at the general points of
$S_{\underline{n}}$. By the behaviour of intersection cohomology under
normalization and étale maps, we obtain that
$R\chi(2,4)_{*}\operatorname{IC}(M_{\\{2,2\\}}(2),\chi(4,2)^{*}\mathcal{L}_{\\{2,2\\}})$
is locally isomorphic to

(48)
$R\chi(2,1)^{2}_{*}\operatorname{IC}({M}(2,1)^{2},\mathbb{Q})\otimes\mathbb{Q}^{\operatorname{rank}\mathcal{L}_{\\{2,2\\}}}\otimes\mathbb{Q}^{d_{\underline{n}}}$

at the general points of $S_{\underline{n}}$. We omit the tensorisation by
$\otimes\mathbb{Q}^{\operatorname{rank}\mathcal{L}_{\\{2,2\\}}}$ since
$\operatorname{rank}\mathcal{L}_{\\{2,2\\}}=1$. Indeed,

(49)
$\mathscr{L}_{\underline{n}}(d)\leq\operatorname{rank}\mathcal{L}_{\underline{n}}=(|\underline{n}|-1)!$

by Theorem 1.1 and [17, cor. 6.20]. By the Künneth formula and Theorem 1.7, we
have

$R\chi(2,1)^{2}_{*}\operatorname{IC}({M}(2,1)^{2},\mathbb{Q})\simeq(\mathscr{S}_{\\{2\\}}\oplus\mathscr{S}_{\\{1,1\\}})\boxtimes(\mathscr{S}_{\\{2\\}}\oplus\mathscr{S}_{\\{1,1\\}}),$

so all the Ngô strings of
$R\chi(2,1)^{2}_{*}\operatorname{IC}({M}(2,1)^{2},\mathbb{Q})$ start with a
local system of rank one. By (47) and (48), this implies that
$\operatorname{rank}\mathscr{L}_{\underline{n}}(2)=\operatorname{rank}\mathcal{L}_{\underline{n}}-d_{\underline{n}}$.
In particular, $S_{\\{2,2\\}}$ is not a support of
$R\chi(2,4)_{*}\operatorname{IC}({M}(2,4),\mathbb{Q})$. We summarise the
results in the following table.

$\operatorname{gcd}(n,d)$ |  {4} |  {3,1} |  {2,2} |  {2,1,1} |  {1,1,1,1}
---|---|---|---|---|---
0 | 1 | 0 | 0 | 0 | 0
1 | 1 | 1 | 1 | 2 | 6
2 | 1 | 1 | 0 | 1 | 3
Table 1. The entries of the table are the rank of the local systems
$\mathscr{L}_{\underline{n}}(d)$ for $n=4$.

Theorem 3.11 says that the complex
$R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})$ is a direct sum of Ngô
strings

$\mathscr{S}(\mathscr{L}_{\underline{n}}(d))\coloneqq\bigoplus^{2\dim
S_{\underline{n}}}_{l=0}\operatorname{IC}(S_{\underline{n}},\Lambda^{l}_{\underline{n}}\otimes\mathscr{L}_{\underline{n}}(d))[-l]\langle\operatorname{codim}S_{\underline{n}}\rangle.$

By 10.1, the summands
$R\chi(n,d)_{*}\operatorname{IC}(M_{\underline{n}}(d),\chi(n,d)^{*}\mathcal{L}_{\underline{n}})$
of the LHS of (46) are the $\mathfrak{S}_{\underline{n}}$-invariant part of

$\operatorname*{\scalerel*{\boxtimes}{\textstyle\sum}}^{r}_{i=1}R\chi(n_{i},d_{i})_{*}\operatorname{IC}(M(n_{i},d_{i}),\mathbb{Q})\otimes\mathbb{Q}^{\operatorname{rank}\mathcal{L}_{\underline{n}}},$

where $n_{i}<n$ and $d_{i}<d$. Therefore, using (46), one could rewrite the
local system $\mathscr{L}_{\underline{n}}(d)$ as a combinations of direct
sums, linear quotients and $\mathfrak{S}_{\underline{n}^{\prime}}$-quotients
of $\mathscr{L}_{\underline{n}^{\prime}}(d^{\prime})$,
$\mathcal{L}_{\underline{n}}$ and $\mathcal{L}_{\underline{n}^{\prime}}$, with
$n^{\prime}<n$ and $d^{\prime}<d$. Recursively, we could remove the dependence
of $\mathscr{L}_{\underline{n}}(d)$ on
$\mathscr{L}_{\underline{n}^{\prime}}(d^{\prime})$, and write
$\mathscr{L}_{\underline{n}}(d)$ as depending only on
$\mathcal{L}_{\underline{k}}$ with $k\leq n$. For any partition
$\underline{n}=(n_{1},\ldots,n_{r})$, 10.1 says that the Ngô string associated
to $\mathcal{L}_{\underline{n}}$, namely

$\mathscr{S}_{\underline{n}}\coloneqq\bigoplus^{2\dim
S_{\underline{n}}}_{l=0}\operatorname{IC}(S_{\underline{n}},\Lambda^{l}_{\underline{n}}\otimes\mathcal{L}_{\underline{n}})[-l]\langle\operatorname{codim}S_{\underline{n}}\rangle,$

is the $\mathfrak{S}_{\underline{n}}$-invariant part of
$(\operatorname*{\scalerel*{\boxtimes}{\textstyle\sum}}^{r}_{i=1}\mathscr{S}_{n_{i}})^{\operatorname{rank}\mathcal{L}_{\underline{n}}}$.
This means that $R\chi(n,d)_{*}\operatorname{IC}({M}(n,d),\mathbb{Q})$ can be
expressed in terms of the Ngô strings $\mathscr{S}_{k}\simeq
R\chi(k,0)_{*}\operatorname{IC}({M}(k,0),\mathbb{Q})$ with $k\leq n$. In other
words, one obtain a recursive (though highly nontrivial) formula for
$IH^{*}({M}^{\text{red}}(n,d),\mathbb{Q})$ in terms of
$IH^{*}({M}^{\text{red}}(k,0),\mathbb{Q})=H^{*}(\mathscr{S}_{k})$ with $k\leq
n$.

###  10.2. Some applications of the Hodge-to-singular correspondence

###### Proof of 1.10.

Taking global cohomology in (17), we have

$I\\!H^{\leq k}(M(n,d))\simeq I\\!H^{\leq k}(M(n,0))$

with $k=\min\\{2(\dim\mathcal{A}_{n}-\dim\mathcal{A}^{\text{red}}_{n}),\dim
M_{\underline{n}}\colon\underline{n}\neq\\{n\\}\\}$. By [4, Lem. 2.2], we have

(50) $\displaystyle k$
$\displaystyle=2(n^{2}(g-1)+1)-2\max\bigg{\\{}r+(g-1)\sum^{r}_{i=1}n^{2}_{i}\,\bigg{|}\,m_{i},n_{i},\sum^{r}_{i=1}m_{i}r_{i}=n\bigg{\\}}$
$\displaystyle=4(g-1)(n-1)-2=O(n).$

This gives the independence of the limits (9) by the degree. The finiteness
follows from the generation of the cohomology of $M(n,e)$ with $\gcd(e,n)=1$
by finitely many tautological classes in [46, Thm 7]. ∎

Recall that the perverse filtration on $I\\!H^{*}(M(n,d))$ associated to the
Hitchin map $\chi(n,d)$ is given by

(51) $P_{j}I\\!H^{k}(M(n,d),\mathbb{Q})=\ker\\{I\\!H^{k}(M(n,d),\mathbb{Q})\to
I\\!H^{k}(\chi(n,d)^{-1}(\Lambda^{k-j-1}),\mathbb{Q})\\},$

where $\Lambda^{s}$ is a general $s$-dimensional affine hyperplane of $A_{n}$;
see [21, Thm 4.1.1]. By (50), $\Lambda^{s}$ does not intersect the supports of
the RHS of (5) which are properly contained in $A_{n}$, provided that
$s<2(g-1)(n-1)-1$. As a result, we obtain the following proposition.

######  Proposition 10.3.

The graded piece of the perverse filtration

$\operatorname{Gr}^{P}_{j}I\\!H^{k}(M(n,d),\mathbb{Q})\coloneqq
P_{j}I\\!H^{k}(M(n,d),\mathbb{Q})/P_{j-1}I\\!H^{k}(M(n,d),\mathbb{Q})$

are independent of the degree $d$ for $j>k-2(g-1)(n-1)+1$.

###### Proof of 1.11.

By (51) we have

$\operatorname{Gr}^{P}_{k}I\\!H^{k}(M(n,d),\mathbb{Q})\simeq\mathrm{Im}\\{I\\!H^{k}(M(n,d),\mathbb{Q})\to
H^{k}(M_{a},\mathbb{Q})\\}.$

Let $e$ be an integer coprime with $n$. Since

$\operatorname{Gr}^{P}_{k}I\\!H^{k}(M(n,d),\mathbb{Q})\simeq\operatorname{Gr}^{P}_{k}H^{k}(M(n,e),\mathbb{Q})$

by 10.3, it suffice to determine the latter. Let $(\mathbb{E},\Phi)$ be the
universal family of Higgs bundles on $C\times M(n,e)$ normalised such that

$\mathrm{ch}_{1}(\mathbb{E})|_{p\times M(n,e)}=0\in
H^{2}(M(n,e),\mathbb{Q}),\qquad\mathrm{ch}_{1}(\mathbb{E})|_{C\times q}=0\in
H^{2}(C,\mathbb{Q}).$

For any $\gamma\in H^{j}(C,\mathbb{Q})$, the tautological class $c(\gamma,k)$
is the integral

$\int_{\gamma}\mathrm{ch}^{k}(\mathbb{E})\in H^{j+2k-2}(M(n,d),\mathbb{Q}).$

By [46, Thm 7], the cohomology $H^{k}(M(n,e),\mathbb{Q})$ is generated by
tautological classes as $\mathbb{Q}$-algebra. The proof of [16, Cor. 5.1.3]
works in arbitrary rank, and shows that the only tautological classes that do
not vanish along $M_{a}$ are $c(\gamma,1)$ with $\gamma\in
H^{1}(C,\mathbb{Q})$ and the $\chi(n,e)$-relative ample class $\alpha\coloneqq
c([c],2)$, where $[c]$ generates $H^{2}(C,\mathbb{Q})$. We conclude that

(52) $\operatorname{Gr}^{P}_{*}H^{*}(M(n,e),\mathbb{Q})=\langle
c(\gamma,1)|_{M_{a}},\alpha|_{M_{a}}\rangle=H^{\bullet}(C,\mathbb{Q})\otimes{\mathbb{Q}[\alpha|_{M_{a}}]}/{(\alpha|_{M_{a}}^{\dim
M(n,d)+1})}.$

The result for $\check{M}(n,d)$ is an immediate corollary of (52) since

$\operatorname{Gr}^{P}_{*}H^{*}(M(n,d),\mathbb{Q})\simeq\operatorname{Gr}^{P}_{*}H^{*}(\check{M}(n,d),\mathbb{Q})\otimes
H^{\bullet}(C,\mathbb{Q});$

see for instance [16, §4.4]. ∎

###  10.3. Mukai moduli spaces

Let $S$ be a K3 surface. Given an effective Mukai vector888i.e. there exists a
coherent sheaf $\mathcal{F}$ on $S$ such that
$v=(rk(\mathcal{F}),c_{1}(\mathcal{F}),\chi(\mathcal{F})-rk(\mathcal{F}))$.
$v\in H^{*}_{\mathrm{alg}}(S,\mathbb{Z})$, we denote by $M(S,v,H)$ the moduli
space of Gieseker semistable sheaves on $S$ with Mukai vector $v$ with respect
to a polarization $H$; see [66, §1]. If $v$ is the Mukai vector of a pure
dimensional one sheaf $\mathcal{F}$, then $v=(0,D,\chi(\mathcal{F}))$, where
$D\in\operatorname{Pic}(S,\mathbb{Z})$ is the class of the curve supporting
$\mathcal{F}$. Taking (Fitting) supports defines a Lagrangian fibration

$M(S,v,H)\to|D|,\qquad\mathcal{F}\mapsto\mathrm{Supp}(\mathcal{F}),$

whose fibers are compactified Jacobians. The Hodge-to-singular correspondence
extends to this context with no change, so does 1.10.

######  Corollary 10.4.

Let $S$ be a K3 surface with $\operatorname{Pic}(S)\simeq\mathbb{Z}$ generated
by the class of a smooth curve $C$ of genus $g\geq 2$. Then
$\lim_{n\to\infty}\dim I\\!H^{p,q}(M(S,(0,nC,\chi),C))$ is finite and
independent of $\chi$.

###### Proof.

The same proof of 1.10 holds in this context too. The assumption on the Picard
rank grants that the codimension of the locus of non-reduced curves in $|nC|$
grows linearly in $n$. ∎

######  Definition 10.5.

[60, Def. 1.15] We say that $(S,v,H)$ is a $(m.k)$-triple such that:

  1. (1)

the polarization $H$ is primitive and $v$-generic in the sense of [60, §2.1];

  2. (2)

we have $v=mw$, where $w$ is primitive and $w^{2}=2k$;

  3. (3)

if $w=(0,w_{1},w_{2})$ and $\rho(S)>1$, then $w_{2}\neq 0$.

Note that the stabilization of the cohomology of $M(S,v,H)$ ([12, Conjecture
1.1]) is known in the literature if $v$ is primitive; see [12, p. 4] and
references therein. We can now drop the primitivity assumption.

######  Corollary 10.6.

Let $v^{2}=mw^{2}$ and $w^{2}=2k$, with $v,w\in
H^{*}_{\mathrm{alg}}(S,\mathbb{Z})$, $m,k>0$. If $H$ is primitive and
$v$-general, the intersection Betti and Hodge numbers of $M(S,v,H)$ stabilize
to the stable Betti and Hodge numbers of the Hilbert scheme of $s$-points as
$s$ tends to infinity, i.e. the limits $\lim_{m\to\infty}\dim
I\\!H^{p,q}(M(S,v,H))$ are finite and independent of $k$.

###### Proof.

Given two $(m,k)$-triples $(S_{1},v_{1},H_{1})$ and $(S_{2},v_{2},H_{2})$,
$M(S_{1},v_{1},H_{1})$ and $M(S_{2},v_{2},H_{2})$ are deformation equivalent,
and the deformation is locally analytically trivial by [60, Thm 1.17]. In
particular, we have $\dim I\\!H^{p,q}(M(S_{1},v_{1},H_{1}))=\dim
I\\!H^{p,q}(M(S_{2},v_{2},H_{2}))$.

Now choose a K3 surface $S_{0}$ with
$\operatorname{Pic}(S_{0})\simeq\mathbb{Z}$ generated by the class of a smooth
curve $C$ of genus $g=k+1$. Since $(S,v,H)$ and $(S_{0},(0,mC,m),C)$ are
$(m,g-1)$-triples, we have

$\dim I\\!H^{p,q}(M(S,v,H))=\dim I\\!H^{p,q}(M(S_{0},(0,mC,m),C)).$

By 10.4, we have

$\lim_{m\to\infty}\dim I\\!H^{p,q}M(S_{0},(0,mC,m),C))=\lim_{m\to\infty}\dim
I\\!H^{p,q}(M(S_{0},(0,mC,1),C)),$

where $M(S_{0},(0,mC,1)$ is deformation equivalent to a Hilbert scheme of
$s\coloneqq 2m^{2}(g-1)$ points on a K3 surface. ∎

## References

  * [1] D. Baraglia.  Monodromy of the $SL(n)$ and $GL(n)$ Hitchin fibrations.  Math. Ann., 370(3-4):1681–1716, 2018.
  * [2] A. Beauville, M.S. Narasimhan, and S. Ramanan.  Spectral curves and the generalised theta divisor.  J. Reine Angew. Math., 398:169–179, 1989.
  * [3] A.A. Beilinson, J.N. Bernstein, and P. Deligne.  Faisceaux pervers.  In Analysis and topology on singular spaces, I (Luminy,1981), volume 100 of Astérisque, pages 5–171. Soc. Math. France, Paris, 1982.
  * [4] G. Bellamy and T. Schedler.  Symplectic resolutions of character varieties.  arXiv:1909.12545, 2019.
  * [5] R. Bielawski and A.S. Dancer.  The geometry and topology of toric hyperkähler manifolds.  Comm. Anal. Geom., 8(4):727–760, 2000.
  * [6] A. Björner.  The homology and shellability of matroids and geometric lattices.  In Matroid applications, volume 40 of Encyclopedia Math. Appl., pages 226–283. Cambridge Univ. Press, Cambridge, 1992.
  * [7] L. Caporaso.  A compactification of the universal Picard variety over the moduli space of stable curves.  J. Amer. Math. Soc., 7(3):589–660, 1994.
  * [8] S. Casalaina-Martin, J.L. Kass, and F. Viviani.  The local structure of compactified Jacobians.  Proc. Lond. Math. Soc. (3), 110(2):510–542, 2015.
  * [9] S. Casalaina-Martin, J.L. Kass, and F. Viviani.  The singularities and birational geometry of the compactified universal Jacobian.  Algebr. Geom., 4(3):353–393, 2017.
  * [10] P.H. Chaudouard and G. Laumon.  Un théorème du support pour la fibration de Hitchin.  Ann. Inst. Fourier (Grenoble), 66(2):711–727, 2016.
  * [11] P. Cook.  Local and Global Aspects of the Module Theory of Singular Curves.  PhD thesis, University of Liverpool, 1993.
  * [12] J. Coskun and M. Woolf.  The stable cohomology of moduli spaces of sheaves on surfaces.  to appear in J. Differential Geom.
  * [13] B. Davison.  Nonabelian Hodge theory for stacks and a stacky P=W conjecture.  arXiv:2112.10830, 2021.
  * [14] B. Davison.  Purity and 2-Calabi-Yau categories.  arXiv:2106.07692, 2021.
  * [15] M.A. de Cataldo.  A support theorem for the Hitchin fibration: the case of ${\rm SL}_{n}$.  Compos. Math., 153(6):1316–1347, 2017.
  * [16] 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(3):1329–1407, 2012.
  * [17] M.A. de Cataldo, J. Heinloth, and L. Migliorini.  A support theorem for the Hitchin fibration: the case of ${\rm GL}_{n}$ and $K_{C}$.  J. Reine Angew. Math., 780:41–77, 2021.
  * [18] M.A. de Cataldo and D. Maulik.  The perverse filtration for the Hitchin fibration is locally constant.  Pure Appl. Math. Q., 16(5):1441–1464, 2020.
  * [19] M.A. de Cataldo, D. Maulik, and J. Shen.  Hitchin fibrations, abelian surfaces, and the P=W conjecture.  J. Amer. Math. Soc., 2021.
  * [20] M.A. de Cataldo and L. Migliorini.  The Hodge theory of algebraic maps.  Ann. Sci. École Norm. Sup. (4), 38(5):693–750, 2005.
  * [21] M.A. de Cataldo and L. Migliorini.  The perverse filtration and the Lefschetz hyperplane theorem.  Ann. of Math. (2), 171(3):2089–2113, 2010.
  * [22] M.A. de Cataldo, A. Rapagnetta, and G. Saccà.  The Hodge numbers of O’Grady 10 via Ngô strings.  J. Math. Pures Appl. (9), 156:125–178, 2021.
  * [23] M.A. de Cataldo and S. Zhang.  A Cohomological Non Abelian Hodge Theorem in Positive Characteristic.  to appear in Alg. Geom., arXiv:2104.12970, 2021.
  * [24] J.M. Drézet.  Luna’s slice theorem and applications.  In Algebraic group actions and quotients, pages 39–89. Hindawi Publ. Corp., Cairo, 2004.
  * [25] E. Esteves.  Compactifying the relative Jacobian over families of reduced curves.  Trans. Amer. Math. Soc., 353(8):3045–3095, 2001.
  * [26] C. Felisetti.  Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve.  J. Inst. Math. Jussieu, pages 1–50, 2021.
  * [27] C. Felisetti and M. Mauri.  P=W conjectures for character varieties with symplectic resolution.  J. Ec. Polytech. Math., 9:853–905, 2022.
  * [28] C. Felisetti, J. Shen, and Q. Yin.  On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties.  Trans. Amer. Math. Soc., 375(4):2987–3001, 2022.
  * [29] M. Gagné.  Compactified jacobians of integral curves with double points.  Ph.D. Thesis, 1997.
  * [30] M. Goresky and R. MacPherson.  Stratified Morse theory, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3).  Springer-Verlag, Berlin, 1988.
  * [31] M. Groechenig, D. Wyss, and P. Ziegler.  Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration.  Invent. Math., 221(2):505–596, 2020.
  * [32] M. Harada and N. Proudfoot.  Properties of the residual circle action on a hypertoric variety.  Pacific J. Math., 214(2):263–284, 2004.
  * [33] T. Hausel and F. Rodriguez-Villegas.  Mixed Hodge polynomials of character varieties.  Invent. Math., 174(3):555–624, 2008.  With an appendix by Nicholas M. Katz.
  * [34] T. Hausel and B. Sturmfels.  Toric hyperKähler varieties.  Doc. Math., 7:495–534, 2002.
  * [35] T. Hausel and M. Thaddeus.  Mirror symmetry, Langlands duality, and the Hitchin system.  Invent. Math., 153(1):197–229, 2003.
  * [36] T. Hausel and M. Thaddeus.  Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles.  Proc. London Math. Soc. (3), 88(3):632–658, 2004.
  * [37] J. Heinloth.  A conjecture of Hausel on the moduli space of Higgs bundles on a curve.  Astérisque, (370):157–175, 2015.
  * [38] J. Heinloth.  The intersection form on moduli spaces of twisted $PGL_{n}$-Higgs bundles vanishes.  Math. Ann., 365(3-4):1499–1526, 2016.
  * [39] N. Hitchin.  Stable bundles and integrable systems.  Duke Math. J., 54(1):91–114, 1987.
  * [40] N.J. Hitchin.  The self-duality equations on a Riemann surface.  Proc. London Math. Soc. (3), 55(1):59–126, 1987.
  * [41] R. Joshua.  The intersection cohomology and derived category of algebraic stacks.  In Algebraic $K$-theory and algebraic topology (Lake Louise, AB, 1991), volume 407 of NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., pages 91–145. Kluwer Acad. Publ., Dordrecht, 1993.
  * [42] D. Kaledin.  Symplectic singularities from the Poisson point of view.  J. Reine Angew. Math., 600:135–156, 2006.
  * [43] T. Kinjo and N. Koseki.  Cohomological $\chi$-independence for Higgs bundles and Gopakumar-Vafa invariants.  arXiv:2112.10053, 2021.
  * [44] Y. Laszlo and M. Olsson.  Perverse t-structure on artin stacks.  Mathematische Zeitschrift, 261:737–748, 04 2009.
  * [45] G. Laumon and L. Moret-Bailly.  Champs algébriques, volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer-Verlag, Berlin, 2000.