1.1 D-Higgs bundles

For our purpose, it is important to consider generalised Higgs bundles $({\mathcal E}, \theta )$ with the Higgs field $\theta $ twisted by a divisor D that is not necessarily the canonical divisor $K_C$. This flexibility also plays a crucial role in the proof of the fundamental lemma [Reference Ngô32, Reference Ngô33].

Let D be either an effective divisor of degree $\mathrm {deg}(D)> 2g-2$ or $D= K_C$. A D-Higgs bundle is a pair $({\mathcal E}, \theta )$, where ${\mathcal E}$ is a vector bundle and $\theta $ is a D-twisted Higgs field

$$ \begin{align*} \theta: {\mathcal E} \rightarrow {\mathcal E} \otimes {\mathcal O}_C(D). \end{align*} $$

We denote by $\mathrm {char}(\theta )$ the tuple of the coefficients for the characteristic polynomial associated with $({\mathcal E}, \theta )$:

$$ \begin{align*} \mathrm{char}(\theta) = (a_1, \dots, a_n), \quad a_i =\mathrm{trace}\left(\wedge^i \theta\right) \in H^0(C, {\mathcal O}_C(iD)). \end{align*} $$

Parallel to the case of $K_C$-Higgs bundles, the stability condition for D-Higgs bundles is with respect to the slope $\mu ({\mathcal E}) = \mathrm {deg}({\mathcal E})/\mathrm {rank}({\mathcal E})$. By [Reference Nitsure34], there is a nonsingular quasi-projective moduli space $\widetilde {{\mathcal M}}_{n,d}^D$ parameterising stable D-Higgs bundles of rank n and degree d, with the Hitchin map

(10)$$ \begin{align} \widetilde{h}^D : \widetilde{{\mathcal M}}_{n,d}^D \rightarrow \widetilde{{\mathcal A}}^D = \oplus_{i=1}^n H^0(C, {\mathcal O}_C(iD)), \quad ({\mathcal E}, \theta) \mapsto \mathrm{char}(\theta), \end{align} $$

which is proper and surjective.

The moduli space of stable $\mathrm {SL}_n\ D$-Higgs bundles is defined to be the subvariety

(11)$$ \begin{align} {\mathcal M}^D_{n,L} = \left\{({\mathcal E}, \theta) \in \widetilde{{\mathcal M}}_{n,d}^D; \ \mathrm{det}({\mathcal E}) \simeq L, \ \mathrm{trace}(\theta) = 0 \right\} \subset \widetilde{{\mathcal M}}_{n,d}^D. \end{align} $$

It is nonsingular and irreducible by [Reference de Cataldo6, Section 2.1], which has a Hitchin map induced from the Hitchin map of the ambient space (10),

(12)$$ \begin{align} h^D: {\mathcal M}^D_{n,L} \rightarrow {\mathcal A}^D = \oplus_{i=2}^n H^0(C, {\mathcal O}_C(iD)). \end{align} $$

It is clear that the variety ${\mathcal M}^D_{n,L}$ is the fibre over the closed point $(L,0)$ of the smooth map

(13)$$ \begin{align} q: \widetilde{{\mathcal M}}_{r,d}^{D} \rightarrow \widetilde{{\mathcal M}}_{1,d}^{D} =\mathrm{Pic}^d(C) \times H^0(C, {\mathcal O}_C(D)), \quad ({\mathcal E}, \theta) \mapsto (\mathrm{det}({\mathcal E}), \mathrm{trace}(\theta)). \end{align} $$

A major difference between the cases $D = K_C$ and $\mathrm {deg}(D)>2g-2$ is that the Hitchin fibration for either $\mathrm {GL}_n$ or $\mathrm {SL}_n$ is Lagrangian with respect to the canonical hyper-Kähler structure for $D = K_C$, but the dimension of the Hitchin base is always larger than the dimension of a fibre in the case of $\mathrm {deg}(D)>2g-2$. In view of the support theorems (Section 2.2), such a difference will substantially influence the study of the topology of Hitchin fibrations.

From now on, all D-Higgs bundles will be uniformly called Higgs bundles for convenience.

1.2 Relative Hitchin moduli spaces

In this section, we study the relative Hitchin moduli space associated with a Galois cover $\pi : C' \to C$. This parameterises stable Higgs bundles with respect to the endoscopic group of $\mathrm {SL}_n$ over C attached to a character $\kappa \in \hat {\Gamma }$ [Reference Ngô32, Reference Ngô33].

Let $C'$ be a nonsingular curve with a cyclic Galois cover

$$ \begin{align*} \pi: C' \to C \end{align*} $$

whose Galois group is $G_{\pi } \cong \mathbb {Z}/m{\mathbb {Z}}$. We denote the divisor $\pi ^*D$ by $D'$. For an element $\sigma \in H^0(C', {\mathcal O}_{C'}(D'))$, the push-forward along $\pi $ gives an element

$$ \begin{align*} \pi_* \sigma \in H^0(C, \pi_* {\mathcal O}_{C'}(D')). \end{align*} $$

The trace of $\pi _*\sigma $ recovers its projection to the direct summand component:

$$ \begin{align*} \mathrm{trace}(\pi_*\sigma) \in H^0(C, {\mathcal O}_C(D)) \subset H^0(C, \pi_* {\mathcal O}_{C'}(D')). \end{align*} $$

The moduli space $\widetilde {{\mathcal M}}_{r,d}^{D'}(C')$ of rank r, degree d stable Higgs bundles on $C'$ admits a map

(14)$$ \begin{align} q_{\pi}: \widetilde{{\mathcal M}}_{r,d}^{D'}(C') \rightarrow \widetilde{{\mathcal M}}_{1,d}^{D}(C), \end{align} $$

which is the composition of equation (13) for the curve $C'$ and the push-forward

(15)$$ \begin{align} \pi_*: \widetilde{{\mathcal M}}_{1,d}^{D'}(C') \rightarrow \widetilde{{\mathcal M}}_{1,d}^D(C), \quad ({\mathcal L}, \sigma) \mapsto (\mathrm{det}(\pi_*{\mathcal L}), \mathrm{trace}(\pi_{\ast} \sigma)).^{2} \end{align} $$

SinceFootnote ² both formulas (13) and (15) are smooth, the composition $q_{\pi }$ is also smooth.

We define the relative Hitchin moduli space of rank r and degree d associated with $\pi $ as the subvariety of $\widetilde {{\mathcal M}}_{r,d}^{D'}(C')$ given as a fibre of formula (14):

$$ \begin{align*} {\mathcal M}_{r,L}^D(\pi) = q_{\pi} ^{-1}(L,0) \subset \widetilde{{\mathcal M}}_{r,d}^{D'}(C'). \end{align*} $$

The variety ${\mathcal M}_{r,L}^D(\pi )$, which recovers the $\mathrm {SL}_n$-Hitchin moduli space (11) when $\pi = \mathrm {id}$, is nonsingular due to the smoothness of $q_{\pi }$.

Next we describe the Hitchin fibration associated with ${\mathcal M}_{r,L}^D(\pi )$ which generalises equation (12). Recall from equation (10) the $\mathrm {GL}_r$-Hitchin fibration $h^{D'}: \widetilde {{\mathcal M}}_{r,d}^{D'}(C') \to \widetilde {{\mathcal A}}^{D'}(C')$ associated with the curve $C'$. The restriction of $h^{D'}$ to ${\mathcal M}_{r,L}^D(\pi )$ induces the Hitchin map

(16)$$ \begin{align} h_{\pi}^D: {\mathcal M}_{r,L}^D(\pi) \to {\mathcal A}^D(\pi), \end{align} $$

which fits into the commutative diagram

(17)

The Hitchin base ${\mathcal A}^D(\pi )$ can be concretely described as

(18)$$ \begin{align} {\mathcal A}^D(\pi) = H^0(C', {\mathcal O}_{C'}(D'))_{\mathrm{var}} \oplus \left(\oplus_{i=2}^r H^0(C', {\mathcal O}_{C'}(iD'))\right), \end{align} $$

where $H^0(C', {\mathcal O}_{C'}(D'))_{\mathrm {var}}$ is the variant part with respect to the natural Galois group $G_{\pi }$-action induced by the $G_{\pi }$-action on $C'$ [Reference Hausel and Pauly21, Section 5]. Since the line bundles ${\mathcal O}_{C'}(iD')$ have canonical $G_{\pi }$-linearisations, there is a natural $G_{\pi }$-action on the Hitchin base (18).

Proof. Recall that ${\mathcal M}_{r,L}^D(\pi )$ is the fibre of

$$ \begin{align*} q_{\pi} = (\pi_*) \circ q \end{align*} $$

(see formula (14)) over the point $(L,0) \in \widetilde {{\mathcal M}}_{1,d}^{D}(C)$. The map q is surjective and smooth, and its fibres are isomorphic to the moduli of stable $\mathrm {SL}_r$-Higgs bundles of degree d on the curve $C'$. In particular, each fibre of q is nonsingular and irreducible.

The morphism $\pi _*$ given in formula (15) respects the product structures

(21)$$ \begin{align} \pi_*: \mathrm{Pic}^d(C') \times H^0(C', {\mathcal O}_{C'}(D'))\rightarrow \mathrm{Pic}^d(C) \times H^0(C, {\mathcal O}_C(D)), \end{align} $$

where the morphism between the second factors form a trivial affine bundle. For the first factors, a fibre of $\pi _*: \mathrm {Pic}^d(C') \rightarrow \mathrm {Pic}^d(C)$ is isomorphic to the degree d Prym variety associated with the Galois cover $\pi : C' \to C$, which is the disjoint union of m isomorphic abelian varieties [Reference Hausel and Thaddeus22, Section 7]. Hence the moduli space ${\mathcal M}_{r,L}^D(\pi )$ has m nonsingular connected components.

Assume $n = mr$. Tensoring with a line bundle ${\mathcal L} \in \Gamma = \mathrm {Pic}^0(C)[n]$ induces an ${\mathcal A}^D(\pi )$-automorphism

(22)$$ \begin{align} \phi_{\mathcal L}: {\mathcal M}_{r,L}^D(\pi) \xrightarrow{\simeq} {\mathcal M}_{r,L}^D(\pi), \quad \phi_{\mathcal L} ({\mathcal E}, \theta) = ({\mathcal L}\otimes {\mathcal E} , \theta). \end{align} $$

Moreover, for a general point $a \in {\mathcal A}^D(\pi )$ corresponding to a degree $n=rm$ spectral cover

$$ \begin{align*} g_a: C^{\prime}_a \to C' \xrightarrow{\pi} C, \end{align*} $$

the fibre ${\left (h_{\pi }^D\right )}^{-1}(a)$ is identical to a fibre of the morphism

$$ \begin{align*} {g_a}_*: \mathrm{Pic}^d\left(C^{\prime}_a\right) \rightarrow \mathrm{Pic}^d(C), \quad {\mathcal L} \mapsto \mathrm{det}\left({g_a}_* {\mathcal L}\right), \end{align*} $$

where $\Gamma $ acts transitively on the set of its connected components (compare [Reference Hausel and Pauly21, Lemmas 2.1 and 2,2]). This ensures that $\Gamma $ acts transitively on $\{M_i\}_{1\leq i\leq m}$. In particular, for any pair $1\leq i,j \leq m$, there exists a line bundle ${\mathcal L}_{ij}\in \Gamma $ such that the isomorphism $\phi _{ij} = \phi _{{\mathcal L}_{ij}}$ given in formula (22) satisfies the commutative diagram (20). Thus (a) and (b) are proved.

1.3 Weil pairing and cyclic covers

Recall from Section 0.1 that the group $\Gamma = \mathrm {Pic}^0(C)[n]$ acts on the $\mathrm {SL}_n$-moduli space ${\mathcal M}^D_{n,L}$ via the tensor product. For $\gamma \in \Gamma $, the $\gamma $-fixed subvariety ${\mathcal M}^D_{\gamma } \subset {\mathcal M}^D_{n,L}$ carries an induced Hitchin map

As indicated by Theorem 0.3, the cohomology of ${\mathcal M}^D_{\gamma }$ is related to a $\kappa $-isotypic component of the cohomology of ${\mathcal M}^D_{n,L}$ with respect to the $\Gamma $-action.

In order to describe this connection, we need a correspondence (3) between an element $\gamma \in \Gamma $ and a character $\kappa \in \hat {\Gamma }$, which we review in the following.

Let $\mu _n \subset {\mathbb {C}}^*$ denote the finite group of the nth roots of unity. We have the Weil pairing on the group of n-torsion points of $\mathrm {Pic}^0(C)$,

$$ \begin{align*} \langle~,~\rangle_{\Gamma}: \Gamma \times \Gamma \rightarrow \mu_n. \end{align*} $$

Under the identification

$$ \begin{align*} \mathrm{Pic}^0(C)[n] = H_1(C, {\mathbb{Z}}/n{\mathbb{Z}}), \end{align*} $$

the Weil pairing coincides with the intersection pairing on $H_1(C, {\mathbb {Z}}/n{\mathbb {Z}})$. In particular, it is nondegenerate, which induces a character

$$ \begin{align*} \kappa = \langle \gamma, ~\rangle_{\Gamma}: \Gamma \rightarrow \mu_n \subset {\mathbb{C}}^* \end{align*} $$

for each $\gamma \in \Gamma $. This gives the identification (3).

We also note that an element $\gamma \in \Gamma $ naturally corresponds to a cyclic Galois cover of C whose degree is the order of $\gamma \in \Gamma $. In fact, for fixed $\gamma \in \Gamma $, we denote by $L_{\gamma }$ the n-torsion line bundle associated with $\gamma $, and let m be its order which divides n. Taking the mth roots of unity fibrewise inside the total space of $L_{\gamma }$ with respect to the zero section C, we obtain a cyclic Galois cover

$$ \begin{align*} \pi: C' \rightarrow C \end{align*} $$

with the Galois group $G_{\pi } \simeq \mathbb {Z}/m{\mathbb {Z}}$. Conversely, every degree m étale cyclic Galois cover arises this way.

1.4 Characters

In this section we give a concrete description of the character $\kappa $ in terms of the Prym variety associated with the corresponding Galois cover $\pi : C' \to C$.

As before, we assume that $\pi :C' \to C$ is a degree m cyclic Galois cover associated with $\kappa \in \hat {\Gamma }$ as in Section 1.3, and we assume that $n = mr$. The character $\kappa \in \hat {\Gamma }$ matches with $\gamma \in \Gamma $ via equation (3). We consider the Prym variety

(23)$$ \begin{align} \mathrm{Prym}(C'/C) = \mathrm{Ker}\left( \mathrm{det}(\pi_*-): \mathrm{Pic}^0(C') \to \mathrm{Pic}^0(C) \right) \end{align} $$

with the component group

$$ \begin{align*} \pi_0(\mathrm{Prym}(C'/C))\simeq {\mathbb{Z}}/m{\mathbb{Z}}. \end{align*} $$

For an n-torsion line bundle ${\mathcal L} \in \Gamma $, the projection formula yields

$$ \begin{align*} \mathrm{det}(\pi_*\pi^*{\mathcal L}) = {\mathcal L}^{\otimes m} \in \mathrm{Pic}^0(C)[r]. \end{align*} $$

In particular, the line bundle $\pi ^*{\mathcal L}^{\otimes r}\in \mathrm {Pic}^0(C')$ represents a point in $\mathrm {Prym}(C'/C))$, which yields a natural group homomorphism

(24)$$ \begin{align} \Gamma \rightarrow \pi_0(\mathrm{Prym}(C'/C)), \quad {\mathcal L} \mapsto \left[\pi^*{\mathcal L}^{\otimes r}\right] \in \pi_0(\mathrm{Prym}(C'/C)). \end{align} $$

The morphism (24) admits a factorisation

$$ \begin{align*} \Gamma =\mathrm{Pic}^0(C)[n] \xrightarrow{[r]} \mathrm{Pic}^0(C)[m] \rightarrow \pi_0(\mathrm{Prym}(C'/C)). \end{align*} $$

The first map is multiplication by r, which is clearly surjective. The second map sends ${\mathcal L} \in \mathrm {Pic}^0(C)[m]$ to the equivalent class of the line bundle $\pi ^*{\mathcal L} \in \mathrm {Prym}(C'/C)$, and its surjectivity is given by the proof of [Reference Hausel and Pauly21, Theorem 1.1 (2)]. Hence formula (24) is surjective.

Recall that $\gamma \in \Gamma $ is of order m, so

$$ \begin{align*} \gamma \in \mathrm{Pic}^0(C)[m] \subset \Gamma. \end{align*} $$

The following lemma is obtained by viewing the Weil pairing on $\mathrm {Pic}^0(C)[i]$ for any $i \in {\mathbb {N}}$, via Poincaré duality, as the intersection pairing

$$ \begin{align*} H_1(C, {\mathbb{Z}}/i{\mathbb{Z}}) \times H_1(C, {\mathbb{Z}}/i{\mathbb{Z}}) \rightarrow {\mathbb{Z}}/i{\mathbb{Z}} \simeq \mu_i.\end{align*} $$

Lemma 1.2. Assume $\gamma ' \in \Gamma $. We let $\langle ~, ~\rangle _{\mathrm {Pic}^0(C)[m]}$ denote the Weil pairing on $\mathrm {Pic}^0(C)[m]$, and we view $r\gamma '$ naturally as an element in $\mathrm {Pic}^0(C)[m]$. We have

$$ \begin{align*} \langle \gamma, \gamma'\rangle_{\Gamma} = \langle \gamma, r\gamma'\rangle_{\mathrm{Pic}^0(C)[m]}. \end{align*} $$

Proposition 1.3. The character $\kappa \in \hat {\Gamma }$ (corresponding to $\pi :C' \to C$) factors through a character of $\pi _0(\mathrm {Prym}(C'/C))$ of order m via the morphism (24):

$$ \begin{align*} \kappa: \Gamma \rightarrow \pi_0(\mathrm{Prym}(C'/C))\left( \simeq {\mathbb{Z}}/m{\mathbb{Z}} \right) \to {\mathbb{C}}^*. \end{align*} $$

Proof. Recall from Section 1.3 that the character $\kappa $ is given by the Weil pairing $\langle \gamma , ~\rangle _{\Gamma }$, where $\gamma \in \Gamma $ corresponds to $\pi $. We have that $\gamma $ is of order m – that is,

$$ \begin{align*} \gamma \in \mathrm{Pic}^0(C)[m] \subset \Gamma. \end{align*} $$

Therefore Lemma 1.2 implies for any element $\gamma ' \in \Gamma $ that

(25)$$ \begin{align} \kappa(\gamma') = \langle \gamma, \gamma'\rangle_{\Gamma} = \langle \gamma, r\gamma'\rangle_{\mathrm{Pic}^0(C)[m]}. \end{align} $$

We consider the subgroup

$$ \begin{align*} K = \mathrm{Ker}\left\{ \pi^*: \mathrm{Pic}^0(C) \to \mathrm{Pic}^0(C') \right\} \subset \mathrm{Pic}^0(C). \end{align*} $$

It is the cyclic finite subgroup $\langle \gamma \rangle $ of $\mathrm {Pic}^0(C)$ generated by $\gamma $. Since $\gamma $ is of order m, we have $K \subset \mathrm {Pic}^0(C)[m]$. By the proof of [Reference Hausel and Pauly21, Theorem 1.1 (1)], there is a canonical isomorphism

(26)$$ \begin{align} \pi_0(\mathrm{Prym}(C'/C)) = \hat{K}, \end{align} $$

which, for any ${\mathcal L}\in \mathrm {Pic}^0(C)[m]$, identifies $[\pi ^*{\mathcal L}]\in \pi _0(\mathrm {Prym}(C'/C))$ with the character of K sending the generator $\gamma \in K$ to

$$ \begin{align*} \langle \gamma, {\mathcal L}\rangle_{\mathrm{Pic}^0(C)[m]} \in \mu_m \subset {\mathbb{C}}^*. \end{align*} $$

Comparing with equation (25), this implies that $\kappa : \Gamma \to {\mathbb {C}}^*$ is the composition of formula (24) and the order m character of $\pi _0(\mathrm {Prym}(C'/C))$ given by

$$ \begin{align*} \gamma \in K = \mathrm{Hom}({\pi_0(\mathrm{Prym}(C'/C))}, {\mathbb{C}}^*). \end{align*} $$

Here we dualise equation (26) in the last identity. This completes the proof.

Since ${\pi _0(\mathrm {Prym}(C'/C))}\cong {\mathbb {Z}}/m{\mathbb {Z}}$, its character group is also cyclic. We obtain the following corollary of Proposition 1.3:

Now we consider the kernel of the morphism (24), which we denote by

(27)$$ \begin{align} \Omega \subset \Gamma. \end{align} $$

The subgroup $\Omega $ acts on the moduli space ${\mathcal M}^D_{r,L}(\pi )$ via the tensor product.

Proof. Recall that ${\mathcal M}_{r,L}^D(\pi )$ is a fibre of

$$ \begin{align*} q_{\pi} = (\pi_*) \circ q: \widetilde{{\mathcal M}}_{r,d}^{D'}(C') \xrightarrow{q} \widetilde{{\mathcal M}}_{1,d}^{D'}(C') \xrightarrow{\pi_*} \widetilde{{\mathcal M}}_{1,d}^{D}(C). \end{align*} $$

The fibres of the first map are connected. The second map respects the product structure (21). Both the maps q and $\pi _*$ are $\Gamma $-equivariant. Hence the m connected components of equation (19) match the connected components of the degree d Prym variety

$$ \begin{align*} \mathrm{Prym}^d(C'/C)= (\pi_*)^{-1}(L),\quad \pi_*: \mathrm{Pic}^d(C') \to \mathrm{Pic}^d(C). \end{align*} $$

Since $\mathrm {Prym}^d(C'/C)$ is a torsor of $\mathrm {Prym}(C'/C)$ (defined in equation (23)), and by definition the group $\Omega $ acts trivially on the component group $\pi _0(\mathrm {Prym}(C'/C))$, we obtain that the action of $\Omega $ preserves each connected component of $\mathrm {Prym}^d(C'/C)$. The proposition then follows from the fact that the restriction of q to ${\mathcal M}^D_{r,d}(\pi )$,

$$ \begin{align*}q|_{\left(\pi_*\right)^{-1}\left(L,0\right)}: {\mathcal M}^D_{r,L}(\pi) \to (\pi_*)^{-1}(L,0) = \mathrm{Prym}^d(C'/C)\times H^0(C', {\mathcal O}_{C'}(D'))_{\mathrm{var}}, \end{align*} $$

is $\Omega $-equivariant.

1.5 Endoscopic moduli spaces and $\gamma $-fixed loci

In this section, we connect the $\gamma $-fixed subvariety ${\mathcal M}^D_{\gamma }$ to the relative Hitchin moduli spaces introduced in Section 1.2.

We fix $\gamma \in \Gamma $ of order m. Let $\pi : C' \rightarrow C$ be the cyclic Galois cover with the Galois group $G_{\pi } \simeq \mathbb {Z}/m{\mathbb {Z}}$ corresponding to $\gamma $, as in Section 1.3. Assume $n=mr$. We consider the relative Hitchin moduli space ${\mathcal M}^D_{r,L}(\pi )$ with the Hitchin fibration

(28)$$ \begin{align} h_{\pi}^D:{\mathcal M}^D_{r,L}(\pi) \rightarrow {\mathcal A}^D(\pi). \end{align} $$

By [Reference Hausel and Thaddeus22, Proposition 7.1], the Galois group $G_{\pi }$ acts freely on the relative moduli space ${\mathcal M}^D_{r,L}(\pi )$ whose quotient recovers the $\gamma $-fixed subvariety ${\mathcal M}^D_{\gamma }$. The group $G_{\pi }$ also acts on the base ${\mathcal A}^D(\pi )$ with the Hitchin map (28) $G_{\pi }$-equivariant. In summary, we have the following commutative diagram:

(29)

where $q_{\mathcal M}$ and $q_{\mathcal A}$ are the quotient maps with respect to the natural actions of the Galois group $G_{\pi }$. Let ${\mathcal A}^D(\pi )^*$ be the largest open subset of ${\mathcal A}^D(\pi )$ upon which the $G_{\pi }$-action is free. Then the left diagram is Cartesian after restricting to

$$ \begin{align*} {\mathcal A}^D(\pi)^* \xrightarrow{~~~q_{\mathcal A}~~~} {{\mathcal A}_{\gamma}^{D}}^*:= {\mathcal A}^D(\pi)^*/G_{\pi}. \end{align*} $$

Remark 1.6. We see from the description (equation (18)) that ${\mathcal A}^D(\pi )^*$ is nonempty. In fact, it suffices to find vectors in the affine space ${\mathcal A}^D(\pi )$ that are not fixed by any nontrivial element of the cyclic group $G_{\pi }$. We consider the push-forward of the structure sheaf ${\mathcal O}_{C'}$ along the Galois cover $\pi : C' \to C$, which admits a splitting $\pi _* {\mathcal O}_{C'} = \oplus _\chi L_\chi $, where $\chi $ runs through all characters of $G_{\pi }$ and $L_\chi $ is a degree $0$ line bundle corresponding to $\chi $. The projection formula yields

$$ \begin{align*} \pi_* {\mathcal O}_{C'}(iD') = \pi_* \pi^* {\mathcal O}_C(iD) = \bigoplus_{\chi} {\mathcal O}_C(iD)\otimes L_\chi,\quad \forall \ i \geq 1. \end{align*} $$

In particular, for any character $\chi $ of $G_{\pi }$, we have

(30)$$ \begin{align} H^0(C',{\mathcal O}_{C'}(iD'))_\chi=H^0(C, {\mathcal O}_{C}(iD)\otimes L_\chi) \neq 0, \end{align} $$

where we used the Riemann–Roch formula. The nonemptiness of ${\mathcal A}^D(\pi )^*$ follows from equations (30) and (18).

Lemma 1.7. The direct image complex

$$ \begin{align*} {\mathrm{Rh}^{\text{D}}_{\pi}}_* {\underline{{\mathbb{C}}}} \in D^b_c\left({\mathcal A}^D(\pi)\right) \end{align*} $$

is $G_{\pi }$-equivariant, and we have a canonical isomorphism

$$ \begin{align*} \left( {q_{\mathcal A}}_* {\mathrm{Rh}^D_{\pi}}_* {\underline{{\mathbb{C}}}} \right)^{G_{\pi}} \xrightarrow{\simeq} {\mathrm{Rh}^D_{\gamma}}_* {\underline{{\mathbb{C}}}} \in D^b_c\left({\mathcal A}^D_{\gamma}\right). \end{align*} $$

Proof. Since the trivial local system on ${\mathcal M}^D_{r,L}(\pi )$ is $G_{\pi }$-equivariant, the push-forward ${q_{\mathcal M}}_* {\underline {{\mathbb {C}}}}$ along the quotient map $q_{\mathcal M}$ admits a natural $G_{\pi }$-action with a canonical isomorphism

(31)$$ \begin{align} \left( {q_{\mathcal M}}_* {\underline{{\mathbb{C}}}} \right)^{G_{\pi}} \xrightarrow{\simeq} {\underline{{\mathbb{C}}}} \in D^b_c\left({\mathcal M}^D_{\gamma}\right). \end{align} $$

The map $h^D_{\pi }$ is $G_{\pi }$-equivalent, and therefore we obtain in $D^b_c\left ({\mathcal A}^D_{\gamma }\right )$ that

$$ \begin{align*} \left({q_{\mathcal A}}_* {\mathrm{Rh}^{\text{D}}_{\pi}}_* {\underline{{\mathbb{C}}}} \right)^{G_{\pi}} =\left({\mathrm{Rh}^{\text{D}}_{\gamma}}_* {q_{\mathcal M}}_* {\underline{{\mathbb{C}}}} \right)^{G_{\pi}} = {\mathrm{Rh}^{\text{D}}_{\gamma}}_*\left( {q_{\mathcal M}}_* {\underline{{\mathbb{C}}}} \right)^{G_{\pi}} \xrightarrow{\simeq} {\mathrm{Rh}^{\text{D}}_{\gamma}}_*{\underline{{\mathbb{C}}}}, \end{align*} $$

where the last isomorphism is induced by formula (31).

2.1 Supports

Let $f: X\to Y$ be a proper morphism between nonsingular quasi-projective varieties. The decomposition theorem of Bernstein, Beilinson, Deligne and Gabber [Reference Beĭlinson, Bernstein and Deligne3] implies that the direct image complex $\mathrm {Rf}_{\ast } {\underline {{\mathbb {C}}}}$ is (noncanonically) isomorphic to a direct sum of shifted simple perverse sheaves:

(32)$$ \begin{align} \mathrm{Rf}_* {\underline{{\mathbb{C}}}} \simeq \bigoplus_{Z_i} \mathrm{IC}_{Z_i}(L_i)[d_i] \in D^b_c(Y), \end{align} $$

where $d_i \in {\mathbb {Z}}$, $Z_i \subset Y$ is an irreducible subvariety and $L_i$ is a local system on an open subset $U_i$ of $Z_i$. Every $Z_i$ here is called a support of $f: X\to Y$. We say that a direct summand ${\mathcal F}$ of the object

$$ \begin{align*} \mathrm{Rf}_* {\underline{{\mathbb{C}}}} = {\mathcal F} \oplus {\mathcal F}' \end{align*} $$

has full support if each perverse constituent $ \mathrm {IC}_{Z_i}(L_i)[d_i]$ of ${\mathcal F}$ has support $Z_i = Y$.

In this section, we analyse the supports of various Hitchin fibrations introduced in Section 1 when $\mathrm {deg}(D)>2g-2$.

2.2 Support theorems

For the Hitchin fibration $h^D$ (resp., $h^D_{\pi }$), we define the elliptic locus of the Hitchin bases ${\mathcal A}^D$ (resp., ${\mathcal A}^D(\pi )$), denoted by ${\mathcal A}^{D,\mathrm {ell}}$ (resp., ${\mathcal A}^{D,\mathrm {ell}}(\pi ))$, to be the open subset consisting of integral spectral curves.

Following the methods of Ngô [Reference Ngô33] and Chaudouard and Laumon [Reference Chaudouard and Laumon5], de Cataldo showed in [Reference de Cataldo6] that all the supports for the $\mathrm {SL}_n$-Hitchin fibration (12) are governed by the elliptic locus ${\mathcal A}^{D,\mathrm {ell}} \subset {\mathcal A}^D$ when $\mathrm {deg}(D)>2g-2$.

Theorem 2.1 [Reference de Cataldo6, Theorem 1.0.2]

Let D be an effective divisor on C of degree $\mathrm {deg}(D)>2g-2$. Then the generic points of the supports of

$$ \begin{align*} h^D: {\mathcal M}^D_{n,L} \to {\mathcal A}^D \end{align*} $$

are contained in ${\mathcal A}^{D,\mathrm {ell}} \subset {\mathcal A}^D$.

Now we consider the $\Gamma $-action on the moduli space ${\mathcal M}^D_{n,L}$. This action is fibrewise with respect to the Hitchin map $h^D$, which induces a canonical decomposition of the direct image complex

(33)$$ \begin{align} \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}} = \bigoplus_{\kappa \in \hat{\Gamma}} \left(\mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}} \right)_{\kappa} \in D^b_c\left({\mathcal A}^D\right) \end{align} $$

(see [Reference Laumon and Ngô28, Lemma 3.2.5]). We define the stable part $\left (\mathrm {Rh}^{\text {D}}_{*} {\underline {{\mathbb {C}}}} \right )_{\mathrm {st}}$ as the component in equation (33) corresponding to the trivial character $\kappa = 0 \in \hat {\Gamma }$.

Combining Theorem 2.1 with Ngô’s support theorems over the elliptic loci [Reference Ngô33, Theorems 7.8.3 and 7.8.5], we obtain the following complete description of the supports for every $\kappa $-part of equation (33):

The trivial character $0 \in \hat {\Gamma }$ corresponds to $0 \in \Gamma $, and ${\mathcal A}^D_{0} = {\mathcal A}^D$. Hence as a special case of Corollary 2.2, the stable part $\left (\mathrm {Rh}^{\text {D}}_{*} {\mathbb {C}} \right )_{\mathrm {st}}$ has full support ${\mathcal A}^D$.

We fix a character $\kappa \in \hat {\Gamma }$ of order m. Let $\pi : C' \to C$ be the cyclic Galois cover associated with $\kappa $, as in Section 1.3. Assume $n = mr$. By the projection formula, we have

$$ \begin{align*} \mathrm{det}\left(\pi_*(\pi^*{\mathcal L} \otimes {\mathcal E})\right) = \mathrm{det}\left( {\mathcal L} \otimes \pi_*{\mathcal E} \right) = {\mathcal L}^{\otimes mr} \otimes \mathrm{det}(\pi_*{\mathcal E}) =\mathrm{det}(\pi_*{\mathcal E}), \end{align*} $$

for ${\mathcal L} \in \Gamma =\mathrm {Pic}^0(C)[n]$ and ${\mathcal E}$ a rank r vector bundle on $C'$. Therefore, the group $\Gamma $ acts on the moduli space ${\mathcal M}^D_{r,L}(\pi )$ via the tensor product:

$$ \begin{align*} {\mathcal L} \cdot ({\mathcal E} , \theta) = (\pi^*{\mathcal L} \otimes {\mathcal E}, \theta),\quad {\mathcal L} \in \Gamma. \end{align*} $$

We have a similar $\kappa $-decomposition to equation (33) for the Hitchin fibration (16) associated with ${\mathcal M}^D_{r,L}(\pi )$:

$$ \begin{align*} {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} = \bigoplus_{\kappa \in \hat{\Gamma}} \left({\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa} \in D^b_c\left({\mathcal A}^D(\pi)\right). \end{align*} $$

The main result of this section is to prove a support theorem for the Hitchin map (16) associated with $\pi : C' \to C$:

The validity of the support theorems (Theorems 2.1 and 2.3) is the main ingredient in the proof of the endoscopic decomposition in the twisted case of $\mathrm {deg}(D)>2g-2$ when $\mathrm {deg}(D)$ is even (see the proof of Theorem 3.3).

2.3 Weak abelian fibrations

We recall the notion of weak abelian fibration introduced in [Reference Ngô33], which is modelled on the properties of Hitchin fibrations.

We follow closely the exposition of [Reference de Cataldo6, Section 2.6]. Let A be an irreducible nonsingular variety. Assume that

$$ \begin{align*} h: M \to A \text{ and } g: P \to A \end{align*} $$

are morphisms of the same relative dimensions $\mathrm {dim}(h) = \mathrm {dim}(g)$ satisfying the following properties:

1. (a) The map $g: P\to A$ is a smooth commutative group scheme with connected fibres.

2. (b) The map h is proper and M is nonsingular.

3. (c) The group scheme P acts on M fibrewise with affine stabilisers for every geometric point of M.

We call a triple $(M,A,P)$ a weak abelian fibration if the Tate module

$$ \begin{align*} T_{\overline{{\mathbb{Q}}}_l}(P) = \mathrm{R}^{\text{2d}_{\text{g}} - 1}\text{g}_!\overline{{\mathbb{Q}}}_l\left(d_g\right) \end{align*} $$

as an l-adic sheaf is polarisable [Reference de Cataldo6, Section 2.6].

Over a closed point $a \in A$, we consider the Chevalley decomposition for the restricted group scheme $P_a$,

$$ \begin{align*} 0 \to P_a^{\mathrm{aff}} \to P_a \to P_a^{\mathrm{ab}} \to 0, \end{align*} $$

where $P_a^{\mathrm {aff}}$ is the maximal connected affine linear subgroup of $P_a$ and $P_a^{\mathrm {ab}}$ is an abelian variety.

We recall in the following the support inequality of Ngô [Reference Ngô33]:

Theorem 2.4 [Reference Ngô33, Theorem 7.2.2]

Let $(M,A,P)$ be a weak abelian fibration. Assume the irreducible subvariety $Z \subset A$ is a support of $h: M \to A$; then

(34)$$ \begin{align} \mathrm{dim}(h) -\mathrm{dim}(A) +\mathrm{dim}(Z) \geq d_Z^{\mathrm{ab}}(P). \end{align} $$

Here $d_Z^{\mathrm {ab}}(P)=\mathrm {dim} \left (P_a^{\mathrm {ab}}\right )$, with $a \in Z$ a general point.

2.4 Hitchin fibrations

Following [Reference Chaudouard and Laumon5, Reference de Cataldo6], we show that the Hitchin fibration

(35)$$ \begin{align} h_{\pi}^D: {\mathcal M}_{r,L}^D(\pi) \to {\mathcal A}^D(\pi) \end{align} $$

associated with $\pi : C' \to C$ admits the structure of a weak abelian fibration.

Recall commutative diagram (17). The $\mathrm {GL}_r$-Hitchin base $\widetilde {{\mathcal A}}^{D'}(C')$ parameterises spectral curves in the total space $V(D')$ of the line bundle ${\mathcal O}_{C'}(D')$. We assume that

(36)$$ \begin{align} {\mathcal C} \to \widetilde{{\mathcal A}}^{D'}(C') \end{align} $$

is the universal spectral curve. Let $g_{\mathcal C}: \mathrm {Pic}^0_{\mathcal C} \to \widetilde {{\mathcal A}}^{D'}(C')$ be the relative degree $0$ Picard scheme associated with formula (36), which acts on $\widetilde {{\mathcal M}}^{D'}_{r,d}(C')$ via the tensor product. The following result is obtained in [Reference Chaudouard and Laumon5]; see also [Reference de Cataldo6, Section 3] for a detailed review:

Proposition 2.5 [Reference Chaudouard and Laumon5]

The triple

(37)$$ \begin{align} \left(\widetilde{{\mathcal M}}^{D'}_{r,d}(C'), \widetilde{{\mathcal A}}^{D'}(C'), \mathrm{Pic}^0_{\mathcal C} \right), \quad \widetilde{h}^{D'}:\widetilde{{\mathcal M}}^{D'}_{r,d}(C') \rightarrow \widetilde{{\mathcal A}}^{D'}(C'), \quad g_{\mathcal C}: \mathrm{Pic}^0_{\mathcal C} \to \widetilde{{\mathcal A}}^{D'}(C') \end{align} $$

is a weak abelian fibration.

In order to study formula (35), we ‘fix the determinant’ after pushing forward along $\pi : C' \to C$. Since the relative Hitchin moduli space ${\mathcal M}_{r,L}^D(\pi )$ is a closed fibre of formula (14), we consider the morphism of ${\mathcal A}^D(\pi )$-group schemes

$$ \begin{align*} N_{\pi}: \mathrm{Pic}^0_{\mathcal C} \times_{\widetilde{{\mathcal A}}^{D'}\left(C'\right)}{\mathcal A}^D(\pi) \to \mathrm{Pic}^0(C)\times {\mathcal A}^D(\pi), \end{align*} $$

given by composition of the ${\mathcal A}^D(\pi )$-morphisms

$$ \begin{align*} \mathrm{Pic}^0_{\mathcal C} \times_{\widetilde{{\mathcal A}}^{D'}\left(C'\right)}A^D(\pi) \to \mathrm{Pic}^0(C') \times {\mathcal A}^D(\pi) \rightarrow \mathrm{Pic}^0(C) \times {\mathcal A}^D(\pi). \end{align*} $$

Here the first map is the restriction of the norm map $N_p$ ([Reference de Cataldo6, (42)]) to $A^D(\pi )$, and the second map is

$$ \begin{align*} ({\mathcal L} , a) \mapsto (\mathrm{det}(\pi_*{\mathcal L}), a). \end{align*} $$

By the discussion of [Reference de Cataldo6, Section 4.1], the map $N_{\pi }$ is smooth. We let

(38)$$ \begin{align} g_{\pi}: P \rightarrow {\mathcal A}^D(\pi) \end{align} $$

be the kernel of $N_{\pi }$, and we let the group scheme $g^0_{\pi }: P^0 \to {\mathcal A}^D(\pi )$ be the identity component of P. The nonsingular group schemes P and $P^0$ act on ${\mathcal M}^D_{r,L}(\pi )$ induced by the $\mathrm {Pic}^0_{\mathcal C}$-action on $\widetilde {{\mathcal M}}^{D'}_{r,d}(C')$.

Proposition 2.6. The triple

(39)$$ \begin{align} \left( {\mathcal M}^D_{r,L}(\pi), A^D(\pi), P^0 \right), \quad h_{\pi}^D: {\mathcal M}^D_{r,L}(\pi)\to A^D(\pi), \quad g^0_{\pi}: P^0 \to A^D(\pi) \end{align} $$

is a weak abelian fibration.

By [Reference Chaudouard and Laumon5, Section 9] (see also [Reference de Cataldo6, Section 5.2]), the $\mathrm {GL}_r$-Hitchin base admits a stratification

$$ \begin{align*} \widetilde{{\mathcal A}}^{D'}(C') = \bigsqcup_{\underline{m}, \underline{n}}\widetilde{{\mathcal A}}_{\underline{m}, \underline{n}}, \end{align*} $$

with $\underline {m} = (m_1, m_2, \dots , m_s)$, $\underline {n}=(n_1, n_2, \dots , n_s)$ satisfying

1. (a) $n_i \geq n_{i+1}$ for any i;

2. (b) $m_i \geq m_{i+1}$ whenever $n_i = n_{i+1}$;

3. (c) $\sum _{i=1}^s m_in_i = r$.

Each $\widetilde {{\mathcal A}}_{\underline {m}, \underline {n}}$ is a locally closed subset formed by spectral curves of the topological type $(\underline {m}, \underline {n})$:

(40)$$ \begin{align} \widetilde{{\mathcal A}}_{\underline{m}, \underline{n}} = \left\{ E \subset V(D'): E = \sum_i m_iE_i, \ E_i \subset V(D')\right\}, \end{align} $$

where $V(D')$ is the total space of ${\mathcal O}_{C'}(D')$ and $E_i$ is an integral spectral curve of degree $n_i$ over $C'$. The stratification (40) induces a stratification on ${\mathcal A}^D(\pi ) \subset {\mathcal A}^{D'}(C')$,

$$ \begin{align*} {\mathcal A}^D(\pi) = \bigsqcup_{\underline{m}, \underline{n}}{{\mathcal A}}(\pi)_{\underline{m}, \underline{n}}, \quad {{\mathcal A}}(\pi)_{\underline{m}, \underline{n}} = {\mathcal A}^D(\pi) \cap \widetilde{{\mathcal A}}_{\underline{m}, \underline{n}}. \end{align*} $$

We have the following multivariable inequality:

Proposition 2.7 compare [Reference de Cataldo6, Corollary 5.4.4]

Let $Z \subset {\mathcal A}^D(\pi )$ be an irreducible subvariety whose general points are of the type $(\underline {m}, \underline {n})$. Then we have

(41)$$ \begin{align} d^{\mathrm{ab}}_Z\left(\mathrm{Pic}^0_{\mathcal C}\right) \geq \sum_i \left( {d_{\widetilde{h}^{D'}_{n_i}\left(C'\right)}}- d_{\widetilde{{\mathcal A}}^{D'}_{n_i}\left(C'\right)} \right) + \mathrm{dim}(Z)+ \left(\mathrm{deg}(D)-g+1\right). \end{align} $$

Here ${d_{\widetilde {h}^{D'}_{n_i}\left (C'\right )}}$ and $d_{\widetilde {{\mathcal A}}^{D'}_{n_i}\left (C'\right )}$ are the dimensions of a fibre and the base, respectively, of the $\mathrm {GL}_{n_i}$-Hitchin fibration (10) associated with the curve $C'$ and the divisor $D'=\pi ^*D$, and

$$ \begin{align*} d^{\mathrm{ab}}_Z\left(\mathrm{Pic}^0_{\mathcal C}\right) = \mathrm{dim}\left(\left(\mathrm{Pic}^{0}_{{\mathcal C},a}\right)^{\mathrm{ab}}\right), \end{align*} $$

with $a \in Z$ a general point.

Proof. When $\pi =\mathrm {id}$, the subspace

(42)$$ \begin{align} {\mathcal A}^D(\mathrm{id}) = \left\{\mathrm{char}(\theta) \in \widetilde{{\mathcal A}}^D: \mathrm{trace}(\theta)=0\right\} \subset \widetilde{{\mathcal A}}^D \end{align} $$

coincides with the $\mathrm {SL}_n$-Hitchin base, and the inequality (41) is the $\delta $-inequality for $\mathrm {SL}_n$ proven in [Reference de Cataldo6, Corollary 5.4.4 (76)]. As explained in the last paragraph of [Reference de Cataldo6, Proof of Corollary 5.4.4], the main ingredient in the proof of enhancing the $\delta $-inequality for $\mathrm {GL}_n$ to that for $\mathrm {SL}_n$ is [Reference de Cataldo6, Theorem 5.4.2], which asserts that the restriction of the $\delta $-regular weak abelian fibrations to their elliptic loci in the sub-Hitchin base (42) remains $\delta $-regular. This follows from the product structure [Reference de Cataldo6, (74)] for the group schemes associated with the spectral curves, which says that the variations of the group schemes associated with the spectral curves are trivial along the $H^0(C, {\mathcal O}_C(D))$-direction.

Now for a general $\pi : C' \to C$ with $D' =\pi ^*D'$ as we consider here, we have the canonical decomposition for the Hitchin base

(43)$$ \begin{align} {\mathcal A}^D(\pi) \oplus H = \widetilde{{\mathcal A}}^{D'}, \end{align} $$

with $H = H^0(C, {\mathcal O}_C(D))$ a direct summand component of $H^0(C', {\mathcal O}_{C'}(D'))$. Applying the product structure [Reference de Cataldo6, (74)] to the curve $C'$ and the divisor $D'$, we obtain an analogous product structure for equation (43), that the variations of the group schemes associated with the spectral curves are trivial along the H-direction. Hence formula (41) holds in the relative case $\pi : C' \to C$, by the same reason as for $\mathrm {SL}_n$.

2.5 Proof of Theorem 2.3(a)

The proof of Theorem 2.3(a) is parallel to the proofs of the main theorems in [Reference Chaudouard and Laumon5, Reference de Cataldo6], which we provide in the following for the reader’s convenience. The crucial ingredient is to combine Ngô’s support inequality (34) and the multivariable inequality (41).

We assume $Z \subset {\mathcal A}^D(\pi )$ is an irreducible support of formula (35) whose general points have types $(\underline {m}, \underline {n})$. By Theorem 2.4 and Propositions 2.6 and 2.7, we have

$$ \begin{align*} &\mathrm{dim}\left({h^D_{\pi}}\right) - \mathrm{dim}\left({A^D(\pi)}\right)+ \mathrm{dim}(Z) \geq d^{\mathrm{ab}}_Z \left(\mathrm{Pic}^0_{\mathcal C}\right) - g \\ &\quad \ \geq \sum_i \left( {d_{\widetilde{h}^{D'}_{n_i}(C')}}- d_{\widetilde{{\mathcal A}}^{D'}_{n_i}(C')} \right) + \mathrm{dim}(Z) +(\mathrm{deg}(D)-g+1)-g, \end{align*} $$

where we use $d^{\mathrm {ab}}\left (P^0_a\right ) = d^{\mathrm {ab}}\left (\mathrm {Pic}^0_{{\mathcal C},a}\right )-g$. Hence

(44)$$ \begin{align} \mathrm{dim}\left({h^D_{\pi}}\right) - \mathrm{dim}\left({{\mathcal A}^D(\pi)}\right) \geq \sum_{i=1}^s \left( {d_{\widetilde{h}^{D'}_{n_i}(C')}}- d_{\widetilde{{\mathcal A}}^{D'}_{n_i}(C')} \right)+(\mathrm{deg}(D)-2g+1). \end{align} $$

Here we recall that s is the number of irreducible components of the spectral curve. We now apply the dimension formulas for $\mathrm {GL}_*$-Hitchin fibrations [Reference de Cataldo6, Section 6.1] to compute both sides of formula (44). The left-hand side is equal to

$$ \begin{align*} &\left({d_{\widetilde{h}^{D'}_{r}(C')}}- d_{\widetilde{{\mathcal A}}^{D'}_{r}(C')} \right) + \left( \mathrm{dim}H^0(C, {\mathcal O}_C(D))-g\right)\\ &\quad \quad \ = \left(-r\mathrm{deg}(D') + 2r(g'-1)+1\right) +\left(\mathrm{deg}(D)-2g+1\right), \end{align*} $$

where $g'$ is the genus of $C'$ and we used the last formula of [Reference de Cataldo6, (77)]. Similarly, the right-hand side is equal to

$$ \begin{align*} &\sum_{i=1}^s\left(-n_i\mathrm{deg}(D') + 2n_i(g'-1)+1\right) +\left(\mathrm{deg}(D)-2g+1\right) \\ &\quad\quad \ = \left( \left(-\sum_{i=1}^sn_i\right)\left((\mathrm{deg}(D')-2(g'-1)\right)+s \right) +\left(\mathrm{deg}(D)-2g+1\right). \end{align*} $$

In particular, formula (44) implies that

(45)$$ \begin{align} 1-s \geq \left(r-\sum_in_i \right)\left(\mathrm{deg}(D')-(2g'-2)\right). \end{align} $$

Since

$$ \begin{align*} \mathrm{deg}(D')-(2g'-2) = \mathrm{deg}(\pi) (\mathrm{deg}(D)-(2g-2))>0 \end{align*} $$

by the assumption on D, inequality (45) forces $s=1$ and $m_1=1$. This implies that the generic point of Z lies in ${\mathcal A}^{D,\mathrm {ell}}(\pi )$, which completes the proof of Theorem 2.3(a).

2.6 Proof of Theorem 2.3(b)

Due to Theorem 2.3(a), it suffices to prove (b) over the elliptic locus with respect to the restricted Hitchin map

(46)$$ \begin{align} h^{\mathrm{ell}}_{\pi} = h_{\pi}^D\Big\rvert_{{\mathcal A}^{D,\mathrm{ell}}(\pi)}: {\mathcal M}^{D,\mathrm{ell}}_{r,L}(\pi) \to {\mathcal A}^{D,\mathrm{ell}}(\pi). \end{align} $$

Recall the group scheme (38) of the relative Prym variety. By the support theorem [Reference Ngô33, Propositions 7.2.2 and 7.2.3], we only need to show that the constructible sheaf of the top degree cohomology

(47)$$ \begin{align} \left(\mathrm{R}^{\text{2d}_{\text{h}_{\pi}^{\text{D}}}} {\text{h}^{\mathrm{ell}}_{\pi}}_* {\underline{{\mathbb{C}}}}\right)_{\mathrm{st}},\quad d_{h_{\pi}^D} = \mathrm{dim}\left(h_{\pi}^D\right), \end{align} $$

is the trivial local system on the elliptic locus ${\mathcal A}^{D,\mathrm {ell}}(\pi )$.

It is clear that formula (47) contains the trivial local system given by the sum of point classes for all irreducible components of the fibres of equation (46). Hence it suffices to prove that the stable part of the degree $2d_{h_{\pi }^D}$ cohomology group is $1$-dimensional for each fibre of equation (46).

Assume $a \in {\mathcal A}^{D,\mathrm {ell}}(\pi )$. Let $C^{\prime }_a$ be the corresponding integral spectral curve with the spectral cover

(48)$$ \begin{align} C^{\prime}_a \xrightarrow{p_a} C' \xrightarrow{\pi} C. \end{align} $$

The Hitchin fibre ${\mathcal M}^{D}_{r,d}(\pi )_a$ contains a Zariski dense open subset

$$ \begin{align*} {\mathcal M}^{\mathrm{reg}}_a \subset {\mathcal M}^{D}_{r,L}(\pi)_a \end{align*} $$

parameterising line bundles on the spectral curve $C^{\prime }_a$, which is a torsor of the group scheme $P_a$. Hence ${\mathcal M}^{D}_{r,L}(\pi )_a$ has $\lvert \pi _0(P_a)\rvert $ irreducible components.

We fix a base point in $x\in {\mathcal M}_a^{\mathrm {reg}}$. Since ${\mathcal M}_a^{\mathrm {reg}}$ is a torsor of the group scheme $P_a$, the base point x trivialises the torsor, and therefore the $\pi _0(P_a)$-action on x yields an isomorphism

$$ \begin{align*} [x]: \pi_0\left({\mathcal M}^{\mathrm{reg}}_a\right) \xrightarrow{\simeq} \pi_0(P_a), \end{align*} $$

which further induces

(49)$$ \begin{align} H^{2d_{h_{\pi}^D}}\left({\mathcal M}^{D}_{r,L}(\pi)_a, {\mathbb{C}} \right) = \bigoplus_{v\in \pi_0\left(P_a\right)} {\mathbb{C}} v. \end{align} $$

The action of $P_a$ on the cohomology $H^*\left ({\mathcal M}^{D}_{r,L}(\pi )_a, {\mathbb {C}} \right )$ factors through the group $\pi _0(P_a)$ of connected components, which acts naturally on the right-hand side of equation (49). In particular, the action of the discrete subgroup $\Gamma \subset P_a$ on equation (49) factors through the natural action of $\pi _0(P_a)$. By the proof of [Reference Hausel and Pauly21, Theorem 1.1 (2)], the morphism

$$ \begin{align*} \Gamma = \mathrm{Pic}^0(C)[n] \twoheadrightarrow \pi_0(P_a) \end{align*} $$

induced by the pullback $p_a^*\circ \pi ^*$ along formula (48) is a surjection. Therefore we obtain

$$ \begin{align*} H^{2d_{h_{\pi}^D}}\left({\mathcal M}^{D}_{r,L}(\pi)_a, {\mathbb{C}} \right)_{\mathrm{st}} = H^{2d_{h_{\pi}^D}}\left({\mathcal M}^{D}_{r,L}(\pi)_a, {\mathbb{C}} \right)^{\Gamma} \subseteq H^{2d_{h_{\pi}^D}}\left({\mathcal M}^{D}_{r,L}(\pi)_a, {\mathbb{C}} \right)^{\pi_0\left(P_a\right)} = {\mathbb{C}}, \end{align*} $$

where the last equality is given by the $\pi _0(P_a)$-equivariant isomorphism (49). This implies

(50)$$ \begin{align} \left(\mathrm{R}^{\text{2d}_{\text{h}_{\pi}^{\text{D}}}} {\text{h}^{\mathrm{ell}}_{\pi}}_* {\underline{{\mathbb{C}}}}\right)_{\mathrm{st}} = {\underline{{\mathbb{C}}}} \end{align} $$

and completes the proof of Theorem 2.3(b).

Remark 2.8. The vector space

$$ \begin{align*} H^{2d_{h_{\pi}^D}}\left({\mathcal M}^{D}_{r,L}(\pi)_a, {\mathbb{C}} \right)_{\mathrm{st}} = H^{2d_{h_{\pi}^D}}\left({\mathcal M}^{D}_{r,L}(\pi)_a, {\mathbb{C}} \right)^{\Gamma} \end{align*} $$

may fail to be $1$-dimensional when $a \in {\mathcal A}^D(\pi ) \setminus {\mathcal A}^{D,\mathrm {ell}}(\pi )$. In particular, the constructible sheaf $\left (\mathrm {R}^{\text {2d}_{\text {h}_{\pi }^{\text {D}}}} {\text {h}^{\text {D}_{\pi }}}_* {\underline {{\mathbb {C}}}}\right )_{\mathrm {st}}$ is not a rank $1$ local system over the total Hitchin base ${\mathcal A}^D(\pi )$. Hence the proof of Theorem 2.3(b) relies heavily on the support theorem – Theorem 2.3(a).

2.7 Transfer from the $\kappa $-part to the stable part

In this section, we assume that D is an effective divisor of degree $\mathrm {deg}(D)>2g-2$ or $D= K_C$. Our main purpose is to prove Proposition 2.10, which allows us to transfer naturally from the $\kappa $-part to the stable part of the complex ${\mathrm {Rh}_{\pi }^{\text {D}}}_* {\underline {{\mathbb {C}}}}$. This extends [Reference Yun39, Proposition 2.3.2] to the total Hitchin base for certain endoscopic Hitchin moduli spaces associated with $\mathrm {SL}_n$. We note that this transfer does not rely on the support theorem.

Recall the decomposition (19) of Proposition 1.1(a). By Lemma 1.5, the group $\Omega $ (introduced in formula (27)) acts on each direct image complex ${\mathrm {Rh}_{i}}_* {\underline {{\mathbb {C}}}}$, and we consider its $\Omega $-invariant part

$$ \begin{align*} \left({\mathrm{Rh}_i}_* {\underline{{\mathbb{C}}}}\right)^{\Omega} \in D_c^b\left({\mathcal A}^D(\pi)\right). \end{align*} $$

For any pair $1 \leq i,j \leq m$, the isomorphism of Proposition 1.1(b),

$$ \begin{align*} \phi_{ij} = \phi_{{\mathcal L}_{ij}}: M_i \xrightarrow{\simeq} M_j, \quad ({\mathcal E}, \phi) \mapsto \left({\mathcal E} \otimes {\mathcal L}_{ij}, \theta\right) \end{align*} $$

induced by a line bundle ${\mathcal L}_{ij} \in \Gamma $, yields an isomorphism

$$ \begin{align*} \phi_{ij}^*: {\mathrm{Rh}_j}_* {\underline{{\mathbb{C}}}} \xrightarrow{\simeq} {\mathrm{Rh}_i}_* {\underline{{\mathbb{C}}}}. \end{align*} $$

It preserves the $\Omega $-invariant parts:

(51)$$ \begin{align} \phi_{ij}^*: \left({\mathrm{Rh}_j}_* {\underline{{\mathbb{C}}}}\right)^{\Omega} \xrightarrow{\simeq} \left({\mathrm{Rh}_i}_* {\underline{{\mathbb{C}}}}\right)^{\Omega}. \end{align} $$

We note that the isomorphism (51) depends only on the class of the line bundle ${\mathcal L}_{ij} \in \Gamma $ in the quotient group

$$ \begin{align*} \pi_0(\mathrm{Prym}(C'/C)) = \Gamma / \Omega. \end{align*} $$

Hence the $\Gamma $-action on $\bigoplus _{j=1}^m \left ({\mathrm {Rh}_j}_* {\underline {{\mathbb {C}}}}\right )^{\Omega }$ passes through $\pi _0(\mathrm {Prym}(C'/C))$. Since the group $\Omega $ preserves each component $M_i$, it follows from Proposition 1.1(b) that the elements of the cyclic group $\pi _0(\mathrm {Prym}(C'/C)) = \Gamma /\Omega $ act transitively on the set $\{M_i\}_{i=1}^m$. We may view $\pi _0(\mathrm {Prym}(C'/C))$ as the group of connected components of ${\mathcal M}^D_{r,L}(\pi )$. For any fixed $1\leq i_0 \leq m$, the isomorphism (51) yields a canonical $\pi _0(\mathrm {Prym}(C'/C))$-equivariant isomorphism

(52)$$ \begin{align} \bigoplus_{j=1}^m \left({\mathrm{Rh}_j}_* {\underline{{\mathbb{C}}}}\right)^{\Omega} = \left({\mathrm{Rh}_{i_0}}_*{\underline{{\mathbb{C}}}}\right)^{\Omega} \otimes \left(\bigoplus_{v\in \pi_0\left(\mathrm{Prym}\left(C'/C\right)\right)} {\mathbb{C}} v \right), \end{align} $$

where the action on the right-hand side is the natural one.

Before stating Proposition 2.10, we introduce the following definition for convenience:

Definition 2.9. Let X be an algebraic variety, and let ${\mathcal F}_1, {\mathcal F}_2 \in D^b_c(X)$ be two objects. We say that two morphisms

$$ \begin{align*} f_1: {\mathcal F}_1 \to {\mathcal F}_2 \text{ and } f_2: {\mathcal F}_1 \to {\mathcal F}_2 \end{align*} $$

are equivalent up to scaling if there exists $\lambda \in {\mathbb {C}}^*$ such that $f_1 = \lambda f_2$. We say that there is an isomorphism between two objects ${\mathcal F}_1$ and ${\mathcal F}_2$,

$$ \begin{align*} f: {\mathcal F}_1 \to {\mathcal F}_2, \end{align*} $$

which is canonical up to scaling if our construction induces a set of isomorphisms $f_i: {\mathcal F}_1 \xrightarrow {\simeq } {\mathcal F}_2$ which are all equivalent up to scaling.

Proposition 2.10. Let D be an effective divisor on C of degree $\mathrm {deg}(D)>2g-2$ or $D= K_C$. Assume that $\pi :C' \to C$ is the degree m Galois cover associated with $\gamma \in \Gamma $, which corresponds to the character $\kappa \in \hat {\Gamma }$ via equation (3). Assume $n = mr$. Then for any two elements $\kappa _1, \kappa _2$ in the cyclic group $\langle \kappa \rangle \subset \hat {\Gamma }$ generated by $\kappa $, there is an isomorphism for the corresponding isotypic components,

(53)$$ \begin{align} \left({ \mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa_1} = \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa_2}, \end{align} $$

which is canonical up to scaling. In particular, equation (53) induces an isomorphism which is canonical up to scaling:

$$ \begin{align*} \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa} = \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\mathrm{st}}. \end{align*} $$

Proof. We consider the $\Omega $-invariant part

(54)$$ \begin{align} \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)^{\Omega} \in D_c^b\left({\mathcal A}^D(\pi)\right) \end{align} $$

of the direct image complex ${\mathrm {Rh}_{\pi }^{\text {D}}}_* {\mathbb {C}}$. On one hand, the group $\Omega $ acts on each complex ${\mathrm {Rh}_i}_* {\mathbb {C}}$, and we have

(55)$$ \begin{align} \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)^{\Omega} = \bigoplus_{j=1}^m \left({\mathrm{Rh}_j}_* {\underline{{\mathbb{C}}}}\right)^{\Omega}. \end{align} $$

On the other hand, by Corollary 1.4, an isotypic component $\left ( {\mathrm {Rh}_{\pi }^{\text {D}}}_* {\underline {{\mathbb {C}}}} \right )_{\kappa '}$ contributes to formula (54) if and only if $\kappa '$ lies in $\langle \kappa \rangle $. Hence

(56)$$ \begin{align} \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)^{\Omega} = \bigoplus_{\kappa' \in \langle \kappa \rangle} \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa'}. \end{align} $$

Combining equations (52), (55) and (56), we obtain a natural $\pi _0(\mathrm {Prym}(C'/C))$-equivariant isomorphism

$$ \begin{align*} \bigoplus_{\kappa' \in \langle \kappa \rangle} \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa'} = \left({\mathrm{Rh}_{i_0}}_*{\underline{{\mathbb{C}}}}\right)^{\Omega} \otimes \left(\bigoplus_{v\in \pi_0(\mathrm{Prym}(C'/C))} {\mathbb{C}} v \right). \end{align*} $$

In particular, if we take the $\kappa '$-parts on both sides, since the $\kappa '$-part of the regular representation is $1$-dimensional, this yields a natural isomorphism up to scaling

(57)$$ \begin{align} f_{i_0,\kappa'}: \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa'} \xrightarrow{\simeq} \left({\mathrm{Rh}_{i_0}}_*{\underline{{\mathbb{C}}}}\right)^{\Omega}. \end{align} $$

This gives isomorphism (53) up to scaling, which a priori still depends on the choice of $1\leq i_0 \leq m$.

Different choices of $i_0$ influence the isomorphism (53) via the action of an element

$$ \begin{align*} g\in \Gamma/\Omega = \pi_0(\mathrm{Prym}(C'/C)) \end{align*} $$

on both objects of formula (57). After isolating the $\kappa '$-isotypic component, we conclude that this changes the isomorphism (53) by only a possible scalar ambiguity.

2.8 Changing the degree

Assume $\mathrm {deg}(D)>2g-2$. As another application of Ngô’s support theorem, we analyse the $G_{\pi }$-equivariant complex

(58)$$ \begin{align} \left({\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa} \in D_c^b\left({\mathcal A}^D(\pi)\right) \end{align} $$

when the degree of the line bundle $L \in \mathrm {Pic}^d(C)$ is changed.

Due to Theorem 2.3(b) and Proposition 2.10, the object (58) has full support ${\mathcal A}^D(\pi )$, and hence is completely determined by its restriction to the open subset $U^{\mathrm {sm}} \subset {\mathcal A}^D(\pi )$ where the spectral curves are nonsingular.

Ngô’s analysis of supports for direct image complexes [Reference Ngô33, Section 7] works for each $\kappa $-part (see [Reference Ngô33, Proposition 7.2.3]). In particular, as a corollary of ‘freeness’ [Reference Ngô33, Proposition 7.4.10], the isomorphism class of the restriction of formula (58) to $U^{\mathrm {sm}}$ is determined by the group scheme $P^0\rvert _{U^{\mathrm {sm}}}$ of formula (39) and the constructible sheaf

(59)$$ \begin{align} \left(\mathrm{R}^{\text{2d}_{\text{h}_{\pi}^{\text{D}}}} {\text{h}^{\text{D}}_{\pi}}_* {\underline{{\mathbb{C}}}}\right)_{\kappa}\Big\rvert_{U^{\mathrm{sm}}} \in \mathrm{Sh}_c(U^{\mathrm{sm}}), \end{align} $$

which are both equipped with the $G_{\pi }$-actions. See [Reference de Cataldo, Rapagnetta and Saccà12, Appendix] for a precise form expressing formula (58) in terms of the direct image complex associated with

$$ \begin{align*} g^0_{\pi}\Big\rvert_{U^{\mathrm{sm}}}: P^0\rvert_{U^{\mathrm{sm}}} \rightarrow U^{\mathrm{sm}} \end{align*} $$

and formula (59).

The following proposition will be used only in Section 5.5:

Proposition 2.11. Assume $\mathrm {deg}(D)>2g-2$. Let q be an integer coprime to n. We have an isomorphism of the $G_{\pi }$-equivariant objects

(60)$$ \begin{align} \left({\mathrm{Rh}_{\pi,\text{L}}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{q\kappa} \simeq \left({\mathrm{Rh}_{\pi,\text{L}^{\otimes \text{q}}}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa} \in D_c^b\left({\mathcal A}^D(\pi)\right). \end{align} $$

Here $h_{\pi ,L^{\otimes q}}^D: {\mathcal M}^D_{r,L^{\otimes q}}(\pi ) \to {\mathcal A}^D(\pi ) $ stands for the Hitchin fibration associated with the line bundle $L^{\otimes q}$.

Proof. For notational convenience, we use e to denote $2d_{h_{\pi }^D}$. After restricting to $U^{\mathrm {sm}}$ we have

(61)$$ \begin{align} \left(\mathrm{R}^{\text{e}} {\text{h}_{\pi,\text{L}^{\otimes \text{q}}}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa'} \simeq {\underline{{\mathbb{C}}}}, \quad \forall \ \kappa' \in \langle \kappa \rangle, \end{align} $$

by Proposition 2.10 and equation (50). We need to analyse the $G_{\pi }$-equivariant structure on the rank $1$ trivial local systems (61).

Now we consider the constructible sheaf

(62)$$ \begin{align} \mathrm{R}^{\text{e}} {\text{h}_{\pi,\text{L}^{\otimes \text{q}}}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \Big{|}_{U^{\mathrm{sm}}} \in \mathrm{Sh}_c(U^{\mathrm{sm}}) \end{align} $$

with the $G_{\pi }$-equivariant structure.

By Proposition 1.1(b), the sheaf (62) is a trivial local system of rank m (corresponding to the m connected components of ${\mathcal M}^D_{r,L^{\otimes q}}(\pi )$). We may view it as an m-dimensional vector space $V_q \simeq {\mathbb {C}}^m$ on which the cyclic groups $G_{\pi }$ and $\Gamma $ act. Therefore, to prove formula (60) we only need to show that the isotypic component $\left (V_q\right )_{\kappa }$ is $G_{\pi }$-equivariantly isomorphic to $(V_1)_{q\kappa }$.

Recall the degree $dq$ Prym variety $\mathrm {Prym}^{dq}(C'/C)$ associated with the line bundle $L^{\otimes q}$, whose m connected components are identified with the m connected components of ${\mathcal M}^D_{r,L^{\otimes q}}(\pi )$. To connect $V_q$ and $V_1$, we consider the ‘multiplication by q’ map

(63)$$ \begin{align} \mathrm{mult}_q: \mathrm{Prym}^{d}(C'/C) \to \mathrm{Prym}^{dq}(C'/C), \quad {\mathcal L} \mapsto {\mathcal L}^{\otimes q}, \end{align} $$

which is clearly $G_{\pi }$-equivariant.

We note that $\mathrm {mult}_q$ induces an identification of the m connected components for the Prym varieties on both sides of formula (63). In fact, choosing base points $x \in \mathrm {Prym}^{q}(C'/C)$ and $qx \in \mathrm {Prym}^{dq}(C'/C)$ trivialises both $\mathrm {Prym}(C'/C)$-torsors, and the map $\mathrm {mult}_q$ induces a ‘multiplication by q’ map on the cyclic group $\pi _0(\mathrm {Prym}(C'/C))\simeq {\mathbb {Z}}/m{\mathbb {Z}}$. The claim follows from the fact that $\mathrm {gcd}(m,q)=1$.

In conclusion, formula (63) induces a $G_{\pi }$-equivariant isomorphism

$$ \begin{align*} \left[\mathrm{mult}_q\right]: V_1 \xrightarrow{\simeq} V_q \end{align*} $$

whose $\Gamma $-action on the right-hand side is given by the $\Gamma $-action on the left-hand side composed with the ‘multiplication by q’ $\Gamma \xrightarrow {\cdot q} \Gamma $. In particular, we have a $G_{\pi }$-equivariant isomorphism between $\left (V_q\right )_{\kappa }$ and $(V_1)_{q\kappa }$. This completes the proof of the proposition.

The constraint $\mathrm {deg}(D)> 2g-2$ will be removed by Remark 4.9, despite the fact that we no longer have full supports in that case.

3.1 Overview: Main results

In Sections 3 and 4, we establish a generalised version of Theorem 0.3 for any effective divisor D with $\mathrm {deg}(D)>2g-2$ or $D = K_C$.

Let D be as before, and let $\pi : C' \to C$ be a degree m cyclic Galois cover with $n=mr$. Recall the Hitchin fibrations

$$ \begin{align*} h^D: {\mathcal M}^D_{n,L} \to {\mathcal A}^D, \quad h_{\pi}^D: {\mathcal M}^D_{r,L}(\pi) \rightarrow {\mathcal A}^D(\pi), \end{align*} $$

the fibrewise $\Gamma $-actions and the corresponding $\kappa $-decompositions. The Galois group $G_{\pi }$ acts naturally on ${\mathcal M}^D_{r,d}(\pi )$ and ${\mathcal A}^D(\pi )$ such that the Hitchin map $h_{\pi }^D$ is $G_{\pi }$-equivariant (see Section 1.5). By Lemma 1.7, the direct image complex ${\mathrm {Rh}_{\pi }^{\text {D}}}_*{\mathbb {C}}$ is $G_{\pi }$-equivariant, and so is each $\kappa $-isotypic part

$$ \begin{align*} \left({\mathrm{Rh}_{\pi}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} \in D^b_c({\mathcal A}^D(\pi)), \end{align*} $$

due to the commutativity of the $\Gamma $- and the $G_{\pi }$-actions. We also note that

$$ \begin{align*} q_{\mathcal A}^* \left( \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}} \right)_{\kappa}\in D^b_c\left({\mathcal A}^D(\pi)\right) \end{align*} $$

is naturally $G_{\pi }$-equivariant, induced by the pullback map from the $G_{\pi }$-quotient

$$ \begin{align*} q_{\mathcal A}: {\mathcal A}^D(\pi) \rightarrow {{\mathcal A}^D_{\gamma}}. \end{align*} $$

Recall the open subsets ${\mathcal A}^D(\pi )^*$ and ${{\mathcal A}^D_{\gamma }}^*$ for the Hitchin bases and the free $G_{\pi }$-quotient map between them from Section 1.5.

The following theorem is a generalisation of Theorem 0.2:

Theorem 3.1. Let $\kappa \in \hat {\Gamma }$ and $\gamma \in \Gamma $ be identified by equation (3), let $\pi :C'\to C$ be the Galois cover associated with $\gamma $ and let $d^D_{\gamma } = \mathrm {codim}_{{\mathcal A}^D}\left ({\mathcal A}^D_{\gamma }\right )$. There are isomorphisms in $D^b_c\left ({\mathcal A}^D(\pi )^*\right )$ which are canonical up to scaling:

(64)$$ \begin{align} q_{\mathcal A}^* \left( \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}} \right)_{\kappa}\Big\rvert_{{\mathcal A}^D(\pi)^*} \xrightarrow{\simeq} \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa}\Big\rvert_{{\mathcal A}^D(\pi)^*}\left[-2d^D_{\gamma}\right] \xrightarrow{\simeq} \left({\mathrm{Rh}_{\pi}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}\Big\rvert_{{\mathcal A}^D(\pi)^*} \left[-2d^D_{\gamma}\right], \end{align} $$

with the first isomorphism $G_{\pi }$-equivariant.

The second isomorphism of formula (64) is obtained immediately from Proposition 2.10, which actually holds over the total space ${\mathcal A}^D(\pi )$:

$$ \begin{align*} \left( {\mathrm{Rh}_{\pi}^{\text{D}}}_* {\underline{{\mathbb{C}}}} \right)_{\kappa}\left[-2d^D_{\gamma}\right] \xrightarrow{\simeq} \left({\mathrm{Rh}_{\pi}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\mathrm{st}}\left[-2d^D_{\gamma}\right]. \end{align*} $$

The following theorem is a sheaf-theoretic enhancement of the Hausel–Thaddeus conjecture:

Theorem 3.2. Let $i^D_{\gamma }: {\mathcal A}^D_{\gamma } \hookrightarrow {\mathcal A}^D$ be the closed embedding. With the same notation as in Theorem 3.1, there is an isomorphism which is canonical up to scaling:

(65)$$ \begin{align} c^D_{\kappa}: \left(\mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \xrightarrow{\simeq} {i^D_{\gamma}}_*\left({\mathrm{Rh}^{\text{D}}_{\gamma}}_* {\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right] \in D_c^b\left({\mathcal A}^D\right). \end{align} $$

We first observe that formula (65) induces the first isomorphism of formula (64). So Theorem 3.1 is recovered by Theorem 3.2. In fact, we restrict $c^D_{\kappa }$ to the open subset ${{\mathcal A}_{\gamma }^D}^* \subset {\mathcal A}_{\gamma }^D$ and pull it back along the free $G_{\pi }$-quotient map

(66)$$ \begin{align} q_{\mathcal A}: {\mathcal A}^D(\pi)^* \to {{\mathcal A}^D_{\gamma}}^*. \end{align} $$

Since the left diagram in diagram (29) is Cartesian restricting to formula (66), we recover the first map of formula (64) via proper base change.

Theorems 3.1 and 3.2 recover Theorems 0.2 and 0.3 when $D = K_C$. When $\mathrm {deg}(D)>2g-2$, Theorem 3.2 provides a concrete description of the contribution of each support ${\mathcal A}^D_{\gamma }$ to

$$ \begin{align*} \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}} \in D_c^b\left({\mathcal A}^D\right). \end{align*} $$

This enhances the main theorem of de Cataldo [Reference de Cataldo6].

As already discussed, to prove Theorems 3.1 and 3.2, we only need to construct $G_{\pi }$-equivariant isomorphisms

$$ \begin{align*} c^D_{\kappa}: \left(\mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \xrightarrow{\simeq} {i^D_{\gamma}}_*\left({\mathrm{Rh}^{\text{D}}_{\gamma}}_* {\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right] \in D_c^b\left({\mathcal A}^D\right), \end{align*} $$

which we treat in this section for the following special cases.

In Section 4, we reduce the other cases of Theorem 3.2 – including possibly the most interesting case, $D=K_C$ – to Theorem 3.3.

3.2 Spectral curves and line bundles

Recall the universal spectral curve (36) for $\mathrm {GL}_r$-Higgs bundles over the curve $C'$. We denote its restriction to the subspace ${\mathcal A}^D(\pi ) \subset \widetilde {{\mathcal A}}^{D'}(C')$ by

(67)$$ \begin{align} {\mathcal C}_{\pi} \rightarrow {\mathcal A}^D(\pi). \end{align} $$

This is $G_{\pi }$-equivariant with respect to the natural Galois group $G_{\pi }$ actions on both the base ${\mathcal A}^{D}(\pi )$ and the universal curve ${\mathcal C}_{\pi }$.

We consider the largest Zariski open subset

$$ \begin{align*} {\mathcal A}^{\heartsuit}(\pi) \subset A^D(\pi) \end{align*} $$

such thatFootnote ³

1. (a) the action of $G_{\pi }$ is free on ${\mathcal A}^{\heartsuit }(\pi )$ and

2. (b) the restriction

   (68)$$ \begin{align} {\mathcal C}^{\heartsuit}_{\pi} \rightarrow {\mathcal A}^{\heartsuit}(\pi) \end{align} $$
   of the spectral curves (67) is smooth.

Taking the $G_{\pi }$-quotients, the family (68) descends to a family of nonsingular curves

(69)$$ \begin{align} {\mathcal C}^{\heartsuit}_{\gamma} \rightarrow {\mathcal A}^{\heartsuit}_{\gamma}, \end{align} $$

where ${\mathcal A}^{\heartsuit }_{\gamma } = {\mathcal A}^{\heartsuit }(\pi )/G_{\pi }$ is an open dense subset of ${\mathcal A}^D_{\gamma } = {\mathcal A}^D(\pi )/G_{\pi }$. We denote by

(70)$$ \begin{align} {\mathcal C}^{\circ}_{\gamma} \rightarrow {\mathcal A}^{\heartsuit}_{\gamma} \end{align} $$

the restriction of the universal $\mathrm {SL}_n$-spectral curves over ${\mathcal A}^D$ to ${\mathcal A}^{\heartsuit }_{\gamma }$. The families (69) and (70) are connected by the following lemma:

Lemma 3.4. There is a natural ${\mathcal A}^{\heartsuit }_{\gamma }$-morphism

(71)

whose restriction to each closed fibre

$$ \begin{align*} u_{a}: {\mathcal C}^{\heartsuit}_{\gamma,a} \rightarrow {\mathcal C}^{\circ}_{\gamma,a}, \quad a\in {\mathcal A}^{\heartsuit}_{\gamma}, \end{align*} $$

is a normalisation of curves.

Proof. We first recall the construction in [Reference Hausel and Pauly21, Section 5.1] that, for a given degree r spectral curve $C^{\prime }_{\alpha } \rightarrow C'$ lying in $V(D')$, there is a natural birational morphism

$$ \begin{align*} C^{\prime}_{\alpha} \to C_{\alpha}, \end{align*} $$

with $C_{\alpha }$ a degree n spectral curve over C lying in the total space $V(D)$. In fact, given $C^{\prime }_{\alpha } \to C'$, let

$$ \begin{align*} g^*C^{\prime}_{\alpha} \rightarrow C' \end{align*} $$

be another degree r spectral cover over $C'$ obtained as the pullback of $C^{\prime }_{\alpha } \rightarrow C'$ along the Galois automorphism

$$ \begin{align*} g: C' \xrightarrow{\simeq} C', \quad g\in G_{\pi}. \end{align*} $$

The $G_{\pi }$-invariant curve

$$ \begin{align*} \widetilde{C}^{\prime}_{\alpha} = \bigcup_{g\in G_{\pi}} g^* C^{\prime}_{\alpha} \end{align*} $$

is a degree $n(=mr)$ spectral cover over $C'$, which descends to a degree n spectral cover $C_{\alpha } \to C$ via taking the $G_{\pi }$-quotient. Moreover, we see from the construction of $\Phi _{\Gamma }$ in [Reference Hausel and Pauly21, Section 5.1] that the point $\left [C^{\prime }_{\alpha }\right ] \in {\mathcal A}^{\heartsuit }(\pi )$ maps to $[C_{\alpha }] \in {\mathcal A}^{\heartsuit }_{\gamma }$ via the natural quotient map $q^D_{\mathcal A}: {\mathcal A}^{\heartsuit }(\pi ) \to {\mathcal A}^{\heartsuit }_{\gamma }$. The composition

$$ \begin{align*} C^{\prime}_{\alpha} \hookrightarrow \widetilde{C}^{\prime}_{\alpha} \rightarrow C_{\alpha} \end{align*} $$

is birational, hence it is a normalisation by the smoothness of $C^{\prime }_{\alpha }$.

This construction works for families of spectral curves over the Hitchin bases. Hence we obtain a commutative diagram

where the left vertical morphism is $G_{\pi }$-equivariant and the horizontal morphisms are $G_{\pi }$-quotient maps. The lemma follows from descending the left vertical arrow.

For a closed point $a \in {\mathcal A}^{\heartsuit }_{\gamma }$, we denote the corresponding spectral curves over $C'$ and C by $C^{\prime }_a$ and $C_a$, respectively, with the morphism

$$ \begin{align*} u_a: C^{\prime}_a \rightarrow C_a \end{align*} $$

given by Lemma 3.4. We consider the commutative diagram

(72)

Here $s_a: C_a \to C$ is the spectral cover over C, and $s^{\prime }_a: C^{\prime }_a \to C$ is the composition of the spectral cover $C^{\prime }_a \to C'$ and the cyclic Galois cover $\pi :C' \to C$. Both $s^{\prime }_a$ and $s_a$ are finite of degree n.

We also consider the line bundles

$$ \begin{align*} {\omega_{\pi,a}} = \mathrm{det}\left({s^{\prime}_a}_* {\mathcal O}_{C^{\prime}_a}\right) \in \mathrm{Pic}(C) \text{ and } \omega_a = \mathrm{det}\left({s_a}_* {\mathcal O}_{C_a}\right) \in \mathrm{Pic}(C). \end{align*} $$

The line bundle $\omega _{\pi ,a}$ is defined for every spectral curve over ${\mathcal A}^D(\pi )$, which gives a family of line bundles over the affine space ${\mathcal A}^D(\pi )$. Hence it is constant over ${\mathcal A}^D(\pi )$ and does not depend on the choice of the spectral curve. Similarly, the line bundle $\omega _a$ is also independent of the spectral curve over C. So we may write

$$ \begin{align*} \omega_{\pi} = \omega_{\pi,a}, \quad \omega = \omega_a. \end{align*} $$

The following lemma is obtained via a direct calculation:

Lemma 3.5. We have

$$ \begin{align*} \mathrm{deg}(\omega_{\pi}) =n(1-r)\frac{\mathrm{deg}(D)}{2}, \quad \mathrm{deg}(\omega) = n(1-n)\frac{\mathrm{deg}(D)}{2}. \end{align*} $$

In particular, if $\mathrm {deg}(D)$ is even, both line bundles $\omega _{\pi }$ and $\omega $ have degrees divisible by n.

Proof. Since the second equality is a special case of the first one, we only prove the degree formula for $\mathrm {deg}(\omega _{\pi })$.

By the discussion before Lemma 3.5, the line bundle $\omega _{\pi }$ does not depend on the choice of the spectral curve. Let

$$ \begin{align*} s^{\prime}_a: C^{\prime}_a \to C' \xrightarrow{\pi} C \end{align*} $$

be the spectral cover where $C^{\prime }_a$ is a nonsingular curve lying in $V(D')$ of genus

$$ \begin{align*} g\left(C^{\prime}_a\right) = r(r-1) \frac{\mathrm{deg}(D')}{2}+r\left(g(C')-1\right)+1 \end{align*} $$

(see the formula for $d_{h_n}$ in [Reference de Cataldo6, Section 6.1]). Then applying the Riemann–Roch formula to $\chi \left (C^{\prime }_a, {\mathcal O}_{C^{\prime }_a}\right ) = \chi (C, \omega _{\pi })$, we obtain

$$ \begin{align*} 1-g\left(C^{\prime}_a\right) = \mathrm{deg}(\omega_{\pi})+(1-g),\end{align*} $$

which implies the lemma.

The following lemma concerns pushing forward a line bundle ${\mathcal N} \in \mathrm {Pic}(C_a)$ and its pullback

$$ \begin{align*} u_a^*{\mathcal N} \in \mathrm{Pic}\left(C_a'\right) \end{align*} $$

to the curve $C_a'$:

Lemma 3.6. With the same notation as in diagram (72), we have

$$ \begin{align*} \mathrm{det}\left({s_a}_* {\mathcal N}\right) = \mathrm{det}\left({s^{\prime}_a}_*u_a^*{\mathcal N}\right)\otimes \omega \otimes \omega_{\pi}^{-1}. \end{align*} $$

Proof. Recall the norm maps

$$ \begin{align*} \mathrm{Nm}: \mathrm{Pic}\left(C^{\prime}_a\right) \rightarrow \mathrm{Pic}(C) \text{ and } \mathrm{Nm}:\mathrm{Pic}(C_a) \rightarrow \mathrm{Pic}(C) \end{align*} $$

from [Reference Hausel and Pauly21, Section 3]. By [Reference Hausel and Pauly21, Lemma 3.4], we have

$$ \begin{align*} \mathrm{Nm}({\mathcal N}) = \mathrm{Nm}\left(u_a^*{\mathcal N}\,\right). \end{align*} $$

Then [Reference Hausel and Pauly21, Corollary 3.12] implies that

$$ \begin{align*} \mathrm{det}\left({s_a}_*{\mathcal N}\,\right) \otimes \omega^{-1} = \mathrm{Nm}({\mathcal N}\,) = \mathrm{Nm}\left(u_a^*{\mathcal N}\,\right) = \mathrm{det}\left({s^{\prime}_a}_*u_a^*{\mathcal N}\,\right)\otimes \omega_{\pi}^{-1}. \end{align*} $$

For $n\geq 1$ and $L \in \mathrm {Pic}(C)$, we consider the regular parts

$$ \begin{align*} {\mathcal M}^{D,\mathrm{reg}}_{n,L} \subset {\mathcal M}^D_{n,L}, \quad {\mathcal M}^{D, \mathrm{reg}}_{r,L}(\pi) \subset {\mathcal M}^{D}_{r,L}(\pi), \end{align*} $$

which are open subvarieties parameterising Higgs bundles obtained as the push-forward of line bundles supported on the spectral curves. We define the line bundle

(73)$$ \begin{align} L' = L \otimes \omega \otimes \omega_{\pi}^{-1} \in \mathrm{Pic}(C). \end{align} $$

The following is a corollary of Lemma 3.6:

Corollary 3.7. The pullback $u^*_{\mathcal C}$ of diagram (71) induces a $G_{\pi }$-equivariant morphism of the regular parts

$$ \begin{align*} g_u: {\mathcal M}^{D,\mathrm{reg}}_{n,L}\times_{{\mathcal A}^D} {\mathcal A}^{\heartsuit}(\pi) \rightarrow {\mathcal M}^{D, \mathrm{reg}}_{r,L'}(\pi)\times_{{\mathcal A}^D(\pi)}{\mathcal A}^{\heartsuit}(\pi), \end{align*} $$

where $L'$ is given by equation (73). The morphism ${\mathcal A}^{\heartsuit }(\pi ) \rightarrow {\mathcal A}^D$ used in the base change of the left-hand side is the composition

$$ \begin{align*} {\mathcal A}^{\heartsuit}(\pi) \xrightarrow{q_{\mathcal A}} {\mathcal A}^{\heartsuit}_{\gamma} \hookrightarrow {\mathcal A}^D. \end{align*} $$

Remark 3.8. Since $\mathrm {gcd}(n, \mathrm {deg}(L)) = 1$ and

$$ \begin{align*} \mathrm{deg}(L') = \mathrm{deg}(L) + \mathrm{deg}(\omega) - \mathrm{deg}(\omega_{\pi}), \end{align*} $$

Lemma 3.5 implies that

$$ \begin{align*} \mathrm{gcd}(r, \mathrm{deg}(L')) = \mathrm{gcd}(n, \mathrm{deg}(L')) = 1 \end{align*} $$

when $\mathrm {deg}(D)$ is even.

Finally, we note that both varieties ${\mathcal M}^{D,\mathrm {reg}}_{n,L}$ and ${\mathcal M}^{D, \mathrm {reg}}_{r,L'}(\pi )$ admit $\Gamma $-actions induced by the tensor product with n-torsion line bundles. The $\Gamma $-actions are clearly fibrewise with respect to the Hitchin maps

(74)$$ \begin{align} {\mathcal M}^{D,\mathrm{reg}}_{n,L} \to {\mathcal A}^D, \quad {\mathcal M}^{D, \mathrm{reg}}_{r,L'}(\pi) \to {\mathcal A}^D(\pi). \end{align} $$

Since the pullback $u_a^*: \mathrm {Pic}(C_a) \to \mathrm {Pic}\left (C_a'\right )$ induced by diagram (72) is compatible with the $\Gamma $-action on both sides, we conclude the following lemma:

3.3 Proof of Theorem 3.3, step 1: Cohomological correspondences

Through Section 3.5, we assume that D is an effective divisor with $\mathrm {deg}(D)$ even and $\mathrm {deg}(D)>2g-2$. Our first step in the proof of Theorem 3.3 is to construct a correspondence between the direct image complexes associated with the two Hitchin maps attached to formula (74). Then we show that this correspondence induces an isomorphism for the $\kappa $-parts following Ngô and Yun ([Reference Yun39, Appendix A]).

We consider the graph of $g_u$ in Corollary 3.7, which gives a subvariety

$$ \begin{align*} \mathrm{Graph}(g_u) \subset \left({\mathcal M}^{D, \mathrm{reg}}_{r,L'}(\pi)\times_{{\mathcal A}^D(\pi)}{\mathcal A}^{\heartsuit}(\pi)\right)\times _{{\mathcal A}^{\heartsuit}(\pi)} \left({\mathcal M}^{D,\mathrm{reg}}_{n,L}\times_{{\mathcal A}^D} {\mathcal A}^{\heartsuit}(\pi)\right). \end{align*} $$

Taking its Zariski closure, we obtain a closed subvariety

$$ \begin{align*} \Sigma = \overline{\mathrm{Graph}(g_u)} \subset {\mathcal M}^D_{r,L'}(\pi) \times_{{\mathcal A}^D(\pi)} \left( {\mathcal M}^D_{n,L} \times_{{\mathcal A}^D} {\mathcal A}^D(\pi) \right), \end{align*} $$

which fits into the commutative diagram

(75)

Here we use $h^D_{\pi ,L'}$ to denote the Hitchin fibration

$$ \begin{align*} h^D_{\pi,L'} := h^D_{\pi}: {\mathcal M}^D_{r,L'}(\pi) \to {\mathcal A}^D(\pi), \end{align*} $$

indicating its dependence on the line bundle $L'$. All the morphisms in the diagram are proper.

By Corollary 3.7 and Lemma 3.9, the morphism $g_u$ is equivariant under the actions of $G_{\pi }$ and $\Gamma $. Hence, as the Zariski closure of the graph of $g_u$, the subvariety $\Sigma $ is invariant under the natural actions of $G_{\pi }$ and $\Gamma $ on the ambient space ${\mathcal M}^D_{r,L'}(\pi ) \times _{{\mathcal A}^D(\pi )} \left ( {\mathcal M}^D_{n,L} \times _{{\mathcal A}^D} {\mathcal A}^D(\pi ) \right )$. Since the projections

$$ \begin{align*} \begin{split} {\mathcal M}^D_{r,L'}(\pi) \times_{{\mathcal A}^D(\pi)} \left( {\mathcal M}^D_{n,L} \times_{{\mathcal A}^D} {\mathcal A}^D(\pi) \right) & \to {\mathcal M}^D_{r,L'}, \\ {\mathcal M}^D_{r,L'}(\pi) \times_{{\mathcal A}^D(\pi)} \left( {\mathcal M}^D_{n,L} \times_{{\mathcal A}^D} {\mathcal A}^D(\pi) \right) & \to {\mathcal M}^D_{n,L} \times_{{\mathcal A}^D} {\mathcal A}^D(\pi), \end{split} \end{align*} $$

to both factors are $G_{\pi }$- and $\Gamma $-equivariant, the projections from the invariant subvariety $\Sigma $ to both factors are also $G_{\pi }$- and $\Gamma $-equivariant.

Theorem 3.10. The correspondence (75) induces a morphism

(76)$$ \begin{align} [\Sigma]_\#: q_{\mathcal A}^* \left( \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right) \rightarrow {\mathrm{Rh}_{\pi,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}} \left[-2d^D_{\gamma}\right], \end{align} $$

which is equivariant under the natural actions of $G_{\pi }$ and $\Gamma $. Assume that the element $\gamma \in \Gamma $ inducing the Galois cover $\pi : C' \to C$ matches with $\kappa $ via equation (3). Then the $G_{\pi }$-equivariant morphism for the $\kappa $-parts

(77)$$ \begin{align} [\Sigma]_{\#,\kappa}: q_{\mathcal A}^* \left( \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \to \left({\mathrm{Rh}_{\pi,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right] \end{align} $$

induced by formula (76) is an isomorphism after restricting to ${\mathcal A}^D(\pi )^*$:

(78)$$ \begin{align} [\Sigma]_{\#,\kappa}\Big\rvert_{{\mathcal A}^D(\pi)^*}: q_{\mathcal A}^* \left( \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa}\Big\rvert_{{\mathcal A}^D(\pi)^*} \xrightarrow{\simeq} \left({\mathrm{Rh}_{\pi,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big\rvert_{{\mathcal A}^D(\pi)^*} \left[-2d^D_{\gamma}\right]. \end{align} $$

Proof. The first part follows from the general theory of cohomological correspondences. We refer to [Reference Yun38, Appendix A] as a reference; see also the paragraph before [Reference Yun39, Proposition 3.3.1]. In particular, since the variety $\Sigma $ is invariant under the $G_{\pi }$- and $\Gamma $-actions, and both projections from $\Sigma $ to ${\mathcal M}^D_{n,L}\times _{{\mathcal A}^D}{\mathcal A}^D(\pi )$ and ${\mathcal M}^D_{r,L'}(\pi )$ are $G_{\pi }$- and $\Gamma $-equivariant, we conclude that the cohomological correspondence (76) is also $G_{\pi }$- and $\Gamma $-equivariant.

Now we restrict $[\Sigma ]_{\#,\kappa }$ to the open subset ${\mathcal A}^D(\pi )^*$ and show that it is an isomorphism. It suffices to prove that the restriction of $[\Sigma ]_{\#,\kappa }$ induces an isomorphism on every perverse cohomology – that is,

(79)$$ \begin{align} {}^p{\mathcal H}^i\left([\Sigma]_{\#,\kappa}\right)\Big\rvert_{{\mathcal A}^D(\pi)^*}: q_{\mathcal A}^* {{}^p{\mathcal H}^i}\left( \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \Big\rvert_{{\mathcal A}^D(\pi)^*} \xrightarrow{\simeq} {{}^p{\mathcal H}^{i-2d^D_{\gamma}}}\left({\mathrm{Rh}_{\pi,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} \Big\rvert_{{\mathcal A}^D(\pi)^*}. \end{align} $$

Here we used the fact that $q_{\mathcal A}$ is étale restricting to ${\mathcal A}^D(\pi )^*$. By Corollary 2.2, the left-hand side has ${\mathcal A}^D(\pi )^*$ as the only support. Theorem 2.3(b) and Proposition 2.10 yield that the right-hand side also has ${\mathcal A}^D(\pi )^*$ as the only support. Therefore, both sides are intermediate extensions of certain local systems defined on an open subset of ${\mathcal A}^D(\pi )$. As a consequence, in order to prove formula (79), we only need to show that

$$ \begin{align*} {}^p{\mathcal H}^i\left([\Sigma]_{\#,\kappa}\right)\Big\rvert_U: q_{\mathcal A}^* {{}^p{\mathcal H}^i}\left( \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \Big\rvert_U \xrightarrow{\simeq} {{}^p{\mathcal H}^{i-2d^D_{\gamma}}}\left({\mathrm{Rh}_{\pi,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big\rvert_U, \end{align*} $$

with $U \subset {\mathcal A}^D(\pi )^*$ a Zariski open subset. This reduces the proof to showing that, for a general point $a \in {\mathcal A}^D(\pi )^*$, the correspondence between the Hitchin fibres

(80)$$ \begin{align} [\Sigma_a]: H^i\left(\left({\mathcal M}^D_{n,L}\right)_a, {\mathbb{C}}\right) \rightarrow H^{i-2d^D_{\gamma}}\left({\mathcal M}^D_{r,L'}(\pi)_a, {\mathbb{C}}\right) \end{align} $$

induced by the fundamental class of $\Sigma _a$ is an isomorphism between the $\kappa $-parts. Here $\Sigma _a$ is the restriction of $\Sigma $ over a. Let a be a general point lying in the open subset ${\mathcal A}^{\heartsuit }(\pi )\subset {\mathcal A}^D(\pi )$. The pullback of the diagram (71) along $\{a\} \rightarrow {\mathcal A}^{\heartsuit }_{\gamma }$ is a normalisation

$$ \begin{align*} u_a: C^{\prime}_a \to C_a \end{align*} $$

of curves where $C_a$ has at worst nodal singularities. In this case, the description of the correspondence (80) is concretely given in [Reference Yun39, Appendix A], and the isomorphism of the $\kappa $-parts follows from a direct calculation [Reference Yun39, Lemma 3.4.1].Footnote ⁴ This completes the proof.

Corollary 3.11. With the same assumption as in Theorem 3.10, the $G_{\pi }$-equivariant isomorphism (78) induces an isomorphism

$$ \begin{align*} [\Sigma]^{G_{\pi}}_{\#,\kappa}: \left(\mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \xrightarrow{\simeq} {i^D_{\gamma}}_*\left({\mathrm{Rh}^{\text{D}}_{\gamma,L'}}_* {\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right] \in D^b_c\left({\mathcal A}^D\right). \end{align*} $$

Proof. We push forward the morphism (77) along the $G_{\pi }$-quotient map $q_{\mathcal A}: {\mathcal A}^D(\pi ) \to {\mathcal A}^D_{\gamma }$ and take the $G_{\pi }$-invariant parts on both sides. This gives a morphism

$$ \begin{align*} [\Sigma]^{G_{\pi}}_{\#,\kappa}: \left(\mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \to {i^D_{\gamma}}_*\left( {\mathrm{Rh}^{\text{D}}_{\gamma,L'}}_* {\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right], \end{align*} $$

where both sides are semisimple with ${\mathcal A}^D_{\gamma }$ the only support, by Corollary 2.2, Theorem 2.3(b) and Proposition 2.10. Hence, similar to the proof of Theorem 3.10, it suffices to check that it is an isomorphism restricting to ${{\mathcal A}^D_{\gamma }}^*$ which is equivalent to formula (78).

Remark 3.12. We cannot conclude that formula (77) is an isomorphism over the total space ${\mathcal A}^D(\pi )$ from the isomorphism (78), for the following reason. For a nonsmooth finite morphism $f: X \to Y$ and a semisimple perverse sheaf ${\mathcal K}$ on Y, the pullback $f^*{\mathcal K}$ may fail to be semisimple. A typical example is the case where

$$ \begin{align*} f: {\mathbb{A}}^1 \to {\mathbb{A}}^1, \quad z \mapsto z^2, \end{align*} $$

and ${\mathcal K}$ is the intermediate extension of a $2$-torsion rank $1$ local system on ${\mathbb {C}}^* \subset {\mathbb {A}}^1$. The object $f^*{\mathcal K}$ is not determined by its restriction to the open subset ${\mathbb {C}}^*$. In particular, although the restriction of the natural morphism $f^*{\mathcal K}\to \underline {{\mathbb {C}}}$ to ${\mathbb {C}}^*$ is an isomorphism, the morphism $f^*{\mathcal K} \to \underline {{\mathbb {C}}}$ itself fails to be an isomorphism on ${\mathbb {A}}^1$. The issue here is caused by the fact that f is not étale at $0\in {\mathbb {A}}^1$.

3.4 Proof of Theorem 3.3, step 2: Changing from $L'$ to L

By Lemma 3.5, we have

(81)$$ \begin{align} \mathrm{deg}(L') - \mathrm{deg}(L) = 0 \mod n. \end{align} $$

Hence there exists a line bundle

(82)$$ \begin{align} {\mathcal N}_0 \in \mathrm{Pic}^k(C), \quad k = \frac{\mathrm{deg}(L') - \mathrm{deg}(L)}{n} \in {\mathbb{Z}}, \end{align} $$

such that $L' = L \otimes {\mathcal N}_0^{\otimes n}$. The line bundle ${\mathcal N}_0$ induces an ${\mathcal A}^D(\pi )$-isomorphism between the relative Hitchin moduli spaces,

(83)

via the tensor product $\phi _{{\mathcal N}_0}({\mathcal E}, \theta ) = ({\mathcal E} \otimes {\mathcal N}_0, \theta )$. It is clear that $\phi _{{\mathcal N}_0}$ is $G_{\pi }$- and $\Gamma $-equivariant.

Proposition 3.13. There is a $G_{\pi }$-equivariant isomorphism

(84)$$ \begin{align} \left({\mathrm{Rh}_{\pi,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} = \left({\mathrm{Rh}_{\pi,\text{L}}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} \in D^b_c\left({\mathcal A}^D(\pi)\right) \end{align} $$

induced by diagram (83). Up to scaling, it is independent of the choice of line bundle (82). In particular, equation (84) induces

$$ \begin{align*} \left({\mathrm{Rh}_{\gamma,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} = \left({\mathrm{Rh}_{\gamma,\text{L}}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} \in D^b_c\left({\mathcal A}_{\gamma}^D\right). \end{align*} $$

Proof. The pullback along $\phi _{{\mathcal N}_0}$ in the diagram (83) induces a $G_{\pi }$- and $\Gamma $- equivariant isomorphism

(85)$$ \begin{align} {\mathrm{Rh}_{\pi,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}} = {\mathrm{Rh}_{\pi,\text{L}}^{\text{D}}}_*{\underline{{\mathbb{C}}}}, \end{align} $$

which gives our desired isomorphism. For another choice ${\mathcal N}^{\prime }_0$ of the line bundle (82), we have

$$ \begin{align*} {\mathcal N}_0^{-1}\otimes {\mathcal N}^{\prime}_0 \in \Gamma. \end{align*} $$

Hence the difference of the isomorphisms $\phi ^*_{{\mathcal N}_0}$ and $\phi ^*_{{\mathcal N}^{\prime }_0}$ is induced by an automorphism of ${\mathcal M}^D_{r,L}$ given by an element $g\in \Gamma $. In particular, our choice of isomorphism is affected only by scaling.

The last claim follows from Lemma 1.7.

3.5 Completing the proof of Theorem 3.3

Theorem 3.3 follows from Corollary 3.11 and Proposition 3.13.

More precisely, we construct an isomorphism

$$ \begin{align*} c^D_{\kappa}: \left(\mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \xrightarrow{\simeq} {i^D_{\gamma}}_*\left({\mathrm{Rh}^{\text{D}}_{\gamma}}_* {\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right] \in D_c^b\left({\mathcal A}^D\right) \end{align*} $$

which is canonical up to scaling as the composition

$$ \begin{align*} \left( \mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \xrightarrow[\simeq]{~~[\Sigma]^{G_{\pi}}_{\#,\kappa}~~} {i^D_{\gamma}}_*\left({\mathrm{Rh}_{\gamma,\text{L}'}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right] \xrightarrow[\simeq]{\text{Prop.~3.13}} {i^D_{\gamma}}_*\left({\mathrm{Rh}_{\gamma,\text{L}}^{\text{D}}}_*{\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right]. \end{align*} $$

Here the first isomorphism $[\Sigma ]^{G_{\pi }}_{\#,\kappa }$ is given by Corollary 3.11, and the second isomorphism is given by Proposition 3.13. This completes the proof of Theorem 3.3.

4.1 Overview

In this section we follow the same notation as in Section 3. We complete the proof of Theorem 3.2 (and therefore of Theorem 3.1, as explained in the text after Theorem 3.2) by constructing an operator

$$ \begin{align*} c^D_{\kappa}: \left(\mathrm{Rh}^{\text{D}}_{*} {\underline{{\mathbb{C}}}}\right)_{\kappa} \xrightarrow{\simeq} {i^D_{\gamma}}_*\left({\mathrm{Rh}^{\text{D}}_{\gamma}}_* {\underline{{\mathbb{C}}}}\right)_{\kappa} \left[-2d^D_{\gamma}\right] \in D_c^b\left({\mathcal A}^D\right), \end{align*} $$

which is constructed up to scaling for any effective divisor D with $\mathrm {deg}(D)>2g-2$ or $D= K_C$. Our main tool is Theorem 4.5, where we apply a vanishing cycle functor to connect the moduli of D-Higgs bundles to the moduli of $(D+p)$-Higgs bundles, with $p\in C$ a closed point. This reduces the general cases of Theorem 3.2 to the special cases already treated in Theorem 3.3.

In this section, it is convenient to work with the moduli stacks of stable $\mathrm {SL}_n$-Higgs bundles and relative stable Higgs bundles associated with $\pi : C' \to C$. As these are nonsingular Deligne–Mumford stacks, gerbes over the coarse moduli spaces, this has no effect on the direct image complexes (64). Therefore, throughout this section, we still use the notation ${\mathcal M}^D_{n,L}$ and ${\mathcal M}^D_{r,L}(\pi )$ to denote the corresponding moduli stacks for stable Higgs bundles and relative stable Higgs bundles.

4.2 Restrictions of Higgs bundles to a point

Let p be an abstract reduced point $\mathrm {Spec}({\mathbb {C}})$. Any rank n vector bundle on p is an n-dimensional vector space. Hence the category of $\mathrm {SL}_n$-Higgs bundles on p can be thought of as the category of matrices in $\mathfrak {sl}_n$ up to $\mathrm {SL}_n$-conjugation, whose moduli stack is given by the quotient

$$ \begin{align*} {\mathcal M}_{n,p} = [\mathfrak{sl}_n/\mathrm{SL}_n]. \end{align*} $$

Here $\mathrm {SL}_n$ acts on $\mathfrak {sl}_n$ via conjugation. The Hitchin fibration associated with ${\mathcal M}_{n,p}$ is

$$ \begin{align*} h_p: {\mathcal M}_{n,p} \to {\mathcal A}_p, \end{align*} $$

where ${\mathcal A}_p = \mathfrak {sl}_n\mathbin {/\mkern -6mu/}\mathrm {SL}_n$ is the affine $\mathrm {GIT}$-quotient parameterising all characteristic polynomials

$$ \begin{align*} (a_2, a_3, \dots, a_n) \in {\mathcal A}_p, \quad a_i = \mathrm{trace}\left(\wedge^i \theta_p\right), \end{align*} $$

of the traceless endomorphism $\theta _p \in \mathrm {End}({\mathbb {A}}^n)_0$ associated with a matrix in ${\mathcal M}_{n,p}$. We refer to [Reference Ngô33, Section 2.2] for more details concerning the stack ${\mathcal M}_{n,p}$ and the morphism $h_p$.

Now we consider p as a closed point on the curve C. We fix a trivialisation

(86)$$ \begin{align} {\mathcal O}_C(D)_p \xrightarrow{\simeq} {\mathbb{C}}. \end{align} $$

Then the restriction map with respect to the closed embedding

$$ \begin{align*} i_p: \{p\} \hookrightarrow C \end{align*} $$

induces the following commutative diagram:

(87)

where the trivialisation (86) induces an evaluation (at p) map $\mathrm {ev}_p$:

$$ \begin{align*} \mathrm{ev}_p({\mathcal E}, \theta) = i_p^*({\mathcal E}, \theta) \in {\mathcal M}_{n,p}. \end{align*} $$

To generalise the diagram (87) for the relative moduli space ${\mathcal M}^D_{r,L}(\pi )$ with $n=mr$, we consider the Lie group

$$ \begin{align*} H_{\pi} = \left\{(g_1, g_2, \dots, g_m) \in \mathrm{GL}_r^{\times m}: \prod_{i} \mathrm{det}(g_i) = 1\right\} \subset \mathrm{GL}_r^{\times m}, \end{align*} $$

which is naturally a subgroup of $\mathrm {SL}_n$ with Lie algebra

$$ \begin{align*} \mathfrak{h}_{\pi} = \left\{(g_1, g_2, \dots, g_m) \in \mathfrak{gl}_r^{\times m}: \sum_{i} \mathrm{trace}(g_i) = 0\right\}. \end{align*} $$

The quotient stack

$$ \begin{align*} {\mathcal M}_{r,p}(\pi) = [\mathfrak{h}_{\pi}/H_{\pi}] \end{align*} $$

is the moduli of $\mathrm {SL}_n$-Higgs bundles on the point p obtained as the push-forward of rank r Higgs bundles on m distinct reduced points $\sqcup _{i=1}^m p_i$ along the projection

$$ \begin{align*} \pi_p: \sqcup_{i=1}^m p_i \to p. \end{align*} $$

Similar to diagram (87), we have the following diagram given by the restriction to $p \in C$:

(88)

Here for a Higgs bundle $({\mathcal E}, \theta ) \in {\mathcal M}^D_{r,L}$ on $C'$, again formula (86) induces an evaluation map:

$$ \begin{align*} \mathrm{ev}_p({\mathcal E}, \theta) = i_p^*\left(\pi_* {\mathcal E}, \pi_* \theta\right) \in {\mathcal M}_{r,p}(\pi), \end{align*} $$

and the ‘Hitchin map over a point’ $h_p$ is the natural projection

$$ \begin{align*} h_{\pi,p}: [\mathfrak{h}_{\pi}/H_{\pi}] \rightarrow {\mathcal A}_p(\pi):= \mathfrak{h}_{\pi}\mathbin{/\mkern-6mu/} H_{\pi}. \end{align*} $$

Diagram (88) recovers diagram (87) when $C' = C$ and $\pi : C' \to C$ is the identity.

Proof. We first review the deformation theory of ${\mathcal M}^D_{r,L}(\pi )$ following [Reference Biswas and Ramanan4] and [Reference Ngô33, Section 4.14]. The deformation theory of a rank r Higgs bundle $({\mathcal E}, \theta )$ on $C'$ is governed by the tangent complex

(89)$$ \begin{align} \left[{\mathcal E}{nd}({\mathcal E}) \xrightarrow{\mathrm{ad}(\theta)} {\mathcal E}{nd}({\mathcal E})\otimes {\mathcal O}_{C'}(D')\right] \end{align} $$

lying in degrees $-1$ and $0$. Since now we only consider rank r Higgs bundles $({\mathcal E}, \theta )$ on $C'$ with the fixed determinant and trace after pushing forward to C, to govern the deformation theory of ${\mathcal M}^D_{r,L}(\pi )$ we need to remove the traces of ${\mathcal E}{nd}({\mathcal E})$ after pushing forward the complex (89) to C. Hence the deformation theory of ${\mathcal M}^D_{r,L}(\pi )$ is governed by

(90)$$ \begin{align} C^{\bullet}({\mathcal E},\theta,D) = \left[(\pi_*{\mathcal E}{nd}({\mathcal E}))_0 \xrightarrow{\pi_*\mathrm{ad}(\theta)} (\pi_*{\mathcal E}{nd}({\mathcal E}))_0\otimes {\mathcal O}_C(D)\right], \quad ({\mathcal E},\theta)\in {\mathcal M}^D_{r,L}(\pi), \end{align} $$

where $(\pi _*{\mathcal E}{nd}({\mathcal E}))_0$ denotes the kernel with respect to the trace

$$ \begin{align*} \mathrm{tr}_C: \pi_* {\mathcal E}{nd}({\mathcal E}) \xrightarrow{\pi_*\mathrm{tr}_{C'}} \pi_*{\mathcal O}_{C'} \to {\mathcal O}_C \end{align*} $$

on C. The complex (90) is the tangent complex of ${\mathcal M}^D_{r,L}(\pi )$. The automorphism space, the tangent space and the obstruction space, respectively, are thus given by the cohomology groups

$$ \begin{align*} H^0(C, C^{\bullet}({\mathcal E},\theta,D)), \quad H^1(C, C^{\bullet}({\mathcal E},\theta,D)) \quad \text{and} \quad H^2(C, C^{\bullet}({\mathcal E},\theta,D)). \end{align*} $$

Since the evaluation map $\mathrm {ev}_p$ is induced by the restriction to the point p via $i_p: \{p\}\hookrightarrow C$, the tangent map of $\mathrm {ev}_p$ is

(91)$$ \begin{align} \mathrm{Tan}_{\mathrm{ev}_p}: H^1(C, C^{\bullet}({\mathcal E},\theta,D)) \rightarrow H^1\left(p, i_p^*C^{\bullet}({\mathcal E},\theta,D)\right), \end{align} $$

induced by the restriction morphism between the tangent complexes. Here $i_p^*C^{\bullet }({\mathcal E},\theta ,D) = \left [\mathfrak {h}_{\pi } \xrightarrow {\mathrm {ad}} \mathfrak {h}_{\pi }\right ]$ recovers the tangent complex of ${\mathcal M}_{r,p}(\pi )$; see [Reference de Cataldo, Heinloth and Migliorini8, Appendix 8.2]. To prove the smoothness of $\mathrm {ev}_p$, we show in the following that formula (91) is surjective.

The restriction map between the tangent complexes

$$ \begin{align*} C^{\bullet}({\mathcal E},\theta,D)) \to {i_p}_*i_p^*C^{\bullet}({\mathcal E},\theta,D)) \end{align*} $$

fits into the exact triangle

(92)$$ \begin{align} E^{\bullet} \to C^{\bullet}({\mathcal E},\theta,D) \to {i_p}_*i_p^*C^{\bullet}({\mathcal E},\theta,D) \xrightarrow {+1}, \end{align} $$

where $E^{\bullet }$ is given by

$$ \begin{align*} E^{\bullet} =C^{\bullet}({\mathcal E},\theta,D)\otimes {\mathcal O}_C(-p) = \left[(\pi_*{\mathcal E}{nd}({\mathcal E}))_0\otimes {\mathcal O}_C(-p) \xrightarrow{\pi_*\mathrm{ad}(\theta)} (\pi_*{\mathcal E}{nd}({\mathcal E}))_0\otimes {\mathcal O}_C\left(D-p\right)\right]. \end{align*} $$

The long exact sequence associated with formula (92) contains

$$ \begin{align*} H^1(C, C^{\bullet}({\mathcal E},\theta,D)) \xrightarrow{\mathrm{Tan}_{\mathrm{ev}_p}} H^1\left(C,{i_p}_* i_p^*C^{\bullet}({\mathcal E},\theta,D)\right) \rightarrow H^2(C, E^{\bullet}). \end{align*} $$

Hence, in view of the Serre duality, it suffices to show the vanishing of

(93)$$ \begin{align} H^2(C, E^{\bullet})^{\vee} = H^0\left(C, (E^{\bullet})^{\vee} \otimes \Omega_C\right), \end{align} $$

where

$$ \begin{align*} (E^{\bullet})^{\vee} \otimes \Omega_C = \left[(\pi_*{\mathcal E}{nd}({\mathcal E}))_0\otimes {\mathcal O}_C(p-D+K_C) \xrightarrow{-\mathrm{ad}(\pi_*\theta)} (\pi_*{\mathcal E}{nd}({\mathcal E})_0\otimes {\mathcal O}_C\left(p+K_C\right)\right]. \end{align*} $$

To calculate equation (93), we consider the hypercohomology group $H^0\left (C, \left (\widetilde {E}^{\bullet }\right )^{\vee } \otimes \Omega _C\right )$ with

$$ \begin{align*} \widetilde{E}^{\bullet} = \left[(\pi_*{\mathcal E}{nd}({\mathcal E}))\otimes {\mathcal O}_C(-p) \xrightarrow{\pi_*\mathrm{ad}(\theta)} (\pi_*{\mathcal E}{nd}({\mathcal E}))\otimes {\mathcal O}_C(D-p)\right]. \end{align*} $$

It is clear that $H^0\left (C, \left (\widetilde {E}^{\bullet }\right )^{\vee } \otimes \Omega _C\right )$ contains equation (93) as a direct sum component, and the complement is contributed by the trace parts on C. The hypercohomology group $H^0\left (C, \left (\widetilde {E}^{\bullet }\right )^{\vee } \otimes \Omega _C\right )$ can be written as

(94)$$ \begin{align} H^0\left(C', \left[{\mathcal E}{nd}({\mathcal E})\otimes \pi^*{\mathcal O}_C(p-D+K_C) \xrightarrow{-\mathrm{ad}(\theta)} {\mathcal E}{nd}({\mathcal E})\otimes \pi^*{\mathcal O}_C(p+K_C)\right] \right) \end{align} $$

via the projection formula associated with $\pi : C' \to C$. By [Reference Gothen and King16, Theorem 5.1] (see also the calculation of [Reference Groechenig, Wyss and Ziegler17, Lemma 7.3]), formula (94) can be interpreted as the group of homomorphisms of stable Higgs bundles

(95)$$ \begin{align} \mathrm{Hom}_{C'}\left( ({\mathcal E}, \theta), ({\mathcal E} \otimes \pi^* {\mathcal O}_C(K_C-D+p), \theta) \right). \end{align} $$

In case (a), these two stable Higgs bundles coincide, so the $\mathrm {Hom}$ space (95) is $1$-dimensional, given by the identity map. Hence we have the vanishing of equation (93) by the removal of the $1$-dimensional trace parts on C from formula (95). In case (b), we have

$$ \begin{align*} \mathrm{deg}({\mathcal E})> \mathrm{deg}\left({\mathcal E} \otimes \pi^*{\mathcal O}_C(K_C-D+p)\right). \end{align*} $$

Hence formula (95) vanishes due to the stability condition, which further implies the vanishing of equation (93).

Assume that the divisor D satisfies Proposition 4.1(a) and (b). The moduli of stable $(D-p)$-Higgs bundles admits a natural closed embedding into the moduli of D-Higgs bundles. More precisely, let $({\mathcal E}, \theta )$ be a stable $(D-p)$-Higgs bundle on C; then we may view it naturally as a D-Higgs bundle $({\mathcal E}, \theta ')$ by setting the new Higgs field as the composition

$$ \begin{align*} \theta': {\mathcal E} \xrightarrow{\theta} {\mathcal E}\otimes {\mathcal O}_C(D-p) \rightarrow {\mathcal E} \otimes{\mathcal O}_C(D), \end{align*} $$

where the second map is induced by ${\mathcal O}_C(-p) \hookrightarrow {\mathcal O}_C$. By definition, the (slope-)stability conditions of $({\mathcal E}, \theta )$ and $({\mathcal E}, \theta ')$ coincide. Hence we obtain a closed embedding

(96)$$ \begin{align} {\mathcal M}^{D-p}_{n,L} \hookrightarrow {\mathcal M}^D_{n,L}, \quad ({\mathcal E},\theta) \mapsto ({\mathcal E}, \theta'). \end{align} $$

Similarly, we also have the relative version with respect to $\pi : C' \to C$:

(97)$$ \begin{align} {\mathcal M}^{D-p}_{r,L}(\pi) \hookrightarrow {\mathcal M}^D_{r,L}(\pi), \quad ({\mathcal E},\theta)\mapsto ({\mathcal E}, \theta'), \end{align} $$

where $\theta ': {\mathcal E} \to {\mathcal E}\otimes \pi ^*{\mathcal O}_{C}(D)$ is given by the composition

$$ \begin{align*} \theta': {\mathcal E}\xrightarrow{\theta} {\mathcal E} \otimes \pi^*{\mathcal O}_C(D-p) \to {\mathcal E} \otimes \pi^*{\mathcal O}_C(D). \end{align*} $$

As before, the relative case (97) recovers formula (96) by setting $\pi = \mathrm {id}$.

A Higgs bundle in ${\mathcal M}^D_{r,L}(\pi )$ which sits inside ${\mathcal M}^{D-p}_{r,L}(\pi )$ can be characterised by the vanishing of the restricted Higgs field over the point p. Therefore it is natural to use the evaluation map $\mathrm {ev}_p$ to describe formulas (96) and (97) as in the following lemma:

Lemma 4.2. Assume that the divisor D satisfies Proposition 4.1(a) and (b). We denote by

$$ \begin{align*} 0_H = [0/H_{\pi}]\hookrightarrow {\mathcal M}_{r,p}(\pi) = [\mathfrak{h}_{\pi}/H_{\pi}] \end{align*} $$

the closed substack corresponding to the $0$ matrix. Then the closed embedding (97) is realised as a closed fibre of $\mathrm {ev}_p$ over $0_H$.

4.3 Functions and critical loci

We consider the quotient map

$$ \begin{align*} \mathfrak{sl}_n \rightarrow \mathfrak{sl}_n\mathbin{/\mkern-6mu/} \mathrm{SL}_n \end{align*} $$

sending a matrix to the coefficients

$$ \begin{align*} (a_2, a_3, \dots, a_n)\in \mathfrak{sl}_n\mathbin{/\mkern-6mu/} \mathrm{SL}_n = \mathrm{Spec}\left({\mathbb{C}}[\mathfrak{sl}_n]^{\mathrm{SL}_n}\right), \quad \mathrm{deg}(a_i) = i, \end{align*} $$

of its characteristic polynomial. The term $a_i$ defines a degree i polynomial function on the Lie algebra $\mathfrak {sl}_n$. We define the quadratic function on the Lie algebra $\mathfrak {sl}_n$ given by $a_2$ as

(98)$$ \begin{align} \mu = a_2: \mathfrak{sl}_n \to {\mathbb{A}}^1, \end{align} $$

which induces a function $\mu _{\pi }: \mathfrak {h}_{\pi } \to {\mathbb {A}}^1$ via the composition

(99)$$ \begin{align} \mu_{\pi} : \mathfrak{h}_{\pi} \hookrightarrow \mathfrak{sl}_n \xrightarrow{\mu} {\mathbb{A}}^1. \end{align} $$

By definition, the functions $\mu $ and $\mu _{\pi }$ are invariant under the conjugation actions by the Lie groups $\mathrm {SL}_n$ and $H_{\pi }$, respectively.

Since any matrix $g \in \mathfrak {h}_{\pi } \subset \mathfrak {sl}_n$ satisfies $\mathrm {trace}(g) = 0$, the quadratic function $\mu _{\pi }$ (up to scaling) can be written as

(100)$$ \begin{align} g \mapsto \mathrm{trace}\left(g^2\right). \end{align} $$

For the semisimple Lie algebra $\mathfrak {sl}_n$, formula (100) is the Killing form, which is clearly nondegenerate. In the following we prove the nondegeneracy for general $\mathfrak {h}_{\pi }$:

Lemma 4.3. The critical locus of the quadratic function $\mu _{\pi }$ is the isolated reduced point $0 \in \mathfrak {h}_{\pi }$ – that is,

$$ \begin{align*} \mathrm{Crit}(\mu_{\pi})\left( = \{d\mu_{\pi}=0\}\right) = \{0\}\subset \mathfrak{h}_{\pi}. \end{align*} $$

Consequently, the perverse sheaf of vanishing cycles $\varphi _{\mu _{\pi }}(\underline {{\mathbb {C}}}[\mathrm {dim}({\mathfrak {h}_{\pi }}])$ [Reference Dimca13, Theorem 5.2.21] is the skyscraper sheaf supported on the closed point $0 \in \mathfrak {h}_{\pi }$.

Proof. To prove the first part of the lemma, it suffices to show that the quadratic form (100) on $\mathfrak {h}_{\pi }$ is nondegenerate.

We consider the decomposition

(101)$$ \begin{align} \mathfrak{h}_{\pi} = \mathfrak{h}^{\prime}_{\pi} \oplus \mathfrak{c}, \end{align} $$

where $\mathfrak {c} \subset \mathfrak {h}_{\pi }$ is the Cartan subalgebra of trace-free diagonal matrices and $\mathfrak {h}^{\prime }_{\pi }$ consists of the matrices in $\mathfrak {h}_{\pi }$ such that the entries of the diagonals vanish. For a matrix $g\in \mathfrak {h}_{\pi }$ with the decomposition

$$ \begin{align*} g = g' + c, \quad g' \in \mathfrak{h}^{\prime}_{\pi}, \quad c\in \mathfrak{c}, \end{align*} $$

a direct calculation yields $\mathrm {trace}(g'c)=0$. Therefore we have

$$ \begin{align*} \mathrm{trace}\left(g^2\right) = \mathrm{trace}\left({g'}^2\right) + \mathrm{trace}\left(c^2\right). \end{align*} $$

So it suffices to show that the quadratic forms (100) are nondegenerate for both $\mathfrak {h}^{\prime }_{\pi }$ and $\mathfrak {c}$.

We notice that the Cartan subalgebra $\mathfrak {c}$ of $\mathfrak {h}_{\pi }$ is the same as that of $\mathfrak {sl}_n$. Also, equipped with the quadratic forms (100), $\mathfrak {h}^{\prime }_{\pi }$ is a direct summand component of the Lie algebra $\mathfrak {sl}_r^{\times m}$ via the decomposition (101) for $\mathfrak {sl}_r^{\times m}$. Since both $\mathfrak {h}_{\pi }'$ and $\mathfrak {c}$ are direct summand components of semisimple Lie algebras where the Killing forms (100) are nondegenerate, we conclude the nondegeneracy of formula (100) for $\mathfrak {h}^{\prime }_{\pi }$ and $\mathfrak {c}$, which further implies the nondegeneracy of $\mathfrak {h}_{\pi }$ through equation (101).

This completes the proof of the first part of the lemma and reduces the second part to the case

$$ \begin{align*} \mu_{\pi}: {\mathbb{A}}^N \to {\mathbb{A}}^1, \quad (z_1, \dots, z_N) \mapsto \sum_{i=1}^N z_i^2. \end{align*} $$

In this case, the Milnor fibre is a sphere [Reference Milnor31] whose reduced homology computes the vanishing cycle [Reference Dimca13, Proposition 4.2.2].