2.1 Deligne–Mumford curves

We recall some basic definitions of Deligne–Mumford curves. For a detailed discussion of these topics, please refer to [Reference Behrend and NoohiBN06].

2.2 Semistable sheaves on Deligne–Mumford stacks

On a Deligne–Mumford stack, there is no very ample line bundle on it unless it is an algebraic space. However, Olsson and Starr [Reference Olsson and StarrOS03] have discovered that under certain hypothesis, there are locally free sheaves, called generating sheaves, which behave like very ample line bundles. In the following, ${\mathcal {X}}$ is a Deligne–Mumford stack with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ .

Example 2.9 Suppose that ${\mathcal {X}}$ is a smooth irreducible stacky curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . Let $\{p_1,\ldots ,p_m\}$ be the set of stacky points of ${\mathcal {X}}$ with the orders of stabilizer groups $\{r_1,\ldots ,r_m\}$ . Then, the locally free sheaf

(2) $$ \begin{align} {\mathcal E}_u=\textstyle{\bigoplus_{i=1}^m\bigoplus_{j=0}^{r_i-1}{\mathcal{O}}_{{\mathcal{X}}}(\frac{j}{r_i}\cdot p_i)} \end{align} $$

is a generating sheaf, since it is $\pi $ -very ample (see [Reference NironiNir08, Definition 2.2 and Proposition 2.7]). Let ${\mathcal {O}}_X(1)$ be a very ample line bundle on X. Then, $({\mathcal E}_u,{\mathcal {O}}_X(1))$ is a polarization on ${\mathcal {X}}$ .

Remark 2.11 In general, the modified Hilbert polynomial is

(3) $$ \begin{align} \textstyle{P_{F}(m)={\sum}_{i=0}^d\frac{a_i(F)}{i!}\cdot m^i}, \end{align} $$

where d is the dimension of the support $\mathrm {supp}(F)$ and $a_i(F)$ are rationals.

Definition 2.14 Suppose that F is a pure sheaf on ${\mathcal {X}}$ . F is said to be semistable (stable) if for every proper coherent subsheaf $F^{\prime }$ of F, we have

$$ \begin{align*} p_{F^{\prime}}(m)\leq(<)p_F(m),\quad \text{for } m\gg0. \end{align*} $$

2.3 Higgs bundles and stability

Let ${\mathcal {X}}$ be a smooth Deligne–Mumford curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ .

Example 2.17 Let ${\mathcal {X}}$ be a smooth irreducible Deligne–Mumford curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . The canonical line bundle is $K_{\mathcal {X}}=\pi ^{*}K_X\otimes L_{\mathcal {X}}$ , for some line bundle $L_{\mathcal {X}}$ on ${\mathcal {X}}$ . In fact, if ${\mathcal {X}}$ is a stacky curve, it is so (see [Reference Voight and Zureick-BrownVB22, Proposition 5.5.6]). Then, we can get the formula for a general Deligne–Mumford curve by Remark (2.2). Let $E=E_1\oplus E_2$ , where $E_1=\pi ^{*}K^{\frac {1}{2}}_{X}$ and $E_2=\pi ^{*}K_{X}^{-\frac {1}{2}}\otimes L_{\mathcal {X}}^{-1}$ . With respect to the decomposition of E, there is a morphism of ${\mathcal {O}}_{{\mathcal {X}}}$ -modules

$$ \begin{align*} \phi= \begin{pmatrix} 0 & 0\\ \mathbf{1} & 0 \end{pmatrix} \in \mathrm{Hom}_{{\mathcal{O}}_{{\mathcal{X}}}}(E,E\otimes K_{{\mathcal{X}}}), \end{align*} $$

where $\mathbf {1}$ is the identity morphism in $\mathrm {Hom}_{{\mathcal {O}}_{{\mathcal {X}}}}(E_1,E_1)$ . The pair $(E,\phi )$ is a Higgs bundle on ${\mathcal {X}}$ .

Using the modified slope, we introduce the notions of semistable (stable) Higgs bundles.

Definition 2.18 Fix a polarization $({\mathcal E},{\mathcal {O}}_{X}(1))$ on ${\mathcal {X}}$ . A Higgs bundle $(E,\phi )$ is said to be semistable (resp. stable) if for all proper nonzero $\phi $ -invariant locally free subsheaf $F\subset E$ (i.e., $\phi (F)\subseteq F\otimes K_{{\mathcal {X}}}$ ), we have

$$\begin{align*}\mu_{\mathcal E}(F)\leq\mu_{\mathcal E}(E)\quad(\text{resp. } \mu_{\mathcal E}(F)<\mu_{\mathcal E}(E)). \end{align*}$$

If $(E,\phi )$ is not semistable, we say that $(E,\phi )$ is unstable.

3.1 Moduli stacks of Higgs bundles

The moduli functor of rank r Higgs bundles is

$$ \begin{align*} \mathcal M_{\mathop{\mathrm{Dol}}\nolimits}(\mathop{\mathbf{GL}}\nolimits_r) : (\mathop{\mathrm{Sch}}\nolimits/{\mathbb C})_{\mathop{\acute{\rm e}{\rm t}}\nolimits}^o\longrightarrow(\mathrm{groupoids}), \end{align*} $$

where ${\mathcal M}_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {GL}}\nolimits _r)(T)$ is the groupoid of T-families of rank r Higgs bundles on ${\mathcal {X}}$ for a test scheme T. Similarly, we can also define the moduli functor $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ of rank r Higgs bundles with modified Hilbert polynomial P on ${\mathcal {X}}$ . Suppose that $\mathcal M_{\mathop {\mathrm {Vec}}\nolimits ,r}$ is the moduli functor of rank r locally free sheaves on ${\mathcal {X}}$ . There is a forgetful functor

(4) $$ \begin{align} {\mathcal F} : \mathcal M_{\mathop{\mathrm{Dol}}\nolimits}(\mathop{ \mathbf{GL}}\nolimits_r)\longrightarrow\mathcal M_{\mathop{\mathrm{Vec}}\nolimits,r} \end{align} $$

defined by forgetting the Higgs fields.

Let $\mathcal Y=\mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})$ be the projective bundle associated with $K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}}$ , and let ${\mathcal {O}}_{\mathcal Y}(1)$ be the relative hyperplane bundle on $\mathcal Y$ . Due to the universal property of coarse moduli spaces, we have the commutative diagram

(5)

where $\pi ^{\prime } : \mathcal Y\rightarrow Y$ is the coarse moduli space of $\mathcal Y$ and the first square is Cartesian. For a polarization $({\mathcal E}_{{\mathcal {X}}},{\mathcal {O}}_X(1))$ on ${\mathcal {X}}$ , there is a polarization $({\mathcal E}_{\mathcal Y},{\mathcal {O}}_{Y}(1))$ on $\mathcal Y$ , where ${\mathcal E}_{\mathcal Y}=\Psi ^{*}{\mathcal E}_{{\mathcal {X}}}$ ( $\Psi ^{*}{\mathcal E}_{{\mathcal {X}}}$ is a generating sheaf on $\mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})$ (see [Reference Olsson and StarrOS03, Proposition 5.3])) and ${\pi ^{\prime }}^{*}{\mathcal {O}}_{Y}(1)={(\pi \circ \Psi )}^{*}{\mathcal {O}}_{X}(1){\otimes } {\mathcal {O}}_{\mathcal Y}(m)$ for some $m\gg 0$ . A Higgs bundle $(E,\phi )$ on ${\mathcal {X}}$ is equivalent to a compactly supported one-dimensional pure sheaf $E_{\phi }$ on $\mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})$ (see Appendix C).

Proof $E_{\phi }$ is a pure sheaf on $\mathcal Y$ with modified Hilbert polynomial $P_{E_{\phi }}(n)=\chi ({\mathcal Y},{{\mathcal E}_{\mathcal Y}^{\vee }}\otimes E_{\phi }\otimes {\pi ^{\prime }}^{*}{\mathcal {O}}_{Y}(n))$ . Since the support of $E_{\phi }$ is contained in ${\mathop {\mathrm {Tot}}\nolimits }(K_{\mathcal {X}})$ , we have

$$ \begin{align*} \chi(\mathcal Y,{{\mathcal E}^{\vee}_{\mathcal Y}}\otimes E_{\phi}\otimes{\pi^{\prime}}^{*}{\mathcal{O}}_{Y}(n))= \chi(\mathop{\mathrm{Tot}}\nolimits(K_{{\mathcal{X}}}),\psi^{*}{\mathcal E}^{\vee}\otimes E_{\phi}\otimes{(\pi\circ\psi)}^{*}{\mathcal{O}}_{X}(n)), \end{align*} $$

where $\psi : \mathop {\mathrm {Tot}}\nolimits ({\mathcal {X}})\rightarrow {\mathcal {X}}$ is the natural projection. Note that $\psi $ is an affine morphism. Hence, the pushforward functor $\psi _{*}$ is exact on the category of quasicoherent sheaves. Then, we have the identity

$$ \begin{align*} \chi({\mathop{\mathrm{Tot}}\nolimits}(K_{{\mathcal{X}}}),\psi^{*}{{\mathcal E}^{\vee}}\otimes E_{\phi}\otimes{(\pi\circ\psi)}^{*}{\mathcal{O}}_{X}(n)) =\chi({\mathcal{X}},{{\mathcal E}^{\vee}}\otimes E\otimes \pi^{*}{\mathcal{O}}_{X}(n))), \end{align*} $$

where we use the fact $\psi _{*}E_{\phi }=E$ . So, $P_{E_{\phi }}=P_{E}$ . On the other hand, the $\phi $ -invariant locally free subsheaves of E are equivalent to the coherent subsheaves of $E_{\phi }$ . We complete the proof.

In order to construct the moduli space of semistable Higgs bundles on ${\mathcal {X}}$ , we recall a lemma (see [Reference Fantechi, Gottsche, Illusie, Kleiman and NitsureFGIKN05, Section 5.6]).

We need a stacky version of the above lemma. First, we state a technical lemma.

Lemma 3.6 Suppose that $\widehat {{\mathcal {X}}}$ is a proper Deligne–Mumford stack over a Noetherian scheme S and E is a coherent sheaf on $\widehat {{\mathcal {X}}}$ . If ${\mathcal E}$ is a generating sheaf on $\widehat {{\mathcal {X}}}$ , then we have

$$ \begin{align*} \mathrm{supp} (F_{\mathcal E}(E))\subseteq\pi(\mathrm{supp}(E)), \end{align*} $$

where $\pi : \widehat {{\mathcal {X}}}\rightarrow \widehat X$ is the coarse moduli space of $\widehat {{\mathcal {X}}}$ and $F_{\mathcal E}(E)=\pi _{*}({\mathcal E}^{\vee }\otimes E)$ . Moreover, $F_{{\mathcal E}}(E)=0$ if and only if $E=0$ .

Proof Let $\pi _{{\mathcal W}}:{\mathcal W}\rightarrow W$ and $\pi _{\widehat {{\mathcal {X}}}}:\widehat {{\mathcal {X}}}\rightarrow \widehat X$ be the coarse moduli spaces of ${\mathcal W}$ and $\widehat {{\mathcal {X}}}$ , respectively. Then, by the universal property of coarse moduli spaces, there is a commutative diagram

(6)

where i is the closed immersion and $i^{\prime }$ is the induced morphism.

Claim The morphism $i^{\prime } : W\rightarrow \widehat X$ in (6) is a closed immersion. In fact, there is a short exact sequence of coherent sheaves

(7)

where $\mathcal I_{{\mathcal W}}$ is the ideal sheaf of ${\mathcal W}$ in $\widehat X$ . By the tameness of $\widehat {{\mathcal {X}}}$ (see Definition 1.1 and Theorem 1.2 in [Reference NironiNir08]), the pushforward of (7) to $\widehat X$ is

(8)

By (6), the following two compositions

(9) $$ \begin{align} {\mathcal{O}}_{\widehat X}\rightarrow\pi_{\widehat{{\mathcal{X}}}*}{\mathcal{O}}_{\widehat{{\mathcal{X}}}}\rightarrow \pi_{\widehat{{\mathcal{X}}}*}(i_{*}({\mathcal{O}}_{{\mathcal W}})) \text{ and } {\mathcal{O}}_{\widehat X}\rightarrow i_{*}^{\prime}{\mathcal{O}}_W\rightarrow i^{\prime}_{*}(\pi_{{\mathcal W}*}({\mathcal{O}}_{{\mathcal W}})) \end{align} $$

are the same. By (8) and the isomorphism ${\mathcal {O}}_{\widehat X}\rightarrow \pi _{\widehat {{\mathcal {X}}}*}{\mathcal {O}}_{\widehat {{\mathcal {X}}}}$ , we have the commutative diagram of short exact sequences

(10)

where $\mathcal I_W$ is the kernel of the composition in (9). Note that the naturel morphism ${\mathcal {O}}_{W}\rightarrow \pi _{{\mathcal W}*}{\mathcal {O}}_{{\mathcal W}}$ is an isomorphism. By (6) and (10), we have the short exact sequence

(11)

On the other hand, the morphism of topological spaces induced by $i^{\prime }$ is a closed embedding. Thus, $W\rightarrow X$ is a closed immersion. By Lemma 3.5, for the coherent sheaf $F_{\mathcal E}(E)$ , there is an open subscheme $S^{\prime }$ of S with the universal property: a morphism $T\rightarrow S$ factors through $S^{\prime }$ if and only if the support of the pullback ${F_{{\mathcal E}}(E)}_T$ of $F_{{\mathcal E}}(E)$ on $\widehat X_T=\widehat X\times _ST$ is disjoint from $W\times _ST$ . On the other hand, for a morphism $T\rightarrow S$ , we consider the Cartesian diagram

By Proposition 1.5 in [Reference NironiNir08], we have $F_{{\mathcal E}}(E)_T\simeq F_{{\mathcal E}_{T}}(E_T)$ , where the pullback ${\mathcal E}_T$ of ${\mathcal E}$ to $T\times _{S}\widehat {{\mathcal {X}}}$ is a generating sheaf (see [Reference Olsson and StarrOS03, Theorem 5.5]). Then, the support of the pullback $E_{T}$ of E on $\widehat {{\mathcal {X}}}_T$ is disjoint from ${\mathcal W}_T$ if and only if the morphism $T\rightarrow S$ factors through the $S^{\prime }$ , by Lemma 3.6.

Recall Theorem 5.1 in [Reference NironiNir08]:

Let $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ be the moduli stack of rank r semistable Higgs bundles with modified Hilbert polynomial P on ${\mathcal {X}}$ . By Proposition 3.4 and Lemma 3.7, we have the following corollary.

We need the following proposition to define the S-equivalence of semistable Higgs bundles.

In general, the algebraic stacks without finite inertia rarely admit coarse moduli spaces. Alper introduced the notion of good moduli spaces in [Reference AlperAlp13].

Proof According to Theorem 6.22 in [Reference NironiNir08], the moduli stack $[R^{ss}_1/{\mathop { \mathbf {GL}}\nolimits _{N}}]$ of semistable one-dimensional pure sheaves with modified Hilbert polynomial P on $\mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})$ has a good moduli space $\mathcal Q_1 : [R^{ss}_1/{\mathop {\mathbf {GL}}\nolimits _N}]\longrightarrow M^{ss}_1$ , where $M_1^{ss}$ is the GIT quotient of $R^{ss}_1$ with respect to the $\mathop {\mathbf {SL}}\nolimits _N$ -action. It also satisfies the properties:

• • Universal property: for every scheme Z and every morphism $g_1 : [R^{ss}_1/{\mathop { \mathbf {GL}}\nolimits _N}]\rightarrow Z$ , there is a unique morphism $\theta _1 : M^{ss}_1\rightarrow Z$ such that $g_1=\theta _1\circ \mathcal Q_1$ .

• • $M^{ss}_1$ is a projective scheme over ${\mathbb C}$ .

Recall $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {GL}}\nolimits _r)=[R^{ss}/\mathop {\mathbf {GL}}\nolimits _N]$ (see Corollary 3.9). Since $R^{ss}$ is a $\mathop {\mathbf {GL}}\nolimits _N$ -invariant open subscheme of $R_1^{ss}$ , $R^{ss}_1\setminus {R}^{ss}$ is a ${\mathop {\mathbf {GL}}\nolimits _N}$ -invariant closed subset. Let $Q : R^{ss}_1\rightarrow M^{ss}_1$ be the GIT quotient, which is a good quotient. Hence, the image $Q(R^{ss}_1\setminus {R}^{ss})$ is a closed subset of $M^{ss}_1$ . And, $Q(R^{ss})\cap Q({R^{ss}_1\setminus R}^{ss})=\emptyset $ . In fact, two semistable sheaves on $\mathcal Y$ represent the same point in $M^{ss}_1$ if and only if they are S-equivalent (see [Reference NironiNir08, Theorem 6.20]). By Remark 3.11, for a semistable $(E,\phi )$ on ${\mathcal {X}}$ , the support of the graded sheaf associated with some Jordan–Hölder filtration of $E_{\phi }$ is contained in $\mathop {\mathrm {Tot}}\nolimits (K_{{\mathcal {X}}})$ . Denote $Q({R}^{ss})$ by ${M}^{ss}$ . The following diagram is Cartesian:

The universal property of $\mathcal Q : [{R}^{ss}/{\mathop {\mathbf {GL}}\nolimits _N}]\rightarrow {M}^{ss}$ is an immediate conclusion, since Q is a universal categorical quotient. ${M}^{ss}$ is the good moduli space of $[{R}^{ss}/{\mathop {\mathbf {GL}}\nolimits _N}]$ (see [Reference AlperAlp13, Remark 6.2]).

3.2 Moduli stack of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles

The moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {SL}}\nolimits _r)$ of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles is the stack whose fiber over a test scheme T is the groupoid of T-families of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles on ${\mathcal {X}}$ . Similarly, we have the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {SL}}\nolimits _r)$ (resp. $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ ) of (resp. semistable) $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with fixed modified Hilbert polynomial P.

Proof The Picard stack $\mathcal Pic({{\mathcal {X}}})$ of ${\mathcal {X}}$ is an algebraic stack locally of finite type over ${{\mathbb C}}$ (see [Reference AokiAok06]). There is a morphism of algebraic stacks $\mathcal Det : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {GL}}\nolimits _r)\rightarrow \mathcal {P}ic({{\mathcal {X}}})$ , which is defined by taking determinants. By taking the traces of Higgs fields, we can define a morphism $\mathcal Tr : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop { \mathbf {GL}}\nolimits _r)\rightarrow {\mathbb H}^0({\mathcal {X}},K_{{\mathcal {X}}})$ , where ${\mathbb H}^0({\mathcal {X}},K_{{\mathcal {X}}})$ is the affine space associated with $H^0({\mathcal {X}},K_{{\mathcal {X}}})$ . On the other hand, L defines a geometric point $[L] : {\mathop {\mathrm {Spec}}\nolimits }({\mathbb C})\rightarrow \mathcal Pic({{\mathcal {X}}})$ and the origin of $H^0({\mathcal {X}},K_{\mathcal {X}})$ defines a closed point $o : \mathrm { Spec}({\mathbb C})\rightarrow \mathbb {H}^0({\mathcal {X}},K_{{\mathcal {X}}})$ . We therefore have the Cartesian diagram

(12)

Hence, $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }({\mathop {\mathbf {SL}}\nolimits _r})$ is an algebraic stack locally of finite type over ${\mathbb C}$ by Proposition 3.2.

In the following, we will show that the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ is a quotient stack. Let $(E_{R^{ss}},\phi _{R^{ss}})$ be the $\mathop {\mathbf {GL}}\nolimits _N$ -equivariant Higgs bundle on ${\mathcal {X}}\times R^{ss}$ , which is the pushforward of the universal quotient sheaf on $\mathrm {Tot}(K_{{\mathcal {X}}})\times R^{ss}$ . Let $\mathrm {det}: R^{ss}\rightarrow \mathrm {Pic}({\mathcal {X}})$ be the classifying morphism defined by $\mathop {\mathrm {det}}\nolimits (E_{R^{ss}})$ , where ${\mathop {\mathrm {Pic}}\nolimits }({\mathcal {X}})$ is the Picard scheme of ${\mathcal {X}}$ (see [Reference BrochardBro12, Corollary 2.3.7(i)]). On the other hand, the trace of the morphism ${\phi _{R^{ss}}}: E_{R^{ss}}\rightarrow E_{R^{ss}}\otimes \mathrm {pr}^{*}_{{\mathcal {X}}}K_{{\mathcal {X}}}$ defines a section $\mathrm {tr}(\phi _{R^{ss}})$ of $\mathrm {pr}^{*}_{{\mathcal {X}}}K_{{\mathcal {X}}}$ . It defines a morphism $\mathop {\mathrm {tr}}\nolimits : R^{ss}\rightarrow {\mathbb H}^0({\mathcal {X}},K_{{\mathcal {X}}})$ . Consider the Cartesian diagram

(13)

Proof First, we consider the Cartesian diagram

(14)

Hence, $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)$ can be presented by $[R^{ss,o}/{\mathop {\mathbf {GL}}\nolimits _N}]$ , where $R^{ss,o}=R^{ss}\times _{{\mathbb H}^0({\mathcal {X}},K_{\mathcal {X}})}\mathop {\mathrm {Spec}}\nolimits ({\mathbb C})$ . Moreover, the moduli stack $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ is the fiber product

(15)

On the other hand, we have the following Cartesian diagram:

(16)

where $\mathop {\mathrm {det}}\nolimits $ is the classifying morphism of the determinant line bundle of the universal Higgs bundle on ${\mathcal {X}}\times \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)$ . The closed substack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {SL}}\nolimits _r)$ of $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)$ can be presented as $[R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}/\mathop {\mathbf {GL}}\nolimits _N]$ , where $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ is the fiber product in the Cartesian diagram (13). Since $\mathcal M^{ss}_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop { \mathbf {SL}}\nolimits _r)\rightarrow \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop { \mathbf {GL}}\nolimits _r)$ in the diagram (15) factors through the closed immersion $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {SL}}\nolimits _r)\rightarrow \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)$ in the diagram (16), it is easy to check that the following commutative diagram is Cartesian

(17)

where $\mathcal Det$ is the restriction of $\mathcal Det : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)\rightarrow \mathcal Pic({\mathcal {X}})$ to $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {SL}}\nolimits _r)$ . On the other hand, $\det (E_{R^{ss}})|_{{\mathcal {X}}\times R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}}\simeq \mathrm { pr}^{*}_{{\mathcal {X}}}L\otimes \mathrm {pr}^{*}_{R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}}W$ for some line bundle W on $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ , where $\mathrm {pr}_{{\mathcal {X}}}$ and $\mathrm { pr}_{R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}}$ are the projections to ${\mathcal {X}}$ and $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ , respectively. Moreover, W is a $\mathop { \mathbf {GL}}\nolimits _N$ -equivariant line bundle on $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ , since $\mathrm { pr}^{*}_{{\mathcal {X}}}L$ is a $\mathop {\mathbf {GL}}\nolimits _N$ -equivariant line bundle with the trivial equivariant structure. By the Cartesian diagram (17), $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ can be represented by $[W^{*}/\mathop { \mathbf {GL}}\nolimits _N]$ , where $W^{*}$ is the frame bundle associated with W.

Proof As the center ${\mathbb C}^{*}$ of $\mathop {\mathbf {GL}}\nolimits _N$ acts trivially on $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ , the $\mathop {\mathbf {GL}}\nolimits _N$ -equivariant morphism $W^{*}\rightarrow R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ induces a morphism of quotient stacks

(18) $$ \begin{align} [W^{*}/{\mathop{\mathbf{GL}}\nolimits_N}]\longrightarrow [R^{ss}_{\mathop{ \mathbf{SL}}\nolimits_r}/{\mathop{\mathbf{PGL}}\nolimits_N}]. \end{align} $$

On the other hand, there is a Cartesian diagram:

(19)

Note that the top morphism in (19) is a $\mu _r$ -gerbe. It follows that the bottom morphism (18) is also a $\mu _r$ -gerbe. So, the good moduli space of $[R^{ss}_{\mathrm {SL_r}}/{\mathop {\mathbf {PGL}}\nolimits _N}]$ coincides with the good moduli space of $[W^{*}/{\mathop {\mathbf {GL}}\nolimits _N}]$ . Note the good moduli space of $[R^{ss}_{\mathrm { SL_r}}/{\mathop {\mathbf {PGL}}\nolimits _N}]$ is a closed subscheme of $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ . This completes the proof.

3.3 Moduli stack of ${\mathop {\mathbf {PGL}}\nolimits _r}$ -Higgs bundles

We first recall some basic facts about principal bundles (or torsors) on ${\mathcal {X}}$ . Our main reference is [Reference GiraudGir71]. For an algebraic group G, the set of isomorphism classes of principal G-bundles is denoted by $H^1({\mathcal {X}}, G)$ (if G is abelian, $H^1({\mathcal {X}},G)$ is equivalent to the étale cohomology group with values in G). For a morphism of algebraic groups $G\rightarrow H$ , we have a morphism of pointed sets

(20) $$ \begin{align} H^1({\mathcal{X}}, G)\longrightarrow H^1({\mathcal{X}}, H),\quad [{\mathcal P}_G]\longmapsto[{\mathcal P}_G\wedge^GH], \end{align} $$

where ${\mathcal P}_G\wedge ^GH$ is also denoted by ${\mathcal P}_G\times ^GH$ in some literatures. In general, the morphism (20) is not surjective. We say a principal H-bundle ${\mathcal P}_H$ can be lifted to a principal G-bundle if ${\mathcal P}_H\simeq {\mathcal P}_G\wedge ^GH$ for some principal G-bundle ${\mathcal P}_G$ . For simplicity, we only consider the case when G is a central extension of H by C, i.e., there is an exact sequence of algebraic groups . The obstruction of lifting ${\mathcal P}_H$ to a principal G-bundle is the so-called lifting gerbe $\mathcal G_{{\mathcal P}_H}$ (or G -lifting gerbe). Recall that there is a natural morphism of classifying stacks $BG\rightarrow BH$ defined by

$$ \begin{align*} BG(T)\longrightarrow BH(T),\quad ({\mathcal P}_G\rightarrow T)\longmapsto({\mathcal P}_G\wedge^GH\rightarrow T), \end{align*} $$

for a test scheme T. The lifting gerbe $\mathcal G_{{\mathcal P}_H}$ is the fiber product ${\mathcal {X}}\times _{BH}BG$ for the Cartesian diagram

where ${\mathcal {X}}\rightarrow BH$ is the classifying morphism of ${\mathcal P}_H$ .

The set of isomorphism classes of C-gerbes is equal to $H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},C)$ . Then, we have a morphism of pointed sets

(21) $$ \begin{align} \partial : H^1({\mathcal{X}},H)\longrightarrow H^2_{\mathop{{\acute{\rm e}{\rm t}}}\nolimits}({\mathcal{X}},C),\quad [{\mathcal P}_H]\longmapsto [\mathcal G_{{\mathcal P}_H}], \end{align} $$

which maps the trivial principal H-bundle to the trivial C-gerbe. Indeed, according to the general theory of [Reference GiraudGir71], we have:

Proposition 3.21 A principal H-bundle ${\mathcal P}_H$ can be lifted to a principal G-bundle if and only if the lifting gerbe $\mathcal G_{{\mathcal P}_H}$ is trivial. Moreover, there is an associated exact sequence of pointed sets

(22)

Recall the Kummer sequence

(23)

For a line bundle L on ${\mathcal {X}}$ , we use $\mathcal G_{L}$ to denote the $\mu _r$ -gerbe defined by the cohomology class $\delta ([L])$ . Consider the central extension

(24)

By Proposition 3.21, we have the following exact sequences of pointed sets:

(25)

The $\mathop {\mathbf {SL}}\nolimits _r$ -lifting gerbe of ${\mathcal P}_{\mathop { \mathbf {PGL}}\nolimits _r}$ is denoted by $\mathcal G_{{\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}}$ .

Proposition 3.22 Let E be a rank r locally free sheaf on ${\mathcal {X}}$ , and let ${\mathcal P}_E$ be the associated frame bundle of E. Then, the two gerbes are equivalent

$$ \begin{align*} \mathcal G_{\mathop{\mathrm{det}}\nolimits(E)^{\vee}}\simeq \mathcal G_{{\mathcal P}_{\mathop{ \mathbf{PGL}}\nolimits_r}}, \end{align*} $$

where ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}={\mathcal P}_{E}\wedge ^{\mathop { \mathbf {GL}}\nolimits _r}{\mathop {\mathbf {PGL}}\nolimits _r}$ .

In order to construct the moduli stack of $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles, we introduce the following notion:

The moduli stack $\mathcal M^{\alpha }_{\mathrm {Dol}}(\mathop {\mathbf {PGL}}\nolimits _r)$ of $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ is defined by: for a test scheme T, $\mathcal M^{\alpha }_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {PGL}}\nolimits _r)(T)$ is the groupoid of T-families of $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles, in which the principal bundles with topological type $\alpha $ . Suppose that there is a principal $\mathop { \mathbf {PGL}}\nolimits _r$ -bundle ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ satisfying $\partial ([{\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}])=\alpha $ . Consider the exact sequence:

The cohomology class in $H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mathbb G_m)$ corresponding the $\mathop {\mathbf {GL}}\nolimits _r$ -lifting gerbe of ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ is the image of $\alpha $ in $H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mathbb G_m)$ (see the proof of [Reference MilneMil80, Proposition 2.7 in Chapter IV]).

Case I: Assume that the image of $\alpha $ in $H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mathbb G_m)$ is zero. Then, there is a locally free sheaf E on ${\mathcal {X}}$ such that the associated frame bundle is a $\mathop {\mathbf {GL}}\nolimits _r$ -lifting of ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ and $\delta ([\mathop {\mathrm {det}}\nolimits (E)^{\vee }])=\alpha $ . Moreover, by the Kummer sequence (23), we see that $\mathop {\mathrm {det}}\nolimits (E)$ is uniquely determined up to an rth power of some line bundle. We therefore have the proposition:

Let $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {SL}}\nolimits _r)$ be the moduli stack of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with fixed determinant L. By Proposition 3.25, there is a surjective morphism of stacks

(26) $$ \begin{align} \mathcal M_{\mathop{\mathrm{Dol}}\nolimits}(\mathop{\mathbf{SL}}\nolimits_r)\longrightarrow\mathcal M_{\mathop{\mathrm{Dol}}\nolimits}^{\alpha}(\mathop{\mathbf{PGL}}\nolimits_r), \end{align} $$

defined by

$$ \begin{align*} \mathcal M_{\mathop{\mathrm{Dol}}\nolimits}(\mathop{\mathbf{SL}}\nolimits_r)(T)\longrightarrow\mathcal M_{\mathrm{Dol}}^{\alpha}(\mathop{\mathbf{PGL}}\nolimits_r)(T),\quad (E_T,\phi_T)\longmapsto({\mathcal P}_{E_T}\wedge^{\mathop{\mathbf{GL}}\nolimits_r}\mathop{\mathbf{PGL}}\nolimits_r,\phi_T), \end{align*} $$

where T is any test scheme and ${\mathcal P}_{E_T}$ is the frame bundle associated with $E_T$ .

Case II: Assume that $\alpha \in H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mu _r)$ is not zero in $H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mathbb G_m)$ . The corresponding $\mu _r$ -gerbe is denoted by $p_{\alpha } : \mathcal G_{\alpha }\rightarrow {\mathcal {X}}$ , which is a Deligne–Mumford curve. By the universal property of $\mathcal G_{\alpha }$ , for any ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ with topological type $\alpha $ , $p_{\alpha }^{*}{\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ has an $\mathop {\mathbf {SL}}\nolimits _r$ -lifting on $\mathcal G_{\alpha }$ , i.e., there exists a locally free sheaf E of rank r with $\mathop {\mathrm {det}}\nolimits (E)\simeq {\mathcal {O}}_{\mathcal G_{\alpha }}$ on $\mathcal G_{\alpha }$ such that the associated principal bundle is an $\mathop {\mathbf {SL}}\nolimits _r$ -lifting of $p_{\alpha }^{*}{\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ . The E is a twisted vector bundle. In what follows, we will give the definition of twisted vector bundles. For a quasicoherent sheaf F on $\mathcal G_{\alpha }$ , it admits an eigendecomposition

(27) $$ \begin{align} F={\bigoplus}_{\lambda\in \mathbb{Z}/{r\mathbb{Z}}}F_{\lambda}, \end{align} $$

where $F_{\lambda }$ is the eigensheaf on $\mathcal G_{\alpha }$ with respect to the character $\lambda $ of $\mu _r$ (see [Reference LieblichLie08, Proposition 3.1.1.4]).

Consider the moduli stack $\mathcal M_{\mathrm {Dol}}^{\alpha }(\mathop {\mathbf {GL}}\nolimits _r)$ of twisted Higgs bundles on $\mathcal G_{\alpha }$ , whose fiber over a test scheme T is the groupoid of T-families of rank r twisted Higgs bundles on $\mathcal G_{\alpha }$ . $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {GL}}\nolimits _r)$ is an open and closed substack of the moduli stack of rank r Higgs bundles on $\mathcal G_{\alpha }$ for the decomposition (27). For a modified Hilbert polynomial P, we can also consider the moduli stack $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{\alpha }(\mathop {\mathbf {GL}}\nolimits _r)$ of rank r twisted Higgs bundles with modified Hilbert polynomial P, which is an open and closed substack of $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ on $\mathcal G_{\alpha }$ . If there is a polarization on $\mathcal G_{\alpha }$ , we can also introduce the notion of stability for twisted Higgs bundles as usual. The moduli stack of semistable twisted Higgs bundle with modified Hilbert polynomial P is denoted by $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{\alpha ,ss}(\mathop {\mathbf {GL}}\nolimits _r)$ .

The moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {SL}}\nolimits _r)$ of twisted $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles is an open and closed substack of the moduli stack of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundle on $\mathcal G_{\alpha }$ . As Proposition 3.28, we have:

For a twisted $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundle on $\mathcal G_{\alpha }$ , the associated $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundle is the pullback of a $\mathop { \mathbf {PGL}}\nolimits _r$ -Higgs bundle with topological data $\alpha $ on ${\mathcal {X}}$ . Then, we have a surjective morphism of algebraic stacks

(28) $$ \begin{align} \mathcal M_{\mathop{\mathrm{Dol}}\nolimits}^{\alpha}(\mathop{\mathbf{SL}}\nolimits_r)\longrightarrow\mathcal M_{\mathrm{ Dol}}^{\alpha}(\mathop{\mathbf{PGL}}\nolimits_r). \end{align} $$

Similar to Proposition 3.26, we also have:

By [Reference LieblichLie09, Lemma 3.4] and Propositions 3.26 and 3.31, we have the following theorem.

3.4 Application to the case of stacky curves

In this subsection, ${\mathcal {X}}$ is assumed to be a genus g hyperbolic stacky curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . Fix a polarization $({\mathcal E},{\mathcal {O}}_X(1))$ on ${\mathcal {X}}$ . Since $H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mathbb G_m)$ is trivial (see [Reference PomaPom13, Proposition 5.3]), every cohomology class of $H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mu _r)$ satisfies the assumption of Case I (see Section 3.3). Suppose that $\alpha \in H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mu _r)$ is in the image of the $\partial $ in (25) and L is a line bundle on ${\mathcal {X}}$ such that $\delta ([L])=-\alpha $ in (23). Note that there are finitely many modified Hilbert polynomials if the rank and the determinant are fixed. Then, the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^s(\mathop { \mathbf {SL}}\nolimits _r)$ of stable $\mathop { \mathbf {SL}}\nolimits _r$ -Higgs bundles with determinant L contains finitely many open-closed substacks indexed by the modified Hilbert polynomials. Therefore, $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^s(\mathop { \mathbf {SL}}\nolimits _r)$ admits good moduli space $M_{\mathop {\mathrm {Dol}}\nolimits }^s(\mathop { \mathbf {SL}}\nolimits _r)$ , which is finite type over ${\mathbb C}$ . After rigidification, we get an action of the group $\Gamma $ of r-torsion points of $\mathop {\mathrm {Pic}}\nolimits ({\mathcal {X}})$ on the moduli space $M_{\mathop {\mathrm {Dol}}\nolimits }^{s}(\mathop {\mathbf {SL}}\nolimits _r)$ of stable $\mathop { \mathbf {SL}}\nolimits _r$ -Higgs bundles (see Proposition 3.26). Specifically, the group $\Gamma $ acts on $M_{\mathop {\mathrm {Dol}}\nolimits }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ via the tensor product

$$ \begin{align*} W\cdot (E,\phi)=(W\otimes E,\phi),\quad W\in\Gamma. \end{align*} $$

We give the definition of moduli space of stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ .

Definition 3.33 The moduli space of stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ is defined to be the quotient stack

$$ \begin{align*} M_{\mathop{\mathrm{Dol}}\nolimits}^{\alpha,s}(\mathop{\mathbf{PGL}}\nolimits_r)=[M_{\mathop{\mathrm{ Dol}}\nolimits}^{s}(\mathop{\mathbf{SL}}\nolimits_r)/\Gamma]. \end{align*} $$

Suppose that the set of stacky points of ${\mathcal {X}}$ is $\{p_1,\ldots ,p_m\}$ and the corresponding stabilizer groups are $\mu _{r_1},\ldots ,\mu _{r_m}$ . For each $p_i$ , the residue gerbe $\iota _i : B\mu _{r_i}\rightarrow {\mathcal {X}}$ is a closed immersion. On the other hand, $K_0(B\mu _{r_i})$ is isomorphic to the representation ring $\mathbf {R}\mu _{r_i}={{\mathbb Z}[x]}/{(x^{r_i}-1)}$ where x represents the representation defined by the inclusion $\mu _{r_i}\hookrightarrow {\mathbb C}^{*}$ . The following proposition is well known (see [Reference Adem and RuanAR03, Example 5.9] or [Reference Marcolli and MathaiMM99, p. 563]).

Proposition 3.35 We have an isomorphism

(29) $$ \begin{align} K_0({\mathcal{X}})_{\mathbb Q}\simeq\mathbb Q\times\mathbb Q\times\mathbb Q^{r_1-1}\times\dots\times\mathbb Q^{r_m-1}. \end{align} $$

Suppose that E is a locally free sheaf on ${\mathcal {X}}$ . If the K-class $[\iota _i^{*}E]=\sum _{k=0}^{r_i-1}m_{i,k}\cdot x^k$ for every i, then the image of $[E]$ under (29) is

$$ \begin{align*} (\mathop{\mathrm{rk}}\nolimits(E),\mathop{\mathrm{ deg}}\nolimits(\pi_{*}(E)),(m_{1,i})_{i=1}^{r_1-1},\ldots,(m_{m,i})_{i=1}^{r_m-1}). \end{align*} $$

According to the rational K-classes of line bundles on ${\mathcal {X}}$ , the Picard group $\mathop {\mathrm { Pic}}\nolimits ({\mathcal {X}})$ is the disjoint union:

(30) $$ \begin{align} \mathop{\mathrm{Pic}}\nolimits({\mathcal{X}})=\textstyle{\coprod_{d\in{\mathbb Z}}\coprod_{i_1=0}^{r_1-1}\coprod_{i_2=0}^{r_2-1}\cdots\coprod_{i_m=0}^{r_m-1}\mathop{\mathrm{ Pic}}\nolimits^{d,(i_1,\ldots,i_m)}({\mathcal{X}})}. \end{align} $$

Then, the line bundle L belongs to a unique $\mathop {\mathrm {Pic}}\nolimits ^{d,(i_1,\ldots ,i_m)}({\mathcal {X}})$ . Suppose $\xi =(r,d^{\prime },(m_{1,i})_{i=1}^{r_1-1},\ldots ,(m_{m,i})_{i=1}^{r_m-1})\in K_0({\mathcal {X}})_{\mathbb Q}$ and r, $d^{\prime }$ , $m_{1,i},\ldots ,m_{m,i}$ are all integers. We further assume that $\xi $ satisfies:

• • $i_k$ is the remainder, when $\sum _{i=1}^{r_1-1}m_{k,i}$ divided by $r_k$ for every $1\leq k\leq m$ ;

• • $d^{\prime } = d+\sum _{k=1}^m\sum _{i=1}^{r_k-1}m_{k,i}\frac {i}{r_k}-\sum _{k=1}^m i_k$ .

Consider the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ (with good moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ ) of stable $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with K-class $\xi $ . We give the following definition.

Definition 3.36 The moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)$ of stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ and K-class $\xi $ is defined to be the quotient stack $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)=[M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{s}/\Gamma _0]$ , where $\Gamma _0$ is the group of r-torsion points of $\mathop {\mathrm { Pic}}\nolimits ^0(X)$ and the action $\Gamma _0$ on $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ is given by

$$ \begin{align*} W\cdot(E,\phi)=(\pi^{*}W\otimes E,\phi),\quad W\in\Gamma_0. \end{align*} $$

By the standard infinitesimal deformation theory of Higgs bundles on stacky curves (see [Reference Kydonakis, Sun and ZhaoKSZ20, Proposition 3.2 and Corollaries 3.3 and 3.4]), we have following proposition.

Proposition 3.39 If $(E,\phi )$ is a stable ${\mathop {\mathbf {SL}}\nolimits _r}$ -Higgs bundle with K-class $\xi $ , the dimension of the tangent space of $M_{\mathrm {Dol},\xi }^{s}(\mathrm {SL_r})$ at $(E,\phi )$ is

$$ \begin{align*} \textstyle{r^2(2g-2)+2-2g+{\sum}_{i=1}^m\big(r^2-(r-\sum_{k=1}^{r_i-1}m_{i,k})^2-{\sum}_{k=1}^{r_i-1}m_{i,k}^2\big)}. \end{align*} $$

Moreover, $M_{\mathrm {Dol,\xi }}^{s}(\mathop {\mathbf {SL}}\nolimits _r)$ is smooth at $(E,\phi )$ .

4.1 Spectral curves

Let $(E,\phi )$ be a Higgs bundle on a hyperbolic Deligne Mumford curve ${\mathcal {X}}$ . The characteristic polynomial of $\phi $ is $\mathrm {{det}}(\lambda -\phi )=\lambda ^r+a_1\lambda ^{r-1}+\cdots +a_r$ , where $\lambda $ is an indeterminate variable and $a_i=(-1)^i\mathrm {tr}(\wedge ^i\phi )$ for $1\leq i\leq r$ . It defines the so-called spectral curve associated with the Higgs bundle $(E,\phi )$ . More precisely, the spectral curve is the zero locus of the section

(31) $$ \begin{align} \tau^{\otimes r}+\psi^{*}a_1\otimes\tau^{\otimes r-1}+\cdots+\psi^{*}a_{r-1}\otimes\tau+\psi^{*}a_r, \end{align} $$

where $\psi : \mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})\rightarrow {\mathcal {X}}$ is the total space of $K_{\mathcal {X}}$ and $\tau $ is the tautological section of $\psi ^{*}K_{{\mathcal {X}}}$ . Since the spectral curve is only dependent on the coefficients of the characteristic polynomial, we can define a spectral curve ${\mathcal {X}}_{\boldsymbol a}$ for any element $\boldsymbol a=(a_1,\ldots ,a_r)\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}}, K_{{\mathcal {X}}}^i)$ . In general, a spectral curve is neither smooth nor integral. Nevertheless, under some mild conditions, for a general element $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the associated spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral (see Proposition 4.3). It is easy to check the following proposition.

There is another method to construct spectral curves, which is used in [Reference Beauville, Narasimhan and RamananBNR89]. Recall that $\Psi : \mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})\rightarrow {\mathcal {X}}$ is the projective bundle associated with $K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}}$ . Since $\Psi _{*}{\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1)= K_{{\mathcal {X}}}^{-1}\oplus {\mathcal {O}}_{{\mathcal {X}}}$ , the section $(0,1)$ of $K_{{\mathcal {X}}}^{-1}\oplus {\mathcal {O}}_{{\mathcal {X}}}$ gives a section y of ${\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1)$ . Meanwhile, since $\Psi _{*}(\Psi ^{*}K_{{\mathcal {X}}}\otimes {\mathcal {O}}_{\mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1))={\mathcal {O}}_{{\mathcal {X}}}\oplus K_{{\mathcal {X}}}$ , $\Psi _{*}(\Psi ^{*}K_{{\mathcal {X}}}\otimes {\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1))$ has a section $(1,0)$ . It gives a section x of $\Psi ^{*}K_{{\mathcal {X}}}\otimes {\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1)$ . For $\boldsymbol a=(a_1,\ldots ,a_r)\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}}, K_{{\mathcal {X}}}^i)$ , there is a section

(32) $$ \begin{align} s:=x^{\otimes r}+\Psi^{*}a_1\otimes x^{\otimes r-1}\otimes y +\cdots+\Psi^{*}a_r\otimes y^{\otimes r} \end{align} $$

of $\Psi ^{*}K_{{\mathcal {X}}}^r\otimes {\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(r)$ . Note that the zero locus of x and y are $\mathbb P({\mathcal {O}}_{\mathcal {X}})$ and $\mathbb P(K_{\mathcal {X}})$ , respectively. Hence, the zero locus of section (32) is the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ associated with $\boldsymbol a$ .

Proposition 4.3 Let ${\mathcal {X}}$ be a hyperbolic Deligne–Mumford curve, and let $r\geq 2$ be an integer. Suppose that $K_{\mathcal {X}}$ satisfies

(33) $$ \begin{align} \begin{aligned} \mathrm{dim}_{\mathbb C}H^0({\mathcal{X}},K^k_{\mathcal{X}})\geq 2\text{ for some } 1\leq k\leq r \quad\text{and}\quad\mathrm{dim}_{\mathbb C}H^0({\mathcal{X}},K^r_{\mathcal{X}})\neq0. \end{aligned} \end{align} $$

Then, for a general element of $\bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the associated spectral curve is integral.

Proof Recall a basic fact: for a gerbe ${\mathcal {X}}_1\rightarrow {\mathcal {X}}_2$ , ${\mathcal {X}}_1$ is integral if and only if ${\mathcal {X}}_2$ is so. By Remark 4.2, we can assume that ${\mathcal {X}}$ is a stacky curve in the following discussion. Since ${\mathcal {X}}$ is a hyperbolic stacky curve, there is a smooth projective algebraic curve $\Sigma $ with an action of a finite group G such that ${\mathcal {X}}=[\Sigma /G]$ (see [Reference Behrend and NoohiBN06, Corollary 7.7]). Suppose that $g : \Sigma \rightarrow {\mathcal {X}}$ is the morphism defined by the trivial G-torsor on $\Sigma $ and the G-action. Then, g is a G-torsor over ${\mathcal {X}}$ . As before, let $\Psi : \mathbb P(K_{\mathcal {X}}\oplus {\mathcal {O}}_{{\mathcal {X}}})\rightarrow {\mathcal {X}}$ be the projective bundle of $K_{\mathcal {X}}\oplus {\mathcal {O}}_{\mathcal {X}}$ . Then, there is a Cartesian diagram:

(34)

Similar to the second method of the construction of a spectral, we use $x^{\prime }$ to denote the section of ${\Psi ^{\prime }}^{*}K_{\Sigma }\otimes {{\mathcal {O}}}_{\mathbb {P}(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma })}(1)$ corresponding to the section $(1,0)$ of ${\mathcal {O}}_{\Sigma }\oplus K_{\Sigma }$ . And, let $y^{\prime }$ be the section of ${\mathcal {O}}_{\mathbb {P}(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma })}(1)$ corresponding to the section $(0,1)$ of $K_{\Sigma }^{-1}\oplus {\mathcal {O}}_{\Sigma }$ . For any $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the associated spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is the zero locus of the section s defined by (32). Since $({g^{\prime }})^{*}x=x^{\prime }$ and $({g^{\prime }})^{*}y=y^{\prime }$ , the pullback section of s is

(35) $$ \begin{align} s^{\prime}:={g^{\prime}}^{*}s=x^{\prime\otimes r}+{\Psi^{\prime}}^{*}a^{\prime}_1\otimes x^{\prime\otimes r-1}\otimes{y^{\prime}}+\cdots+ {\Psi^{\prime}}^{*}a^{\prime}_r\otimes y^{\prime\otimes r}, \end{align} $$

where $a_i^{\prime }=g^{*}a_i$ for all $1\leq i \leq r$ . The zero locus $\Sigma _{\boldsymbol a}$ of $s^{\prime }$ fits into the Cartesian diagram

(36)

where the vertical morphisms are closed immersions. Then, $\widehat g : \Sigma _{\boldsymbol a}\rightarrow {\mathcal {X}}_{\boldsymbol a}$ is a G-torsor and ${\mathcal {X}}_a=[\Sigma _{\boldsymbol a}/G]$ . Under the hypothesis of Proposition 4.3, we will show that the $\Sigma _{\boldsymbol a}$ is integral, for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ . Consider the injective linear map of complex vector spaces

(37) $$ \begin{align} \begin{aligned} \textstyle{ \bigoplus_{i=1}^rH^0({\mathcal{X}},K^i_{\mathcal{X}})}&\rightarrow H^0(\mathbb P(K_{\Sigma}\oplus{\mathcal{O}}_{\Sigma}),\Psi^{\prime*} K^r_{\Sigma}\otimes{\mathcal{O}}_{\mathbb P(K_{\Sigma}\oplus{\mathcal{O}}_{\Sigma})}(r)),\\ (a_1,\ldots,a_r)&\mapsto\textstyle{\sum_{i=1}^r(\Psi^{\prime}\circ g)^{*}a_i\otimes x^{\prime\otimes(r-i)}\otimes y^{\prime\otimes i}}. \end{aligned} \end{align} $$

Let V be the vector subspace of $H^0(\mathbb P(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma }),\Psi ^{\prime *} K^r_{\Sigma }\otimes {\mathcal {O}}_{\mathbb P(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma })}(r))$ generated by the section $x^{\prime \otimes r}$ and the image of (37). Note that the zero loci of $x^{\prime }$ and $y^{\prime }$ are disjoint. Since $H^0({\mathcal {X}},K^r_{\mathcal {X}})\neq 0$ , the base locus $\mathcal B$ of the linear system corresponding to V is codimension $2$ . Then, there is a morphism

(38) $$ \begin{align} \Phi_V : \mathbb P(K_{\Sigma}\oplus{\mathcal{O}}_{\Sigma})\setminus\mathcal B\rightarrow\mathbb P(V^{\vee}), \end{align} $$

where $\mathbb P(V^{\vee })$ is the projective space associated with the dual $V^{\vee }$ of V.

Claim The dimension of the image of $\Phi _V$ is 2. We only need to show that the dimension of the image of the restriction

(39) $$ \begin{align} \Phi_V|_{\Pi} : \Pi\rightarrow\mathbb P(V^{\vee}) \end{align} $$

is 2, where $\Pi =\mathbb P(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma })\setminus (\mathbb P(K_{\Sigma })\cup \mathbb P({\mathcal {O}}_{\Sigma }))$ . For any closed point $x\in \Sigma $ , the fiber of $\Psi ^{\prime }|_{\Pi } : \Pi \rightarrow \Sigma $ over x is

$$ \begin{align*} (\Psi^{\prime}|_{\Pi})^{-1}(x)=\mathbb A^1\setminus\{0\}. \end{align*} $$

And, the restriction of the morphism $\Phi _V|_{\Pi }$ to $(\Psi ^{\prime }|_{\Pi })^{-1}(x)$ is

(40) $$ \begin{align} \begin{aligned} \mathbb A^1\setminus\{0\}\rightarrow\mathbb P(V^{\vee}),\quad z\mapsto [1,c_{11}z,\ldots,c_{1n_1}z,\ldots,c_{r1}z^r,\ldots,c_{rn_r}z^r], \end{aligned} \end{align} $$

where all the $c_{\bullet \bullet }\in {\mathbb C}$ and $n_i=\mathrm {dim}_{\mathbb C}H^0({\mathcal {X}},K^i_{\mathcal {X}})$ for all $1\leq i\leq r$ . If the image $g(x)$ of x in ${\mathcal {X}}$ is not in the base locus $\widetilde {\mathcal B}$ of the complete linear system $|K^r_{\mathcal {X}}|$ , the coefficients of $z^r$ in (40) are not all zero. In this case, the image of the fiber $(\Psi ^{\prime }|_{\Pi })^{-1}(x)$ under the morphism $\Phi _V|_{\Pi }$ is dimension 1. On the other hand, if $K_{\mathcal {X}}$ satisfies the condition (33), there exist two closed points $y_1,y_2\in {\mathcal {X}}^o\setminus (\widetilde {\mathcal B}\cup \widehat {\mathcal B})$ and a section $a\in H^0({\mathcal {X}},K^k_{\mathcal {X}})$ such that

(41) $$ \begin{align} a(y_1)=0\quad\text{and}\quad a(y_2)\neq 0, \end{align} $$

where ${\mathcal {X}}^o$ is the non-stacky locus of ${\mathcal {X}}$ and $\widehat {\mathcal B}$ is the base locus of the complete linear system $|K^k_{\mathcal {X}}|$ (if $r=k$ , then $\widehat {\mathcal B}=\widetilde {\mathcal B}$ ). Therefore, for any $x_1\in g^{-1}(y_1)$ and $x_2\in g^{-1}(y_2)$ , we have

(42) $$ \begin{align} &(\Psi^{\prime}\circ g)^{*}a\otimes x^{\prime\otimes k}\otimes y^{\prime\otimes (r-k)}|_{(\Psi^{\prime}|_{\Pi})^{-1}(x_1)}=0 \quad\text{and}\nonumber\\& \quad(\Psi^{\prime}\circ g)^{*}a\otimes x^{\prime\otimes k}\otimes y^{\prime\otimes (r-k)}|_{(\Psi^{\prime}|_{\Pi})^{-1}(x_2)}\neq0. \end{align} $$

It means that the images of the two fibers $(\Psi ^{\prime }|_{\Pi })^{-1}(x_1)$ and $(\Psi ^{\prime }|_{\Pi })^{-1}(x_2)$ do not coincide. Hence, the image of $\Phi _V$ has dimension 2.

By Theorem 3.3.1 in [Reference LazarsfeldLaz04], for a general element $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^{r}H^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the zero locus $\Sigma _{\boldsymbol a}$ of

$$ \begin{align*} {x^{\prime}}^{\otimes r}+(\Psi^{\prime}\circ g)^{*}a_1\otimes{x^{\prime}}^{\otimes(r-1)}\otimes y^{\prime}+\cdots+(\Psi^{\prime}\circ g)^{*}a_r\otimes{y^{\prime}}^{\otimes r} \end{align*} $$

is integral. Therefore, ${\mathcal {X}}_{\boldsymbol a}=[\Sigma _{\boldsymbol a}/G]$ is integral for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ .

By the proof of Proposition 4.3, we get an immediate corollary.

Corollary 4.5 If a hyperbolic Deligne–Mumford curve ${\mathcal {X}}$ satisfies the conditions

(43) $$ \begin{align} \mathrm{dim}_{\mathbb C} H^0({\mathcal{X}},K^k_{\mathcal{X}})\geq 2 \text{ for some } 2\leq k \leq r\quad\text{and}\quad H^0({\mathcal{X}},K^r_{\mathcal{X}})\neq 0, \end{align} $$

then for a general element of $\bigoplus _{i=2}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the corresponding spectral curve is integral.

4.2 The Hitchin morphism

Let $(E_T,\phi _T)$ be a T-family of rank r Higgs bundles on ${\mathcal {X}}$ for a scheme T. Its characteristic polynomial is

(44) $$ \begin{align} \mathop{\mathrm{det}}\nolimits(\lambda-\phi_T)=\lambda^r+a_1(T)\lambda^{r-1}+\cdots+a_r(T), \end{align} $$

where $a_i(T)=(-1)^i\wedge ^i\phi _T\in H^0({\mathcal {X}}_T,\mathrm {pr_{{\mathcal {X}}}}^{*}K^i_{\mathcal {X}})$ . The zero locus of (44) in the total space of $\mathrm {pr_{\mathcal {X}}}^{*}K_{{\mathcal {X}}}$ is a flat family of spectral curves over T. The affine space ${\mathbb H}(r,K_{\mathcal {X}})$ associated with vector space ${\bigoplus }_{i=1}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ parametrizes the universal family of spectral curves. Indeed, it represents the functor

(45) $$ \begin{align} (\mathop{\mathrm{Sch}}\nolimits/{\mathbb C})^o\rightarrow (sets)\quad T\mapsto\textstyle{{\bigoplus}_{i=1}^{r}H^0({\mathcal{X}}_T,\mathrm{pr_{{\mathcal{X}}}}^{*}K_{{\mathcal{X}}}^i)}, \end{align} $$

since there is a canonical isomorphism

$$ \begin{align*} H^0(T, H^0({\mathcal{X}},K^i_{{\mathcal{X}}})\otimes_{{\mathbb C}}{\mathcal{O}}_{T})\xrightarrow{\sim}H^0({\mathcal{X}}_T,\mathrm{pr_{{\mathcal{X}}}}^{*}K^i_{{\mathcal{X}}})\quad\text{for each } 1\leq i\leq r \end{align*} $$

(see [Reference BrochardBro12, Corollary A.2.2]). We therefore have a morphism of stacks

(46) $$ \begin{align} \mathcal H : \mathcal M_{\mathop{\mathrm{Dol}}\nolimits,P}(\mathop{ \mathbf{GL}}\nolimits_r)\rightarrow\mathbb{H}(r,K_{{\mathcal{X}}}),\quad(E_T,\phi_T)\mapsto \mathop{\mathrm{ det}}\nolimits(\lambda-\phi_T)\quad\text{for any test scheme } T, \end{align} $$

which is called the Hitchin morphism. The following proposition describes the fibers of the Hitchin morphism.

Proposition 4.6 For a nonzero element $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K_{{\mathcal {X}}}^i)$ , let $\underline {\boldsymbol a}:\mathrm {{Spec}}({\mathbb C})\rightarrow \mathbb {H}(r,K_{{\mathcal {X}}})$ be the closed point defined by $\boldsymbol a$ . Consider the Cartesian diagram

(47)

If the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ associated with $\boldsymbol a$ is integral, then $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop { \mathbf {GL}}\nolimits _r)\times _{\mathbb {H}(r,K_{{\mathcal {X}}})}\mathop {\mathrm {Spec}}\nolimits ({\mathbb C})$ is the moduli stack of rank one torsion-free sheaves on ${\mathcal {X}}_{\boldsymbol a}$ with modified Hilbert polynomial P.

Since the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ of semistable Higgs bundles is an open substack of $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ , we can restrict the Hitchin morphism $\mathcal H : \mathcal M_{\mathrm {Dol}}(\mathop { \mathbf {GL}}\nolimits _r)\rightarrow \mathbb {H}(r,K_{{\mathcal {X}}})$ to $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ , which is also denoted by $\mathcal H$ . Let $\mathcal Q : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)\rightarrow M^{ss}_{\mathop {\mathrm { Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ be the good moduli space of $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ . By the universal property of ${M}^{ss}_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ , there exists a unique morphism $h : M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {GL}}\nolimits _r)\rightarrow \mathbb {H}(r,K_{{\mathcal {X}}})$ satisfying $\mathcal H=h\circ \mathcal Q$ . h is also called Hitchin morphism.

Restricting the Hitchin morphism h to $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {SL}}\nolimits _r)$ of $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ , we get the Hitchin morphism for the moduli space of semistable $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles $h_{\mathop {\mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {SL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ , where ${\mathbb H}^o(r,K_{\mathcal {X}})$ is the affine space associated with $\bigoplus _{i=2}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ . Let $\xi \in K_0({\mathcal {X}})_{\mathbb Q}$ be a K-class such that the modified Hilbert polynomial is P. The restriction of $h_{\mathop { \mathbf {SL}}\nolimits _r}$ to $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop { \mathbf {SL}}\nolimits _r)$ is also denoted by $h_{\mathop {\mathbf {SL}}\nolimits _r}$ . Since $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ is an open and closed subscheme of $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ , the restriction $h_{\mathop {\mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop { \mathbf {SL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ is also proper. Since $h_{\mathop { \mathbf {SL}}\nolimits _r}$ is invariant under the action of the group $\Gamma _0$ of r-torsion points of $\mathop {\mathrm { Pic}}\nolimits ^0(X)$ , we have the morphism $h_{\mathop {\mathbf {PGL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ , which is called the Hitchin morphism of $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)$ .

4.3 Classification of spectral curves

In the following, ${\mathcal {X}}$ is a stacky curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . The set of stacky points is $\{p_1,\ldots ,p_m\}$ and the stabilizer groups are $\mu _{r_1},\ldots ,\mu _{r_m}$ . Every smooth stacky curve can be obtained by applying root constructions (see [Reference CadmanCad07]) to its coarse moduli space. Recall Theorem 3.63 in [Reference BehrendBeh14].

For each $1\leq k\leq m$ , let $(L_k,s_k)$ be the pair, which consists of the universal line bundle $L_k$ and section $s_k$ of $L_k$ on $\sqrt [r_k]{p_k}$ . And, let s be the section $\bigotimes _{k=1}^m\mathrm {pr}_k^{*}s_k$ of $\bigotimes _{k=1}^m\mathrm {pr}_k^{*}L_k$ , where $\mathrm {pr}_k : {\mathcal {X}}\rightarrow \sqrt [r_k]{p_k}$ is the projection to $\sqrt [r_k]p_k$ for every $1\leq k\leq m$ .

Corollary 4.10 [Reference Voight and Zureick-BrownVB22]

Under the hypothesis of Theorem 4.9, we have

$$ \begin{align*} K_{{\mathcal{X}}}=\textstyle{\pi^{*}K_X\otimes\bigotimes_{k=1}^m\mathrm{pr}_k^{*}L_k^{r_k-1}}\quad\text{and}\quad \pi^{*}{{\mathcal{O}}_X(D))}=\bigotimes_{k=1}^m\mathrm{pr}_k^{*}L_k^{r_k}, \end{align*} $$

where $D=\sum _{k=1}^mp_k$ .

Lemma 4.11 If the natural number r satisfies $r\leq r_k$ for every $1\leq k \leq m$ , then for any element $\boldsymbol a=(a_1,\ldots ,a_r)\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}},K_{{\mathcal {X}}}^i)$ , there is an element

$$ \begin{align*} \textstyle{\overline{\boldsymbol a}=(\overline{a}_1,\ldots,\overline{a}_r)\in{\bigoplus}_{i=1}^rH^0(X,K_X^i\otimes{\mathcal{O}}((i-1)D))}, \end{align*} $$

satisfying

(48) $$ \begin{align} a_i=\textstyle{\pi^{*}\overline{a}_i\otimes\bigotimes_{k=1}^m\mathrm{pr}_k^{*}s_k^{\otimes(r_k-i)}\quad\text{for each}\quad 1\leq i\leq m}. \end{align} $$

Proof By Corollary 4.10, we have $K_{{\mathcal {X}}}^i=\pi ^{*}K_X^i\otimes \bigotimes _{k=1}^m\mathrm { pr}_k^{*}L_k^{(i-1)r_k+(r_k-i)}$ for any integer i. If i satisfies $i\leq r_k$ for every $1\leq k \leq m$ , then we have

$$ \begin{align*} H^0(X,K^i_X\otimes{\mathcal{O}}_X((i-1)D)\rightarrow H^0({\mathcal{X}},K^i_{\mathcal{X}}),\quad\overline{a}\mapsto\textstyle{\pi^{*}\overline{a}\otimes\bigotimes_{k=1}^m\mathrm{ pr}_k^{*}s_k^{\otimes(r_k-i)}} \end{align*} $$

is an isomorphism, where $\overline {a}\in H^0(X,K^i_X\otimes {\mathcal {O}}_X((i-1)D)$ .

Let t be the global section of the line bundle ${\mathcal {O}}_X(D)$ such that $\pi ^{*}t=\textstyle {\bigotimes _{k=1}^m\mathrm {pr}_k^{*}s_k^{\otimes r_k}}$ . Then, we have the following corollary.

Corollary 4.12 Assume that the natural number r satisfies $r\leq r_k$ for every $1\leq k \leq m$ . Then there is an injection of vector spaces

(49) $$ \begin{align}\quad \textstyle{\bigoplus_{i=1}^rH^0({\mathcal{X}},K^i_{\mathcal{X}})}\longrightarrow\textstyle{\bigoplus_{i=1}^rH^0(X,(K_X(D))^i)},\quad \boldsymbol a=(a_1,\ldots,a_r)\longmapsto\widetilde{\boldsymbol a}=(\widetilde a_1,\ldots,\widetilde a_r), \end{align} $$

where $\widetilde a_i=\overline {a}_i\otimes t$ and $\overline {a}_i\in H^0(X,K^i_X\otimes {\mathcal {O}}_X((i-1)D))$ is the section associated with $a_i$ in Lemma 4.11, for every $1\leq i\leq r$ .

Proof The section t defines an injection $K_X^i\otimes {\mathcal {O}}_X((i-1)D)\hookrightarrow (K_X(D))^i\quad \text {for every } 1\leq i\leq r$ . Then, we get the injective linear map

(50) $$ \begin{align} \textstyle{\bigoplus_{i=1}^rH^0(X,K^i_X\otimes{\mathcal{O}}_X((i-1)D))\hookrightarrow\bigoplus_{i=1}^rH^0(X,(K_X(D))^i).} \end{align} $$

Under the morphism (50), the image of $\overline {\boldsymbol a}=(\overline {a}_1,\ldots ,\overline {a}_r)\in \bigoplus _{i=1}^rH^0(X,K^i_X\otimes {\mathcal {O}}_X((i-1)D))$ is

$$ \begin{align*} \widetilde{\boldsymbol a}=(\widetilde{a}_1,\ldots,\widetilde{a}_m)\in\textstyle{{\bigoplus}_{i=1}^rH^0(X,(K_X(D))^i)}, \end{align*} $$

where $\widetilde {a}_i=\overline {a}_i\otimes t$ for every $1\leq i\leq m$ .

Proof The section $s=\bigotimes _{k=1}^m\mathrm {pr}_k^{*}s_k$ of $\bigotimes _{k=1}^m\mathrm {pr}_k^{*}L_k$ defines an injection $K_{{\mathcal {X}}}\hookrightarrow \pi ^{*}K_{X}(D)$ . Let $\pi ^{\prime \prime \prime } : \mathop {\mathrm { Tot}}\nolimits (K_{\mathcal {X}})\rightarrow \mathop {\mathrm {Tot}}\nolimits (\pi ^{*}K_X(D))$ be the corresponding morphism between total spaces. In general, $\pi ^{\prime \prime \prime }$ is not injective. It satisfies the commutative diagram

(51)

On the other hand, there is a Cartesian diagram

(52)

Composing the diagrams (51) and (52), we get a new commutative diagram

(53)

where $\pi ^{\prime }=\pi ^{\prime \prime }\circ \pi ^{\prime \prime \prime }$ . The curve $X_{\widetilde {\boldsymbol a}}$ is the zero locus of the section $\overline {\tau }^{\otimes r}+\varphi ^{*}\widetilde {a}_1\otimes \overline {\tau }^{\otimes (r-1)}+\cdots +\varphi ^{*}\widetilde {a}_r$ on the total space $\mathop {\mathrm {Tot}}\nolimits (K_X(D))$ . And, the spectral curve ${\mathcal {X}}_{\alpha }$ is the zero locus of section $\tau ^{\otimes r}+\psi ^{*}a_1\otimes \tau ^{\otimes {(r-1)}}+\cdots +\psi ^{*}a_r$ , where $\tau $ is the tautological section of $\psi ^{*}K_{{\mathcal {X}}}$ . Since $(\pi ^{\prime })^{*}\overline {\tau }=\tau \otimes \psi ^{*}s$ , we have

$$ \begin{align*} &(\pi^{\prime})^{*}(\overline{\tau}^{\otimes r}+\varphi^{*}\widetilde{a}_1\otimes\overline{\tau}^{\otimes (r-1)}+\cdots+\varphi^{*}\widetilde{a}_r)\\&\quad= (\tau^{\otimes r}+\psi^{*}a_1\otimes\tau^{\otimes(r-1)}+\cdots+\psi^{*}a_r)\otimes\psi^{*}s^{\otimes r}. \end{align*} $$

We get the commutative diagram

In order to show that $X_{\widetilde \alpha }$ is the coarse moduli space of ${\mathcal {X}}_{\alpha }$ , we only need to check this locally. For each stacky point $p_i$ , there is an affine open subset $U_i=\text {Spec}(A_i)$ of X, such that $U_i$ contains only $p_i$ and $p_i =(f_i)$ as a divisor on $U_i$ for some $f_i\in A_i$ . In the following, we consider the commutative diagram

where

$$ \begin{align*} \textstyle{X_{\widetilde\alpha}{\times}_X U_i={\mathop{\mathrm{ Spec}}\nolimits}\left(\frac{A_i[x]}{(x^r+\overline{a}_1f_ix^{r-1}+\overline{a}_2f_ix^+\cdots+\overline{a}_rf_i)}\right)}. \end{align*} $$

Since

$$ \begin{align*} \textstyle{{\mathcal{X}}{\times}_X U_i=\left[\mathop{\mathrm{ Spec}}\nolimits\left(\frac{A_i[t]}{(t^{r_i}-f_i)}\right)\bigg/\mu_{r_i}\right]} \end{align*} $$

(see [Reference OlssonOls16, Theorem 10.3.10(ii)]), we have

$$ \begin{align*} \textstyle{{\mathcal{X}}_{\alpha}{\times}_X U_i=\left[\text{Spec}\left(\frac{A_i[t,y]}{(y^r+\overline{a}_1t^{r_i-1}y^{r-1}+\cdots+\overline{a}_rt^{r_i-r},t^{r_i}-f_i)}\right)\bigg/\mu_{r_i}\right]}, \end{align*} $$

where the action of $\mu _{r_i}=\text {Spec}(\mathbb {C}[z]/(z^{r_i}-1))$ is defined by

$$ \begin{align*} t\mapsto z\otimes t,\quad y\mapsto z^{-1}\otimes y. \end{align*} $$

Hence, the coarse moduli space of ${\mathcal {X}}_{\alpha }{\times }_X U_i$ is ${\mathop {\mathrm { Spec}}\nolimits }\left (\frac {A_i[ty]}{((ty)^r+ \overline {a}_1f_i(ty)^{r-1}+\overline {a}_2f_i(ty)^{r-2}+\cdots +\overline {a}_rf_i)}\right )$ .

Suppose that j is a natural number. It can be written uniquely in the form $j=h_{jk}\cdot r_{k}-q_{jk}$ , where $h_{jk},q_{jk}\in {\mathbb Z}$ satisfy $0\leq q_{jk} <r_k$ for every $0\leq k\leq m$ . More precisely, we have

(54) $$ \begin{align} h_{jk}=\textstyle{\left\lceil\frac{j}{r_k}\right\rceil}\quad\text{and}\quad q_{jk}=\textstyle{r_k\left(\left\lceil\frac{j}{r_k}\right\rceil-\frac{j}{r_k}\right)} \end{align} $$

for every $0\leq k\leq m$ . Let $\widetilde h_{jk}=j-h_{jk}$ for every $1\leq k\leq m $ . Then, we have

(55) $$ \begin{align} \widetilde h_{jk}=\textstyle{j-\left\lceil \frac{j}{r_k}\right\rceil}\quad\text{for every } 0\leq k\leq m. \end{align} $$

We therefore have $K_{{\mathcal {X}}}^j=\textstyle {\pi ^{*}K^j_X\otimes \bigotimes _{k=1}^m\mathrm {pr}_k^{*}L_k^{j(r_{k}-1)}= \pi ^{*}K_X^j\otimes \bigotimes _{k=1}^m\mathrm {pr}_k^{*}L_k^{\widetilde h_{jk}\cdot r_k+q_{jk}}}$ . Then, the pushforward of $K^j_{\mathcal {X}}$ is

(56) $$ \begin{align} \pi_{*}K_{{\mathcal{X}}}^j=\textstyle{K_X^j\otimes{\mathcal{O}}_X(\sum_{k=1}^m\widetilde h_{jk}\cdot p_k)}. \end{align} $$

Hence, there is an isomorphism

(57) $$ \begin{align} \begin{aligned} H^0(X,K_X^j\otimes{\mathcal{O}}_X(\textstyle{\sum_{k=1}^m\widetilde h_{jk}\cdot p_k}))\longrightarrow H^0({\mathcal{X}},K_{{\mathcal{X}}}^j),\quad a\longmapsto\pi^{*}a\otimes\textstyle{\bigotimes_{k=1}^m\pi_1^{*}s_k^{\otimes q_{jk}}}. \end{aligned} \end{align} $$

Lemma 4.15 Suppose that the aforementioned ${\mathcal {X}}$ is hyperbolic with genus g and the natural number $r\geq 2$ . Furthermore, we assume that

(58) $$ \begin{align} q_{rk}=0\quad \text{or}\quad 1\quad\text{for all } 1\leq k\leq m. \end{align} $$

If $\mathop {\mathrm {deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^r))\geq 2g$ and the condition (33) hold, then the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth for a general element $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{{\mathcal {X}}})$ .

Lemma 4.16 We assume that the aforementioned hyperbolic stacky curve ${\mathcal {X}}$ and the natural number r do not satisfy the condition (58). For that, we make the assumption:

(59) $$ \begin{align} \text{If } q_{rk}\geq 2 \text{ for some } 1\leq k\leq m, \text{ then }q_{(r-1)k}=0. \end{align} $$

If $\mathop {\mathrm {deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^{r-1}))\geq 2g$ and the condition (33) holds, then a general spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth.