The Poincaré line bundle and autoduality of Hitchin fibers

Abstract

In this paper we construct the Poincare line bundle for the stack of Higgs bundles on smooth projective curves and show that it induces a fully-faithful Fourier-Mukai transform on the category of quasi-coherent sheaves.

Appendices

Appendix A: Review about Poincaré line bundles on ${\text {Bun} }_{T}(C)\times {\text {Bun} }_{{\check{T}}}(C)$

In this section we gather some facts about the Poincaré line bundle on ${\text {Bun} }_{T}(C)\times {\text {Bun} }_{{\check{T}}}(C)$ where C is an integral projective curve. The main references are [2, 12] and [14].

First let us look at the Poincaré line bundle on ${\mathcal {P}}ic(C)\times {\mathcal {P}}ic(C)$ where ${\mathcal {P}}ic(C)$ is the Picard stack of C.

Proposition A.0.1

• (1) There exists a Poincaré line bundle ${\mathcal {P}}$ on ${\mathcal {P}}ic(C)\times {\mathcal {P}}ic(C)$ such that the fiber of ${\mathcal {P}}$ at the point $(L_{1},L_{2})\in {\mathcal {P}}ic(C)\times {\mathcal {P}}ic(C)$ is given by:

  $$\begin{aligned} {\mathcal {P}}= & {} {\text {det} }(R\Gamma (C,L_{1}\otimes L_{2}))\otimes {\text {det} }(R\Gamma (C,O_{C}))\otimes {\text {det} }(R\Gamma (C,L_{1}))^{-1}\\&\otimes {\text {det} }(R\Gamma (C,L_{2}))^{-1} \end{aligned}$$

  where $L_{1}$ and $L_{2}$ are line bundles on C. Moreover, ${\mathcal {P}}$ is canonically a biextension on ${\mathcal {P}}ic(C)\times {\mathcal {P}}ic(C)$.

• (2) If $C^{n}$ is the normalization of C, then pullback of line bundles induces ${\mathcal {P}}ic(C)\rightarrow {\mathcal {P}}ic(C^{n})$. Moreover, the Poincaré line bundle on ${\mathcal {P}}ic(C)\times {\mathcal {P}}ic(C)$ is isomorphic to the pullback of the Poincaré line bundle on ${\mathcal {P}}ic(C^{n})\times {\mathcal {P}}ic(C^{n})$ under the morphism ${\mathcal {P}}ic(C)\rightarrow {\mathcal {P}}ic(C^{n})$.

• (3) Let ${\mathcal {P}}ic^{0}(C)$ be the neutral component of ${\mathcal {P}}ic(C)$ and ${\text {Pic} }(C)$ be the Picard scheme of C. Then restriction of ${\mathcal {P}}$ to ${\mathcal {P}}ic^{0}(C)\times {\mathcal {P}}ic(C)$ descends to ${\mathcal {P}}ic^{0}(C)\times {\text {Pic} }(C)$.

• (4) Let ${\text {Div} }^{n}$ be the scheme parameterizing degree n divisors on C. Consider the natural morphism ${\text {Div} }^{n}\xrightarrow {AJ}{\mathcal {P}}ic(C)$ given by $D\mapsto O_{C}(D)$. If we pullback ${\mathcal {P}}$ along ${\text {Div} }^{n}\times {\mathcal {P}}ic(C)\rightarrow {\mathcal {P}}ic(C)\times {\mathcal {P}}ic(C)$, then the fiber of $(AJ\times {\text {id} })^{*}({\mathcal {P}})$ at (D, L) is given by ${\text {det} }(L_{D})\otimes {\text {det} }(O_{D})^{-1}$ where $L_{D}$ stands for the restriction of L to D.

• (5) If C is smooth of genus g, then we have $Rp_{1*}{\mathcal {P}}\simeq e_{*}(k)[-g]$ where e is the unit ${\text {Spec} }(k)\xrightarrow {e}{\mathcal {P}}ic(C)$.

Now we look at the Poincaré line bundle on ${\text {Bun} }_{T}(C)\times {\text {Bun} }_{{\check{T}}}(C)$. When $\phi \in X_{*}(T)$ and L is a line bundle on C, we will use the notation $\phi _{*}(L)$ to denote the T bundle induced from L and the group homomorphism ${\mathbb {G}}_{m}\xrightarrow {\phi } T$. Also, if ${\mathcal {F}}_{{\check{T}}}$ is a ${\check{T}}$ bundle, then $\phi ({\mathcal {F}}_{{\check{T}}})$ denotes the line bundle induced from ${\mathcal {F}}_{{\check{T}}}$ and the morphism ${\check{T}}\xrightarrow {\phi }{\mathbb {G}}_{m}$.

Proposition A.0.2

• (1) There exists a Poincaré line bundle ${\mathcal {P}}_{T}$ on ${\text {Bun} }_{T}(C)\times {\text {Bun} }_{{\check{T}}}(C)$ with the following property: Let $L_{1}$ and $L_{2}$ be line bundles and $\phi \in X_{*}(T)$, $\varphi \in X_{*}({\check{T}})$, then the fiber of ${\mathcal {P}}_{T}$ at $(\phi _{*}(L_{1}),\varphi _{*}(L_{2}))$ is given by ${\mathcal {P}}^{\otimes <\phi ,\varphi >}_{L_{1},L_{2}}$ where ${\mathcal {P}}_{L_{1},L_{2}}$ is the fiber of the Poincaré line bundle on ${\mathcal {P}}ic(C)\times {\mathcal {P}}ic(C)$ at $(L_{1}.L_{2})$. ${\mathcal {P}}_{T}$ is naturally a biextension on ${\text {Bun} }_{T}(C)\times {\text {Bun} }_{{\check{T}}}(C)$.

• (2) Let $C^{n}$ be the normalization of C. Consider the morphisms ${\text {Bun} }_{T}(C)\rightarrow {\text {Bun} }_{T}(C^{n})$ and ${\text {Bun} }_{{\check{T}}}(C)\rightarrow {\text {Bun} }_{{\check{T}}}(C^{n})$ induced by pullback. Then the Poincaré line bundle on ${\text {Bun} }_{T}(C)\times {\text {Bun} }_{{\check{T}}}(C)$ is isomorphic to the pullback of the Poincaré line bundle on ${\text {Bun} }_{T}(C^{n})\times {\text {Bun} }_{{\check{T}}}(C^{n})$ via the morphisms above.

• (3) Let ${\text {Bun} }_{T}^{0}(C)$ be the neutral component of ${\text {Bun} }_{T}(C)$ Then the restriction of ${\mathcal {P}}_{T}$ to ${\text {Bun} }_{T}^{0}(C)\times {\text {Bun} }_{{\check{T}}}(C)$ descends to ${\text {Bun} }_{T}^{0}(C)\times X_{*}({\check{T}})\otimes {\text {Pic} }(C)$.

• (4) Let $T_{1}\rightarrow T_{2}$ be a group homomorphism and ${\check{T}}_{2}\rightarrow {\check{T}}_{1}$ be the induced morphism on the dual torus. Consider the diagram:

  

  Then the pullback of ${\mathcal {P}}_{T_{1}}$ to ${\text {Bun} }_{T_{1}}(C)\times {\text {Bun} }_{{\check{T}}_{2}}(C)$ is isomorphic to the pullback of ${\mathcal {P}}_{T_{2}}$ to ${\text {Bun} }_{T_{1}}(C)\times {\text {Bun} }_{{\check{T}}_{2}}(C)$.

• (5) Consider the morphism ${\text {Div} }^{n}(C)\times X_{*}(T)\xrightarrow {AJ}{\text {Bun} }_{T}(C)$ given by $(D,\phi )\mapsto \phi _{*}(O_{C}(D))$. Then the fiber of $(AJ\times {\text {id} })^{*}({\mathcal {P}}_{T})$ at $((D,\phi ),{\mathcal {F}}_{{\check{T}}})$ is given by ${\text {det} }(\phi ({\mathcal {F}}_{{\check{T}}})\mid _{D})\otimes {\text {det} }(O_{D})^{-1}$.

• (6) If C is smooth of dimension g, then we have $Rp_{1*}{\mathcal {P}}_{T}\simeq e_{*}(k)[-rg]$ where r is the dimension of T.

Appendix B: Detailed analysis of ${\mathcal {H}}iggs_{G}({\widetilde{X}})$ when ${\widetilde{X}}$ has nodal singularities

In this section we give a detailed analysis about the structure of ${\mathcal {H}}iggs_{G}({\widetilde{X}})$ when ${\widetilde{X}}$ has nodal singularities over a unique point of X. Here are the precise assumptions on ${\widetilde{X}}$:

Assumption

• (1) ${\widetilde{X}}$ is integral.

• (2) For any pair of roots $\alpha$ and $\beta$ such that $\alpha \ne \pm \beta$, the ramification divisors $D_{\alpha }$ dose not intersect with $D_{\beta }$

• (3) There exists a unique point $x\in X$ such that ${\widetilde{X}}$ is smooth over the preimage of $X-\{x\}$. Moreover, ${\widetilde{X}}$ has nodal singularities over the preimage of x.

One can also reformulate it as follows:

Assumption

${\widetilde{X}}$ is integral and it is smooth away from x. Moreover, if we restrict ${\widetilde{X}}$ to the formal disk ${\text {Spec} }({\widehat{O}}_{x})$ and look at the induced morphism ${\text {Spec} }({\widehat{O}}_{x})\rightarrow {\mathfrak {c}}$, then the image of x lies in the smooth part of the discriminant divisor in ${\mathfrak {c}}$ and that the image of the formal curve ${\text {Spec} }({\widehat{O}}_{x})$ intersects the discriminant divisor with multiplicity equals to two.

Let us denote the normalization of ${\widetilde{X}}$ by ${\widetilde{X}}^{n}$. Notice that in this case ${\widetilde{X}}^{n}$ is also an abstract cameral cover over X in the sense of [6]. Hence one can define the group scheme $J_{G,{\widetilde{X}}^{n}}$ associated to the cameral cover ${\widetilde{X}}^{n}$. One has a natural morphism $J_{G,{\widetilde{X}}}\rightarrow J_{G,{\widetilde{X}}^{n}}$ and hence a morphism ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})\rightarrow {\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$. Notice that since ${\widetilde{X}}^{n}$ is smooth, the moduli space of ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ is projective. Let us denote the moduli spaces of ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ and ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ by ${\text {Tors} }(J_{G,{\widetilde{X}}})$ and ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$. ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ and ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ are Z(G) gerbes over their moduli spaces.

The main result of this section is the following:

Theorem B.0.1

Assuming G is simply connected. There exists a pushout diagram:

Let us remind the reader that ${\text {Tors} }(J_{G,{\widetilde{X}}})$ and ${\text {Higgs} }_{{\check{G}}}({\widetilde{X}})$ stands for the moduli space of ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ and ${\mathcal {H}}iggs_{{\check{G}}}({\widetilde{X}})$.

We will prove the theorem in several steps. To begin with, we need to determine the structure of ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})$. Let ${\widetilde{x}}$ be a point in ${\widetilde{X}}$ in the preimage of $x\in X$ and let ${\widetilde{x}}_{1}$ and ${\widetilde{x}}_{2}$ be the two points in ${\widetilde{X}}^{n}$ above ${\widetilde{x}}$. Assume ${\widetilde{x}}$ corresponds to the ramification divisor $D_{\alpha }$. One has $s_{\alpha }({\widetilde{x}}_{1})={\widetilde{x}}_{2}$. By part (1) of Proposition 2.2.2 we have a natural morphism ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})\rightarrow {\text {Bun} }_{T}({\widetilde{X}}^{n})$. Denote the universal T-bundle on ${\widetilde{X}}^{n}\times {\text {Bun} }_{T}({\widetilde{X}}^{n})$ by ${\mathcal {F}}^{n}_{T}$.

Lemma B.0.2

${\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ is a ${\mathbb {G}}_{m}$ torsor over ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$. More precisely, ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ is isomorphic to the ${\mathbb {G}}_{m}$ torsor corresponds to the pullback of the line bundle $\alpha ({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}$ on ${\text {Bun} }_{T}({\widetilde{X}}^{n})$ via ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})\rightarrow {\text {Bun} }_{T}({\widetilde{X}}^{n})$. The group homomorphism ${\mathbb {G}}_{m}\rightarrow {\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ can be identified with ${\mathbb {G}}_{m}\rightarrow {\text {Gr} }_{J_{G,{\widetilde{X}}},x}\rightarrow {\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ where the first arrow is obtained by ${\mathbb {G}}_{m}\simeq J_{G,{\widetilde{X}}^{n}}({\widehat{O}}_{x})/J_{G,{\widetilde{X}}}({\widehat{O}}_{x})\subseteq {\text {Gr} }_{J_{G,{\widetilde{X}}},x}$.

Proof

To begin with, let us recall that ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ parameterizes W-equivariant T-bundles on ${\widetilde{X}}^{n}$ together with some additional structures at ramification points. For details, see Subsection 3.1 of [5]. Let $T_{\alpha }$ be $T/(s_{\alpha }-1)$, the torus of coinvariants. The pullback of the T bundle ${\mathcal {F}}^{n}_{T}$ on ${\widetilde{X}}^{n}\times {\text {Bun} }_{T}({\widetilde{X}}^{n})$ to ${\widetilde{X}}^{n}\times {\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ is a W-equivariant T-bundle on ${\widetilde{X}}^{n}\times {\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$. To simplify the notations, let us still denote it by ${\mathcal {F}}^{n}_{T}$. The W-equivariant structure implies that we have an isomorphism of T-bundles on ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$:

$$\begin{aligned} T\times ^{T,s_{\alpha }}(s^{-1}_{\alpha })^{*}({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}\simeq {\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}} . \end{aligned}$$

Take the associated $T_{\alpha }$ bundles on both sides and use the fact that

$$\begin{aligned} (s^{-1}_{\alpha })^{*}({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}\simeq {\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{2}}, \end{aligned}$$

one concludes that there exists an isomorphism of $T_{\alpha }$-bundles:

$$\begin{aligned} T_{\alpha }\times ^{T}{\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{2}}\simeq T_{\alpha }\times ^{T}{\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}} . \end{aligned}$$

So one has a trivialization of the $T_{\alpha }$ bundle $T_{\alpha }\times ^{T}({\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}}\otimes ({\mathcal {F}}^{n}_{T})^{-1}\mid _{{\widetilde{x}}_{2}})$. Let $K_{\alpha }$ be the kernel of $T\rightarrow T_{\alpha }$. Then the discussions above implies that the T-bundle ${\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}}\otimes ({\mathcal {F}}^{n}_{T})^{-1}\mid _{{\widetilde{x}}_{2}}$ on ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ has a canonical reduction to a $K_{\alpha }$. Denote the $K_{\alpha }$ bundle by ${\mathcal {F}}_{K_{\alpha }}$. By definition, a point in ${\mathcal {F}}_{K_{\alpha }}$ corresponds to a point in ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ together with a trivialization of ${\mathcal {F}}_{K_{\alpha }}$. By our discussions above, a trivialization of ${\mathcal {F}}_{K_{\alpha }}$ gives a trivialization of ${\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}}\otimes ({\mathcal {F}}^{n}_{T})^{-1}\mid _{{\widetilde{x}}_{2}}$, so we get a gluing data between ${\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}}$ and ${\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{2}}$. Using the W-equivariance one concludes that we actually get gluing datas between ${\mathcal {F}}^{n}_{T}\mid _{w({\widetilde{x}}_{1})}$ and ${\mathcal {F}}^{n}_{T}\mid _{w({\widetilde{x}}_{2})}$ for all $w\in W$ and these are compatible with the W-equivariance structure. Since ${\widetilde{X}}$ has nodal singularities, we see that these gluing data determines a W-equivariant T-torsor ${\mathcal {G}}_{T}$ on ${\widetilde{X}}$. Moreover, one checks that the W-equivariance structure induces the identity automorphism on $T_{\alpha }\times ^{T}{\mathcal {G}}_{T}\mid _{{\widetilde{x}}}$. And since ${\widetilde{X}}^{n}$ is isomorphic to ${\widetilde{X}}$ at points away from the preimage of $x\in X$, the W-equivariant T-torsor ${\mathcal {G}}_{T}$ is actually determines a $J_{G,{\widetilde{X}}}$ torsor away from the preimage of x.

Assume first that ${\check{\alpha }}$ is primitive. Then we can conclude by Lemma 3.1.2 of [5] that any point in ${\mathcal {F}}_{K_{\alpha }}$ corresponds to a $J_{G,{\widetilde{X}}}$ torsor. So this defines a morphism ${\mathcal {F}}_{K_{\alpha }}\rightarrow {\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ and it is not hard to see the procedure above actually gives an isomorphism between them. We also claim that in this case ${\mathcal {F}}_{K_{\alpha }}$ is isomorphic to the pullback of the line bundle $\alpha ({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}$ on ${\text {Bun} }_{T}({\widetilde{X}}^{n})$. Indeed, since the coroot ${\check{\alpha }}$ is primitive, ${\check{\alpha }}$ induces an isomorphism:

$$\begin{aligned} {\mathbb {G}}_{m}\xrightarrow {{\check{\alpha }}}K_{\alpha }\subseteq T . \end{aligned}$$

We shall now identify $K_{\alpha }$ with ${\mathbb {G}}_{m}$ via ${\check{\alpha }}$. By the definition of ${\mathcal {F}}_{K_{\alpha }}$, we have

$$\begin{aligned} {\check{\alpha }}_{*}({\mathcal {F}}_{K_{\alpha }})\simeq {\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}}\otimes ({\mathcal {F}}^{n}_{T})^{-1}\mid _{{\widetilde{x}}_{2}} . \end{aligned}$$

We claim that

$$\begin{aligned} {\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}}\otimes ({\mathcal {F}}^{n}_{T})^{-1}\mid _{{\widetilde{x}}_{2}}\simeq {\check{\alpha }}(\alpha ({\mathcal {F}}_{T}))\mid _{{\widetilde{x}}_{1}} . \end{aligned}$$

This will prove that ${\mathcal {F}}_{K_{\alpha }}\simeq \alpha ({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}$ since ${\check{\alpha }}$ is primitive. It is easy to see that we have:

$$\begin{aligned} T\times ^{T,s_{\alpha }}(s^{-1}_{\alpha })^{*}({\mathcal {F}}^{n}_{T})\simeq {\check{\alpha }}(\alpha (s^{-1 *}_{\alpha }({\mathcal {F}}^{n}_{T})))\otimes s^{-1*}_{\alpha }({\mathcal {F}}^{n}_{T}). \end{aligned}$$

So the isomorphism:

$$\begin{aligned} T\times ^{T,s_{\alpha }}(s^{-1}_{\alpha })^{*}({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{2}}\simeq {\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{2}} \end{aligned}$$

implies that

$$\begin{aligned} {\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}}\otimes ({\mathcal {F}}^{n}_{T})^{-1}\mid _{{\widetilde{x}}_{2}}\simeq {\check{\alpha }}(\alpha ({\mathcal {F}}^{n}_{T}))\mid _{{\widetilde{x}}_{1}} . \end{aligned}$$

Next we shall deal with the case when ${\check{\alpha }}$ is not primitive. In this case we claim that we still have a natural isomorphism ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})\simeq \alpha ({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}$. Indeed, since ${\check{\alpha }}$ is not primitive, one concludes that $\alpha$ induces an isomorphism between $K_{\alpha }$ and ${\mathbb {G}}_{m}$. Since

$$\begin{aligned} T\times ^{K_{\alpha }}{\mathcal {F}}_{K_{\alpha }}\simeq {\mathcal {F}}^{n}_{T}\mid _{{\widetilde{x}}_{1}}\otimes ({\mathcal {F}}^{n}_{T})^{-1}\mid _{{\widetilde{x}}_{2}} , \end{aligned}$$

we conclude that

$$\begin{aligned} {\mathcal {F}}_{K_{\alpha }}\simeq \alpha ({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}\otimes \alpha ({\mathcal {F}}^{n}_{T})^{-1}\mid _{{\widetilde{x}}_{2}}\simeq \alpha ({\mathcal {F}}^{n}_{T})^{\otimes 2}\mid _{{\widetilde{x}}_{1}}. \end{aligned}$$

where the last isomorphism comes from $s_{\alpha }$ equivariance. So if we have a point in $\alpha ({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}$, we get a trivialization of $\alpha ({\mathcal {F}}^{n}_{T})\mid _{{\widetilde{x}}_{1}}$. This induces a trivialization of ${\mathcal {F}}_{K_{\alpha }}$, so by our previous discussions this gives rise to a W-equivariant T-torsor ${\mathcal {G}}_{T}$ on ${\widetilde{X}}$ such that the W-equivariance structure induces the identity automorphism on $T_{\alpha }\times ^{T}{\mathcal {G}}_{T}\mid _{{\widetilde{x}}}$. And that ${\mathcal {G}}_{T}$ corresponds to a $J_{G,{\widetilde{X}}}$ torsor away from x. Moreover, since $\alpha ({\mathcal {G}}_{T})|_{{\widetilde{x}}}\simeq \alpha ({\mathcal {F}}^{n}_{T})|_{{\widetilde{x}}_{1}}$, the trivialization of $\alpha ({\mathcal {F}}^{n}_{T})|_{{\widetilde{x}}_{1}}$ induces a trivialization of $\alpha ({\mathcal {G}}_{T})|_{{\widetilde{x}}}$ that is compatible with the trivialization of ${\check{\alpha }}(\alpha ({\mathcal {G}}_{T}))\mid _{{\widetilde{x}}}$. By Lemma 3.1.3 of [5], this lifts ${\mathcal {G}}_{T}$ to a ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ torsor.

The last claim follows from an easy calculation of the cokernel of the group scheme $J_{G,{\widetilde{X}}}\rightarrow J_{G,{\widetilde{X}}^{n}}$, which is left the reader. $\square$

Remark 2

It is possible to prove the Lemma B.0.2 by showing that the quotient $J_{G,{\widetilde{X}}^{n}}/J_{G,{\widetilde{X}}}$ is isomorphic to $i_{x *}({\mathbb {G}}_{m})$ directly.

To proceed further, one needs to analyze the structure of ${\mathcal {H}}iggs_{{\check{G}}}({\widetilde{X}})$. To do so we need to study the local situation. That is, one needs to analyze the category of Higgs bundles on the formal disk at $x\in X$. We first recall the following well-known lemma:

Lemma B.0.3

Let L be a Levi subgroup of G. Denote the Chevalley base of G by ${\mathfrak {c}}_{G}$ and the Chevalley base of L by ${\mathfrak {c}}_{L}$. Let ${\mathfrak {c}}^{\circ }_{L}$ be the open subscheme of ${\mathfrak {c}}_{L}$ such that the natural morphism ${\mathfrak {c}}_{L}\rightarrow {\mathfrak {c}}_{G}$ is etale over ${\mathfrak {c}}^{\circ }_{L}$. Denote the lie algebra of L by ${\mathfrak {l}}$ and lie algebra of G by ${\mathfrak {g}}$. Then we have a Cartesian diagram:

where $({\mathfrak {l}}/L)^{\circ }$ is the preimage of ${\mathfrak {c}}^{\circ }_{L}$ in ${\mathfrak {l}}/L$.

Remark 3

The open set ${\mathfrak {c}}^{\circ }_{L}$ can be described as follows: Let x be a semisimple element in ${\mathfrak {l}}$. Then the conjugacy class of x lies in ${\mathfrak {c}}^{\circ }_{L}$ iff $Z_{G}(x)\subseteq L$.

Here is a useful corollary:

Corollary B.0.4

Let $x\in X$ be a closed point and ${\widehat{O}}_{x}$ be the formal completion at x. Let ${\widetilde{X}}_{G}$ be a cameral cover of ${\text {Spec} }({\widehat{O}}_{x})$ for G and ${\widetilde{X}}_{L}$ be a cameral cover of ${\text {Spec} }({\widehat{O}}_{x})$ for L. Assuming that there exists an isomorphism of $W_{G}$-covers: ${\widetilde{X}}_{G}\simeq W_{G}\times ^{W_{L}}{\widetilde{X}}_{L}$ where $W_{G}$ and $W_{L}$ denotes the Weyl group of G and L, respectively. Then the category of G-Higgs bundles on ${\text {Spec} }({\widehat{O}}_{x})$ with cameral cover ${\widetilde{X}}_{G}$ is equivalent to the category of L-Higgs bundles on ${\text {Spec} }({\widehat{O}}_{x})$ with cameral cover ${\widetilde{X}}_{L}$.

Proof

The assumptions on the cameral covers implies that there exists a morphism ${\text {Spec} }({\widehat{O}}_{x})\xrightarrow {s}{\mathfrak {c}}^{\circ }_{L}$ such that ${\widetilde{X}}_{L}$ corresponds to s and ${\widetilde{X}}_{G}$ corresponds to the composition ${\text {Spec} }({\widehat{O}}_{x})\xrightarrow {s}{\mathfrak {c}}^{\circ }_{L}\rightarrow {\mathfrak {c}}_{G}$. A L-Higgs bundle with cameral cover ${\widetilde{X}}_{L}$ is equivalent to a factorization:

While G-Higgs bundles with cameral cover ${\widetilde{X}}_{G}$ is equivalent to a factorization:

Since the image of s lies in ${\mathfrak {c}}^{\circ }_{L}$, the previous lemma implies the claim. $\square$

From Lemma B.0.2 we can derive the following:

Lemma B.0.5

Let G be simply connected. There exists a surjective birational morphism:

$$\begin{aligned} {\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})\rightarrow {\text {Higgs} }_{G}({\widetilde{X}}) . \end{aligned}$$

Moreover, this realizes ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})$ as the normalization of ${\text {Higgs} }_{G}({\widetilde{X}})$.

Proof

First we claim that there exists a morphism ${\mathbf {P}}^{1}\rightarrow {\mathcal {H}}iggs_{G}({\widetilde{X}})$. Indeed, by our assumptions on ${\widetilde{X}}$, if we choose a point ${\widetilde{x}}\in D_{\alpha }$ above $x\in X$, we see that over the formal disk at x, the cameral cover ${\widetilde{X}}$ is induced from a cameral cover for L where L is the minimal Levi subgroup of G corresponds to the root $\alpha$. Hence Corollary B.0.4 applies. Since G is simply connected, ${\check{\alpha }}$ is primitive, so L is isomorphic to either ${\text {GL} }_{2}\times T_{0}$ or ${\text {SL} }_{2}\times T_{0}$ where $T_{0}$ is a torus, see Section 3 of [8]. In either case, our assumptions about the cameral cover implies that the cameral cover for L over ${\text {Spec} }({\widehat{O}}_{x})$ is isomorphic to

$$\begin{aligned} {\text {Spec} }({\widehat{O}}_{x}[T]/(T^{2}-t^{2})) \end{aligned}$$

where t is the uniformizer for ${\widehat{O}}_{x}$. Now consider the matrix

$$\begin{aligned} \gamma = \left( \begin{array}{cc} t &{} 0 \\ 0 &{} -t \\ \end{array} \right) \end{aligned}$$

View it as an element in ${\text {SL} }_{2}$. Denote the affine springer fiber of $\gamma$ by ${\text {Spr} }(\gamma )$. It is well -known that ${\text {Spr} }(\gamma )$ parameterizes ${\text {SL} }_{2}$ Higgs bundles on ${\text {Spec} }({\widehat{O}}_{x})$ with cameral cover ${\text {Spec} }({\widehat{O}}_{x}[T]/(T^{2}-t^{2}))$ together with a trivialization over ${\text {Spec} }({\mathcal {K}}_{x})$. It is well-known that affine springer fibers of $\gamma$ is an infinite chain of ${\mathbf {P}}^{1}$’s and regular Higgs bundles corresponds to ${\mathbb {G}}_{m}\subseteq {\mathbf {P}}^{1}$. Consider ${\text {SL} }_{2}$ as a subgroup of L and fix a copy of ${\mathbf {P}}^{1}$, we conclude from Corollary B.0.4 that we have a morphism ${\mathbf {P}}^{1}\rightarrow {\text {Higgs} }_{G}({\widetilde{X}})$. Moreover, one checks that in our case, the neutral component of the reduced part of ${\text {Gr} }_{J_{G,{\widetilde{X}}}}$ is isomorphic to ${\mathbb {G}}_{m}$. So from the product formula (see Proposition 4.15.1 of [12]) we get the desired morphism. When we restrict to the open part ${\mathbb {G}}_{m}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})$ this is an isomorphism. Moreover, by Lemma B.0.2, we conclude that ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})$ is a ${\mathbf {P}}^{1}$ fibration over ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$, hence it is projective. So ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})\rightarrow {\text {Higgs} }_{G}({\widetilde{X}})$ is surjective. This implies the claim. $\square$

Corollary B.0.6

Let ${\mathcal {Q}}$ be the line bundle on ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$ corresponds to the ${\mathbb {G}}_{m}$ torsor ${\text {Tors} }(J_{G,{\widetilde{X}}})$ on ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$, see Lemma B.0.2. Then the morphism ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})\rightarrow {\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$ identifies ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})$ with the projective bundle ${\text {Proj} }(O\oplus {\mathcal {Q}})$ over ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$.

Proof

This follows directly from Lemma B.0.2. $\square$

We also have the following description about the complement of ${\mathcal {H}}iggs_{G}^{reg}({\widetilde{X}})$ in ${\mathcal {H}}iggs_{G}({\widetilde{X}})$:

Lemma B.0.7

The category of irregular Higgs bundles with cameral cover ${\widetilde{X}}$ over k form a single orbit under the action of the Picard stack ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})(k)$. Moreover, it is a torsor over ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})(k)$.

Proof

Let $\lambda _{1}$ and $\lambda _{2}$ be two irregular Higgs bundles on X with cameral cover ${\widetilde{X}}$. We will prove they are locally isomorphic and that the sheaf of isomorphisms ${\text {Iso} }(\lambda _{1},\lambda _{2})$ is a $J_{G,{\widetilde{X}}^{n}}$ torsor. This will imply the claim of the lemma.

Since ${\widetilde{X}}$ is smooth away from the preimage of x, we see that all Higgs bundles with cameral cover ${\widetilde{X}}$ are regular over $X-x$, hence they form a torsor for the Picard stack ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})(X-x)\simeq {\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})(X-x)$. So the claim is true over $X-x$. Now we pick an open set U containing x. Using Lemma 3.0.3, one may shrink U so that $\lambda _{1}$ and $\lambda _{2}$ are isomorphic over $U-x$. To show they are isomorphic over U we look at the formal disk ${\text {Spec} }({\widehat{O}}_{x})$. We claim that all irregular Higgs bundles with cameral cover ${\widetilde{X}}$ over ${\text {Spec} }({\widehat{O}}_{x})$ are isomorphic and that the set of isomorphisms between $\lambda _{1}$ and $\lambda _{2}$ over ${\text {Spec} }({\widehat{O}}_{x})$ is a $J_{G,{\widetilde{X}}^{n}}({\widehat{O}}_{x})$ torsor. Assuming this for the moment, we will finish the proof. Indeed, $\lambda _{1}$ and $\lambda _{2}$ are isomorphic on $U-x$ as well as on ${\text {Spec} }({\widehat{O}}_{x})$, the obstruction for them to be isomorphic is an element in $\xi \in J_{G,{\widetilde{X}}^{n}}(U-x){\setminus } J_{G,{\widetilde{X}}^{n}}({\mathcal {K}}_{x})/J_{G,{\widetilde{X}}^{n}}({\widehat{O}}_{x})$. Now we can shrink U so that it becomes zero. Hence $\lambda _{1}$ and $\lambda _{2}$ are isomorphic over U and it is not hard to check that the set of isomorphisms between them is a $J_{G,{\widetilde{X}}^{n}}(U)$ torsor.

Now we analyze the local situation. Let us recall that if we have a cameral cover ${\widetilde{X}}\xrightarrow {\pi } X$ for ${\text {GL} }_{2}$, then the category of ${\text {GL} }_{2}$ Higgs bundles with cameral cover ${\widetilde{X}}$ is equivalent to the category of torsion free rank on sheaves on ${\widetilde{X}}$. Similarly, if ${\widetilde{X}}\xrightarrow {\pi } X$ is a cameral cover for ${\text {SL} }_{2}$, then the category of ${\text {SL} }_{2}$ Higgs bundles with cameral cover ${\widetilde{X}}$ is equivalent to the pair (F, s) where F is a torsion free rank on sheaf on ${\widetilde{X}}$ and s is an isomorphism ${\text {det} }(\pi _{*}(F))\simeq O_{X}$. In the local situation, using Corollary B.0.4 as we did in the proof of Lemma B.0.5, we reduce the question to ${\text {SL} }_{2}$ or ${\text {GL} }_{2}$ case. Denote the cameral cover and its normalization by ${\text {Spec} }({\widetilde{O}}_{x})$ and ${\text {Spec} }({\widetilde{O}}^{n})$. Since ${\text {Spec} }({\widetilde{O}}_{x})\simeq {\text {Spec} }({\widehat{O}}_{x}[T]/(T^{2}-t^{2}))$ has nodal singularity, it is well known that every torsion-free rank one sheaves on it that are not line bundles are isomorphic. So in the ${\text {GL} }_{2}$ case we conclude that irregular Higgs bundles over ${\text {Spec} }({\widehat{O}}_{x})$ are all isomorphic. In the ${\text {SL} }_{2}$ case one observe that if F is a torsion free rank one sheaf that is not a line bundle, then $F\simeq {\widetilde{O}}^{n}$. Since ${\text {Spec} }({\widetilde{O}}^{n})$ is an unramified cameral cover over ${\text {Spec} }({\widehat{O}}_{x})$, the determinant map ${\widetilde{O}}^{n \times }\xrightarrow {{\text {det} }}{\widehat{O}}^{\times }_{x}$ is surjective. Hence this shows that in the ${\text {SL} }_{2}$ case all irregular Higgs bundles are isomorphic. The claim that the set of isomorphisms between $\lambda _{1}$ and $\lambda _{2}$ over ${\text {Spec} }({\widehat{O}}_{x})$ is a $J_{G,{\widetilde{X}}^{n}}({\widehat{O}}_{x})$ torsor also follows from this. $\square$

Combining these together, we can show:

Lemma B.0.8

There exists a pushout diagram (in our case the pushout exists in the category of schemes exists, see 36.59 of [16]):

Moreover, there exists a natural morphism ${\mathcal {X}}\rightarrow {\text {Higgs} }_{G}({\widetilde{X}})$. It is a finite surjective morphism which induces a bijection on closed points.

Proof

In Lemma B.0.5 we already constructed the morphism:

$$\begin{aligned} {\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})\rightarrow {\text {Higgs} }_{G}({\widetilde{X}}). \end{aligned}$$

It comes from a morphism ${\mathbf {P}}^{1}\rightarrow {\text {Higgs} }_{G}({\widetilde{X}})$ by realizing ${\mathbf {P}}^{1}$ as a component of the affine springer fiber of a Levi subgroup. As we already mentioned in the proof of Lemma B.0.5, the open set ${\mathbb {G}}_{m}\subseteq {\mathbf {P}}^{1}$ corresponds to regular Higgs bundles and the two points 0 and $\infty$ correspond to irregular Higgs bundles. Denote the image of 0 and $\infty$ in ${\text {Higgs} }_{G}({\widetilde{X}})$ by $\lambda _{0}$ and $\lambda _{\infty }$. By Lemma B.0.7 we see that there exists a $J_{G,{\widetilde{X}}}$ torsor ${\mathcal {T}}$ such that $\lambda _{0}\otimes {\mathcal {T}}\simeq \lambda _{\infty }$. Now we have two morphisms $i_{0}$ and $i_{\infty }$ from ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$ to ${\text {Higgs} }_{G}({\widetilde{X}})$ induced by:

$$\begin{aligned}&{\text {Tors} }(J_{G,{\widetilde{X}}^{n}})\simeq 0\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})\rightarrow {\text {Higgs} }_{G}({\widetilde{X}}) \\&\quad {\text {Tors} }(J_{G,{\widetilde{X}}^{n}})\simeq \infty \times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})\rightarrow {\text {Higgs} }_{G}({\widetilde{X}}). \end{aligned}$$

Moreover, if ${\mathcal {T}}^{n}$ is the induced $J_{G,{\widetilde{X}}^{n}}$ torsor from ${\mathcal {T}}$, we see from Lemma B.0.7 that $i_{\infty }(-)=i_{0}(-\otimes {\mathcal {T}}^{n})$. Now let us identify the irregular Higgs bundles with ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$ via Lemma B.0.7 using the ${\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ action on $\lambda _{0}$. Then by our discussion above, the following diagram commutes:

Hence if we denote the pushout by ${\mathcal {X}}$, then we get ${\mathcal {X}}\rightarrow {\text {Higgs} }_{G}({\widetilde{X}})$. And by our discussions above, it is easy to see that it induces an isomorphism on closed points. By Lemma 36.59.4 of [16] and the fact that ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})\rightarrow {\mathcal {X}}$ is surjective, we see that ${\mathcal {X}}$ is proper. Hence ${\mathcal {X}}\rightarrow {\text {Higgs} }_{G}({\widetilde{X}})$ is a proper surjective morphism which induces an isomorphism on closed points, hence it is finite. $\square$

Corollary B.0.9

The gluing morphism between ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$ in the pushout diagram in Lemma B.0.8 can be described as follows: let us fix a point ${\widetilde{x}}\in {\widetilde{X}}$ in the preimage of $x\in X$ that lies in the ramification divisor $D_{\alpha }$. Let ${\widetilde{x}}_{1}$ be a point in ${\widetilde{X}}^{n}$ above ${\widetilde{x}}$. Then the gluing morphism is induced by translation by $Nm({\check{\alpha }}_{*}O({\widetilde{x}}_{1}))$. Here Nm is the morphism ${\widetilde{X}}^{n}\times X_{*}(T)\rightarrow {\text {Bun} }_{T}({\widetilde{X}}^{n})\rightarrow {\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ defined in Sect. 3 (Notice that since ${\widetilde{X}}^{n}$ is a smooth cameral cover over X, ${\widetilde{X}}^{n}\times X_{*}(T)\rightarrow {\text {Bun} }_{T}({\widetilde{X}}^{n})$ is defined over the entire ${\widetilde{X}}^{n}$).

Proof

Recall from the proof of Lemma B.0.5 that $\alpha$ determines a subgroup ${\text {SL} }_{2}\subseteq G$ and that the morphism ${\mathbf {P}}^{1}\rightarrow {\mathcal {H}}iggs_{G}({\widetilde{X}})$ is obtained by identifying ${\mathbf {P}}^{1}$ as a component of the affine springer fiber of the matrix $\gamma$ for the group ${\text {SL} }_{2}$:

$$\begin{aligned} \gamma = \left( \begin{array}{c@{\quad }c} t &{} 0 \\ 0 &{} -t \\ \end{array} \right) . \end{aligned}$$

The full affine springer fiber is a chain of ${\mathbf {P}}^{1}$’s equipped with an action of the group of integers ${\mathbb {Z}}$ where $1\in {\mathbb {Z}}$ acts on the lattices fixed by $\gamma$ via the matrix:

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} t &{} 0 \\ 0 &{} t^{-1} \\ \end{array} \right) . \end{aligned}$$

$0\in {\mathbf {P}}^{1}$ and $\infty \in {\mathbf {P}}^{1}$ are identified under this action. Notice that this ${\mathbb {Z}}$ action comes from the action of the affine grassmannian of the regular centralizer group scheme for ${\text {SL} }_{2}$. Moreover, if we denote the restriction of ${\widetilde{X}}$ to ${\text {Spec} }({\widehat{O}}_{x})$ by ${\widetilde{X}}_{x}$, then from the proof of Lemma B.0.5 and Corollary B.0.4, we see that ${\widetilde{X}}_{x}$ is induced from a cameral cover ${\widetilde{Y}}_{x}$ for ${\text {SL} }_{2}$: ${\widetilde{X}}_{x}\simeq W\times ^{S_{\alpha }}{\widetilde{Y}}_{x}$ where $S_{\alpha }$ stands for the subgroup generated by the reflection $s_{\alpha }$. We may assume ${\widetilde{x}}\in {\widetilde{Y}}_{x}$. This induces a morphism ${\text {Gr} }_{J_{{\text {SL} }_{2},{\widetilde{Y}}_{x},x}}\rightarrow {\text {Gr} }_{J_{G,{\widetilde{X}}_{x},x}}$. Passing to normalizations we get ${\text {Gr} }_{J_{{\text {SL} }_{2},{\widetilde{Y}}^{n}_{x},x}}\rightarrow {\text {Gr} }_{J_{G,{\widetilde{X}}^{n}_{x},x}}$. One can view

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} t &{} 0 \\ 0 &{} t^{-1} \\ \end{array} \right) \end{aligned}$$

as an element in ${\text {Gr} }_{J_{{\text {SL} }_{2},{\widetilde{Y}}_{x},x}}$. Hence we get the corresponding elements in ${\text {Gr} }_{J_{G,{\widetilde{X}}_{x},x}}$ and ${\text {Gr} }_{J_{G,{\widetilde{X}}^{n}_{x},x}}$. It is not hard to check that the element

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} t &{} 0 \\ 0 &{} t^{-1} \\ \end{array} \right) \end{aligned}$$

in ${\text {Gr} }_{J_{{\text {SL} }_{2},{\widetilde{Y}}^{n}_{x},x}}$ corresponds to the line bundle $O({\widetilde{x}}_{1}-{\widetilde{x}}_{2})$ on ${\widetilde{Y}}^{n}_{x}$ with its natural trivialization at the generic point. So this is equal to $Nm_{{\text {SL} }_{2}}({\check{\alpha }}_{*}O({\widetilde{x}}_{1}))$ via the morphism:

$$\begin{aligned} {\widetilde{Y}}^{n}_{x}\times {\mathbb {Z}}{\check{\alpha }}\xrightarrow {Nm_{{\text {SL} }_{2}}}{\text {Gr} }_{J_{{\text {SL} }_{2},{\widetilde{Y}}^{n}_{x},x}} . \end{aligned}$$

Unraveling the definitions, we see that its image under

$$\begin{aligned} {\text {Gr} }_{J_{{\text {SL} }_{2},{\widetilde{Y}}^{n}_{x},x}}\rightarrow {\text {Gr} }_{J_{G,{\widetilde{X}}^{n}_{x},x}}\rightarrow {\mathcal {T}}ors(J_{G,{\widetilde{X}}})\end{aligned}$$

is equal to $Nm({\check{\alpha }}_{*}O({\widetilde{x}}_{1}))$. $\square$

In order to show the morphism ${\mathcal {X}}\rightarrow {\text {Higgs} }_{{\check{G}}}({\widetilde{X}})$ in Lemma B.0.8 is an isomorphism, one has to analyze the local structure of ${\mathcal {H}}iggs_{G}({\widetilde{X}})$ near the points correspond to irregular Higgs bundles. This is provided by the following:

Lemma B.0.10

Let ${\widetilde{X}}$ be a cameral cover of X satisfying the assumptions at the beginning of this section. Let ${\widetilde{X}}_{x}$ be the induced cameral cover over the formal completion ${\text {Spec} }({\widehat{O}}_{x})$ and let ${\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$ be the stack of Higgs bundles on ${\text {Spec} }({\widehat{O}}_{x})$ with cameral cover ${\widetilde{X}}_{x}$. Then :

• (1) ${\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$ admits versal deformations.

• (2) The natural morphism ${\mathcal {H}}iggs_{G}({\widetilde{X}})\rightarrow {\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$ is formally smooth.

Proof

For part (1), it is not hard to check that the deformation functor ${\text {Def} }_{{\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})}$ attached to ${\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$ satisfies the Schlessinger’s conditions S1 and S2, see Section 87.16 of [16]. Moreover, its tangent spaces are also of finite dimension. From this one concludes that it admits versal deformations, see Lemma 87.13.4 of [16].

For part (2), let ${\text {Spec} }(A)\hookrightarrow {\text {Spec} }(A')$ be a small extension. Assume we have a commutative diagram:

where ${\text {Spec} }(A)\rightarrow {\mathcal {H}}iggs_{G}({\widetilde{X}})$ is given by a Higgs bundle $x_{A}$ on X and ${\text {Spec} }(A')\rightarrow {\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$ is given by a Higgs bundle $x^{loc}_{A'}$ on ${\text {Spec} }(A'[[t]])$. We want to lift the $A'$ point of ${\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$ to an $A'$ point of ${\mathcal {H}}iggs_{G}({\widetilde{X}})$. To do this, we interpret a Higgs bundle on $X\times {\text {Spec} }(A')$ with cameral cover ${\widetilde{X}}$ as a Higgs bundle on $(X-x)\times {\text {Spec} }(A')$, a Higgs bundle on ${\text {Spec} }(A'[[t]])$ together with an isomorphism of the two over the formal punctured disk ${\text {Spec} }(A'((t)))$. So what we need to is a deformation of $x_{A}\mid _{(X-x)\times {\text {Spec} }(A)}$ to $(X-x)\times {\text {Spec} }(A')$ and then a deformation of the isomorphism between $x_{A}\mid _{{\text {Spec} }(A((t)))}$ and $x^{loc}_{A'}\mid _{{\text {Spec} }(A((t)))}$ to ${\text {Spec} }(A'((t)))$. But since ${\widetilde{X}}$ is smooth away from x, deformations of $x_{A}\mid _{(X-x)\times {\text {Spec} }(A)}$ is the same as deformations of $J_{G,{\widetilde{X}}}$ torsors over $X-x$. Using the fact that $X-x$ is affine, we see that such deformations exists and all deformations are isomorphic. Similarly, ${\widetilde{X}}$ is smooth over the formal punctured disk k((t)), so any two deformations over ${\text {Spec} }(A((t)))$ will be isomorphic. This finishes the proof. $\square$

Remark 4

Similar arguments can be used to show that if ${\widetilde{X}}$ is a reduced cameral cover over X, then ${\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$ admits versal deformations. Moreover, if we denote ${\mathcal {H}}iggs_{G}^{0}({\widetilde{X}})$ to be the open substack of ${\mathcal {H}}iggs_{G}({\widetilde{X}})$ corresponds to Higgs bundles that are regular away from x, then the morphism ${\mathcal {H}}iggs_{G}^{0}({\widetilde{X}})\rightarrow {\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$ is formally smooth.

Now we can finish the proof of Theorem B.0.1:

Proof

First we claim that if y is a closed point in ${\text {Higgs} }_{G}({\widetilde{X}})$ corresponds to an irregular Higgs bundle and $O_{{\text {Higgs} }_{G}({\widetilde{X}}),y}$ is the local ring of ${\text {Higgs} }_{G}({\widetilde{X}})$ at y, then the formal completion ${\widehat{O}}_{{\text {Higgs} }_{G}({\widetilde{X}}),y}$ is isomorphic to $k[[t_1,t_2,\ldots ,t_n]]/(t_{1}t_{2})$. Indeed, since ${\mathcal {H}}iggs_{G}({\widetilde{X}})$ is a Z(G) gerbe over ${\text {Higgs} }_{G}({\widetilde{X}})$ by Lemma 4.1.8, we only need to prove this for versal deformations of ${\mathcal {H}}iggs_{G}({\widetilde{X}})$. Using Lemma B.0.10 we conclude that we only need to prove it for versal deformations of ${\mathcal {H}}iggs_{G}({\widetilde{X}}_{x})$. We can further reduced the problem to versal deformations of ${\mathcal {H}}iggs_{{\text {GL} }_{2}}({\widetilde{C}}_{x})$ or ${\mathcal {H}}iggs_{{\text {SL} }_{2}}({\widetilde{C}}_{x})$ by Lemma B.0.4, where ${\widetilde{C}}_{x}$ is a nodal $S_{2}$ cover over ${\text {Spec} }({\widehat{O}}_{x})$. This can again be reduced to the versal deformations of rank two Higgs bundles on any smooth curve C with a nodal cameral cover ${\widetilde{C}}$ using Lemma B.0.10. In this case Higgs bundles with cameral cover ${\widetilde{C}}$ are parameterized by the compactified Jacobian ${\overline{J}}_{{\widetilde{C}}}$. It is well-known that the natural morphism ${\widetilde{C}}\times J_{{\widetilde{C}}}\rightarrow {\overline{J}}_{{\widetilde{C}}}$ is smooth. So we are done.

Now we can prove ${\mathcal {X}}\simeq {\text {Higgs} }_{G}({\widetilde{X}})$. Since ${\mathcal {X}}\rightarrow {\text {Higgs} }_{G}({\widetilde{X}})$ is birational and finite and that ${\text {Higgs} }_{G}({\widetilde{X}})$ is Cohen-Macaulay (See Lemma 2.1.2), we only need to prove that $O_{{\mathcal {X}}}\simeq O_{{\text {Higgs} }_{G}({\widetilde{X}})}$ at all codimension one points of ${\text {Higgs} }_{G}({\widetilde{X}})$. Since ${\mathcal {X}}$ is isomorphic to ${\text {Higgs} }_{G}({\widetilde{X}})$ over the open set of regular Higgs bundles, we only need to look at generic points of the closed subvariety of ${\text {Higgs} }_{G}({\widetilde{X}})$ formed by irregular Higgs bundles. Let $\eta$ be the generic point. Denote the morphism ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})\rightarrow {\text {Higgs} }_{G}({\widetilde{X}})$ by q. Our discussions above the formal completions of ${\text {Higgs} }_{G}({\widetilde{X}})$ at closed points implies that the cokernel of $O_{{\text {Higgs} }_{G}({\widetilde{X}})}\rightarrow q_{*}(O_{{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})})$ is supported on ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$ and it is isomorphic to a line bundle on ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$. Let us look at the local ring $O_{{\text {Higgs} }_{G}({\widetilde{X}}),\eta }$. We conclude from the discussion above that the cokernel of $O_{{\text {Higgs} }_{G}({\widetilde{X}}),\eta }\rightarrow q_{*}(O_{{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})}))_{\eta }$ is isomorphic to the residue field of $O_{{\text {Higgs} }_{G}({\widetilde{X}}),\eta }$. Since we have injections

$$\begin{aligned} O_{{\text {Higgs} }_{G}({\widetilde{X}}),\eta }\hookrightarrow O_{{\mathcal {X}},\eta }\hookrightarrow q_{*}(O_{{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})})_{\eta } , \end{aligned}$$

and that $O_{{\mathcal {X}},\eta }$ is not equal to $q_{*}(O_{{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G,{\widetilde{X}}})})_{\eta }$, we are done. $\square$

As a corollary, we will prove the following result about $H^{1}(O_{{\mathcal {H}}iggs_{G}})$.

Corollary B.0.11

Let G be simply connected and H be the Hitchin base. Let ${\text {Lie} }({\text {Tors} }(J_{G}))$ be the vector bundle on H corresponds to the lie algebra of ${\text {Tors} }(J_{G})$. To simplify the notation, let us also denote the restrictions of ${\mathcal {H}}iggs_{G}$ and ${\text {Lie} }({\text {Tors} }(J_{G}))$ to $H_{int}$ still by ${\mathcal {H}}iggs_{G}$ and ${\text {Lie} }({\text {Tors} }(J_{G}))$. Then we have $R^{1}\pi _{*}(O_{{\mathcal {H}}iggs_{G}})\simeq {\text {Lie} }({\text {Tors} }(J_{G}))$ as coherent sheaves on $H_{int}$. Here $\pi$ is the morphism ${\mathcal {H}}iggs_{G}\xrightarrow {\pi } H_{int}$.

Proof

Let us first construct the morphism ${\text {Lie} }({\text {Tors} }(J_{G}))\rightarrow R^{1}\pi _{*}(O_{{\mathcal {H}}iggs_{G}})$. Pick a line bundle ${\mathcal {M}}$ on ${\mathcal {H}}iggs_{G}$ which is ample when restricted to the semistable locus. Pick a local section $\xi \in {\text {Lie} }({\text {Tors} }(J_{G}))$. $\xi$ induces an infinitesimal automorphism of ${\mathcal {H}}iggs_{G}$ via the action of ${\mathcal {T}}ors(J_{G})$ on ${\mathcal {H}}iggs_{G}$. Consider the line bundle $\xi ^{*}({\mathcal {M}})\otimes {\mathcal {M}}^{-1}$ on ${\mathcal {H}}iggs_{G}\times D$ where D denotes the ring of dual numbers. This line bundle is canonically trivial when restricted to ${\mathcal {H}}iggs_{G}$, hence it defines an element in $R^{1}\pi _{*}(O_{{\mathcal {H}}iggs_{G}})$. Next we will prove this induces an isomorphism between them. Consider the open set U of $H_{int}$ defined in Lemma C.0.2 in Appendix C. Denote the restriction of $\pi$ to U by $\pi _{U}$. We will first prove that the morphism we constructed above induces an isomorphism $R^{1}\pi _{U*}O_{{\mathcal {H}}iggs_{G}\mid _{U}}\simeq {\text {Lie} }({\text {Tors} }(J_{G}))\mid _{U}$ as coherent sheaves on U. To do this we fix a cameral cover ${\widetilde{X}}$ and look at the cohomology $H^{1}(O_{{\mathcal {H}}iggs_{G}({\widetilde{X}})})$. By Lemma C.0.2, we either have ${\mathcal {H}}iggs_{G}({\widetilde{X}})={\mathcal {H}}iggs_{G}^{reg}({\widetilde{X}})$ or ${\widetilde{X}}$ satisfies the assumptions given at the beginning of this section. If ${\mathcal {H}}iggs_{G}({\widetilde{X}})={\mathcal {H}}iggs_{G}^{reg}({\widetilde{X}})$, then ${\mathcal {H}}iggs_{G}({\widetilde{X}})\simeq {\mathcal {T}}ors(J_{G,{\widetilde{X}}})$ and that ${\text {Tors} }(J_{G,{\widetilde{X}}})$ is an abelian variety. Since ${\mathcal {M}}$ also restricts to an ample line bundle on ${\text {Tors} }(J_{G,{\widetilde{X}}})$, the claim is well-known, see [11] or [14]. Now let us assume ${\widetilde{X}}$ satisfies the assumptions given at the beginning of this section. Using Theorem B.0.1 we can describe $H^{1}(O_{{\text {Higgs} }_{G}({\widetilde{X}})})$ as follows. First we have an exact sequence (To simplify the notations we will write ${\text {Tors} }(J_{G})$ and ${\text {Tors} }(J^{n}_{G})$ instead of ${\text {Tors} }(J_{G,{\widetilde{X}}})$ and ${\text {Tors} }(J_{G,{\widetilde{X}}^{n}})$ in the rest of the proof):

$$\begin{aligned} 0\rightarrow O_{{\text {Higgs} }_{G}({\widetilde{X}})}\rightarrow O_{{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})}\oplus O_{{\text {Tors} }(J^{n}_{G})}\rightarrow O_{{\text {Tors} }(J^{n}_{G})}\oplus O_{{\text {Tors} }(J^{n}_{G})}\rightarrow 0. \end{aligned}$$

Since ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})$ is a ${\mathbf {P}}^{1}$ bundle over ${\text {Tors} }(J^{n}_{G})$, we have $H^{i}(O_{{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})})\simeq H^{i}(O_{{\text {Tors} }(J^{n}_{G})})$. Since ${\text {Tors} }(J^{n}_{G})$ is an abelian variety, translation by an element induces the trivial isomorphism on $H^{i}$. Using Corollary B.0.9 we see that the induced morphism

$$\begin{aligned} H^{1}(O_{{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})})\rightarrow H^{1}(O_{{\text {Tors} }(J^{n}_{G})})\oplus H^{1}(O_{{\text {Tors} }(J^{n}_{G})}) \end{aligned}$$

can be identified with the diagonal embedding:

$$\begin{aligned} H^{1}(O_{{\text {Tors} }(J^{n}_{G})})\xrightarrow {\vartriangle } H^{1}(O_{{\text {Tors} }(J^{n}_{G})})\oplus H^{1}(O_{{\text {Tors} }(J^{n}_{G})}). \end{aligned}$$

The same is true for

$$\begin{aligned} H^{1}(O_{{\text {Tors} }(J^{n}_{G})})\rightarrow H^{1}(O_{{\text {Tors} }(J^{n}_{G})})\oplus H^{1}(O_{{\text {Tors} }(J^{n}_{G})}). \end{aligned}$$

Hence we see that $H^{1}(O_{{\text {Higgs} }_{G}({\widetilde{X}})})$ fits into an exact sequence:

$$\begin{aligned} 0\rightarrow H^{0}(O_{{\text {Tors} }(J^{n}_{G})})\rightarrow H^{1}(O_{{\text {Higgs} }_{G}({\widetilde{X}})})\rightarrow H^{1}(O_{{\text {Tors} }(J^{n}_{G})})\rightarrow 0. \end{aligned}$$

(B.1)

Since G is simply connected, ${\text {Tors} }(J^{n}_{G})$ is connected. From this one concludes that . Since we also have , we see that $H^{1}(O_{{\mathcal {H}}iggs_{G}({\widetilde{X}})})$ and ${\text {Lie} }({\text {Tors} }(J_{G}))$ has the same dimension. To prove they are isomorphic we need to show the morphism is injective. Let ${\mathcal {M}}$ be the ample line bundle on ${\mathcal {H}}iggs_{G}({\widetilde{X}})$ we pick at the beginning. Then the restriction of ${\mathcal {M}}$ to ${\text {Tors} }(J^{n}_{G})$ via the embedding ${\text {Tors} }(J^{n}_{G})\rightarrow {\mathcal {H}}iggs_{G}({\widetilde{X}})$ in Theorem B.0.1 is also an ample line bundle on the abelian variety ${\text {Tors} }(J^{n}_{G})$. Hence if $\xi \in {\text {Lie} }({\text {Tors} }(J_{G}))$ and that $\xi ^{*}({\mathcal {M}})\otimes {\mathcal {M}}^{-1}$ is trivial, then its restriction to ${\text {Tors} }(J^{n}_{G})$ is also trivial. Because ${\mathcal {M}}$ is ample and ${\text {Tors} }(J^{n}_{G})$ is an abelian variety, we see that the image of $\xi$ in ${\text {Lie} }({\text {Tors} }(J^{n}_{G}))$ is trivial. It is not hard to verify that the following diagram commutes:

Hence by the exact sequence B.1, we see that if $\xi$ lies in the kernel of ${\text {Lie} }({\text {Tors} }(J_{G}))\rightarrow H^{1}(O_{{\mathcal {H}}iggs_{G}({\widetilde{X}})})$, then $\xi$ is in kernel of ${\text {Lie} }({\text {Tors} }(J_{G}))\rightarrow {\text {Lie} }({\text {Tors} }(J^{n}_{G}))$, which can be identifies with an element in the Lie algebra of ${\mathbb {G}}_{m}$ by Lemma B.0.2. To show $\xi$ is zero, let us look at the pullback of ${\mathcal {M}}$ to ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})$. Let O(1) be the universal line bundle on the projective bundle ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})$ over ${\text {Tors} }(J^{n}_{G})$, see Corollary B.0.6. The pullback of ${\mathcal {M}}$ to ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})$ is isomorphic to $O(n)\otimes f^{*}({\mathcal {M}}')$ where ${\mathcal {M}}'$ is an ample line bundle on ${\text {Tors} }(J^{n}_{G})$, f is the morphism ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})\rightarrow {\text {Tors} }(J^{n}_{G})$ and $n>0$. Now let us notice that by Theorem B.0.1, a line bundle on ${\mathcal {H}}iggs_{G}({\widetilde{X}})$ is the same a line bundle on ${\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})$ together with a gluing map when restricted to ${\text {Tors} }(J^{n}_{G})\amalg {\text {Tors} }(J^{n}_{G})$. We have two sections:

$$\begin{aligned}&0\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})\simeq {\text {Tors} }(J^{n}_{G})\xrightarrow {i_{0}}{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})\\&\quad \infty \times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G})\simeq {\text {Tors} }(J^{n}_{G})\xrightarrow {i_{\infty }}{\mathbf {P}}^{1}\times ^{{\mathbb {G}}_{m}}{\text {Tors} }(J_{G}). \end{aligned}$$

Using Corollary B.0.6 one may assume that

$$\begin{aligned}&i_{0}^{*}(O(n)\otimes f^{*}({\mathcal {M}}'))\simeq {\mathcal {M}}' \\&i_{1}^{*}(O(n)\otimes f^{*}({\mathcal {M}}'))\simeq {\mathcal {Q}}^{\otimes n}\otimes {\mathcal {M}}'. \end{aligned}$$

Since we have shown that $\xi$ belongs to the lie algebra of ${\mathbb {G}}_{m}$, one may view it as an element in the unit of the ring of dual numbers $k[\epsilon ]/\epsilon ^{2}$. From the expression of the restriction of $O(n)\otimes f^{*}({\mathcal {M}}')$ to $i_{0}({\text {Tors} }(J^{n}_{G}))$ and $i_{1}({\text {Tors} }(J^{n}_{G}))$, we conclude that the gluing maps for $\xi ^{*}(O(n)\otimes f^{*}({\mathcal {M}}'))$ and $O(n)\otimes f^{*}({\mathcal {M}}')$ differs by $\xi ^{n}$ where we view $\xi$ as a unit in $k[\epsilon ]/\epsilon ^{2}$. This shows that $\xi ^{*}({\mathcal {M}})\otimes {\mathcal {M}}^{-1}$ gives a nontrivial element in $H^{1}(O_{{\mathcal {H}}iggs_{G}({\widetilde{X}})})$ as long as $\xi$ is nonzero.

The analysis above shows that if ${\widetilde{X}}$ is a cameral cover corresponds to a point in $U\subseteq H_{int}$, then the morphism ${\text {Lie} }({\text {Tors} }(J_{G,{\widetilde{X}}}))\rightarrow H^{1}(O_{{\mathcal {H}}iggs_{G}({\widetilde{X}})})$ is an isomorphism. Since U is smooth, Grauert’s theorem on cohomology and base change shows that $R^{1}\pi _{U *}O_{{\mathcal {H}}iggs_{G}\mid _{U}}$ is locally free and isomorphic to ${\text {Lie} }({\text {Tors} }(J_{G}))\mid _{U}$. Let j be the open embedding $U\xrightarrow {j}H_{int}$. Then one gets a morphism $R^{1}\pi _{*}O_{{\mathcal {H}}iggs_{G}}\rightarrow j_{*}({\text {Lie} }({\text {Tors} }(J_{G}))\mid _{U})$. Since the complement of U has codimension greater than or equals to two, we conclude that $j_{*}({\text {Lie} }({\text {Tors} }(J_{G}))\mid _{U})\simeq {\text {Lie} }({\text {Tors} }(J_{G}))$. Now arguing as in Theorem 6.5 of [9] we get the result. $\square$

Let us also note the following:

Lemma B.0.12

Let G be simply connected and ${\widetilde{X}}$ satisfies the assumptions given at the beginning of this section. Let ${\widetilde{X}}^{n}$ be its normalization. Then ${\mathcal {T}}ors(J_{G,{\widetilde{X}}^{n}})$ is connected.

Proof

We will use Proposition 4.10.3 of [12]. First note that ${\widetilde{X}}^{n}$ is a smooth connected cameral cover over X, hence for all ramification points of the morphism ${\widetilde{X}}^{n}\rightarrow X$ have ramification index 2 and that the discriminant divisor on X is multiplicity free(for the construction of the discriminant divisor, see Remark 2.6.2 of [5]). Since ${\widetilde{X}}$ is smooth away from $x\in X$ and has nodal singularities over X, using the construction of the discriminant divisor and the assumption that $l>2g$ and $g\ge 2$, we see that for each root $\alpha$, the ramification divisor $D_{\alpha }$ on ${\widetilde{X}}^{n}$ is nonempty. Now Proposition 4.10.3 of [12] implies the claim. $\square$

Appendix C: An open locus of the Hitchin base

In this section we determine an open subset U of the Hitchin base H whose complement has codimension greater than or equals to two, see Lemma C.0.2. Lemma C.0.1 is due to Professor Dima Arinkin and the author would like to thank him for giving the permission to publish it.

Lemma C.0.1

Let G be a reductive group and r be the semisimple rank of G. Let $x=x_{0}+tx_{1}+\cdots$ be an element in ${\mathfrak {g}}(k[[t]])$ with $x_{0}$ nilpotent. Let ${\text {Disc} }$ be the discriminant function on the Chevalley base ${\mathfrak {c}}$ and denote its pullback to ${\mathfrak {g}}$ via the Chevalley map ${\mathfrak {g}}\rightarrow {\mathfrak {c}}$ also by ${\text {Disc} }$. Then we have:

• (1) ${\text {Disc} }(x)\in t^{r}k[[t]]$.

• (2) If the order of vanishing of ${\text {Disc} }(x)$ is exactly equal to r, then $x_{0}$ is regular.

• (3) Let ${\text {Spec} }({\widehat{O}}_{x})\xrightarrow {x} {\mathfrak {g}}$ be a morphism. Let $\chi (x_{0})$ be the image of the closed point of ${\text {Spec} }({\widehat{O}}_{x})$ under the morphism ${\mathfrak {g}}\xrightarrow {\chi }{\mathfrak {c}}$. If the image of the tangent space of ${\text {Spec} }({\widehat{O}}_{x})$ in $T_{\zeta }{\mathfrak {c}}$ under the morphism ${\text {Spec} }({\widehat{O}}_{x})\xrightarrow {x} {\mathfrak {g}}\xrightarrow {\chi }{\mathfrak {c}}$ does not lies inside the tangent cone of ${\text {Disc} }$ in $T_{\chi (x_{0})}{\mathfrak {c}}$, then x factors through ${\mathfrak {g}}_{reg}$.

Proof

We may assume G is semisimple. Let us recall that if $a\in {\mathfrak {g}}$, then we have that the ${\text {Disc} }(a)$ is equal to the coefficient of $T^{r}$ of the polynomial ${\text {det} }(T-ad(a))$. We apply this to $a=x$. If $n={\text {dim} }{\mathfrak {g}}$, then it is well-known that the coefficient of $T^{r}$ is equal to the sum of $(n-r)\times (n-r)$ principle minors of the matrix corresponds to the linear transform ad(x) on ${\mathfrak {g}}$. Now observe that if A is a $n\times n$ nilpotent matrix such that the dimension of the kernel of A is greater than or equals to r. Then for any $n\times n$ matrix $B=B_{0}+tB_{1}+\cdots$, all $(n-r)\times (n-r)$ principle minors of $A+tB$ are power series whose initial terms have degree greater than or equals to the dimension of the kernel of A. This is easy to check by reducing to the case when A has Jordan normal form. We apply this to the case when $A=ad(x_{0})$ and $B=ad(x_{1}+tx_{2}+\cdots )$. This proves part (1). Moreover, using the same argument, it is not hard to see that if the assumption of part (2) holds, then the dimension of the kernel of $x_{0}$ must equal to r, hence $x_{0}$ is regular.

For part (3), write x as $x=x_{0}+tx_{1}+\cdots$. Let us first consider the case when $x_{0}$ is nilpotent. Then the claim a reformulation of part (2), using the fact that the multiplicity of the divisor ${\text {Disc} }$ at $0\in {\mathfrak {c}}$ is equal to r, see Section 3.18 of [10]. The general case can be reduced to the nilpotent case as follows. Let us consider the Jordan decomposition $x_{0}=t+n$ where t is semisimple and n is nilpotent. Let $L=Z_{G}(t)$. Then we see that $\chi (x_{0})$ is in the image of ${\mathfrak {c}}^{\circ }_{L}$ (here we will denote the Chevalley base of G and L by ${\mathfrak {c}}_{G}$ and ${\mathfrak {c}}_{L}$ in order to distinguish between them. Similarly for the discriminant functions ${\text {Disc} }_{G}$ and ${\text {Disc} }_{L}$), see our notations in Lemma B.0.3. Since ${\widehat{O}}_{x}$ is complete, use Lemma B.0.3 we can assume we have a lift x to an element in $y\in {\mathfrak {l}}({\widehat{O}}_{x})$. We claim that y satisfies the same assumption as x. Indeed, it is not hard to check that the quotient ${\text {Disc} }_{G}/{\text {Disc} }_{L}$ is an invertible function on ${\mathfrak {c}}^{\circ }_{L}$. Since ${\mathfrak {c}}^{\circ }_{L}\rightarrow {\mathfrak {c}}_{G}$ is etale, the tangent cone of ${\text {Disc} }_{G}$ at $\chi _{G}(x_{0})\in {\mathfrak {c}}_{G}$ can be identified with the tangent cone of ${\text {Disc} }_{L}$ at $\chi _{L}(y_{0})\in {\mathfrak {c}}_{L}$. Now write $y=y_{[L,L]}+y_{Z(L)}$ where $y_{[L,L]}$ lies in the lie algebra of [L, L] and $y_{Z(L)}$ lies in the lie algebra of the center of L. Using the discussions above one can reduce everything to the semisimple group [L, L]. Also notice that the initial term of $y_{[L,L]}$ is nilpotent, so we are done. $\square$

Let us consider the following subset U of $H_{int}$, which consists of points such that the corresponding discriminant divisor on X is multiplicity free except at one point $x\in X$, where it has order less than or equals to 2.

Lemma C.0.2

• (1) U is an open subset of H whose complement has codimension greater than or equals to two.

• (2) If ${\widetilde{X}}$ is a cameral cover corresponds to a point in U, then either we have ${\mathcal {H}}iggs_{G}({\widetilde{X}})={\mathcal {H}}iggs_{G}^{reg}({\widetilde{X}})$ or ${\widetilde{X}}$ satisfies the assumptions on cameral covers we give at the beginning of Appendix B.

Proof

For part (1), let us recall that the morphism $X\times H\rightarrow {\mathfrak {c}}\times ^{{\mathbb {G}}_{m}}{L}$ is smooth, this follows from the expression ${\mathfrak {c}}\times ^{{\mathbb {G}}_{m}}{L}\simeq L^{e_{1}}\oplus \cdots \oplus L^{e_{r}}$ where $e_{i}$ are the degrees of the W invariant fundamental polynomials on ${\mathfrak {t}}$, see Subsection 4.13 of [12]. By definition, the complement of U corresponds to points in H such that either there exists a point $x\in X$ such that the discriminant divisor has multiplicity greater than or equals to three at x, or there exists two points $x,y\in X$ such that the discriminant divisor has multiplicities greater than or equals to two at x and y. So part (1) follows from the following claims:

• (1) The set of pairs consisting of $(x,h)\in X\times H$ such that the discriminant divisor has multiplicity greater than or equals to three at x has codimension greater than or equals to 3 in $X\times H$.

• (2) The set of triples $(x,y,h)\in X^{2}\times H$ such that the discriminant divisor has multiplicities greater than or equals to two at x and y has codimension greater than or equals to 4 in $X^{2}\times H$.

Let us look at the first claim. To simplify the notation, we will denote the bundle ${\mathfrak {c}}\times ^{{\mathbb {G}}_{m}}{L}$ by ${\mathfrak {c}}^{L}$ in the rest of the proof. We will prove that if we fix x, then the set of h such that the discriminant divisor has multiplicity greater than or equals to three at x has codimension greater than or equals to 3 in H. Let us look at the smooth morphism $H\rightarrow {\mathfrak {c}}^{L}_{x}$ obtained by restricting $X\times H\rightarrow {\mathfrak {c}}^{L}$ to x. If $(x,h)\in X\times H$ is such that x lies in the locus of ${\mathfrak {c}}^{L}_{x}$ where ${\text {Disc} }_{G}$ has multiplicity greater than or equals to 3, then by Lemma C.0.3 conclude that all such h form a subset of codimension greater than or equals to 3. Now assume x lies in the locus of ${\mathfrak {c}}^{L}_{x}$ where the multiplicity of ${\text {Disc} }_{G}$ is equal to two. Since the discriminant divisor has multiplicity greater than or equals to three at x, it implies that if we look at the map ${\text {Spec} }({\widehat{O}}_{x})\rightarrow {\mathfrak {c}}$ induced by h (Plus a trivialization of the line bundle L on ${\text {Spec} }({\widehat{O}}_{x})$), then the image of the tangent space of ${\text {Spec} }({\widehat{O}}_{x})$ lies inside the tangent cone of the discriminant function. Notice that the morphism $H\rightarrow {\mathfrak {c}}^{L}/m^{2}_{x}{\mathfrak {c}}^{L}$ is a surjective morphism of vector spaces (See Lemma 4.7.2 of [12]). Combine this with Lemma C.0.3, we see that the condition that x lies inside the locally closed subset of ${\mathfrak {c}}^{L}_{x}$ where ${\text {Disc} }_{G}$ has multiplicity two plus the condition that the image of the tangent space of ${\text {Spec} }({\widehat{O}}_{x})$ lies inside the tangent cone of ${\text {Disc} }_{G}$ gives a codimension three subset in H. Finally let us assume that x lies in the locus of ${\mathfrak {c}}^{L}_{x}$ where ${\text {Disc} }_{G}$ has multiplicity one. In this case x actually lies in the open set ${\mathfrak {c}}^{L,sm}_{x}$ where the morphism ${\mathfrak {c}}^{L,sm}_{x}\xrightarrow {{\text {Disc} }} {\mathbb {A}}^{1}$ is smooth. So if we identify ${\mathfrak {c}}^{L}_{x}/{\mathfrak {m}}^{3}_{x}{\mathfrak {c}}^{L}_{x}$ with ${\mathfrak {c}}(O_{x}/{\mathfrak {m}}^{3}_{x})$ and consider the smooth surjective morphism $H\rightarrow {\mathfrak {c}}^{L}_{x}/{\mathfrak {m}}^{3}_{x}{\mathfrak {c}}^{L}_{x}\simeq {\mathfrak {c}}(O_{x}/{\mathfrak {m}}^{3}_{x})$, then h lies inside the open set ${\mathfrak {c}}^{sm}(O_{x}/{\mathfrak {m}}^{3}_{x})$. Since

$$\begin{aligned} {\mathfrak {c}}^{sm}(O_{x}/{\mathfrak {m}}^{3}_{x})\rightarrow {\mathbb {A}}^{1}(O/{\mathfrak {m}}^{3}_{x})\simeq O/{\mathfrak {m}}^{3}_{x} \end{aligned}$$

is a smooth morphism, we conclude that the condition that x lies in the locus of ${\mathfrak {c}}^{L}_{x}$ where ${\text {Disc} }_{G}$ has multiplicity one and that the discriminant divisor has order greater than or equals to three at x cuts out a locally closed subset of H with codimension greater than or equals to three. Combining these together, we have established the first claim. The proof for the second claim is similar. Namely, assume first that the image of both x and y lies in the locus of ${\mathfrak {c}}^{L}_{x}$ and ${\mathfrak {c}}^{L}_{y}$ where ${\text {Disc} }_{G}$ has multiplicity equals to one. In this case, the induced morphisms ${\text {Spec} }({\widehat{O}}_{x})\rightarrow {\mathfrak {c}}$ and ${\text {Spec} }({\widehat{O}}_{y})\rightarrow {\mathfrak {c}}$ has the property that the image of the tangent spaces of ${\text {Spec} }({\widehat{O}}_{x})$ and ${\text {Spec} }({\widehat{O}}_{y})$ lies inside the tangent cone of ${\text {Disc} }_{G}$. Since the morphism $H\rightarrow {\mathfrak {c}}^{L}_{x}/{\mathfrak {m}}^{2}_{x}{\mathfrak {c}}^{L}_{x}\oplus {\mathfrak {c}}^{L}_{y}/{\mathfrak {m}}^{2}_{y}{\mathfrak {c}}^{L}_{y}$ is smooth and surjective, this cuts out a codimension 4 locally closed subset of H. Now assume that the image of x lies in the locus of ${\mathfrak {c}}^{L}_{x}$ while the image of y lies in the closed subset where ${\text {Disc} }_{G}$ has multiplicity greater than or equals to two. Using Lemma C.0.3 and the argument above, we see that this also cuts out a locally closed subset of codimension greater than or equals to 4. The last case is when both x and y lies in the locus of ${\mathfrak {c}}^{L}_{x}$ and ${\mathfrak {c}}^{L}_{y}$ where the multiplicity of ${\text {Disc} }_{G}$ is greater than or equals to two. Arguing as above we get a codimension 4 closed subset in H. This finishes the proof for claim (2).

Now let us look at part (2). Under our assumptions, the discriminant divisor is either multiplicity free or there exists a unique point $x\in X$ where it is not multiplicity free and the multiplicity at x is equal to two. In the case when it is multiplicity free everywhere, it is well-known that the cameral cover is smooth and we have ${\mathcal {H}}iggs_{G}({\widetilde{X}})={\mathcal {H}}iggs_{G}^{reg}({\widetilde{X}})$, see Proposition 4.7.7 of [12]. When it has multiplicity two at $x\in X$, ${\widetilde{X}}$ is smooth away from x. Over the formal disk ${\text {Spec} }({\widehat{O}}_{x})$, ${\widetilde{X}}$ is determined by a morphism ${\text {Spec} }({\widehat{O}}_{x})\rightarrow {\mathfrak {c}}_{G}$. Choose an element $t\in {\mathfrak {t}}$ such that the image of t in ${\mathfrak {c}}_{G}$ is equal to the image of the closed point x in ${\mathfrak {c}}_{G}$. Let $L=Z_{G}(t)$. Then arguing as in the proof of part (3) of Lemma C.0.1, we see that the image of x in ${\mathfrak {c}}_{G}$ lies in the image of ${\mathfrak {c}}^{\circ }_{L}\rightarrow {\mathfrak {c}}_{G}$. Hence ${\text {Spec} }({\widehat{O}}_{x})\rightarrow {\mathfrak {c}}_{G}$ can be lifted to ${\text {Spec} }({\widehat{O}}_{x})\rightarrow {\mathfrak {c}}_{L}$ and that the multiplicity of the pullback of ${\text {Disc} }_{L}$ to ${\text {Spec} }({\widehat{O}}_{x})$ has multiplicity two. Moreover, if we identify ${\mathfrak {c}}_{L}\simeq {\mathfrak {c}}_{[L,L]}\times {\mathfrak {z}}_{L}$, then the induced morphism ${\text {Spec} }({\widehat{O}}_{x})\rightarrow {\mathfrak {c}}_{[L,L]}$ has the property that the image of x is the origin. By part (1) of Lemma C.0.1 we conclude that L has semisimple rank 1 or 2. If it has semisimple rank two, then by part (2) of Lemma C.0.1 we conclude that whenever we have a lift of the morphism ${\text {Spec} }({\widehat{O}}_{x})\rightarrow {\mathfrak {c}}_{L}$ to ${\text {Spec} }({\widehat{O}}_{x})\xrightarrow {\varphi }{\mathfrak {l}}$, then $\varphi$ is a regular element. Since ${\widetilde{X}}$ is smooth away from x, combining this with Lemma B.0.3 we see that every Higgs bundle with cameral cover ${\widetilde{X}}$ is regular at x, hence we have ${\mathcal {H}}iggs_{G}({\widetilde{X}})={\mathcal {H}}iggs_{G}^{reg}({\widetilde{X}})$. If L has semisimple rank one, then ${\widetilde{X}}$ satisfies the conditions at the beginning of Appendix B. This finishes the proof. $\square$

Lemma C.0.3

Let G be a reductive algebraic group and ${\mathfrak {g}}$ be its lie algebra. Let ${\mathfrak {c}}_{G}$ be the Chevalley base. Then the closed subset of ${\mathfrak {c}}_{G}$ where the discriminant function ${\text {Disc} }_{G}$ has multiplicity greater than or equals to m has codimension greater than or equals to m. Here we recall that for a smooth variety Y, the multiplicity of an effective divisor D at a point $y\in Y$ is defined to be the unique integer k such that if ${\mathfrak {m}}_{y}$ is the maximal ideal of the local ring at y, then the local equation of D belongs ${\mathfrak {m}}^{k}_{y}{\setminus }{\mathfrak {m}}^{k+1}_{y}$

Proof

We will first prove that if G has semisimple rank less than m, then the multiplicity of the discriminant function at any point in ${\mathfrak {c}}_{G}$ is always less than m. It is enough to prove this when G is semisimple. Let us consider:

$$\begin{aligned} {\mathfrak {t}}\rightarrow {\mathfrak {c}}_{G}\xrightarrow {{\text {Disc} }_{G}}{\mathbb {A}}^{1} . \end{aligned}$$

The natural ${\mathbb {G}}_{m}$ action on ${\mathfrak {t}}$ induces ${\mathbb {G}}_{m}$ actions on ${\mathfrak {c}}_{G}$ and ${\mathbb {A}}^{1}$ such that ${\text {Disc} }_{G}$ is ${\mathbb {G}}_{m}$ equivariant. So multiplicity of ${\text {Disc} }_{G}$ stays constant on each ${\mathbb {G}}_{m}$ orbit. Consider the closed subset where ${\text {Disc} }$ has multiplicity greater than or equals to m. Since it is invariant under ${\mathbb {G}}_{m}$ action, if it is nonempty, it must contains $0\in {\mathfrak {c}}_{G}$. On the other hand, it is well-known that the multiplicity of ${\text {Disc} }_{G}$ at $0\in {\mathfrak {c}}_{G}$ is equal to the rank of G (See Section 3.18 of [10]), which is less than m by our assumption.

Now we shall consider the general case. We may assume the semisimple rank of G is greater than or equals to m by our discussions above. Suppose the multiplicity of ${\text {Disc} }$ is greater than or equals to m at $x\in {\mathfrak {c}}_{G}$ and let t be an element in ${\mathfrak {t}}$ that maps to x. Consider the levi subgroup $L=Z_{G}(t)$ and the morphisms:

$$\begin{aligned} {\mathfrak {t}}\rightarrow {\mathfrak {c}}_{L}\rightarrow {\mathfrak {c}}_{G}. \end{aligned}$$

Denote the image of t in ${\mathfrak {c}}_{L}$ by $x'$. Since $L=Z_{G}(t)$, we see that $x'\in {\mathfrak {c}}^{\circ }_{L}$ (See the notation in Lemma B.0.3). Moreover, it is not hard to see that ${\text {Disc} }_{G}/{\text {Disc} }_{L}$ is an invertible function on ${\mathfrak {c}}^{\circ }_{L}$. Hence the multiplicity of ${\text {Disc} }_{G}$ at x is equal to the multiplicity of ${\text {Disc} }_{L}$ at $x'$. Since we take x to be a point where ${\text {Disc} }_{G}$ has multiplicity greater than or equals to m, we see that the multiplicity of ${\text {Disc} }_{L}$ at $x'$ is also greater than or equals to m. Our discussions above implies that $L=Z_{G}(t)$ must have semisimple rank greater than or equals to m. All such elements t form a closed subset with codimension greater than or equals to m. This finishes the proof. $\square$