Picard group and fundamental group of the moduli of Higgs bundles on curves

Let $X$ be an irreducible smooth projective curve of genus $g \geq 2$ over $\mathbb{C}$. Let $G$ be a connected reductive affine algebraic group over $\mathbb{C}$. Let $\mathrm{M}_{G, {\rm Higgs}}^{\delta}$ be the moduli space of semistable principal $G$--Higgs bundles on $X$ of topological type $\delta \in \pi_1(G)$. In this article, we compute the fundamental group and Picard group of $\mathrm{M}_{G, {\rm Higgs}}^{\delta}$.


INTRODUCTION
Let X be an irreducible smooth projective curve of genus g ≥ 2 over C. Let G be a connected reductive affine algebraic group over C. The topological types of holomorphic principal G-bundles on X are parametrized by π 1 (G). Let M δ G be the moduli space of semistable holomorphic principal G-bundles on X of topological type δ ∈ π 1 (G). This is an irreducible normal complex projective variety (generally nonsmooth) of dimension (g − 1) · dim(G) + dim(Z(G)), where Z(G) is the center of the group G. Geometry of moduli spaces of bundles over projective curves is an important topic to study in algebraic geometry. Picard group of M δ G was studied in [KN] for simply connected semisimple complex affine algebraic groups. Later A. Beauville, Y. Laszlo and C. Sorger studied the Picard group of M δ G case by case for almost all classical semisimple complex affine algebraic groups (see [BLS]). The case of reductive groups was studied in [BH] for moduli stacks. The fundamental group of moduli space (and stack) of G-bundles was studied in [BMP]. The case of moduli of G-Higgs bundles was open. In this article we study the fundamental group and Picard group of the moduli spaces of semistable G-Higgs bundles over X.
Topological type of a holomorphic principal G-Higgs bundle on X is defined by the topological type of the underlying principal G-bundle on X. Let M δ G,Higgs be the moduli space of semistable holomorphic principal G-Higgs bundles on X of topological type δ ∈ π 1 (G). The space M δ G,Higgs is nonempty and connected, for all δ ∈ π 1 (G), [GO,Theorem 1.1,p. 791]. Let M δ G be the moduli space of semistable principal G-bundles on X of topological type δ ∈ π 1 (G). For any C-scheme Z, we denote by Pic(Z) (resp., π 1 (Z)) the Picard group (resp., fundamental group) of Z.
Therefore, it follows from [BMP,Theorem 1.1] that π 1 (M δ G,Higgs ) ∼ = Z 2gd , where d = dim Z(G) is the dimension of the center of G. In particular, M δ G,Higgs is simply connected, whenever G is connected and semisimple. Therefore, together with the results of [KN, BLS], the above theorem determines Picard group of M δ G,Higgs for essentially all classical semisimple complex affine algebraic groups.
We further remark that, our method of determining the fundamental group and Picard group generalize to the case of moduli stack of G-Higgs bundles over X.

PRELIMINARIES
2.1. Some results on locally trivial fibrations. Let Z be a smooth quasi-projective variety over C. We first recall some well-known results related to fundamental groups and Picard groups of vector bundles over Z. Let p : E −→ Z be a vector bundle of finite rank over Z.
Proof. Let n be the rank of the vector bundle E over Z. Note that the fibers of p are affine n-spaces A n C . We have a short exact sequence of groups where the second map is given by the restriction of a line bundle to the fiber A n C . Since Pic(A n C ) is trivial, p * is surjective. To show p * injective, let L ∈ Pic(Z) be such that p * L is a trivial bundle on E. Since E is Zariski locally trivial over Z, we must have L is trivial on Z.
Proof. Since the morphism p is locally trivial, we have an exact sequence of homotopy groups . Since the fibers A n C are connected and contractible, we conclude that p * is an isomorphism. Proof. The first isomorphism follows from [Ha,Proposition 1.6,p. 126], and for the second isomorphism, see [Mi,p. 42].

Principal G-Higgs bundles.
Let Y be an irreducible smooth projective variety over C. Let Ω 1 Y be the sheaf of differential 1-forms on Y . For any i ≥ 1, let Ω i Y = ∧ i Ω 1 Y . Let G be a connected reductive affine algebraic group over k. Let g = Lie(G) be the Lie algebra of G. The adjoint action of G on its Lie algebra g is denoted by ad : G −→ End(g) .
Let p : E G −→ Y be a principal G-bundle on Y . The right G-action on E G and the adjoint action of G on g produces a G-action on E G × g defined by Then the associated quotient θ) consisting of a principal G-bundle E G on Y and a Higgs field θ on E G .

Definition 2.4.
A principal G-Higgs bundle (E G , θ) on Y is said to be semistable (respectively, stable) if for any reduction σ : U −→ E G /P of the structure group of E G to a proper parabolic subgroup P ⊂ G over an open subset U ⊂ Y whose complement in Y has codimension at least 2, such that θ ∈ H 0 (X, ad(E where T rel is the relative tangent bundle of the projection E G /P −→ Y .
Note that, any principal G-bundle E G on Y is a principal G-Higgs bundle on Y with Higgs field θ = 0. Taking θ = 0 in the above definition, we get the definition of semistability and stability of principal G-bundles E G on Y .

FUNDAMENTAL GROUP AND PICARD GROUP OF M δ G,Higgs
Fix δ ∈ π 1 (G). Let M δ G be the moduli space of semistable holomorphic principal Gbundles on X of topological type δ. This is a normal complex projective variety (generally non-smooth) of dimension (g − 1) · dim(G) + dim(Z(G)). This contains a smooth open subvariety M s,δ G parametrizing stable principal G-bundles on X of topological type δ. Let M δ G,Higgs be the moduli space of semistable holomorphic principal G-Higgs bundles on X of topological type δ, and let M s,δ G,Higgs be the subvariety parametrizing the stable principal G-Higgs bundles on X of topological type δ. Note that M s,δ G,Higgs is a smooth quasi-projective variety over C.
( * ) Assume that either g = genus(X) ≥ 3 or there is no nontrivial homomorphism of G onto PGL(2, C). Proof. Let z ∈ M s,δ G (C) be a closed point of the moduli space represented by a stable principal G-bundle E G on X of topological type δ. Since G is reductive, there an isomorphism of G-modules g ≃ −→ g * . This gives an isomorphism of the adjoint vector bundle ad(E G ) with its dual ad(E G ) * . It follows from the deformation theory that the tangent space of M s,δ G at z is given by where the second isomorphism is given by Serre duality. So T * z (M s,δ G ) is isomorphic to the space of all Higgs fields on E G . Therefore, a closed point of T * (M s,δ G ) is given by a pair (E G , θ), for some E G ∈ M s,δ G (C) and θ ∈ H 0 (X, ad(E G ) ⊗ Ω 1 X ). This defines an open embedding φ : where T * (M s,δ G ) is the cotangent bundle of M s,δ G . It follows from [F,Theorem II.6 (iii), p. 534] that M s,δ G,Higgs \ φ(T * (M s,δ G )) has codimension ≥ 2 in M s,δ G,Higgs (here we are using the condition ( * )). Since p : T * M s,δ G −→ M s,δ G is a vector bundle over a smooth base M s,δ G , from Lemma 2.2 we have an isomorphism π 1 (M s,δ G ) ∼ = π 1 (T * M rs,δ G ) .  Proof. It follows from [KN,Proposition 3.4] and openness of stability that M s,δ G is a smooth open subscheme of M δ G , and the complement M δ G \ M s,δ G has codimension ≥ 2. Then from Lemma 2.3, we have isomorphisms π 1 (M s,δ G ) ∼ = π 1 (M δ G ) and Pic(M s,δ G ) ∼ = Pic(M δ G ) . (3.6) Similarly, M δ G,Higgs contains M s,δ G,Higgs as a smooth open subscheme such that the complement M δ G,Higgs \M s,δ G,Higgs has codimension ≥ 2 (see [F,Theorem II.6]). Then from Lemma 2.3, we have isomorphisms π 1 (M s,δ G,Higgs ) ∼ = π 1 (M δ G,Higgs ) and Pic(M s,δ G,Higgs ) ∼ = Pic(M δ G,Higgs ) . (3.7) Then the corollary follows from Theorem 3.2 and the above isomorphisms in (3.6) and (3.7).