Automorphisms of ${\mathbb C}^*$ Moduli Spaces Associated to a Riemann Surface

We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group ${\mathbb C}^*$ associated to a compact connected Riemann surface.


Introduction
Let X be a compact connected Riemann surface of genus g, where g ≥ 1. Let C * = C\{0} be the multiplicative group. We are interested in studying the automorphism groups of certain C * -moduli spaces associated to X, arising from non-abelian Hodge theory. Namely these are the de Rham, Betti and Dolbeault moduli spaces M C , M R , M H parametrizing holomorphic C * -connections, representations of the fundamental group into C * and degree zero Higgs line bundles respectively. While these three moduli spaces are all homeomorphic, their algebraic structures are quite different (M C and M H are not even biholomorphic) and we find that their automorphism groups are also quite different.
In [2], a classification was obtained of the analytic automorphism groups of the moduli space of SL(n, C)-Higgs bundles, i.e., the SL(n, C) Dolbeault moduli space. It remains an open question to determine which of the analytic automorphisms found in [2] are algebraic and also to determine the corresponding automorphism groups for the SL(n, C) de Rham and Betti moduli spaces (note that de Rham and Betti moduli spaces are analytically but not algebraically isomorphic). As mentioned above, the goal of this paper is to address this classification problem for the corresponding C * -moduli spaces. We leave the task of extending our results to noncommutative reductive groups as an interesting and challenging open problem.
Motivation for studying the automorphisms of these moduli spaces arises from mirror symmetry, the geometric Langlands program and their relation to physics, as promoted in the celebrated work of Kapustin and Witten [9]. Namely, one is interested in the construction of examples of naturally defined subvarieties of these moduli spaces, known as branes in the language of physics. One way of constructing such subvarieties which has proved fruitful is as the fixed point set of an automorphism of the moduli space, as seen in [3,4]. This has lead us to consider the problem of determining the automorphism groups of these moduli spaces in order to see how general our constructions are. In what follows we shall describe the structure and results of this paper. We begin this paper by studying in Section 2 the structure of the de Rham moduli space M C of holomorphic C * -connections on X up to gauge equivalence, i.e., pairs (L, D) where L is a holomorphic line bundle and D is a holomorphic connection on L. After recalling properties of the space, we give in Proposition 2.2 a gauge theoretic proof of the known result that every algebraic function on M C is constant.
The moduli space M C is a complex algebraic group with multiplication given by taking the tensor product of line bundles with connections, and thus M C acts on itself by translations giving an injective homomorphism where Aut(M C ) denote the group of algebraic automorphisms of M C . This map is considered in Section 3, where we show the following (see Theorem 3.1): ) is a countable group. In particular, the image of ρ is the connected component of Aut(M C ) containing the identity element.
Let J(X) be the Jacobian of X and let ρ 0 : J(X) −→ Aut(J(X)) be the homomorphism given by letting J(X) act on itself by translation. In Section 3, it is found that the quotient Aut(M C )/(ρ(M C )) can be identified with a subgroup of Aut(J(X))/ρ 0 (J(X)).
From non-abelian Hodge theory it is seen that the moduli space M C carries a naturally defined algebraic symplectic form [1,7]. Let θ ∈ H 2 (M C , C) denote the cohomology class of the symplectic form and let Aut θ (M C ) be the subgroup of Aut(M C ) preserving θ. In Section 3 we study this subgroup, and give its complete characterization in Theorem 3.2. For this, consider the homomorphism defined by sending h ∈ Aut(X) to the automorphism of M C given by (L, D) −→ (h * L, h * D). We show in Section 3.1 that ρ C is injective if g ≥ 2. Let G denote the subgroup of Aut(M C ) generated by ρ C (Aut(X)) together with the inversion (L, D) −→ (L ∨ , D ∨ ) of the group M C ; we denote the dual of a vector bundle, a vector space or a homomorphism by the superscript "∨". Using the actions of G and ρ C (Aut(X)) on M C , consider the semi-direct products (X)) and Through these groups we can characterize Aut θ (M C ) (see Theorem 3.2): As a Corollary, we deduce that any automorphism of M C preserving the cohomology class θ actually preserves the symplectic form and so the above theorem also gives the group of algebraic symplectomorphisms of M C .
In Section 4 we consider the Betti moduli space M R of representations of π 1 (X) into the multiplicative group C * (following [12]). The space M R = Hom(π 1 (X), C * ), which is isomorphic to (C * ) 2g . The group Γ of automorphisms of the Z-module H 1 (X, Z) is isomorphic to GL(2g, Z), and thus there is a natural map Automorphisms of C * Moduli Spaces Associated to a Riemann Surface 3 that sends an automorphism of M R to its induced action on H 1 (X, Z). In Section 4, we show that f admits a right-splitting so that Aut(M R ) = kernel(f ) Γ. Moreover, since the kernel of f is given by the natural action of M R = (C * ) 2g on itself by translations, we obtain that (see Theorem 4.2): As with the de Rham moduli space, non-abelian Hodge theory determines a natural symplectic form on M R . We find that the subgroup of Aut(M R ) preserving this form is given by M R Γ Sp , where Γ Sp is the subgroup of Γ preserving the cap product on H 1 (X, Z), so Γ Sp is isomorphic to the symplectic group Sp(2g, Z).
Finally, in Section 5 we study the Dolbeault moduli space M H of degree zero Higgs line bundles, that is pairs (L, Φ), where L is a degree zero line bundle on X and Φ is a holomorphic 1-form on X. This moduli space is the holomorphic cotangent bundle T ∨ J(X) of the Jacobian J(X). Considering the isomorphism T ∨ J(X) = J(X) × H 0 (X, K X ), where K X is the holomorphic cotangent bundle of X we obtain that (see Lemma 5.1): where f 1 ∈ Aut(J(X)) and f 2 ∈ Aut(H 0 (X, K X )).
2 Structure of the moduli space of C * -connections Let X be a compact connected Riemann surface of genus g ≥ 1, and K X its holomorphic cotangent bundle. The Jacobian of X, which parametrizes all the isomorphism classes of holomorphic line bundles on X of degree zero, is denoted by J(X). Let M C be the moduli space of holomorphic connections on X of rank one. Therefore, M C parametrizes the isomorphism classes of pairs of the form (L, D), where L is a holomorphic line bundle on X and D is a holomorphic connection on L. Since there are no nonzero (2, 0)-forms on X, any holomorphic connection on X is automatically integrable.
The adjoint action of the algebraic group C * on its Lie algebra Lie(C * ) = C is trivial. Consequently, for any (L, D) ∈ M C , the holomorphic tangent bundle to M C at the point (L, D) is (2.1) Therefore, the real tangent bundle T R (L,D) M C is identified with H 1 (X, C), and the almost com- Since any holomorphic connection on X is flat, the degree of any holomorphic line bundle admitting a holomorphic connection is zero. Therefore, we have an algebraic morphism This map ϕ is surjective because any holomorphic line bundle L on X of degree zero admits a holomorphic connection. More precisely, the space of all holomorphic connections on L is an affine space for the vector space H 0 (X, K X ). Therefore, ϕ makes M C an algebraic principal H 0 (X, K X )-bundle over J(X).
Let V denote the trivial holomorphic vector bundle J(X) × H 0 (X, K X ) over J(X) with fiber H 0 (X, K X ). The isomorphism classes of algebraic principal H 0 (X, K X )-bundles over J(X) are parametrized by H 1 (J(X), V). We will calculate the cohomology class corresponding to M C . Note that ϕ does not admit any holomorphic section because J(X) is compact and M C is biholomorphic to (C * ) 2g thus ruling out the existence of any nonconstant holomorphic map from J(X) to M C . Consequently, the class in H 1 (J(X), V) corresponding to M C is nonzero.
We will briefly describe the Dolbeault type construction of cohomological invariants for principal H 0 (X, K X )-bundles.
Take an algebraic principal H 0 (X, for q; such a section exists because the fibers of the projection q are contractible. If s is holomorphic, then the holomorphic principal H 0 (X, K X )-bundle E is trivial. The invariant for E is a measure of the failure of s to be holomorphic. To explain this, let J 1 and J 2 denote the almost complex structures on J(X) and E respectively. Let ds : T R J(X) −→ T R E be the differential of the map s. For any x ∈ J(X) and y ∈ E x , consider the homomorphism • the map q is holomorphic, it follows that the tangent vector ds(J 1 (v)) − J 2 (ds(v)) in (2.3) is vertical for q. Using the action of the group H 0 (X, K X ) on E, the vertical tangent bundle for q is the trivial vector bundle with fiber H 0 (X, K X ). Consequently, the homomorphism in (2.3) defines a section Then, the Dolbeault cohomological class The Lie algebra Lie(J(X)) of J(X) is the abelian algebra H 1 (X, O X ). The Serre duality theorem says that H 1 (X, O X ) = H 0 (X, K X ) ∨ . Therefore, the vector bundle V is identified with the holomorphic cotangent bundle Ω J(X) . Consequently, we have Automorphisms of C * Moduli Spaces Associated to a Riemann Surface 5 Hence the isomorphism classes of holomorphic principal H 0 (X, K X )-bundles on J(X) are parametrized by H 1 (J(X), Ω J(X) ). We note that every element of H 1 (J(X), Ω J(X) ) is the invariant (2.4) for some holomorphic principal H 0 (X, K X )-bundles on J(X).
Let M R := Hom(π 1 (X, x 0 ), C * ) = Hom(H 1 (X, Z), C * ) be the space of 1-dimensional representations. Sending a flat connection to its monodromy representation, we get a holomorphic isomorphism We have Hom (H 1 (X, Z), U(1)) → M R using the inclusion of U(1) = S 1 in C * . From Hodge theory it follows that every L ∈ J(X) admits a unique holomorphic connection such that the monodromy lies in U(1), and thus the composition is a diffeomorphism, where ϕ is constructed in (2.2). We note that the above composition ϕ•f −1 is a diffeomorphism because it is bijective and homomorphism of groups. We shall denote by the C ∞ section of ϕ given by the inverse of the composition in (2.5). Given any L ∈ J(X), we consider ∇ = ∇ 1,0 + ∇ 0,1 the unique unitary flat connection on L such that (0, 1)-type component ∇ 0,1 is the Dolbeault operator on L. The real tangent space T R ξ(L) M C is H 1 (X, C), and the almost complex structure on T R ξ(L) M C coincides with the multiplication by √ −1 on H 1 (X, C) (see (2.1) and the sentence following it). Therefore, the holomorphic tangent space to M C is identified with H 1 (X, C). The inclusion of the Lie group U(1) → C * , identifies the Lie algebra Lie(U(1)) with the subspace √ −1R ⊂ Lie(C * ) = C.
Therefore, the subspace coincides with H 1 (X, √ −1R) equipped with its natural inclusion The anti-holomorphic tangent space T 0,1 L J(X) is identified with H 0 (X, K X ) by sending any α ∈ H 0 (X, K X ) to the flat unitary connection From the above, the complex structure on T 0,1 L J(X) coincides with multiplication by √ −1 on H 0 (X, K X ). If we identify T 0,1 L J(X) with T R L J(X) by sending any (0, 1)-tangent vector to its real part, then the isomorphism given by the differential of the composition map in (2.5) sends any α ∈ H 0 (X, K X ) to the element in The cup product produces a 2-form ω on J(X). The form ω is closed because the translation action of J(X) on itself preserves ω, and any translation invariant form on a torus is closed. In fact, ω is a Kähler form on J(X). We shall let ω ∈ H 1 (J(X), Ω J(X) ) (2.7) be the Dolbeault cohomology class represented by ω. From the above, the anti-holomorphic tangent space T 0,1 L J(X) is identified with a subspace of H 1 (X, C) which in turn gives the holomorphic tangent space to M C . Hence, consider the almost complex structures obtained for J(X) and M C , combined with the above description of the differential of ξ, one has that the class in H 1 (J(X), V) corresponding to the principal H 0 (X, K X )-bundle M C coincides with ω in (2.7). Let be the extension of O J(X) by Ω J(X) associated to the extension class ω in (2.7). The section of O J(X) given by the constant function 1 will be denoted by 1 J(X) . We note that for the projection σ in (2.8), the inverse image σ −1 (1 J(X) (J(X))) ⊂ E is a principal H 0 (X, K X )-bundle on J(X) (recall that the dual vector space Lie(J(X)) ∨ is identified with H 0 (X, K X )). Since the class in H 1 (J(X), Ω J(X) ) corresponding to the principal H 0 (X, K X )-bundle M C coincides with ω, we have the following: Lemma 2.1. The variety M C is algebraically isomorphic to the inverse image σ −1 (1 J(X) (J(X))).
Through the above lemma, we can recover the following result, which from a different perspective can be deduced since the universal vector extension of the Jacobian parametrizes line bundles with connections [10, Chapter 1], and the universal vector extension of any abelian variety is anti-affine [5, Proposition 2.3(i)]. Proof . In view of Lemma 2.1 it suffices to show that the variety σ −1 (1 J(X) (J(X))) does not admit any nonconstant algebraic function. We will first express σ −1 (1 J(X) (J(X))) as a hyperplane complement Y in a projective bundle over J(X) in Step 1. Then in Step 2 we shall study associated bundles, which in turn allow us to study H 0 (Y, O Y ) in Step 3. From the description of the cohomology group that we obtain, we see that Y does not admit any nonconstant algebraic function if and only if certain natural inclusion is surjective. Hence, in Step 4 we study this inclusion, by taking the dual exact sequence to (2.8). Surjectivity of the inclusion can be then seen equivalent to injectivity of an associated map β. We conclude the proof of the proposition by showing in Step 5 that this map is indeed injective.
Automorphisms of C * Moduli Spaces Associated to a Riemann Surface 7 The divisor ι(P (Ω J(X) )) ⊂ P (E) will be denoted by D. We have by sending any v ∈ σ −1 (1 J(X) (z)) and z ∈ J(X), to the line in the fiber E z generated by v.
Step 2. Consider now the natural projection For L −→ P (E) the dual of the tautological line bundle, the fiber of L over any y ∈ P (E) is the dual of the line in E p(y) represented by y.
Note that for any point z ∈ J(X), the two line bundles L| p −1 (z) and O P (E) (D)| p −1 (z) on p −1 (z) are isomorphic. Therefore, from the seesaw theorem (see [11,p. 51,Corollary 6]) it follows that there is a holomorphic line bundle L 0 on J(X) such that (2.10) By the adjunction formula [8, p. 146], the restriction of O P (E) (D) to D is the normal bundle N D to the divisor D ⊂ P (E). This normal bundle N D is identified with is the restriction of p. Now, since the quotient E/Ω J(X) is the trivial line bundle (see (2.8)), it follows that N D is isomorphic to L| D . Consequently from (2.10) it follows that the line bundle L 0 is trivial. This in turn implies that Step 3. To calculate H 0 (Y, O Y ), note that (see (2.9) and (2.11)). Since D is an effective divisor, from (2.9) and (2.12) we conclude that σ −1 (1 J(X) (J(X))) does not admit any nonconstant algebraic function if and only if the natural inclusion is surjective for all i ≥ 0. Note that Step 4. To prove that the homomorphism in (2.13) is indeed surjective, consider the dual of the exact sequence in (2.8): Taking its (i + 1)-th symmetric power, we have where Sym i+1 (ι ∨ ) is the homomorphism of symmetric products induced by the homomorphism ι ∨ ; the above homomorphism σ is the symmetrization of the homomorphism be the connecting homomorphism in the long exact sequence of cohomologies associated to the short exact sequence in (2.14). Consider the homomorphisms where γ is induced by the homomorphism Sym i (ι ∨ ) (see (2.14)). From the long exact sequence of cohomologies for (2.14) it follows immediately that the homomorphism in (2.13) is surjective if β in (2.15) is injective. To prove that β is injective, it is enough to show that the composition γ • β in (2.15) is injective.
Step 5. Since the extension class for (2.8) is the cohomology class ω, the extension class for (2.14) is −(i + 1) ω. Consequently, the homomorphism γ • β sends any to the Dolbeault cohomology class of the contraction of ω ⊗ η of Ω 1,0 J(X) and T (X); note that the tensor product ω ⊗ η is a section of Ω 0,1 J(X) ⊗ Ω 1,0 J(X) ⊗ Sym i+1 (T J(X)) and hence its contraction ω ⊗ η is a section of Ω 0,1 J(X) ⊗ Sym i (T J(X))). Since both ω and η are invariant under translations of J(X), it follows that ω ⊗ η is also invariant under translations of J(X), and hence represents a nonzero cohomology class. The section ω ⊗ η is nonzero because ω is pointwise nondegenerate (recall that it is a Kähler form). Therefore, we conclude that the homomorphism γ • β is injective. Hence the homomorphism in (2.13) is surjective, and the proof is complete.

Automorphisms of the moduli of C * -connections
The group of algebraic automorphisms of the variety M C will be denoted by Aut(M C ). The moduli space M C is an algebraic group, with group operation The algebraic map ϕ in (2.2) is a homomorphism of algebraic groups.
The translation action of M C on itself produces an injective homomorphism Proof . We will show that any automorphism of M C descends to J(X). For that, first note that there is no nonconstant algebraic map from C to an abelian variety. Indeed, such a map would extend to a nonconstant algebraic map from CP 1 , and therefore some holomorphic 1-form on the abelian variety would pull back to a nonzero holomorphic 1-form on CP 1 , but CP 1 does not have any nonzero holomorphic 1-form. Since there is no nonconstant algebraic map from C to J(X), there is no nonconstant algebraic map from a fiber of ϕ (see (2.2)) to the variety J(X), because the fibers of ϕ are isomorphic to C g . This immediately implies that any automorphism of M C descends to an automorphism of J(X).
We shall denote by the homomorphism given by the translation action of J(X) on itself.
The first statement follows from the fact that H 0 (J(X), T J(X)) = Lie(J(X)). In what follows we will prove the second statement.

Automorphisms preserving cohomology class
As mentioned previously, the moduli space M C is equipped with an algebraic symplectic form (see [1,7]). The cohomology class in H 2 (M C , C) defined by the symplectic form will be denoted by θ. The pullback of the symplectic form on M C by the section ξ in (2.6) coincides with the Kähler form on J(X). Therefore, the cohomology class θ on M C coincides with the pullback ϕ * ω of the Kähler class on J(X) (see (2.2) and (2.7)). Let Aut θ (M C ) denote the group of all τ ∈ Aut(M C ) such that τ * θ = θ. Our aim in this subsection is to compute Aut θ (M C ). The group of all holomorphic automorphisms of X will be denoted by Aut(X). Let Aut 0 (X) ⊂ Aut(X) be the connected component containing the identity element. If g ≥ 2, then we have Aut 0 (X)=e. Let be the homomorphism that sends any h ∈ Aut(X) to the automorphism of M C defined by (L, D) −→ (h * L, h * D). If g ≥ 2, then ρ C is injective. Indeed, the homomorphism Aut(X) −→ Aut(J(X)) that sends any h ∈ Aut(X) to the automorphism L −→ h * L is injective if g ≥ 2.
If g = 1 then X is an elliptic curve and Aut 0 (X) = X, acting on itself by translations. If τ : X → X is any such translation then for any line bundle with holomorphic connection (L, D), we have (τ * L, τ * D) ∼ = (L, D) since the corresponding flat connections have the same monodromy. Therefore the homomorphism ρ C | Aut 0 (X) is trivial, and ρ C produces an embedding of Aut(X)/ Aut 0 (X) in Aut(J(X)). Let G denote the subgroup of Aut(M C ) generated by ρ C (Aut(X)) together with the inversion (L, D) −→ (L ∨ , D ∨ ) of the group M C . Using the actions of G and ρ C (Aut(X)) on M C , we construct the semi-direct products G 0 := M C ρ C (Aut(X)) and G := M C G.
Note that using the action of ρ C (Aut(X)) (respectively, G) and the translation action of M C on itself, the group G 0 (respectively, G) acts on M C . 2) Aut θ (M C ) = G 0 if X is hyperelliptic.
Proof . As mentioned before, we have θ = ϕ * ω. From this it follows that for any element of G, the corresponding automorphism of M C preserves θ. Let Aut ω (J(X)) be the group of all automorphisms of the variety J(X) that preserve the cohomology class ω. From [13, Hauptsatz, p. 35] one has the following: 1. Assume that X is not hyperelliptic. Then Aut ω (J(X)) is generated by translations of J(X), Aut(X) and the inversion L −→ L ∨ of J(X).

2.
Assume that X is hyperelliptic. Then Aut ω (J(X)) is generated by translations of J(X) and Aut(X). (The hyperelliptic involution of X induces the inversion of J(X).) Consider the homomorphism δ in (3.1). In the proof of Theorem 3.1 it was shown that the inclusion ρ(kernel(ϕ)) → kernel(δ) (3.2) is surjective. First assume that X is not hyperelliptic. Using ϕ in (2.2), we get a homomorphism G −→ Aut(J(X)).
From the above result of [13] we know that this homomorphism is injective, its image is a normal subgroup of Aut(J(X)) and the composition G −→ Aut(J(X)) −→ Aut(J(X))/J(X) is an isomorphism. Therefore, from the surjectivity of the homomorphism in (3.2) we conclude that Aut θ (M C ) = G. If X is hyperelliptic, then Aut(J(X))/ Aut(X) = J(X) by the above theorem of [13]. Therefore, by the above argument it follows that Aut θ (M C ) = G 0 .
From the definitions of G 0 and G, it is straightforward to verify that these groups preserve the algebraic symplectic form on M C . Therefore, Theorem 3.2 gives the following: