On the Automorphisms of a Rank One Deligne-Hitchin Moduli Space

Let $X$ be a compact connected Riemann surface of genus $g \geq 2$, and let ${\mathcal M}_{\rm DH}$ be the rank one Deligne-Hitchin moduli space associated to $X$. It is known that ${\mathcal M}_{\rm DH}$ is the twistor space for the hyper-K\"ahler structure on the moduli space of rank one holomorphic connections on $X$. We investigate the group $\operatorname{Aut}({\mathcal M}_{\rm DH})$ of all holomorphic automorphisms of ${\mathcal M}_{\rm DH}$. The connected component of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ containing the identity automorphism is computed. There is a natural element of $H^2({\mathcal M}_{\rm DH}, {\mathbb Z})$. We also compute the subgroup of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ that fixes this second cohomology class. Since ${\mathcal M}_{\rm DH}$ admits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that ${\mathcal M}_{\rm DH}$ is Moishezon.


Introduction
The moduli spaces of Higgs bundles on a compact Riemann surface arise in various contexts and are extensively studied. One of the reasons for their usefulness is the nonabelian Hodge correspondence which identifies the moduli space of semi-stable Higgs bundles of rank r and degree zero on a compact Riemann surface X with the character variety Hom(π 1 (X), GL(r, C))/ /PGL(r, C) (see [4,5,8,10]). This identification is a C ∞ diffeomorphism between the open dense subset consisting of stable Higgs bundles and the open dense subset consisting of irreducible representations. However, this diffeomorphism is not holomorphic. In other words, there are two different complex structures on a moduli space of Higgs bundles, one from being the moduli space of Higgs bundles and the other given by the complex structure of the character variety via the above mentioned identification. In fact, these two complex structures are a part of a natural hyper-Kähler structure on a moduli spaces of Higgs bundles over X [8]. It was noticed by Deligne and Hitchin that the twistor space associated to this hyper-Kähler manifold also has an interpretation as a moduli space associated to X. These twistor spaces are known as the Deligne-Hitchin moduli space; see [11,12] for many properties of these spaces.
In a recent interesting paper [2], Baraglia computed the various automorphism groups associated to a moduli space of Higgs bundles on X (like holomorphic automorphisms preserving the holomorphic structure, holomorphic isometries, hypercomplex automorphisms, hyper-Kähler arXiv:1704.04924v3 [math.AG] 6 Sep 2017 automorphisms et cetera). In [2], the structure group is SL(r, C), hence the moduli space parametrizes Higgs bundles of fixed determinant with the trace of the Higgs field being zero. Inspired by [2], in [3] the case of C * -bundles was investigated, where C * is the multiplicative group of nonzero complex numbers. More precisely, the automorphism groups of the moduli spaces of rank one Higgs bundles, rank one holomorphic connections and C * -character varieties were studied.
Our main aim here is to investigate the automorphisms of the rank one Deligne-Hitchin moduli space. The Deligne-Hitchin moduli spaces are not algebraic varieties; they are complex manifolds. However, they are built out of Hodge moduli spaces which are complex quasiprojective varieties. The rank one Hodge moduli space M Hod = M X Hod (1) parametrizes all rank one λ-connections on a compact Riemann surface X with λ running over C. The rank one Deligne-Hitchin moduli space M DH = M DH (X) is constructed by gluing M X Hod (1) and M X Hod (1) over the inverse image of C * ⊂ C, where X is the Riemann surface conjugate to X.
It is natural to ask for the automorphism group of the rank one Deligne-Hitchin moduli space as it is the only case missing for rank one, see [3]. In general, automorphisms of moduli spaces associated to a Riemann surface play an important role in mirror symmetry and help to improve our understanding of these objects. The rank one case serves as a motivating and more simple example, where we can gain intuition for what should be expected in general. On the other hand, this task is also interesting for itself as we cannot use an algebraic structure of the moduli space, and algebraicity was at the heart of the proofs in [3]. As a nice additional observation we would like to mention that the automorphisms of the Deligne-Hitchin moduli space are algebraic when restricted to some of its subspaces, i.e., to the Dolbeault, the de Rham and the Hodge moduli space. Finally, and most importantly, we hope that our techniques can be generalized to study automorphism groups of the Deligne-Hitchin moduli space for higher rank. This is of particular interest as certain sections thereof correspond to solutions to important geometric PDE's such as harmonic maps including Hitchin's self-duality equations, and certain automorphisms might give rise to transformations between some classes of PDE's.
We now describe the results proved here. Let Aut(M Hod ) be the group of all algebraic automorphisms of M Hod , and let Aut(M Hod ) 0 be the connected component of it containing the identity automorphism. The holomorphic cotangent bundle of X will be denoted by K X . Theorem 1.1. The group Aut(M Hod ) 0 fits in a short exact sequence of groups where V is the space of all algebraic maps from C to the affine space H 0 (X, K X ).
Define Γ X := Aut(X) if X is hyperelliptic, and Γ X = Aut(X) ⊕ (Z/2Z) if X is nonhyperelliptic. This group Γ X has a natural action on M Hod via holomorphic automorphisms; see Section 2.2. Let Γ X := Aut(M Hod ) 0 Γ X be the corresponding semi-direct product (see (2.13)). There is a natural cohomology class θ ∈ H 2 (M Hod , Z), and the action of Γ X on M Hod fixes θ. Proposition 1.2. Assume that genus(X) > 1. The group Γ X coincides with the subgroup of Aut(M Hod ) that fixes θ.
Let M DH = M DH (X) be the rank one Deligne-Hitchin moduli space for a compact Riemann surface X. Let Aut(M DH ) be the group of all holomorphic automorphisms of M DH , and let Aut(M DH ) 0 ⊂ Aut(M DH ) be the connected component of it containing the identity map. The moduli space of rank one connections on X will be denoted by M dR (X); it is an algebraic group.

Theorem 1.3. There is an isomorphism
The above group Γ X acts on M DH . Let Γ DH,X := Aut(M DH ) 0 Γ X be the corresponding semi-direct product. There is a natural cohomology class and the action of Γ DH,X on M DH fixes θ X .
Theorem 1.4. Assume that genus(X) > 1. The group Γ DH,X coincides with the subgroup of Aut(M DH ) that fixes θ X .
Although M DH is noncompact, being a twistor space it contains ample rational curves. Verbitsky has shown that in such a context the notion of algebraic dimension continues to hold [13].
We prove the following: 2 Automorphisms of the rank one Hodge moduli space

Connected component of the automorphism group
Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 1. The holomorphic cotangent bundle of X will be denoted by K X .
Definition 2.1. For any λ ∈ C, a rank one λ-connection on X is a pair of the form (L, D), where L is a holomorphic line bundle on X and D : L −→ L ⊗ K X is a holomorphic differential operator of order no more than one such that for every locally defined holomorphic section s of L and every locally defined holomorphic function f 0 on X.
So, if (L, D) is a λ-connection with λ = 0, then D/λ is a holomorphic connection on L; a 0-connection D on L is an element of H 0 (X, K X ) because such a D is O X -linear. Note that for a λ-connection (L, D), with λ = 0 we have degree(L) = 0 because L admits a holomorphic connection. Also, note that any holomorphic connection on a Riemann surface is automatically flat.
If (L, D) is a 0-connection we will always impose the condition that degree(L) = 0. Let be the Hodge moduli space of rank one λ-connections. So, the points of M Hod parametrize triples of the form (λ, L, D), where λ ∈ C, L is a holomorphic line bundle on X of degree zero and D is a λ-connection on L.
Define an algebraic morphism For any λ ∈ C, let be the fiber over λ. By definition, the fiber f 0 is the moduli space of Higgs line bundles of degree zero Pic 0 (X) × H 0 (X, K X ) on X, and the fiber f 1 is the moduli space of rank one holomorphic connections on X. The latter space is also called the de Rham moduli space. For λ = 0, the fiber f λ is canonically identified with f 1 by the map (L, D) −→ (L, D/λ).
We note that f 1 is biholomorphic to the Betti moduli space Hom(π 1 (X), C * ) = (C * ) 2g by sending a holomorphic connection to its monodromy representation, and hence f 1 does not admit a compact submanifold of positive dimension. Since Pic 0 (X) × H 0 (X, K X ) has the projective variety Pic 0 (X) of positive dimension as a subvariety, it follows that f 0 is not biholomorphic to f 1 . The group of all algebraic automorphisms of the quasiprojective variety M Hod will be denoted by Aut(M Hod ). Let Aut(M Hod ) 0 ⊂ Aut(M Hod ) be the connected component containing the identity automorphism and let V := Morphisms C, H 0 (X, K X ) be the infinite dimensional complex vector space parametrizing all algebraic maps from C to the affine space H 0 (X, K X ). Then we have the following theorem.

Proof . Let
T : M Hod −→ M Hod be any algebraic automorphism that lies in Aut(M Hod ) 0 . It is known that the fiber f 1 does not admit any nonconstant algebraic function [3, Proposition 2.2]. Since f λ is isomorphic to f 1 for every λ ∈ C * , it follows that f λ does not admit any nonconstant algebraic function if λ ∈ C * . Hence the composition is a constant function for all λ ∈ C * . Note that the point f • T (f λ ) lies in C * because f 0 is not isomorphic to f λ . Since this also holds for T −1 , it follows that the composition in (2.2) is a constant function for each λ ∈ C. This implies that there is an algebraic automorphism Since f 0 is not isomorphic to any f λ , λ ∈ C * , it follows that τ 0 (0) = 0. Therefore, τ 0 coincides with the multiplication of C by a fixed nonzero number; let The group of all automorphisms of the variety Pic 0 (X) will be denoted by Aut(Pic 0 (X)) (since the variety Pic 0 (X) is projective, any holomorphic automorphism of it is algebraic). The connected component of Aut(Pic 0 (X)) containing the identity automorphism will be denoted by Aut(Pic 0 (X)) 0 . The automorphisms of Pic 0 (X) given by translations of the group Pic 0 (X) lie in Aut(Pic 0 (X)) 0 . In fact, this way Pic 0 (X) gets identified with Aut(Pic 0 (X)) 0 ; note that the Lie algebra H 0 (Pic 0 (X), T Pic 0 (X)) of Aut(Pic 0 (X)) 0 coincides with the Lie algebra of Pic 0 (X) by evaluating sections of T Pic 0 (X) at the identity element.
For any λ ∈ C, let φ λ be the morphism defined by The fibers of φ λ are isomorphic to H 0 (X, K X ), because the space of all holomorphic connections on a line bundle L ∈ Pic 0 (X) is an affine space for the vector space H 0 (X, K X ), and H 0 (X, K X ) is the space of all Higgs fields on any line bundle. There is no nonconstant algebraic map from an affine space to an abelian variety. Hence, there is no nonconstant algebraic map from a fiber of φ λ to Pic 0 (X). Consequently, we get a map which is uniquely determined by the condition that the following diagram is commutative for all λ ∈ C, where τ is the complex number in (2.3). Note that since T ∈ Aut(M Hod ) 0 , it follows that the image of Φ lies in Aut Pic 0 (X) 0 = Pic 0 (X) ⊂ Aut Pic 0 (X) . Thus, Φ is a constant map, again because there is no nonconstant algebraic map from C to Pic 0 (X). Let It is straight-forward to check that h is a group homomorphism.
We will now show that h is surjective. For this, first note that the multiplicative action of C * on C has a natural lift to an action of C * on M Hod . Indeed, any c ∈ C * acts on M Hod as For all these automorphisms of M Hod given by the action of C * , the corresponding elements of Pic 0 (X) (see (2.5)) coincide with the identity element. Therefore, to prove that h is surjective, it suffices to show that the composition of h with the projection Pic 0 (X) × C * −→ Pic 0 (X) is surjective. Take any L 0 ∈ Pic 0 (X) and fix a holomorphic connection D 0 on L 0 . Define It is straight-forward to check that β ∈ Aut(M Hod ); its inverse is the corresponding map for (L * 0 , D * 0 ). Since the moduli space of rank one connections on X is connected, it follows that β ∈ Aut(M Hod ) 0 ; note that β for the trivial connection (O X , d) is the identity map of M Hod . The element in Pic 0 (X) corresponding to β (see (2.5)) is clearly L 0 . Therefore, the composition of h with the projection Pic 0 (X) × C * −→ Pic 0 (X) and hence h itself are surjective. Next Clearly, we have ι v ∈ Aut(M Hod ) 0 , and also ι v ∈ kernel(h). Therefore, there is an injective homomorphism We will prove that image(ι) = kernel(h).
In order to prove that image(ι) = kernel(h), take any T ∈ Aut(M Hod ) 0 such that We have T (f λ ) = f λ for all λ ∈ C * because the constant in (2.3) for T is 1. Since the element in (2.5) corresponding to T is the trivial line bundle, we get a morphism note that since the fibers of φ λ (constructed in (2.4)) are affine spaces for H 0 (X, K X ), and As there are no nonconstant algebraic functions on f λ [3, Proposition 2.2], it follows that the above function δ λ is a constant one. All automorphism of the affine space H 0 (X, K X ) are of the form u −→ A(u) + u 0 , where A ∈ GL(H 0 (X, K X )) and u 0 ∈ H 0 (X, K X ). Considering the restrictions of T to open subsets of the form h −1 (V × U ), where U is an analytic neighborhood of 0 ∈ C and V is an analytic open subset of Pic 0 (X), it now follows that the above map extends across 0 ∈ C as a holomorphic map from C to H 0 (X, K X ). In other words, there is a unique element v ∈ V such that v(λ) = image(δ λ ) for all λ ∈ C * . Clearly, we have ι v = T , where ι v is constructed in (2.6). This implies that image(ι) = kernel(h). This completes the proof of the theorem.

Automorphisms preserving a cohomology class
In this subsection we assume that g = genus(X) > 1.
Since any holomorphic line bundle of degree zero on the compact Riemann surface X admits a unique flat connection with unitary monodromy, the Picard group Pic 0 (X) is identified with There is a natural symplectic form on the character variety Hom(π 1 (X), U(1)) [1,6]. The cohomology class defined by θ is integral. Let be the cohomology class given by θ in (2.7). This θ 0 is a principal polarization on Pic 0 (X). More precisely, it is the class of a theta divisor on Pic 0 (X). Consider the projection Define the cohomology class The moduli space f 1 of holomorphic rank one connections (see (2.2)) has a natural holomorphic symplectic form θ h . The restriction of θ h to the moduli space of holomorphic connections Hom(π 1 (X), U(1)) with unitary monodromy coincides with the above symplectic form θ. On the other hand f 1 has a deformation retraction onto Hom(π 1 (X), U(1)); for example such a deformation retraction is given by a deformation retraction of C * to U(1). Therefore, we conclude that the cohomology class of θ h coincides with the restriction of θ to f 1 . Let be the subgroup consisting of all algebraic automorphisms of M Hod that fixes the cohomology class θ in (2.9). Let Aut(X) denote the group of all holomorphic automorphisms of X. We have a homomorphism We note that the restriction µ| Aut(X) is injective (recall that g ≥ 2), and µ is injective if and only if X is non-hyperelliptic. If X is hyperelliptic, then the hyperelliptic involution acts on Pic 0 (X) as L −→ L * . Define We also have a homomorphism defined by µ(t, 0)(λ, L, D) = (λ, t * L, t * D) and µ(t, 1)(λ, L, D) = (λ, t * L * , t * D * ). Note that if λ = 0, then D * = −D ∈ H 0 (X, K X ). The homomorphism µ in (2.12) factors through the quotient Γ X of Aut(X) ⊕ (Z/2Z) in (2.11). Consider the action of Γ X on Aut(M Hod ) given by the composition of µ with the adjoint action of Aut(M Hod ) on itself. This action preserves the connected component Aut(M Hod ) 0 . Define the semi-direct product for the above action of Γ X on Aut(M Hod ) 0 . Consider the action of Γ X on M Hod given by µ in (2.12). This action clearly fixes the cohomology class θ in (2.9). The action of Aut(M Hod ) 0 on M Hod fixes θ because Aut(M Hod ) 0 is connected. Also, Aut(M Hod ) 0 is a normal subgroup of Aut(M Hod ). Therefore, we get a homomorphism where Aut θ (M Hod ) and Γ X are constructed in (2.10) and (2.13) respectively.
Proof . We will first prove that ρ is injective. For this note that φ in (2.8) induces an isomorphism of first cohomologies with coefficients in C. Indeed, the fibers of φ are diffeomorphic to Let γ ∈ Γ X be the image of γ by the natural projection of Γ X to Γ X . Let γ ∈ Γ X be the image of γ by the natural inclusion Γ X → Γ X . So, γ and γ differ by an element of Aut(M Hod ) 0 . Since γ acts trivially on H 1 (M Hod , C) (as it acts trivially on M Hod ), and Aut(M Hod ) 0 acts trivially on H 1 (M Hod , C) as it is connected, we conclude that γ acts trivially on H 1 (M Hod , C). This implies that γ acts trivially on H 1 (Pic 0 (X), C) = H 1 (M Hod , C). But, as noted earlier, this implies that γ = 1. Hence, γ ∈ Aut(M Hod ) 0 . Now we conclude that γ = 1 because ρ(γ) is the trivial automorphism of M Hod . Hence ρ is injective. Let T ∈ Aut θ (M Hod ). The image for some ω ∈ H 0 (X; K X ), as can be deduced analogously to the proof of Theorem 2.2. By applying a suitable automorphism T 0 ∈ Aut(M Hod ) 0 we get The group of automorphisms of the abelian variety Pic 0 (X) that fix the principal polarization θ 0 is generated by Γ X and the translations of Pic 0 (X) (see [14,p. 35,Hauptsatz]). Because of the discussion in the beginning of Section 2.2 it follows that we can apply an automorphism T 1 ∈ Γ X such that is the identity whence restricted to {0}×Pic 0 (X)×{0}. Clearly, this implies first that T 1 •T 0 •T restricted to f 0 is homotopic to the identity and finally that T 1 • T 0 • T is homotopic to the identity, which finishes the proof.
3 Holomorphic automorphisms of the Deligne-Hitchin moduli space

Connected component of the holomorphic automorphism group
The smooth locus of the moduli space of flat connections on a compact Riemann surface is equipped with a natural hyper-Kähler structure; see [8] for the case of SL(2, C)-connections, and [7] for a detailed treatment of the case of flat line bundles. A hyper-Kähler manifold is a Riemannian manifold (M, g) which is Kähler for three complex structures I, J and K satisfying the quaternionic relations where I, J, K are the three original complex structures on M . The twistor space Z is a complex manifold with a holomorphic surjective submersion to CP 1 such that the fiber over every z ∈ CP 1 is identified with (M, I z ). The twistor lines are the "constant" sections of the above fibration Z = M × CP 1 −→ CP 1 . These sections are holomorphic and the normal bundle of a twistor line is isomorphic to This normal bundle is equipped with additional structures which provide all of the information needed to reconstruct the hyper-Kähler structure on M ; see [9] for details.
For a complex reductive group G C , the hyper-Kähler structure on the moduli space of flat G C -connections on a compact Riemann surface X is given by non-abelian Hodge theory. The complex structure J is induced by the complex Lie group G C through the identification of the moduli space with the Betti moduli space Hom(π 1 (X), G C )/ /G C . Via the solutions of Hitchin's self-duality equation, the moduli space of stable G C -Higgs bundles, i.e., the Dolbeault moduli space, gets identified with the moduli space of irreducible flat connections, yielding the complex structure I. The reverse map M dR −→ M Dol arises from Donaldson's twisted harmonic maps construction [5] for the case of G C = SL(2, C), see [4] for the general case. It was first shown by Hitchin in the case of SL(2, C) that I, J and K := IJ give rise to a hyper-Kähler structure for the natural Riemannian metric. In the abelian case of G C = C * the theory simplifies considerably boiling down to the classical abelian Hodge theory [7].
It was first noticed by Deligne that the twistor space of the moduli space of flat connections on a Riemann surface X has a convenient description by gluing the Hodge moduli spaces for X and X via the Riemann-Hilbert isomorphism; see [11,12] for details. We recall below this construction for the rank one case (the case of our interests).
Take a point (λ, L, D) ∈ M Hod (X) = M X Hod (1) with λ = 0 and consider the flat connection D/λ on L. Let ρ : π 1 (X) −→ H 1 (X, Z) −→ C * be the monodromy representation for D/λ. After identifying H 1 (X, Z) with H 1 (X, Z) by the identity map of the underlying C ∞ manifolds, the homomorphism ρ gives a holomorphic rank one 1-connection (1, E, D) on X. Consider the morphism f in (2.1). Let be the morphism obtained by substituting X in place of X in (2.1). Define a holomorphic map of the restrictions of the fiber bundles M Hod (X) and M Hod (X) to C * ⊂ C. This ϕ is clearly a biholomorphism.
Definition 3.1. The rank one Deligne-Hitchin moduli space We note that the map f | f −1 (C * ) coincides with the composition Therefore, f and the composition patch together to produce a morphism For any λ ∈ CP 1 , the fiber f −1 (λ) will be denoted by f λ . The fiber f ∞ is the moduli space of Higgs line bundles of degree zero on X. This f is the twistor fibration mentioned earlier.
where g X is the genus of X.
Proof . As in the computation in [9, pp. 555-556] it can be shown that the tangent bundle of the complex manifold M DH is holomorphically isomorphic to the pull-back where the first summand is the tangent bundle of CP 1 and the direct sum O CP 1 (1) ⊕2g X corresponds to the hyper-Kähler structure on M dR . Now the lemma follows immediately from the fact that degree(i * f * O CP 1 (1)) = degree(i).
We will compute Aut(M DH ) 0 by proving a series of lemmas. Proof . The first part of the lemma is easily verified. In order to prove the converse direction we start with a lift of the family of holomorphic structures on C ⊂ CP 1 : As C is simply connected, there exists a map for all λ ∈ C, where φ is the projection in (2.8) and [−] denotes gauge equivalence class. Let β ∈ H 0 (X, K X ) be the (well-defined) Higgs field of s(0). A lift of s is then given by for some holomorphic map The image of α 1 is contained in H 0 (X, K X ) ⊂ Ω (1,0) (X) because the λ-connections are integrable. Thus, λ −→ s(λ), λ ∈ C * , produces the following family of flat connections on X: We can repeat this over CP 1 \ {0} for the Riemann surface X. Indeed, there exists a lift s of the form s(µ) = (µ, ∂ + α 2 (µ), µ(∂ + ω 2 (µ)) + η) with µ = 1 λ . The corresponding flat connections are denoted by ∇ λ . By the gluing condition in Definition 3.1 there exists a gauge transformation g(λ) : X −→ C * for every λ ∈ C * such that this gauge transformation is unique up to a constant scalar. Since the connection 1-forms of ∇ λ and ∇ λ are both harmonic 1-forms, their difference is a lattice point As the connection 1-forms depend continuously on λ, it follows that χ is independent of λ. By construction, the connection 1-form of λ −→ ∇ λ has a first order pole at λ = ∞ whose residue is η. This implies that ω 1 (λ) must be a polynomial of degree at most one and that α 1 (λ) is a constant α.
Lemma 3.5. Let T : M DH −→ M DH be a holomorphic automorphism. Then T maps fibers of f to fibers of f . Moreover, it covers the automorphism λ −→ τ λ or λ −→ τ λ of CP 1 for some τ ∈ C * ; if the latter case occurs, then the Jacobian of X is isomorphic to the Jacobian of X.
Proof . We first prove that two points in the same fiber p 1 , p 2 ∈ f λ 0 = f −1 (λ 0 ) are mapped by T into a single fiber f λ 1 . To this end, we claim that any two points p 1 , p 2 in different fibers can be joined by a holomorphic section CP 1 −→ M DH of the map f . This can be proved quite easily by linear interpolation with sections of the form given in Lemma 3.4.
If two points x 1 , x 2 of one fiber of f are mapped by T into two different fibers of f , we consider a holomorphic section s of f passing through T (x 1 ) and T (x 2 ). Then λ −→ T −1 (s(λ)) is a holomorphic immersion of CP 1 to M DH , and the degree of its normal bundle is the same as the degree of the normal bundle of the section s. From Lemma 3.3 we obtain which yields deg(i) = 1. On the other hand, its degree is at least two because it passes through two points x 1 , x 2 lying in one fiber. In view of this contradiction we conclude that T maps fibers of f to fibers of f . Therefore, there is a unique holomorphic map As in the proof of Theorem 2.2, T must satisfy T ({0, ∞}) = {0, ∞}, and hence T is of the form λ −→ τ λ or λ −→ τ λ for some τ ∈ C * . If T (0) = ∞ the rank one Higgs moduli spaces for X and X are holomorphically isomorphic. Since there is no nonconstant holomorphic map from an abelian variety to an affine space, any biholomorphism between the rank one Higgs moduli spaces for X and X produces a biholomorphism between Pic 0 (X) and Pic 0 (X).
Let T : M DH −→ M DH be a holomorphic automorphism that maps each fiber of f to itself, meaning T (f λ ) = f λ for all λ ∈ CP 1 . Consider the projection Then, the automorphism T restricted to f 0 maps fibers of π 2 to fibers of π 2 since Pic 0 (X) does not have any nonconstant holomorphic map to H 0 (X, K X ). A corresponding statement is true for the fiber f ∞ . Thus, the assumptions in the following lemma are natural. where d = ∂ + ∂ is the decomposition into types. From Lemma 3.4 and the assumption that T restricted to Pic 0 (X) × {0} ⊂ f 0 is the identity map it follows that T • s must be of the form for some γ 1 , γ 2 ∈ Λ. By applying the gauge exp − γ 1 , the identity in (3.5) is equivalent to Since T is continuous it follows that for every constant section of f defined by s D (λ) = (λ, L, λD), where D is a holomorphic connection on L, the equality holds. It should be mentioned that this is equivalent to Note that for any λ ∈ C \ {0} and for every point p = (λ, E, D) ∈ f λ , there exists a constant section which goes through p. This completes the proof.
be a section of f , where α, β, η, ω ∈ H 0 (X, K X ); define where the tensor product of two λ-connections D, D on two holomorphic line bundles E, E respectively is the λ-connection Theorem 3.7. There is an isomorphism with the group structure of the right-hand side given by The group M dR (X) acts via the automorphisms T s in (3.6), H 0 (X, K X )×H 0 (X, K X ) acts using addition of forms to connections and Higgs bundles, and C * acts via multiplication.
Proof . Using Lemma 3.5 and the lift of the C * action on CP 1 we can restrict to automorphisms T ∈ Aut(M DH ) 0 such that T f λ = f λ for all λ ∈ CP 1 . We claim that there exists a section s : CP 1 −→ M DH of f such that T s • T is of the form given in Lemma 3.6, where T s is constructed in (3.6).
Recall that T acts fiberwise on the fibers of π 2 in (3.4). An automorphism of Pic 0 (X) which is homotopic to the identity map is a translation by an element of Pic 0 (X). Hence, there exists α, ω ∈ H 0 (X, K X ) such that for all (0, L, 0) ∈ π −1 2 (0) ⊂ f 0 . The same argument yields that there are β, η ∈ H 0 (X, K X ) such that the automorphism T s , for the section s defined by has the desired properties.
Moreover, by applying the arguments in the proof of Lemma 3.6, we see that a section s of f such that T s • T is of the form given in Lemma 3.6 is unique up to tensoring with a constant section of the form λ −→ (λ, ∂, λ(∂ + γ )) for some γ ∈ H 1 (X, Z) ⊂ Harm(X, C).
Altogether, we obtain that any element in Aut(M DH ) 0 is of the form (∇, α, η, τ ) ∈ M dR (X)× H 0 (X, K X ) × H 0 (X, K X ) C * for the actions of the components M dR (X), H 0 (X, K X ), H 0 (X, K X ) and C * on M DH as described above. The group structure of Aut(M DH ) 0 is easily verified to be as in (3.7).

Automorphisms of M DH preserving a cohomology class
In this subsection we assume that g = genus(X) > 1.
Till now we have restricted to automorphism of M DH lying in Aut(M DH ) 0 . In this subsection we define a natural cohomology class on M DH and compute all automorphisms which preserve this particular class.

Now define
where θ 0 is the integral cohomology class in (2.9). The symplectic form θ on the Hom(π 1 (X), U(1)) (see (2.9)) depends on the orientation of X, i.e., if the orientation of the topological surface underlying X is changed, then the corresponding symplectic form is − θ. Therefore, with respect to the isomorphism M DH (X) where θ X ∈ H 2 (M DH (X), Z) is obtained by substituting X in place of X in the construction of θ X . Let Aut(M DH (X)) θ X ⊂ Aut(M DH (X)) (3.12) be the subgroup that fixes the cohomology class θ X . For any holomorphic automorphism t ∈ Aut(Pic 0 (X)), the corresponding automorphism T t of M DH (X) fixes the cohomology class θ X if and only if t ∈ Γ X (defined in (2.11)); see the proof of Proposition 2.3.
As in Section 2.2 we have an action of Γ X on Aut(M DH ) which preserves the identity component Aut(M DH ) 0 ⊂ Aut(M DH ). Consider the corresponding semi-direct product Then, we obtain the following: where Aut(M DH (X)) θ X is defined in (3.12), is an isomorphism.
Proof . Any holomorphic automorphism of M DH (X) takes the union f 0 ∪ f ∞ to itself. Indeed, this follows from the following two facts: • all compact complex submanifolds of M DH (X) of dimension at least two are contained in f 0 ∪ f ∞ , and • the union f 0 ∪ f ∞ is covered by compact complex submanifolds of dimension g ≥ 2.
The algebraic dimension of a compact connected complex manifold Y is the transcendence degree of the field of global meromorphic functions on Y over the field C. The algebraic dimension is bounded above by the complex dimension. If the algebraic dimension of Y coincides with the complex dimension of Y , then Y is called Moishezon.
The definition of algebraic dimension does not make sense in general if Y is not compact. For example the transcendence degree, over the field C, of the field of global meromorphic functions on the open unit disk is infinite. However, a theorem of Verbitsky says that the notion of algebraic dimension continues to remain valid, and it is bounded above by the complex dimension, if Y contains an ample rational curve [13,p. 329,Theorem 1.4]. In particular, the notion of algebraic dimension extends to the twistor spaces. Proof . Consider the compactification P O CP 1 (1) ⊕2g ⊕ O CP 1 of O CP 1 (1) ⊕2g . Since it is a complex projective manifold, it is Moishezon. Now from [13, p. 337, Theorem 3.4] we know that any analytic neighborhood of the zero section of O CP 1 (1) ⊕2g is Moishezon, because the zero section is an ample rational curve. Therefore, from Lemma 4.1 and [13, p. 337, Theorem 3.4] it follows that M DH (X) is Moishezon.