Torelli Theorem for the

Deligne–Hitchin Moduli Space, II

Indranil Biswas, Tomás L. Gómez, and Norbert Hoffmann^{1}

Received: April 9, 2012

Communicated by Ulf Rehmann

Abstract. Let X and X′ be compact Riemann surfaces of genus at least three. Let G and G′ be nontrivial connected semisimple linear algebraic groups over ℂ. If some components M_{DH}^{d}(X, G) and M_{DH}^{d′}(X′, G′) of the associated Deligne–Hitchin moduli spaces are biholomorphic, then X′ is isomorphic to X or to the conjugate Riemann surface \overline{X}.

2010 Mathematics Subject Classification: 14D20, 14C34

Keywords and Phrases: Principal bundle, Deligne–Hitchin moduli space, Higgs bundle, vector field

1. Introduction

Let X be a compact connected Riemann surface of genus g ≥ 3. Let \overline{X} denote the conjugate Riemann surface; by definition, it consists of the real manifold underlying X and the almost complex structure J\overline{X} := −J_{X}. Let G be a nontrivial connected semisimple linear algebraic group over ℂ. The topological types of holomorphic principal G–bundles E over X correspond to elements of π_{1}(G). Let M_{Higgs}^{d}(X, G) denote the moduli space of semistable Higgs G– bundles (E, θ) over X with E of topological type d ∈ π_{1}(G). The Deligne–Hitchin moduli space [Si3] is a complex analytic space M_{DH}^{d}(X, G) associated to X, G and d. It is the twistor space for the hyper-Kähler structure on M_{Higgs}^{d}(X, G); see [Hi2, §9]. Deligne [De] has constructed it together with a surjective holomorphic map

M_{DH}^{d}(X, G) −↠ ℂℙ^{1} = ℂ ∪ {∞}.

1This work was supported by the grant MTM2010-17389, the ICMAT Severo Ochoa project SEV-2011-0087 of the Spanish Ministerio de Economía y Competitividad, and by the German SFB 647: Raum - Zeit - Materie. The first-named author acknowledges the support of the J. C. Bose Fellowship.

1178 Indranil Biswas, Tomás L. Gómez, and Norbert Hoffmann

The inverse image of ℂ ⊆ ℂℙ^{1} is the moduli space M_{Hod}^{d}(X, G) of holomorphic principal G–bundles over X endowed with a λ–connection. In particular, every fiber over ℂ^{∗} ⊂ ℂℙ^{1} is isomorphic to the moduli space of holomorphic G– connections over X. The fiber over 0 ∈ ℂℙ^{1} is M_{Higgs}^{d}(X, G), and the fiber over ∞ ∈ ℂℙ^{1} is M_{Higgs}^{−d}(\overline{X}, G). In this paper, we study the dependence of these moduli spaces on X. Our main result, Theorem 5.3, states that the complex analytic space M_{DH}^{d}(X, G) determines the unordered pair \{X, \overline{X}\} up to isomorphism. We also prove that M_{Higgs}^{d}(X, G) and M_{Hod}^{d}(X, G) each determine X up to isomorphism; see Theorem 5.1 and Theorem 5.2. The key technical result is Proposition 3.1, which says the following: Let Z be an irreducible component of the fixed point locus for the natural ℂ^{∗}–action on a moduli space M_{Higgs}^{d}(X, G) of Higgs G–bundles. Then,

dim Z ≤ (g − 1) · dim_{ℂ} G,

with equality holding only if Z is the moduli space M^{d}(X, G) of holomorphic principal G-bundles of topological type d over X. In [BGHL], the case of G = SL(r, ℂ) was considered. If g = 2 and G = SL(2, ℂ), then the above theorems are no longer valid. So we have assumed that g ≥ 3.

2. Some moduli spaces associated to a compact Riemann surface

Let X be a compact connected Riemann surface of genus g ≥ 3. Let G be a nontrivial connected semisimple linear algebraic group defined over ℂ, with Lie algebra 𝔤.

2.1. Principal G–bundles. We consider holomorphic principal G–bundles E over X. Recall that the topological type of E is given by an element d ∈ π_{1}(G) [Ra]; this is a finite abelian group. The adjoint vector bundle of E is the holomorphic vector bundle

ad(E) := E ×^{G} 𝔤

over X, using the adjoint action of G on 𝔤. E is called stable (respectively, semistable) if

(1)  degree(ad(E_{P})) < 0 (respectively, ≤ 0)

for every maximal parabolic subgroup P ⫋ G and every holomorphic reduction of structure group E_{P} of E to P; here ad(E_{P}) ⊂ ad(E) is the adjoint vector bundle of E_{P}. Let M^{d}(X, G) denote the moduli space of semistable holomorphic principal G– bundles E over X of topological type d ∈ π_{1}(G). It is known that M^{d}(X, G) is an irreducible normal projective variety of dimension (g − 1) · dim_{ℂ} G over ℂ.

2.2. Higgs G–bundles. The holomorphic cotangent bundle of X will be denoted by K_{X}. A Higgs G–bundle over X is a pair (E, θ) consisting of a holomorphic principal G–bundle E over X and a holomorphic section

θ ∈ H^{0}(X, ad(E) ⊗ K_{X}),

the so-called Higgs field [Hi1, Si1]. The pair (E, θ) is called stable (respectively, semistable) if the inequality (1) holds for every holomorphic reduction of structure group E_{P} of E to a maximal parabolic subgroup P ⫋ G such that θ ∈ H^{0}(X, ad(E_{P}) ⊗ K_{X}). Let M_{Higgs}^{d}(X, G) denote the moduli space of semistable Higgs G–bundles (E, θ) over X such that E is of topological type d ∈ π_{1}(G). It is known that M_{Higgs}^{d}(X, G) is an irreducible normal quasiprojective variety of dimension 2(g − 1) · dim_{ℂ} G over ℂ [Si2]. We regard M^{d}(X, G) as a closed subvariety of M_{Higgs}^{d}(X, G) by means of the embedding

M^{d}(X, G) ⟶ M_{Higgs}^{d}(X, G) , E ⟼ (E, 0) .

There is a natural algebraic symplectic structure on M_{Higgs}^{d}(X, G); see [Hi1, BR].

2.3. Representations of the surface group in G. Fix a base point x_{0} ∈ X. The fundamental group of X admits a standard presentation

π_{1}(X, x_{0}) ≅ ⟨a_{1}, …, a_{g}, b_{1}, …, b_{g}| ∏_{i=1}^{g} a_{i}b_{i}a_{i}^{−1}b_{i}^{−1} = 1⟩

which we choose in such a way that it is compatible with the orientation of X. We identify the fundamental group of G with the kernel of the universal covering G̃ −↠ G. The type d ∈ π_{1}(G) of a homomorphism ρ : π_{1}(X, x_{0}) ⟶ G is defined by

d := ∏_{i=1}^{g} α_{i}β_{i}α_{i}^{−1}β_{i}^{−1} ∈ π_{1}(G) ⊂ G̃

for any choice of lifts α_{i}, β_{i} ∈ G̃ of ρ(a_{i}), ρ(b_{i}) ∈ G. This is also the topological type of the principal G–bundle E_{ρ} over X given by ρ. The space Hom^{d}(π_{1}(X, x_{0}), G) of all homomorphisms ρ : π_{1}(X, x_{0}) ⟶ G of type d ∈ π_{1}(G) is an irreducible affine variety over ℂ, and G acts on it by conjugation. The GIT quotient

M_{Rep}^{d}(X, G) := Hom^{d}(π_{1}(X, x_{0}), G)//G

doesn’t depend on x_{0}. It is an affine variety of dimension 2(g − 1) · dim_{ℂ} G over ℂ, which carries a natural symplectic form [AB, Go]. Its points represent equivalence classes of completely reducible homomorphisms ρ. There is a natural bijective map

M_{Rep}^{d}(X, G) ⟶ M_{Higgs}^{d}(X, G)

given by a variant of the Kobayashi–Hitchin correspondence [Si1]. This bijective map is not holomorphic.

2.4. Holomorphic G–connections. Let p : E ⟶ X be a holomorphic principal G–bundle. Because the vertical tangent space at every point of the total space E is canonically isomorphic to 𝔤, there is a natural exact sequence

0 ⟶ E × 𝔤 ⟶ T E ⟶^{dp} p^{∗}T X ⟶ 0

of G-equivariant holomorphic vector bundles over E. Taking the G-invariant direct image under p, it follows that the Atiyah bundle for E

At(E) := p_{∗}(T E)^{G} ⊂ p_{∗}(T E)

sits in a natural exact sequence of holomorphic vector bundles

(2)  0 ⟶ ad(E) ⟶ At(E) ⟶^{dp} T X ⟶ 0

over X. This exact sequence is called the Atiyah sequence. A holomorphic connection on E is a splitting of the Atiyah sequence, or in other words a holomorphic homomorphism

D : T X ⟶ At(E)

such that dp ∘ D = id_{T X} [At]. It always exists if E is semistable [AzBi, p. 342, Theorem 4.1], [BG, p. 20, Theorem 1.1]. The curvature of D is a holomorphic 2–form with values in ad(E), so D is automatically flat. A holomorphic G–connection is a pair (E, D) where E is a holomorphic principal G–bundle over X, and D is a holomorphic connection on E. Such a pair is automatically semistable, because the degree of a flat vector bundle is zero. Let M_{conn}^{d}(X, G) denote the moduli space of holomorphic G–connections (E, D) over X such that E is of topological type d ∈ π_{1}(G). It is known that M_{conn}^{d}(X, G) is an irreducible quasiprojective variety of dimension 2(g − 1) · dim_{ℂ} G over ℂ. Sending each holomorphic G–connection to its monodromy defines a map

(3)  M_{conn}^{d}(X, G) ⟶ M_{Rep}^{d}(X, G)

which is biholomorphic, but not algebraic; it is called Riemann–Hilbert correspondence. The inverse map sends a homomorphism ρ : π_{1}(X, x_{0}) ⟶ G to the associated principal G–bundle E_{ρ}, endowed with the induced holomorphic connection D_{ρ}.

2.5. λ–connections. Let p : E ⟶ X be a holomorphic principal G–bundle. For any λ ∈ ℂ, a λ–connection on E is a holomorphic homomorphism of vector bundles

D : T X ⟶ At(E)

such that dp ∘ D = λ · id_{T X} for the epimorphism dp in the Atiyah sequence (2). Therefore, a 0–connection is a Higgs field, and a 1–connection is a holomorphic connection. If D is a λ–connection on E with λ ≠ 0, then λ^{−1}D is a holomorphic connection on E. In particular, the pair (E, D) is automatically semistable in this case.

Let M_{Hod}^{d}(X, G) denote the moduli space of triples (λ, E, D), where λ ∈ ℂ, E is a holomorphic principal G–bundle over X of topological type d ∈ π_{1}(G), and D is a semistable λ–connection on E; see [Si2]. There is a canonical algebraic map

(4)  pr = pr_{X} : M_{Hod}^{d}(X, G) −↠ ℂ,  (λ, E, D) ⟼ λ.

Its fibers over λ = 0 and λ = 1 are M_{Higgs}^{d}(X, G) and M_{conn}^{d}(X, G), respectively. The Riemann–Hilbert correspondence (3) allows to define a holomorphic open embedding

j = j_{X} : ℂ^{∗} × M_{Rep}^{d}(X, G) ⟶ M_{Hod}^{d}(X, G), (λ, ρ) ⟼ (λ, E_{ρ}, λD_{ρ})

with image pr^{−1}(ℂ^{∗}). This map commutes with the projections onto ℂ^{∗}.

2.6. The Deligne–Hitchin moduli space. The compact Riemann surface X provides an underlying real C^{∞} manifold X_{ℝ}, and an almost complex structure J_{X} : T X_{ℝ} ⟶ T X_{ℝ}. Since any almost complex structure in real dimension two is integrable,

\overline{X} := (X_{ℝ}, −J_{X})

is a compact Riemann surface as well. It has the opposite orientation, so

(5)  M_{Rep}^{d}(X, G) = M_{Rep}^{−d}(\overline{X}, G).

The Deligne–Hitchin moduli space M_{DH}^{d}(X, G) is the complex analytic space obtained by gluing M_{Hod}^{d}(X, G) and M_{Hod}^{−d}(\overline{X}, G) along their common open subspace

M_{Hod}^{d}(X, G) ⟵^{j_{X}} ℂ^{∗} × M_{Rep}^{d}(X, G) ≅ ℂ^{∗} × M_{Rep}^{−d}(\overline{X}, G) ⟶^{j\overline{X}} M_{Hod}^{−d}(\overline{X}, G)

where the isomorphism in the middle sends (λ, ρ) to (1/λ, ρ); see [Si3, De]. The projections pr_{X} on M_{Hod}^{d}(X, G) and 1/pr\overline{X} on M_{Hod}^{−d}(\overline{X}, G) patch together to a holomorphic map

M_{DH}^{d}(X, G) −↠ ℂℙ^{1} = ℂ ∪ {∞}.

Its fiber over any λ ∈ ℂ^{∗} is biholomorphic to the representation space (5), whereas its fibers over λ = 0 and λ = ∞ are M_{Higgs}^{d}(X, G) and M_{Higgs}^{−d}(\overline{X}, G), respectively.

3. Fixed points of the natural ℂ^{∗}–action

The group ℂ^{∗} acts algebraically on the moduli space M_{Higgs}^{d}(X, G), via the formula

(6)  t · (E, θ) := (E, tθ).

The fixed point locus M_{Higgs}^{d}(X, G)^{ℂ^{∗}} contains the closed subvariety M^{d}(X, G).

Proposition 3.1. Let Z be an irreducible component of M_{Higgs}^{d}(X, G)^{ℂ^{∗}}. Then one has

dim Z ≤ (g − 1) · dim_{ℂ} G,

with equality holding only for Z = M^{d}(X, G).

Proof. Let (E, θ) be a stable Higgs G–bundle over X. Its infinitesimal deformations are, according to [BR, Theorem 2.3], governed by the complex of vector bundles

(7)  C^{0} := ad(E) ⟶^{ad(θ)} ad(E) ⊗ K_{X} =: C^{1}

over X. Since (E, θ) is stable, it has no infinitesimal automorphisms, so

ℍ^{0}(X, C^{•}) = 0 .

The Killing form on 𝔤 induces isomorphisms 𝔤^{∗} ≅ 𝔤 and ad(E)^{∗} ≅ ad(E). Hence the vector bundle ad(E) has degree 0. Serre duality allows us to conclude

ℍ^{2}(X, C^{•}) = 0 .

Using all this, the Riemann–Roch formula yields

(8)  dim ℍ^{1}(X, C^{•}) = 2(g − 1) · dim_{ℂ} G .

From now on, we assume that the point (E, θ) is fixed by ℂ^{∗}, and we also assume θ ≠ 0. Then (E, θ) ≅ (E, tθ) for all t ∈ ℂ^{∗}, so the sequence of complex algebraic groups

1 ⟶ Aut(E, θ) ⟶ Aut(E, ℂθ) ⟶ Aut(ℂθ) = ℂ^{∗} ⟶ 1

is exact. Because (E, θ) is stable, Aut(E, θ) is finite. Consequently, the identity component of Aut(E, ℂθ) is isomorphic to ℂ^{∗}. This provides an embedding

ℂ^{∗} ⟶ Aut(E) , t ⟼ φ_{t} ,

and an integer w ≠ 0 with φ_{t}(θ) = t^{w} · θ for all t ∈ ℂ^{∗}. We may assume that w ≥ 1. Choose a point e_{0} ∈ E. Then there is a unique group homomorphism

ι : ℂ^{∗} ⟶ G

such that φ_{t}(e_{0}) = e_{0} · ι(t) for all t ∈ ℂ^{∗}. The conjugacy class of ι doesn’t depend on e_{0}, since the space of conjugacy classes Hom(ℂ^{∗}, G)/G is discrete. The subset

E_{H} := {e ∈ E : φ_{t}(e) = e · ι(t) for all t ∈ ℂ^{∗}}

of E is a holomorphic reduction of structure group to the centralizer H of ι(ℂ^{∗}) in G. Let

(9)  𝔤 = ⨁_{n∈ℤ} 𝔤_{n}

denote the eigenspace decomposition given by the adjoint action of ℂ^{∗} on 𝔤 via ι.

Let N ∈ ℤ be maximal with 𝔤_{N} ≠ 0. Let P ⊂ G be the parabolic subgroup with

Lie(P) = ⨁_{n≥0} 𝔤_{n} ⊂ 𝔤 .

Since H ⊂ G has Lie algebra 𝔤_{0}, it is a Levi subgroup in P. Choose subgroups

ι(ℂ^{∗}) ⊆ T ⊆ B ⊆ P ⊂ G

such that T is a maximal torus in H ⊂ G and B is a Borel subgroup in G. Let

α_{j} : T ⟶ ℂ^{∗}  and  α_{j}^{∨} : ℂ^{∗} ⟶ T

be the resulting simple roots and coroots of G. We denote by ⟨ , ⟩ the natural pairing between characters and cocharacters of T. Let α_{j} be a simple root of G with ⟨α_{j}, ι⟩ > 0, and let β be a root of G with ⟨β, ι⟩ = N. Then the elementary reflection

s_{j}(β) = β − ⟨β, α_{j}^{∨}⟩α_{j}

is a root of G, so ⟨s_{j}(β), ι⟩ ≤ N; this implies that ⟨β, α_{j}^{∨}⟩ ≥ 0. The sum of all such roots β with ⟨β, ι⟩ = N is the restriction χ|_{T} of the determinant

(10)  χ : P ⟶ Aut(𝔤_{N}) ⟶^{det} ℂ^{∗}

of the adjoint action of P on 𝔤_{N}. Hence we conclude ⟨χ|_{T} , α_{j}^{∨}⟩ ≥ 0 for all simple roots α_{j} with ⟨α_{j}, ι⟩ > 0. This means that the character χ of P is dominant. The decomposition (9) of 𝔤 induces a vector bundle decomposition

ad(E) = ⨁_{n∈ℤ} E_{H} ×^{H} 𝔤_{n} .

Since ℂ^{∗} acts with weight w on the Higgs field θ by construction, we have

(11)  θ ∈ H^{0}(X, (E_{H} ×^{H} 𝔤_{w}) ⊗ K_{X}) .

In particular, θ ∈ H^{0}(X, ad(E_{P}) ⊗ K_{X}) for the reduction E_{P} := E_{H} ×^{H} P ⊆ E of the structure group to P. The Higgs version of the stability criterion [Ra, Lemma 2.1] yields

degree(E_{H} ×^{H} 𝔤_{N}) ≤ 0

since P acts on det(𝔤_{N}) via the dominant character χ in (10). Now Riemann– Roch implies the following:

(12)  dim H^{1}(X, E_{H} ×^{H} 𝔤_{N}) ≥ (g − 1) · dim_{ℂ} 𝔤_{N} > 0 .

The complex C^{•} in (7) is, due to (11), the direct sum of its subcomplexes C_{n}^{•} given by

C_{n}^{0} := E_{H} ×^{H} 𝔤_{n} ⟶^{ad(θ)} (E_{H} ×^{H} 𝔤_{n+w}) ⊗ K_{X} =: C_{n}^{1} .

Thus the hypercohomology of C^{•} decomposes as well; in particular, we have

ℍ^{1}(X, C^{•}) = ⨁_{n∈ℤ} ℍ^{1}(X, C_{n}^{•}) .

In the last nonzero summand C_{N}^{•}, we have C_{N}^{1} = 0 and hence

dim ℍ^{1}(X, C_{N}^{•}) = dim H^{1}(X, E_{H} ×^{H} 𝔤_{N}) > 0 due to (12). Since 𝔤_{n}^{∗} ≅ 𝔤_{−n} via the Killing form on 𝔤, Serre duality yields in particular

dim ℍ^{1}(X, C_{0}^{•}) = dim ℍ^{1}(X, C_{−w}^{•}).

Taken together, the last three formulas and the equation (8) imply that

dim ℍ^{1}(X, C_{0}^{•}) < \frac{1}{2} dim ℍ^{1}(X, C^{•}) = (g − 1) · dim_{ℂ} G .

But ℍ^{1}(X, C_{0}^{•}) parameterizes infinitesimal deformations of pairs (E_{H}, θ) consisting of a principal H–bundle E_{H} and a section θ as in (11); see [BR, Theorem 2.3]. This proves that

dim Z < (g − 1) · dim_{ℂ} G

for every irreducible component Z of the fixed point locus M_{Higgs}^{d}(X, G)^{ℂ^{∗}} such that Z contains stable Higgs G–bundles (E, θ) with θ ≠ 0. The non-stable points in M_{Higgs}^{d}(X, G) correspond to polystable Higgs G– bundles (E, θ). Polystability means that E admits a reduction of structure group E_{L} to a Levi subgroup L ⫋ G of a parabolic subgroup in G such that θ is a section of the subbundle

ad(E_{L}) ⊗ K_{X} ⊂ ad(E) ⊗ K_{X}

and the pair (E_{L}, θ) is stable. Let C ⊆ L be the identity component of the center, and let 𝔠 ⊆ 𝔩 be their Lie algebras. Then E_{L/C} := E_{L}/C is a principal (L/C)–bundle over X, and

ad(E_{L}) ≅ (𝔠 ⊗ O_{X}) ⊕ ad(E_{L/C})

since 𝔩 = 𝔠 ⊕ [𝔩, 𝔩], where the subalgebra [𝔩, 𝔩] ⊆ 𝔩 is also the Lie algebra of L/C. We have

dim_{ℂ} G − dim_{ℂ} L ≥ 2 dim_{ℂ} C

because maximal Levi subgroups in G have 1-dimensional center and at least one pair of opposite roots less than G; the other Levi subgroups can be reached by iterating this. Now suppose that ℂ^{∗} fixes the point (E, θ). Then (E_{L}, θ) ≅ (E_{L}, tθ) for all t ∈ ℂ^{∗}. But the action of Aut(E_{L}) on the direct summand 𝔠 ⊗ O_{X} of ad(E_{L}) is trivial, since the adjoint action of L on 𝔠 is trivial. So θ lives in the other summand of ad(E_{L}), meaning

θ ∈ H^{0}(X, ad(E_{L/C}) ⊗ K_{X}) .

The Higgs (L/C)–bundle (E_{L/C}, θ) is still stable and fixed by ℂ^{∗}; we have already proved that the locus of such has dimension ≤ (g − 1) · dim_{ℂ}(L/C). The abelian variety M^{0}(X, C) acts simply transitively on lifts of E_{L/C} to a principal L–bundle E_{L}, so these lifts form a family of dimension g · dim_{ℂ} C. Hence the pairs (E_{L}, θ) in question have at most

(g − 1) · dim_{ℂ}(L/C) + g · dim_{ℂ} C < (g − 1) · dim_{ℂ} G

moduli. This implies that dim Z < (g −1)·dim_{ℂ} G for each non-stable component Z of the fixed point locus, since there are only finitely many possibilities for L up to conjugation. □

The algebraic ℂ^{∗}–action (6) on M_{Higgs}^{d}(X, G) extends naturally to an algebraic ℂ^{∗}–action on M_{Hod}^{d}(X, G), which is given by the formula

(13)  t · (λ, E, D) := (tλ, E, tD).

A point (λ, E, D) can only be fixed by this action if λ = 0, so Proposition 3.1 yields the following corollary:

Corollary 3.2. Let Z be an irreducible component of M_{Hod}^{d}(X, G)^{ℂ^{∗}}. Then one has

dim Z ≤ (g − 1) · dim_{ℂ} G,

with equality only for Z = M^{d}(X, G).

The algebraic ℂ^{∗}–action (13) on M_{Hod}^{d}(X, G) extends naturally to a holomorphic ℂ^{∗}–action on M_{DH}^{d}(X, G), which is on the other open patch M_{Hod}^{−d}(\overline{X}, G) given by the formula

t · (λ, E, D) := (t^{−1}λ, E, t^{−1}D).

Applying Corollary 3.2 to both M_{Hod}^{d}(X, G) and M_{Hod}^{−d}(\overline{X}, G), one immediately gets

Corollary 3.3. Let Z be an irreducible component of M_{DH}^{d}(X, G)^{ℂ^{∗}}. Then one has

dim Z ≤ (g − 1) · dim_{ℂ} G,

with equality only for Z = M^{d}(X, G) and for Z = M^{−d}(\overline{X}, G).

4. Vector fields on the moduli spaces

A stable principal G–bundle E over X is called regularly stable if the automorphism group Aut(E) is just the center of G. The regularly stable locus

M^{d,rs}(X, G) ⊆ M^{d}(X, G)

is open, and coincides with the smooth locus of M^{d}(X, G); see [BH, Corollary 3.4].

Proposition 4.1. There are no nonzero holomorphic vector fields on M^{d,rs}(X, G).

Proof. This statement is contained in [Fa, p. 549, Corollary III.3]. □

Proposition 4.2. There are no nonzero holomorphic 1–forms on M^{d,rs}(X, G).

Proof. The moduli space of Higgs G–bundles is equipped with the Hitchin map

M_{Higgs}^{d}(X, G) ⟶ ⨁_{i=1}^{rank(G)} H^{0}(X, K_{X}^{⊗n_{i}})

where the n_{i} are the degrees of generators for the algebra Sym(𝔤^{∗})^{G}; see [Hi1, § 4], [La]. Any sufficiently general fiber of this Hitchin map is a complex abelian variety A (see [Do], [Fa], [DP] for the details), and

\xymatrix{φ : A \ar@{-->}[r] & M^{d,rs}(X, G), & (E, θ) ⟼ E,}

is a dominant rational map. This rational map φ is defined outside a closed subscheme of codimension at least two; see [Fa, p. 534, Theorem II.6]. Let ω be a holomorphic 1–form on M^{d,rs}(X, G). Then φ^{∗}ω extends to a holomorphic 1–form on A by Hartog’s theorem. As any holomorphic 1–form on A is closed, it follows that ω is closed. Since H^{1}(M^{d,rs}(X, G), ℂ) = 0 by [AB, Ch. 10], we conclude ω = df for a holomorphic function f on M^{d,rs}(X, G). But any such function f is constant, so ω = 0. □

We denote by M_{Higgs}^{d,rs}(X, G) ⊆ M_{Higgs}^{d}(X, G) the open locus of Higgs G– bundles (E, θ) for which E is regularly stable. The forgetful map

(14)  M_{Higgs}^{d,rs}(X, G) ⟶ M^{d,rs}(X, G),  (E, θ) ⟼ E,

is an algebraic vector bundle with fibers H^{0}(X, ad(E) ⊗ K_{X}) ≅ H^{1}(X, ad(E))^{∗}, so it is the cotangent bundle of M^{d,rs}(X, G).

Corollary 4.3. The restriction of the algebraic tangent bundle

T M_{Higgs}^{d,rs}(X, G) ⟶ M_{Higgs}^{d,rs}(X, G)

to the subvariety M^{d,rs}(X, G) ⊆ M_{Higgs}^{d,rs}(X, G) has no nonzero holomorphic sections.

Proof. The subvariety in question is the zero section of the vector bundle (14). Given a vector bundle V ⟶ M with zero section M ⊆ V, there is a natural isomorphism

(15)  (T V)|_{M} ≅ T M ⊕ V

of vector bundles over M. In our situation, both summands have no nonzero holomorphic sections, according to Proposition 4.1 and Proposition 4.2. □

Let M_{conn}^{d,rs}(X, G) ⊆ M_{conn}^{d}(X, G) denote the open locus of holomorphic G– connections (E, D) for which E is regularly stable.

Proposition 4.4. There are no holomorphic sections for the forgetful map

(16)  M_{conn}^{d,rs}(X, G) ⟶ M^{d,rs}(X, G),  (E, D) ⟼ E.

Proof. The map (16) is a holomorphic torsor under the cotangent bundle of M^{d,rs}(X, G). As such, it is isomorphic to the torsor of holomorphic connections on the line bundle

L ⟶ M^{d,rs}(X, G)

with fibers det H^{1}(X, ad(E)); see [Fa, p. 554, Lemma IV.4]. Since L is ample [KNR], its first Chern class is nonzero, so L admits no global holomorphic connections. □

Let M_{Hod}^{d,rs}(X, G) ⊆ M_{Hod}^{d}(X, G) denote the open locus of triples (λ, E, D) for which E is regularly stable. The forgetful maps in (14) and (16) extend to the forgetful map

(17)  M_{Hod}^{d,rs}(X, G) ⟶ M^{d,rs}(X, G),  (λ, E, D) ⟼ E,

which is an algebraic vector bundle. It contains the cotangent bundle (14) as a subbundle; the quotient is a line bundle, which is trivialized by the projection pr in (4).

Corollary 4.5. The vector bundle (17) has no nonzero holomorphic sections.

Proof. Let s be a holomorphic section of the vector bundle (17). Then pr ∘ s is a holomorphic function on M^{d,rs}(X, G), and hence constant. This constant vanishes because of Proposition 4.4. So pr ∘ s = 0, which implies that s = 0 using Proposition 4.2. □

Corollary 4.6. The restriction of the algebraic tangent bundle

T M_{Hod}^{d,rs}(X, G) ⟶ M_{Hod}^{d,rs}(X, G)

to the subvariety M^{d,rs}(X, G) ⊆ M_{Hod}^{d,rs}(X, G) has no nonzero holomorphic sections.

Proof. Use the decomposition (15), Proposition 4.1, and Corollary 4.5. □

5. Torelli theorems

Let X, X′ be compact connected Riemann surfaces of genus ≥ 3. Let G, G′ be nontrivial connected semisimple linear algebraic groups over ℂ. Fix d ∈ π_{1}(G) and d′ ∈ π_{1}(G′).

Theorem 5.1. If M_{Higgs}^{d′}(X′, G′) is biholomorphic to M_{Higgs}^{d}(X, G), then X′ ≅ X.

Proof. Corollary 4.3 implies that the subvariety M^{d}(X, G) is fixed pointwise by every holomorphic ℂ^{∗}–action on M_{Higgs}^{d}(X, G). All other complex analytic subvarieties with that property have smaller dimension, due to Proposition 3.1. Thus we get a biholomorphic map from M^{d′}(X′, G′) to M^{d}(X, G) by restriction. Using [BH], this implies that X′ ≅ X. □

Theorem 5.2. If M_{Hod}^{d′}(X′, G′) is biholomorphic to M_{Hod}^{d}(X, G), then X′ ≅ X.

Proof. The argument is exactly the same as in the previous proof. It suffices to replace Corollary 4.3 by Corollary 4.6, and Proposition 3.1 by Corollary 3.2. □

Theorem 5.3. If M_{DH}^{d′}(X′, G′) is biholomorphic to M_{DH}^{d}(X, G), then X′ ≅ X or X′ ≅ \overline{X}.

Proof. The argument is similar. Corollary 4.6 implies that the two subvarieties M^{d}(X, G) and M^{−d}(\overline{X}, G) are fixed pointwise by every holomorphic ℂ^{∗}–action on M_{DH}^{d}(X, G). All other complex analytic subvarieties with that property have smaller dimension, due to Corollary 3.3. Thus we get a biholomorphic map from M^{d′}(X′, G′) to either M^{d}(X, G) or M^{−d}(\overline{X}, G) by restriction. Using [BH], this implies that either X′ ≅ X or X′ ≅ \overline{X}. □

References

[At] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.

[AB] M. F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. 308 (1982), 523–615.

[AzBi] H. Azad and I. Biswas, On holomorphic principal bundles over a compact Riemann surface admitting a flat connection, Math. Ann. 322 (2002), 333–346.

[BG] I. Biswas and T. L. Gómez, Connections and Higgs fields on a principal bundle, Ann. Global Anal. Geom. 33 (2008), 19–46.

[BGHL] I. Biswas, T. L. Gómez, N. Hoffmann and M. Logares, Torelli theorem for the Deligne–Hitchin moduli space, Commun. Math. Phy. 290 (2009), 357–369.

[BH] I. Biswas and N. Hoffmann, A Torelli theorem for moduli spaces of principal bundles over a curve, Ann. Inst. Fourier 62 (2012), 87–106.

[BR] I. Biswas and S. Ramanan, An infinitesimal study of the moduli of Hitchin pairs, Jour. London Math. Soc. 49 (1994), 219–231.

[De] P. Deligne, Letter to C. T. Simpson (March 20, 1989).

[Do] R. Donagi, Decomposition of spectral covers, Astérisque 218 (1993), 145–175.

[DP] R. Donagi and T. Pantev, Langlands duality for Hitchin systems, Invent. Math. 189 (2012), 653–735.

[Fa] G. Faltings, Stable G–bundles and projective connections, Jour. Alg. Geom. 2 (1993), 507–568.

[Go] W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200–225.

[Hi1] N. J. Hitchin, Stable bundles and integrable systems, Duke Math. Jour. 54 (1987), 91–114.

[Hi2] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987), 59–126.

[KNR] S. Kumar, M. S. Narasimhan and A. Ramanathan, Infinite Grassmannians and moduli spaces of G-bundles, Math. Ann. 300 (1994), 41–75.

[La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671.

[Ra] A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152.

[Si1] C. T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95.

[Si2] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5–79.

[Si3] C. T. Simpson, A weight two phenomenon for the moduli of rank one local systems on open varieties, From Hodge theory to integrability and TQFT tt*-geometry, 175–214, Proc. Sympos. Pure Math., 78, Amer. Math. Soc., Providence, RI, 2008.

Indranil Biswas School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400005 India indranil@math.tifr.res.in Tomás L. Gómez Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) Nicolás Cabrera 15 Campus Cantoblanco UAM 28049 Madrid Spain tomas.gomez@icmat.es

Norbert Hoffmann

Department of Mathematics and Computer Studies Mary Immaculate College South Circular Road Limerick Ireland norbert.hoffmann@mic.ul.ie 1190

Documenta Mathematica 18 (2013)