R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Indranil BISWAS & Norbert HOFFMANN A Torelli theorem for moduli spaces of principal bundles over a curve Tome 62, no 1 (2012), p. 87-106. <http://aif.cedram.org/item?id=AIF_2012__62_1_87_0> © Association des Annales de l’institut Fourier, 2012, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 62, 1 (2012) 87-106 A TORELLI THEOREM FOR MODULI SPACES OF PRINCIPAL BUNDLES OVER A CURVE by Indranil BISWAS & Norbert HOFFMANN (*) 0 Abstract. — Let X and X be compact Riemann surfaces of genus > 3, and 0 d let G and G be nonabelian reductive complex groups. If one component MG(X) of the coarse moduli space for semistable principal G–bundles over X is isomorphic d0 0 0 to another component MG0 (X ), then X is isomorphic to X . Résumé. — Soient X et X0 des surfaces de Riemann compactes de genre au moins 3, et G et G0 des groupes complexes réductifs non abéliens. Si une compo- d sante MG(X) de l’espace des modules de G–fibrés principaux semi-stables sur X d0 0 0 est isomorphe à une composante MG0 (X ), alors X est isomorphe à X . Introduction Let X be a compact connected Riemann surface of genus gX . The clas- sical Torelli theorem says that the isomorphism class of X is uniquely de- termined by the isomorphism class of the Jacobian Pic0(X), together with its canonical principal polarization Θ. Several authors have studied nonabelian analogues, replacing Jacobians by moduli spaces of vector bundles. Suppose gX > 3. Given a line bundle L on X, let Mn,L(X) denote the moduli space of semistable vector bundles E over X of rank n with fixed determinant Vn E =∼ L. Then the isomorphism Keywords: Principal bundle, moduli space, Torelli theorem. Math. classification: 14D20, 14C34. (*) We are very grateful to the referee for pointing out a gap in an earlier version of this paper, and also for some valuable suggestions. We thank D. Prasad for a useful conversation. The first author wishes to thank the Freie Universität Berlin and the Institute for Mathematical Sciences at Chennai for hospitality. The second author was supported by the SFB 647: Raum - Zeit - Materie. He thanks TIFR (Mumbai) and IMSc (Chennai) for hospitality while some part of this work was done. 88 Indranil BISWAS & Norbert HOFFMANN class of X is uniquely determined by the isomorphism class of the projective variety Mn,L(X). This was first proved for n = 2 and deg(L) odd, by Mumford-Newstead [10, p. 1201, Corollary] and Tyurin [19, Theorem 1]. It was then proved more generally for n coprime to deg(L), by Tyurin [20, Theorem 1] and Narasimhan-Ramanan [12, Theorem 3]. Finally, this statement was proved in full generality by Kouvidakis- Pantev [6, Theorem E], and then later for gX > 4 also by Hwang-Ramanan [5, Theorem 5.1] and by Sun [18, Corollary 1.3]. These proofs used either an intermediate Jacobian of the moduli space [10, 12], or Higgs bundles and the Hitchin map [6,5], or rational curves on the moduli spaces [19, 20, 18]. Our aim here is to address a similar question for the moduli spaces of principal bundles. Given a connected reductive complex linear algebraic d group G, let MG(X) denote the moduli space of semistable principal G– bundles over X of topological type d ∈ π1(G). Theorem 0.1. — Let G and G0 be nonabelian connected reductive com- 0 0 0 plex groups. Let d ∈ π1(G) and d ∈ π1(G ) be given. Let X and X be com- d d0 0 pact Riemann surfaces of genus > 3. If MG(X) is isomorphic to MG0 (X ), then X is isomorphic to X0. For the proof, our strategy is fundamentally different from the earlier ones. Whereas [6] and [5] start from the stable locus, our starting point is the strictly semistable locus. We show that it lies in the singular locus and is characterized by its type of singularities. Using powers of its anticanonical line bundle, we map it to a projective space. The fibers of this map allow us to reconstruct the Jacobian of X and its principal polarization. Then the usual Torelli theorem applies. This method does not work if the strictly semistable locus is empty. But this case is rare, and can in fact be reduced to the earlier results on moduli spaces of vector bundles in the coprime case. The structure of this paper is as follows. Section1 deals with quotient singularities arising from a group action. We give a criterion to distinguish the case of a finite group from the torus C∗. d In Section2, we collect some basic facts about the moduli spaces MG(X). We use several tools valid only in characteristic 0, like representations of π1(X), and the stability of induced bundles. d Section3 describes the strictly semistable locus in MG(X). We relate it to moduli spaces of principal H–bundles for Levi subgroups H of maximal ANNALES DE L’INSTITUT FOURIER TORELLI FOR MODULI OF PRINCIPAL BUNDLES 89 parabolic subgroups in G. Furthermore, we characterize it by the singulari- d ties it contains. As a byproduct, we show that the smooth locus of MG(X) coincides with the regularly stable locus. In the final Section4, we prove Theorem 0.1, by reconstructing the com- d pact Riemann surface X from the projective variety MG(X). In fact we prove a slightly stronger result, namely that X is determined up to isomor- d phy by the smooth locus of MG(X); see Theorem 4.1. 1. Some quotient singularities Let S and S0 be schemes of finite type over C. We say that S is near 0 a closed point s0 ∈ S analytically isomorphic to S near a closed point 0 0 0 0 s0 ∈ S0 if for some open neighborhoods s0 ∈ U ⊆ S and s0 ∈ U ⊆ S in the Euclidean topology, there exists a biholomorphic isomorphism U → U 0 0 that maps s0 to s0. We always denote by Ssing ⊆ S the singular locus of S. Let G be a reductive linear algebraic group over C. We assume that G acts linearly on a finite-dimensional complex vector space V . Let p : V −→ S := V//G = Spec(Sym(V ∗)G) be the GIT–quotient, and put s0 := p(0) ∈ S. Lemma 1.1. — Let G be finite and nontrivial. Suppose that the fixed g locus V ⊆ V has codimension > 2 for all g ∈ G with g 6= 1. i) The quotient S = V//G is singular in s0 = p(0). ii) If s0 ∈ U ⊆ S is an open neighborhood in the Euclidean topology sing sing such that U \ S is connected, then π1(U \ S ) is nontrivial. Proof. — Using the assumption on codim V g,[11, Lemma 4.4] implies [ p−1(Ssing) = V g. g6=1 This proves (i). In the situation of (ii), p−1(U \ Ssing) is connected because −1 sing s0 has only one inverse image. Hence p (U \ S ) is a nontrivial covering sing of U \ S , which is therefore not simply connected. Lemma 1.2. — Let G = C∗ act linearly on the finite-dimensional com- plex vector space V , with associated weight space decomposition M V = Vm. m∈Z Suppose that V−1 and V1 both have dimension > 2. TOME 62 (2012), FASCICULE 1 90 Indranil BISWAS & Norbert HOFFMANN ∗ i) The quotient S = V//C is singular in s0 = p(0). ii) Every neighborhood s0 ∈ U ⊆ S in the Euclidean topology contains 0 0 sing an open neighborhood s0 ∈ U ⊆ U such that U \S is connected 0 sing and π1(U \ S ) is trivial. ∗ Proof. — It is easy to verify S = (V+ ⊕ V−)//C × V0 with M M V+ := Vm and V− := Vm. m>0 m<0 Replacing V by V+ ⊕ V− if necessary, we may thus assume V0 = 0 without loss of generality. Then the GIT–stable locus V stab ⊆ V is stab V = V \ (V+ ∪ V−), stab and the image p(V+ ∪ V−) of V \ V is just the point s0 ∈ S. Let µn M V = Vm ⊆ V m∈nZ ∗ be the fixed locus of the finite subgroup µn ⊆ C . We claim −1 sing [ µn (1.1) p (S ) = V+ ∪ V− ∪ V . n>2 To check this, we first consider a stable point v ∈ V stab. Its image p(v) ∈ S is smooth if and only if its stabilizer in C∗ is trivial, according to Luna’s étale slice theorem [8] and Lemma 1.1. µ stab Suppose next that V n ∩ V is non-empty for some n > 2. Then it is dense in V µn , but also contained in the closed subset p−1(Ssing). Hence µn −1 sing sing −1 sing 0 ∈ V ⊆ p (S ), which implies s0 ∈ S . It follows that p (S ) contains the unstable locus V+ ∪ V− in this case.