Torelli Theorem for the Moduli Space of Framed Bundles

Let X be an irreducible smooth complex projective curve of genus g > 2, and let x ∈ X be a fixed point. A framed bundle is a pair (E,φ), where E is a vector bundle over X, of rank r and degree d, and φ : Ex −→ C r is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ– semistable framed bundles M . We prove a Torelli theorem for M , for τ > 0 small enough, meaning, the isomorphism class of the one–pointed curve (X ,x), and also the integer r, are uniquely determined by the isomorphism class of the variety M . Let X be an irreducible smooth projective curve defined over C. Fix a point x ∈ X. Fix a line bundle ξ over X, and let d denote its degree. We consider pairs of the form (E,φ : Ex −→ C ), where E is a vector bundle of fixed rank r and determinant ξ, and φ is a C–linear homomorphism (Ex is the fiber of E over the point x). This is a particular case of the framed bundles of Huybrechts and Lehn [HL]. In our situation the reference sheaf is the torsion sheaf supported at x with fiber C. In [HL], the notion of a semistable framed bundle is introduced which depends on a real parameter τ , and the corresponding moduli space is constructed, which is a complex projective variety. Let τ > 0 be a real number. A pair (E,φ : Ex −→ C ) is called τ–stable (respectively, τ–semistable) if, for all proper subbundles E′ ⊂ E of positive rank, degE′ − ǫ(E′, φ)τ rkE′ < degE − τ rkE (respectively, degE −ǫ(E,φ)τ rkE ≤ degE−τ rkE ), where ǫ(E, φ) = { 1 if φ|E′ x 6= 0, 0 if φ|E′ x = 0. Let MX,x,r,ξ be the moduli space of τ–semistable pairs of rank r and determinant ξ. When the data is clear from the context, we will also use the shortened notation M instead of MX,x,r,ξ. We prove the following Torelli theorem for this moduli space when τ is sufficiently small. Theorem 0.1. Let X be a smooth projective curve of genus g > 2 and x ∈ X a point. Let r > 1 be an integer and ξ a line bundle over X. Let τ > 0 be a real number with τ < τ(r) (cf. Lemma 1.1). Let X ′, g′, x′, r′, ξ′ and τ ′ be another set of data satisfying the same conditions. If the moduli space MX,x,r,ξ is isomorphic to 2000 Mathematics Subject Classification. Primary 14D22, Secondary 14D20. Supported by grant MTM2007-63582 from the Spanish Ministerio de Educación y Ciencia and grant 200650M066 from Comunidad Autónoma de Madrid. 1 2 I. BISWAS, T. GÓMEZ, AND V. MUÑOZ M ′ X′,x′,r′,ξ′, then there is an isomorphism between X and X ′ sending x to x′, and also r = r′. In other words, if we are given MX,x,r,ξ as an abstract variety, we can recover the curve X, the point x, and the rank r. We will first prove some facts about the geometry of this moduli space M := MX,x,r,ξ, and, using these, in the last section we prove Theorem 0.1. 1. Forgetful morphism The following lemma relates τ–semistability of a framed bundle with the usual semistability of its underlying vector bundle. Lemma 1.1. There is a constant τ(r) that depending only on the rank r such that for all τ ∈ (0 , τ(r)) the following hold: (1) (E,φ) is τ–semistable ⇒ E is semistable. (2) E is stable ⇒ (E,φ) is τ–stable. (3) Any τ–semistable pair is τ–stable. Proof. The expression |ǫ/r′−1/r|, where ǫ = 0, 1 and r′ is an integer with 0 < r′ < r, takes only a finite number of values, so there is a largest positive number τ(r) such that the following holds: if 0 < τ < τ(r), then (1.1) 0 < τ ∣∣ ǫ r′ − 1 r ∣∣ < 1 r! . Assume that (E,φ) is τ–semistable but E is not semistable as a vector bundle. This means that there is a proper subbundle E′ such that d r < d′ r′ ≤ d r + τ (ǫ(E,φ) r′ − 1 r ) , where d = degE and d′ = degE′. But this is impossible because the slopes of all subbundles of E are in (1/r!)Z. Hence d′/r′ ≥ d/r + 1/r!, which contradicts (1.1). Now assume E is stable but (E,φ) is not τ–stable. There is a proper subbundle E′ such that d r > d′ r′ ≥ d r + τ (ǫ(E′, φ) r′ − 1 r ) , but this pair of inequalities again contradict (1.1). Finally, if (E,φ) is τ–semistable, then for all proper subbundles E′ of E, d′ r′ ≤ d r + τ (ǫ(E′, φ) r′ − 1 r ) . We note that d ′ r 6= d r + τ ( ǫ(E,φ) r − 1 r ) because d′/r′ and d/r belong to (1/r!)Z, and τ ( ǫ(E,φ) r − 1 r ) does not lie in (1/r!)Z. Hence any τ–semistable pair is τ–stable. This completes the proof of the lemma. Henceforth, we will always assume that 0 < τ < τ(r). Let M denote the moduli space of semistable vector bundles E over X of rank r with ∧r E = ξ. Since the underlying vector bundle of a τ–semistable pair is semistable, we have a forgetful morphism