Torelli theorem for the moduli space of framed bundles

Abstract

Let X be an irreducible smooth complex projective curve of genus g ≥ 2, and let x ∈ X be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, φ), where E is coherent sheaf on X of rank r and fixed determinant ξ_r, and φ : E_x→ 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^τ. We prove a Torelli theorem for M^τ, 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^τ.