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 is an element of X be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, phi), where E is coherent sheaf on X of rank r and fixed determinant xi, and phi: E(x) > C(r) is a non-zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter tau > 0, which gives rise to the moduli space of tau-semistable framed bundles M(tau). We prove a Torelli theorem for M(tau), for tau > 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(tau).