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 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:E_x\to 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.