Torelli theorem for the moduli spaces of pairs

Abstract

Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,\phi) over X consists of an algebraic vector bundle E over X and a holomorphic section \phi of E. There is a concept of stability for pairs which depends on a real parameter \tau. Here we prove that the third cohomology groups of the moduli spaces of \tau-stable pairs with fixed determinant and rank at least two are polarised pure Hodge structures, and they are isomorphic to H^1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank at least two and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H^1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.