Hodge structures of the moduli spaces of pairs.

Abstract

Let X be a smooth projective curve of genus g >= 2 over C. Fix n >= 2, d epsilon Z. A pair (E, phi) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section phi epsilon H(0)(E). There is a concept of stability for pairs which depends on a real parameter tau. Let M(T) (n, d) be the moduli space of tau-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of M(T) (n, d) are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H(1)(X). This implies a similar result for the moduli spaces of stable vector bundles over X.