Hodge polynomials of the moduli spaces of rank 3 pairs

Abstract

Let X be a smooth projective curve of genus g >= 2 over the complex numbers. A holomorphic triple (E(1), E(2), phi) on X consists of two holomorphic vector bundles E(1) and E(2) over X and a holomorphic map phi: E(2) -> E(1). There is a concept of stability for triples which depends on a real parameter sigma. In this paper, we determine the Hodge polynomials of the moduli spaces of sigma-stable triples with rk(E(1)) = 3, rk(E(2)) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincare polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.