HODGE POLYNOMIALS OF THE MODULI SPACES OF TRIPLES OF RANK (2, 2)

Abstract

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple (E1, E2, ϕ) on X consists of two holomorphic vector bundles E1 and E2 over X and a holomorphic map ϕ : E2 → E1. There is a concept of stability for triples which depends on a real parameter σ. In this paper, we determine the Hodge polynomials of the moduli spaces of σ-stable triples with rk(E1) = rk(E2) = 2, using the theory of mixed Hodge structures (in the cases that these moduli spaces are smooth and compact). This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we also give the Hodge polynomial of the moduli space of even degree rank 2 stable vector bundles.