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 $${\phi \colon E_{2}\to E_{1}}$$ . 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(E 1) = 3, rk(E 2) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincaré 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.