Hodge polynomials of the moduli spaces of rank 3 pairs

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(E_1, E_2, \phi)}$$\end{document} on X consists of two holomorphic vector bundles E1 and E2 over X and a holomorphic map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi \colon E_{2}\to E_{1}}$$\end{document} . 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) = 3, rk(E2) = 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.