Hodge polynomials of the moduli spaces of triples of rank (2,2)

Abstract

Let $X$ be a smooth projective curve of genus $g\ge 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,\varphi )$ on $X$ consists of two holomorphic vector bundles $E_1$ and $E_2$ over $X$ and a holomorphic map $\varphi :E_2\to E_1$. 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 $\mathrm{rk}(E_1)=\mathrm{rk}(E_2)=2$, using the theory of mixed Hodge structures (in the cases that they are smooth and compact). This gives in particular the Poincar{é} 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.

28 pages