Hodge polynomials of the moduli spaces of rank 3 pairs

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 $\text{rk}(E_1)=3$, $\text{rk}(E_2)=1$, using the theory of mixed Hodge structures. This gives in particular the Poincar'e 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.