Hodge polynomials of the SL(2,C)-character variety of an elliptic curve with two marked points

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2,C). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.

Hodge polynomials of the moduli spaces of rank 3 pairs

Let $X$ be a smooth projective curve of genus $ggeq 2$ 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 $phi:E_2 to E_1$. 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 $rk(E_1)=3$, $rk(E_2)=1$, using the theory of mixed Hodge structures. This gives in particular the Poincare 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.

Hodge-Deligne polynomials of SL(2,C)-character varieties for curves of small genus

We compute the Hodge-Deligne polynomials of the moduli spaces of representations of the fundamental group of a complex surface into SL(2,C), for the case of small genus g, and allowing the holonomy around a fixed point to be any matrix of SL(2,C), that is Id, -Id, diagonalisable, or of either of the two Jordan types. For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the Hodge-Deligne polynomial under fibrations.