Topology of the moduli spaces of Higgs bundles over abelian varieties

Abstract. Let G be a complex reductive group and A be an Abelian variety of dimension d over $\mathbb{C}$. We determine the Poincaré polynomials and also the mixed Hodge polynomials of the moduli space $\mathcal{M}_{A}^{H}(G)$ of G-Higgs bundles over A. We show that these are normal varieties with symplectic singularities, when G is a classical semisimple group. For $G=GL_{n}(\mathbb{C})$, we also compute Poincaré polynomials of natural desingularizations of $\mathcal{M}_{A}^{H}(G)$ and of G-character varieties of free abelian groups, in some cases. In particular, explicit formulas are obtained when dim A=d=1, and also for rank 2 and 3 Higgs bundles, for arbitrary d>1.