Topology of the moduli spaces of Higgs bundles over abelian varieties

TL;DR

This paper computes topological invariants like Poincaré and mixed Hodge polynomials for moduli spaces of G-Higgs bundles over abelian varieties, revealing their geometric structure and singularities.

Contribution

It provides explicit formulas for topological invariants of Higgs bundle moduli spaces over abelian varieties, including desingularizations and character varieties, for various groups and dimensions.

Findings

Moduli spaces are normal with symplectic singularities for classical semisimple G.

Explicit formulas for Poincaré and mixed Hodge polynomials in low-dimensional cases.

Desingularizations of moduli spaces have computable topological invariants.

Abstract

Abstract. Let G be a complex reductive group and A be an Abelian variety of dimension d over $\mathbb{C}$. We determine the Poincar\'e 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 Poincar\'e 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.