Cohomology of U(2,1) representation varieties of surface groups

TL;DR

This paper computes the equivariant cohomology of U(2,1) and SU(2,1) Higgs bundle moduli spaces on Riemann surfaces using Morse theory, revealing new topological insights into their character varieties.

Contribution

It provides new calculations of equivariant Poincaré polynomials and a Kirwan surjectivity theorem for non-coprime cases, extending previous work on Higgs bundles.

Findings

Calculated equivariant Poincaré polynomials for Higgs bundle spaces

Established Kirwan surjectivity in non-fixed determinant cases

Described Torelli group action on cohomology

Abstract

In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2,1) and SU(2,1) Higgs bundles with fixed Toledo invariant. In the non-coprime case this gives new results about the topology of the U(2,1) and SU(2,1) character varieties of surface groups. The main results are a calculation of the equivariant Poincare polynomials, a Kirwan surjectivity theorem in the non-fixed determinant case, and a description of the action of the Torelli group on the equivariant cohomology of the character variety. This builds on earlier work for stable pairs and rank 2 Higgs bundles.