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

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 Poincaré 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.