Geometry of the moduli space of Higgs bundles

TL;DR

This thesis explores the topology, geometry, and algebraic structure of the moduli space of rank 2 Higgs bundles over Riemann surfaces, providing new compactification techniques, cohomology descriptions, and homotopy stability results.

Contribution

It offers a detailed analysis of the moduli space's topology, introduces a compactification via symplectic cutting, and proves the Mumford conjecture using algebraic geometric methods.

Findings

Vanishing of intersection numbers via Morse flow analysis

Generators and explicit description of the cohomology ring

Homotopy equivalence of the resolution tower's limit with the classifying space

Abstract

This thesis contains work which appeared in several papers. Additionally to the results in the papers it contains a detailed introduction and some further proofs and remarks. The dissertation gives a description of the topology and symplectic and algebraic geometry of Hitchin's hyperkaehler moduli space M of rank 2 Higgs bundles with fixed determinant of odd degree over a fixed Riemann surface. After the long introduction it describes a compactification of M in great detail, using symplectic cutting (math.AG/9804083). Examining the downward Morse flow of a natural circle action on M it shows the vanishing of intersection numbers (math.AG/9805071). Examining the upward Morse flow it explains a set of generators of the cohomology ring and a conjectured explicit description of the cohomology ring (which was proven in math.AG/0003094). Then finally it introduces the resolution tower for M, and shows that its direct limit is homotopically equivalent with the classifying space of the gauge group. In turn it yields another proof of the generation theorem (as in math.AG/0003093) and also yields a purely algebraic geometric proof of the Mumford conjecture about the cohomology ring of the moduli space of rank 2 stable bundles on curves. It finishes by proving homotopy stabilizations in the resolution tower analogously to the Atiyah-Jones conjecture.