AEI-DFG: Topology of moduli of Hodge bundles, G-Higgs bundles and character varieties of surface groups

• Applicant: Professor Dr. Jochen Heinloth
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541511946

Since their introduction by Hitchin more than 35 years ago, moduli spaces of Higgs bundles over a compact Riemann surface have been of tremendous interest in geometry, topology, theoretical physics and their role in the Langlands program intensified this interest also in algebraic geometry and number theory. In this project we study the topology of the closely related moduli spaces of G-Higgs bundles on compact Riemann surfaces, moduli spaces of Hodge bundles and character varieties of surface group representations in real Lie groups G. These spaces have been studied intensively in the case of the complex general linear group, but for general real groups G even the most basic invariants have not been determined. Our approach to this problem relies on a combination of the recent proof of Lie group theoretic analogues of the Milnor-Wood inequality for G-Higgs bundles for groups of Hermitian type, with a stack-theoretic point of view on the space of stability conditions on moduli stacks of Hodge bundles. Surprisingly this indicates that the strategy used to compute cohomology and even motives of stacks of Hodge bundles for the general linear group, and which allowed us to determine topological invariants of the moduli spaces of U(p,q)-Higgs bundles, admits a Lie group theoretic variant that at the same time could provide resolutions of the usually singular moduli spaces.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partner: Professor Oscar García-Prada