Costa Rican-German Collaboration in Research: Fundamental groups, character varieties, and Higgs bundles

• Applicant: Professor Dr. Alexander Schmitt
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 524596398

Character varieties make it possible to study finitely generated groups with geometric methods. These geometric methods may be of analytic or algebraic origin. If one would like to apply algebro geometric tools, one faces the problem that character varieties are affine and, so, the toolbox for studying them is limited. However, if the group under investigation is the fundamental group of a projective algebraic manifold, then the non-abelian Hodge correspondence - building on seminal work by Hitchin - establishes a homeomorphism between the character variety and a moduli space of Higgs bundles. The latter is endowed with a projective morphism to an affine variety and a C*-action and, therefore, has a richer geometry. This has been the basis for several important and interesting developments in algebraic geometry. We would like to investigate character varieties, Higgs bundles, and fundamental groups in new or emerging directions. This includes the study of representations of fundamental groups of elliptic surfaces and the corresponding moduli spaces of Higgs bundles, the computation of homotopy groups of moduli spaces of Higgs bundles on curves and the description of their properties, and the geometry of moduli spaces of circular Higgs bundles on curves. In positive characteristic, we would like to study invariants of the locus of Frobenius destabilized bundles and the cohomology of the stack of vector bundles with integrable lambda-connections on curves.

• DFG Programme: Research Grants
• International Connection: Costa Rica
• Partner Organisation: Consejo Nacional de Rectores
• Cooperation Partner: Professor Dr. Ronald Alberto Zuniga-Rojas