Wall-crossing and hyperkähler geometry of moduli spaces

• Applicant: Claudio Meneses, Ph.D.
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441868307

Moduli spaces of parabolic Higgs bundles on compact Riemann surfaces are a vast and geometrically rich class of non-compact hyperkähler manifolds. While originally described as infinite-dimensional hyperkähler quotients parametrizing equivalence classes of solutions to Hitchin's equations, they also admit a parallel and independent construction in terms of the powerful algebro-geometric techniques of Geometric Invariant Theory. An interesting geometric phenomenon that characterizes them is their dependence on stability parameters called parabolic weights, which are points in a given convex polytope with a prescribed combinatorial structure, leading to the idea of "wall-crossing". Although wall-crossing for moduli spaces of parabolic Higgs bundles on Riemann surfaces is fairly understood from the algebro-geometric perspective, understanding its relation to their hyperkähler structures is an analytic problem that could be better approached and understood from a slightly different perspective. The construction of complex-analytic geometric models for these moduli spaces in the genus 0 case is a recent proposal of the author which is specially designed to address analytic questions related to their geometry and topology, providing natural "charts" and "coordinates" that allow their reformulation in a more convenient setup. Moreover, the explicit nature of the geometric models provides a comprehensive understanding of wall-crossing in a relatively elementary formulation. The goal of this project is to explore several analytic structures on moduli spaces of parabolic Higgs bundles in genus 0 in terms of their geometric models, and most prominently, to provide an explicit description of their asymptotic hyperkähler geometry at infinity, and their explicit behavior under variations of parabolic weights and wall-crossing. These ideas would provide an alternative and complementary perspective towards the understanding of the complex geometry of these moduli spaces, which would broaden their understanding and applicability.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity