Asymptotic geometry of moduli spaces of curves

• Applicant: Professor Roger Bielawski
• Subject Area: Mathematics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 339871735

Several important geometric structures can be constructed and studied via their twistor space, i.e. as a parameter space of (real) rational curves in a complex manifold. Naturally arising examples of these geometries, which include hyperkaehler metrics, are of great significance in several branches of mathematics and mathematical physics: e.g. quiver varieties in representation theory, Hitchin's moduli spaces in algebraic geometry and integrable systems theory, gauge-theoretic moduli spaces of monopoles and instantons in mathematical physics. Many of the above-mentioned examples can also be constructed as moduli spaces of higher genus curves in a complex 3-fold equipped with an antiholomorphic involution.The aim of this project is to investigate the geometry of such moduli spaces; more precisely, we propose to study the global and asymptotic differential geometry of smooth loci of Hilbert schemes of real algebraic curves (satisfying certain stability conditions) in complex (non-compact) manifolds, particularly in 3-folds.The main research goals are as follows:1) to obtain new examples of several interesting differential-geometric structures, including hyperkaehler and quaternion-Kaehler metrics and pluricomplex structures.2) to investigate global properties of these new examples, in particular their completeness.3) to study the asymptotic geometry and geometric compactifications of manifolds arising as such moduli spaces of curves via compactification of the relevant $3$-fold. We hope that this approach will allow to answer open questions (e.g. the Sen and the Vafa-Witten conjectures) related to the asymptotic behaviour of physically relevant manifolds, such as monopole or Hitchin's moduli spaces.

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