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 hyperkähler 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 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 global geometry of Kodaira moduli spaces via twistor methods; more precisely, the project aims to investigate completeness and the asymptotic geometry of natural metrics on manifolds arising as smooth loci of Hilbert schemes of real algebraic curves (satisfying certain stability conditions) in complex (non-compact) manifolds, particularly in 3-folds.