Restriction of stable bundles on complete manifolds, Higher order Kodaira-Spencer maps, Moduli of Instantons - analytic approach

• Applicant: Professor Dr. Georg Schumacher, since 10/2024
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 559896723

The aim of this proposal is to conclude studies of moduli spaces and degenerations. Degenerations typically occur at the boundaries of moduli spaces. In a series of articles with N. Buchdahl the degenerations of stable vector bundles on compact Kahler manifolds to polystable bundles were considered. Introducing analytic GIT spaces we constructed the coarse moduli space of polystable bundles containing the moduli space of stable bundles, and showed that the Weil-Petersson metric extended as a positive current, which possesses locally continuous Kähler potentials. Over complex surfaces, in terms of instantons very specic results can be expected, also based on our recent symbolic computations. Ongoing project with I. Biswas: For stable bundles on canonically polarized framed manifolds and their corresponding complete Kahler-Einstein/Hermite-Einstein objects (and the respective moduli spaces) an even better understanding of degenerations are of interest - open questions will be answered. Project with Y.-J. Choi: Twisted relative Hodge sheaves naturally occur, when studying the curvature of Weil-Petersson metrics with unwanted terms to be cancelled by the contributions of higher order Kodaira-Spencer mappings. In the case of moduli canonically polarized manifolds the proposer applied this to the hyperbolicity of the moduli space. The curvature of most general Hodge sheaves was computed with Y.-J. Choi. Properties of higher order Kodaira-Spencer mappings and curvature are the aim of an ongoing joint project.

• DFG Programme: Research Grants