Singular Hermitian metrics for vector bundles and extension of twisted canonical sections

• Applicant: Professor Mihai Paun, Ph.D.
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 456461425

The project we are presenting here is structured in two main themes whose common denominator is the systematic treatment of fundamental problems arising in algebraic geometry via analytic methods. This point of view has a very long and successful history, starting e.g. with Riemann characterization of abelian varieties and its far reaching generalization due to Kodaira (the famous embedding theorem). Also, the proposed project has extensive training components. We hope that the elegance and the strength of analytic methods will continue to attract many bright students during the following years. We expect to achieve significant progress in the research directions we will now briefly evoke.In the first part, we are aiming at the generalization of the celebrated Ohsawa-Takegoshi theorem in the context of singular varieties. This is a formidable, widely open problem whose solution would have an important impact on algebraic geometry. The main object of study in the second part is the push-forward of relative canonical bundle of an algebraic fiber space. In many important contexts this sheaf admits a singular Hermitian metric with positive curvature. The solution of the specific problems we intend to consider here could follow from a better understanding of the positivity and regularity properties of the metric structure.

• DFG Programme: Research Grants