Positivity in Complex Geometry

University Of Connecticut, Storrs CT

Investigators

Abstract

This project studies fundamental problems on the characterization of positivity in complex geometry. Complex geometry is a branch of geometry that concerns objects defined over the complex number field. The positivity implies the existence of plenty of global objects called holomorphic sections, which play a central role in geometry. The research will develop and bring in techniques from different fields including differential geometry, algebraic geometry, and nonlinear partial differential equations. Some of the answers will shed light on the deep connection between the complex geometric structure and algebraic structure. The proposal will also generate appropriate research problems for graduate students and junior researchers. Positivity of canonical bundles in complex geometry gives rise to intriguing connections between complex geometry, nonlinear partial differential equations, and algebraic geometry. The PI will undertake three research projects to better understand the positivity of canonical bundles from a differential geometric approach using holomorphic curvature, Kahler-Einstein metrics, and the complex Monge-Ampere equation. A class of fully nonlinear equations generalizing the complex Monge-Ampere equation will be also investigated. The first project will study the positivity of canonical bundles by holomorphic curvature, hyperbolicity, and the Bergman metric. The second project will investigate the asymptotic expansion of the canonical metric on the positive logarithmic canonical bundle. The third project seeks to establish a Liouville-type theorem for some fully-nonlinear equations.