Invariant Metrics on Complex Manifolds

As one of the oldest branches in mathematics, geometry concerns the properties of spaces related to sizes, shapes, and distances of objects. Modern mathematics makes use of powerful analytic tools such as calculus to study geometry, which creates the field differential geometry. Differential geometry has been one of the most active fields in mathematics, with interconnections and applications to other mathematical fields like topology and differential equations, and also with physics - in Einstein's theory of relativity, gravity is the curvature of a metric.

This project investigates the complex aspects of differential geometry, especially the important metrics on complex spaces called invariant metrics, with tools from complex analysis and differential equations. A main goal is to understand the shape or curvature of the invariant metrics, and their underlying mathematical structures. This study of complex geometry interplays with the algebraic sides of mathematics, including algebraic geometry, representation theory, and number theory, and has applications to physics such as string theory and to scientific computing via conformal maps. The project includes training through research involvement for graduate students.

There are four classical invariant metrics on complex manifolds, the Bergman metric, the Caratheodory-Reiffen metric, the Kobayashi-Royden metric, and the complete Kähler-Einstein metric with negative scalar curvature. They are invariant under biholomorphisms, and hence depend only on the underlying complex structure of the complex manifold. The study of invariant metrics give rise to intriguing connections between differential geometry, several complex variables, topology, nonlinear partial differential equations, and algebraic geometry.

For instance, the invariant metrics are closely related to several long-standing conjectures such as that a simply-connected, negatively curved, complete Kähler manifold must be biholomorphic to a bounded complex Euclidean domain. The Kähler-Einstein metric and the Kobayashi-Royden metric play key roles in understanding the positivity of canonical bundle.

Combining methods from partial differential equations and several complex variables, the project aims to provide deeper understanding on the geometry of the invariant metrics, for a large class of complex manifolds, including complete noncompact Kähler manifolds, compact complex manifolds, and quasi-projective manifolds. The research will also lead to applications in algebraic geometry and number theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.