Invariants in Several Complex Variables and Complex Geometry

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The study of invariant geometries is of fundamental importance in mathematics. Such geometries arise naturally in areas of mathematics such as several complex variables, partial differential equations (PDE), and algebraic, complex, and differential geometry. This project investigates a particular geometry – CR geometry -- that arises in the study of several complex variables and complex geometry. It has deep connections with current topics in mathematical physics, including quantum field theory, general relativity, and string theory, as well as applications in, e.g., systems engineering and control theory. For example, the study of obstruction flatness has a direct link to the equations of motion in conformal gravity. The notion of Kähler-Einstein metric is highly relevant for these investigations; such metrics are measurements of distance and angle in higher-dimensional spaces (manifolds), which curve according to Einstein’s equations of relativity in vacuum. One way in which CR manifolds arise is as the boundaries of regions in high-dimensional complex spaces which curve in a well-behaved fashion in complex directions; such regions are known as pseudoconvex domains. Techniques to be used in this project come from a variety of mathematical areas: complex analysis/geometry, PDE, and differential geometry. At the same time, techniques and tools developed in this project will influence these areas as well. The project will also provide interesting research topics for graduate students and postdocs. The seminar activity resulting from the project will inspire and stimulate both students and other researchers. The goal of this project is to study geometric, analytic, and algebraic aspects of real submanifolds in complex varieties, together with their symmetries. Such aspects include, for instance, the geometric consequences of global vanishing on a compact CR manifold of a higher order local invariant that arises as an obstruction in the study of a complex Monge--Ampere equation. In three dimensions, this invariant coincides with the trace on the boundary of the log-term in an asymptotic expansion of the Bergman kernel. Solutions to this Monge-Ampere equation give rise to complete Kähler-Einstein metrics with negative curvature, and a major goal of the project is to characterize domains and complex manifolds in terms of properties of various metrics and kernels, and their relations. As an example, one aim is to understand what assumptions ensure that the Bergman metric on a domain is Kähler-Einstein if and only if the domain is biholomorphically equivalent to the ball. Another topic to be considered is the existence of CR umbilical points on compact CR 3-manifolds. Umbilical points on a manifold are locations where the manifold exhibits an unexpectedly high degree of symmetry and smoothness. It is not known whether such points always exist on bounded strictly pseudoconvex domains in complex 2-dimensional space; this question remains open if the domain is diffeomorphic to the ball. Finally, the project considers existence, uniqueness, and regularity questions for CR maps. These include the study of CR maps between generic submanifolds of infinite type and the recently discovered phenomenon that for infinite type hypersurfaces the biholomorphic, formal, and smooth CR equivalence classifications are distinct. These investigations are expected to shed new light on the nature of CR maps between infinite type manifolds, and to lead to a better understanding of the Pfaffian systems arising in this context. In conclusion, this project will significantly enhance the understanding of the role of invariant objects and symmetries in several complex variables and complex geometry, which in turn will increase the utility of the theory in physical models and applications. 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.