Geometry of Invariants and Mappings in Several Complex Variables

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The study of invariant geometries is fundamental in mathematics. Invariant geometries arise in many different areas, such as several complex variables, partial differential equations, and algebraic, complex, and differential geometry. This research project investigates a particular geometry that arises in the study of several complex variables and 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. The ideas and techniques needed for this study are drawn from a variety of mathematical areas, including complex analysis and geometry, partial differential equations, and differential geometry; at the same time, the tools developed in this project will inform these areas as well. The project also provides interesting research topics for graduate students and postdocs. The goal of this mathematics research project is to study geometric, analytic, and algebraic aspects of generic real submanifolds in complex varieties (more generally, of CR structures) and their mappings. The work will investigate the geometric consequences of global vanishing on a compact CR manifold of a higher order local invariant that arises as the obstruction to smooth extension to the boundary of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation. In three dimensions, this invariant coincides with the trace on the boundary of the log-term in the asymptotic expansion of the Bergman kernel; hence this problem is also connected with a strong form of the Ramadanov Conjecture. The PI will study the existence of CR umbilical points on compact CR 3-manifolds, especially a revised version of a question by Chern-Moser, which asks if such points always exist on bounded strictly pseudoconvex domains in complex 2-space; although the answer is 'no' in general, the question is still open if the domain is diffeomorphic to the ball. The PI will also consider questions regarding existence, uniqueness, and regularity of CR maps, as well as related questions. He will consider the context of CR maps of a Levi nondegenerate hypersurface into a nondegenerate hyperquadric of higher dimension. This study will enhance our understanding of the CR submanifold structure of the hyperquadrics, which constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The work should also provide insight into how the local CR geometry of such hypersurfaces affects various properties, such as notions of nondegeneracy and rigidity, of their CR maps. The PI will also continue study of CR maps between generic submanifolds of infinite type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. He will focus on understanding the recently discovered phenomenon that for infinite type hypersurfaces the biholomorphic, formal, and smooth CR equivalence classifications are different. These investigations are expected to shed new light on the nature of CR maps between infinite type manifolds and lead to a better understanding of the Pfaffian systems arising in this context.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.