Global invariants for complex varieties with isolated singularities and applications

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Abstract for NSF Proposal 0802803 "Global invariants for complex varieties with isolated singularities and applications" Yau's research proposal contains projects in seven areas: (1) Fundamental problems in complex geometry on complex varieties and on C^N and their relationship. (2) Higher order Bergman functions and their explicit computation. (3) Explicit computation of biholomorphic maps between complete Reinhardt domains in complex varieties with only quotient singularities. (4) Construction of infinitely many continuous numerical invariants for complete Reinhardt domains in complex varieties with only quotient singularities. (5) Construction of the moduli spaces of complete Reinhardt domains or strictly pseudoconvex domains in complex varieties. (6) Geometry of the moduli spaces of complete Reinhardt domains in complex varieties. (7) Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds. Yau introduces higher order Bergman functions for domains in complex varieties with only isolated normal singularities. These Bergman functions are invariant under biholomorphic maps. He intends to use these Bergman functions to study the problem of biholomorphic equivalence of domains in complex varieties. Yau observes that his higher order Bergman functions put a lot of restriction on biholomorphic maps between two complete Reinhardt domains in a variety, from which these biholomorphic maps can be determined explicitly. He develops a new technique to construct a family of infinite number of continuous numerical invariants on complete Reinhardt domains lying in the same variety. He shows that this family of infinite number of continuous numerical invariants is actually a complete set of invariants for either the set of all strictly pseudoconvex complete Reinhardt domains in the variety or the set of all pseudoconvex complete Reinhardt domains with real analytic boundaries in the variety. In particular, the moduli spaces of these domains in the variety are constructed explicitly as the images of this complete family of numerical invariants. He illustrates how this works in a concrete example of A_1-variety. It is well known that A_1-variety is the quotient of cyclic group of order 2 on C^2. Yau proves that the moduli space of complete Reinhardt domains in A_1-variety coincides with the moduli space of the corresponding complete Reinhardt domains in C^2. Since his complete family of numerical invariants are computable he has solved the biholomorphically equivalent problem for a large family of domains in C2. He proposes to continue his work for any quotient singularities. He also proposes to study the rigidity problem of CR morphisms between two strongly pseudoconvex compact CR embeddable manifolds of the same dimension. If the dimension is at least five and the codimension of the target manifold is relatively small, he shows that non-constants CR morphisms are necessarily CR biholomorphisms. He plans to prove the most general rigidity theorem of CR morphisms between two strongly pseudoconvex compact CR embeddable CR manifolds of the same dimension. It is well known that singularities theory plays an essential role in main stream of mathematics as well as many branches in Science. For example, by taking a cone over a projective manifold, one can get an isolated singularity at the origin. The classification of projective manifolds can be achieved via the classification of islated singularites. Therefore in some sense algebraic geometry is contained in the theory of singularities. The Black Holes can also be viewed as singularities of our universe. We encounter singularities in our daily life. Anythings which are not smooth (for example table corner) can be think of as singularities. Hence it is very important for us to understand singularities structures. This project proposes a new way to understand the global structures of singularities. Yau's NSF grant was used to partially support the Midwest Workshop on Complex Analysis and Complex Geometry, April 13--15, 2007 at University of Illinois at Chicago. There were 11 speakers, two of them are female. Yau and Song-Ying Li have organized a Special Session ``Analysis and CR Geometry'' for AMS meeting at De Paul University Oct. 5--6, 2007. There are 23 speakers. The P.I. was the adviser of a high school student Letian Zhang who was selected as the final 40 in the Intel Science competition. Yau and Zhang paper was published in Math Research Letter. Currently the P.I. is advising two high school students for the Intel Science Competition. One of them is a female student. Yau has a female student who finished her Ph.D. this year. He still has 7 graduate students working for their Ph.D., one of them is an African American. Yau has established research and education collaborations with Chicago State University (a non Ph.D. granting institution with African American students as the majority) and John Tyler Community College at Virginia.