Global invariants for complex varieties with isolated singularities and applications

(1)复杂几何中关于复杂变异和C^N及其关系的基本问题；(1)关于复杂变异和C^N及其关系的基本问题；(2)高阶Bergman函数及其显式计算。(3)仅商奇点复变中完全Reinhardt域间生物全纯映射的显式计算。(4)仅商奇点的复变完全Reinhardt域无穷多连续数值不变量的构造。(5)复变中完全Reinhardt域或严格伪凸域的模空间构造。(6)复变中完全Reinhardt域的模空间几何。(7)紧强伪凸CR流形间CR态射的刚性。Yau在只有孤立正规奇点的复变域中引入了高阶Bergman函数。这些Bergman函数在生物全纯映射下是不变的。他打算利用这些Bergman函数来研究复变域中的生物全纯等价问题。Yau注意到他的高阶Bergman函数在一个变量的两个完全Reinhardt域之间的生物全纯映射上施加了很多限制，这些生物全纯映射可以从中明确地确定。他发展了一种新的技术，在完全Reinhardt域上构造无穷多个连续数值不变量族。他证明了这个无穷多个连续数值不变量族实际上是变量中所有严格伪凸完全Reinhardt域的集合或变量中所有具有实解析边界的伪凸完全Reinhardt域的集合的不变量的完全集合。特别地，这些域中的模空间被明确地构造为这个完整的数值不变量族的像。他用一个具体的a_1品种的例子说明了这是如何工作的。众所周知，a_1是C^2上的2阶循环群的商。Yau证明了a_1变化中的完全Reinhardt域的模空间与C^2中相应的完全Reinhardt域的模空间重合。由于他的完整的数值不变量族是可计算的，他解决了C2中一个大族域的生物全纯等价问题。他打算继续他对任何商奇点的研究。他还提出了两个相同维数的强拟凸紧CR可嵌入流形之间的CR态射的刚性问题。如果维数至少为5且目标流形的余维数相对较小，他证明了非常数CR态必然是CR生物全纯。他计划证明两个相同维数的强伪凸紧CR可嵌入CR流形之间的CR态射的最一般刚性定理。众所周知，奇点理论在数学的主流和科学的许多分支中都起着至关重要的作用。例如，在射影流形上取一个圆锥，我们可以在原点得到一个孤立的奇点。投影流形的分类可以通过孤立奇点的分类来实现。因此，从某种意义上说，代数几何包含在奇点理论中。黑洞也可以被看作是我们宇宙的奇点。我们在日常生活中会遇到奇点。任何不光滑的东西（例如桌角）都可以被认为是奇点。因此，了解奇点结构对我们来说是非常重要的。这个项目提出了一种理解奇点全局结构的新方法。丘的NSF资助部分用于支持2007年4月13日至15日在芝加哥伊利诺伊大学举行的中西部复分析和复几何研讨会。共有11位发言人，其中两位是女性。2007年10月5日至6日，丘成桐和李松英在德保罗大学为AMS会议组织了“分析和CR几何”专题会议。有23位发言人。这位私家侦探是在英特尔科学竞赛中进入40强的高中生张乐天的指导老师。Yau和Zhang的论文发表在《数学研究快报》上。目前，私家侦探正在指导两名高中生参加英特尔科学竞赛。其中一个是女学生。丘成桐有一名女学生今年完成了博士学位。他还有7个研究生在攻读博士学位，其中一个是非裔美国人。丘与芝加哥州立大学（一所非裔美国学生占多数的非博士学位授予机构）和弗吉尼亚的约翰泰勒社区学院建立了研究和教育合作关系。

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.