Global invariants for complex varieties with isolated singularities and applications

针对国家自然科学基金会0802803号提案《具有孤立奇点的复变环的全局不变量及其应用》，邱晓华的研究提案包括七个方面的项目：(1)复变环和C^N上的复几何的基本问题；(2)高阶Bergman函数及其显式计算；(3)仅具有商奇性的复变环中完全Reinhardt域之间双全纯映射的显式计算；(4)仅具有商奇性的复变环中完全Reinhardt域的无穷多个连续数值不变量的构造；(5)完全Reinhardt域或模空间的构造复簇中的严格伪凸域。(6)复簇中完备Reinhardt域的模空间的几何。(7)紧致强伪凸CR流形之间的CR态射的刚性。Yau为仅具有孤立正规奇点的复簇中域引入了高阶Bergman函数。这些Bergman函数在双全纯映射下是不变的。他打算利用这些Bergman函数来研究复簇中域的双全纯等价问题。Yau观察到，他的高阶Bergman函数对簇中两个完全Reinhardt域之间的双全纯映射施加了很大的限制，由此可以显式地确定这些双全纯映射。他发展了一种新的方法，在同一簇中的完备Reinhardt域上构造一族无限个连续的数值不变量。他证明了这一无穷无尽的连续数值不变量族实际上是簇中所有严格伪凸完备Reinhardt域的集合，或者是簇中所有具有实解析边界的伪凸完备Reinhardt域的集合。特别地，簇中这些域的模空间被显式地构造为这一完整的数值不变量族的映象。他用一个A_1变种的具体例子说明了这是如何起作用的。众所周知，A_1-簇是C^2上2阶循环群的商。Yau证明了A_1-簇中完备Reinhardt域的模空间与C^2中相应完备Reinhardt域的模空间重合。由于他的完备族的数值不变量是可计算的，他解决了C2中一大类域的双全纯等价问题。他提议继续他对任何商奇点的研究。他还建议研究同维强伪凸紧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，且目标流形的余维相对较小，他证明了非常数CR态射必然是CR双全纯。他计划证明两个相同维度的强伪凸紧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，奇点理论在数学主流和科学的许多分支中都扮演着重要的角色。例如，在射影流形上取一个锥体，就可以得到一个在原点的孤立奇点。射影流形的分类可以通过孤岛奇点的分类来实现，因此在某种意义上，代数几何包含在奇点理论中。黑洞也可以被视为我们宇宙的奇点。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，理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的全球结构的新方法。Yau的NSF赠款被用来部分支持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杂分析和复杂几何研讨会。有11位发言者，其中两位是女性。邱松英和李松英于2007年10月5-6日在德保罗大学为AMS会议组织了“分析与CR几何”特别会议。有23位演讲者。这位私人侦探是高中生张乐天的顾问，张乐天被选为英特尔科学竞赛的最后40名。邱和张的论文发表在《数学研究通讯》上。目前，私家侦探正在为参加英特尔科学竞赛的两名高中生提供建议。其中一人是一名女学生。Yau有一名女学生今年完成了博士学位，他还有7名研究生正在攻读博士学位，其中一名是非裔美国人。Yau与芝加哥州立大学(一家非博士授予机构，非裔美国学生占多数)和弗吉尼亚州约翰·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.