Global invariants for complex varieties with isolated singularities and applications

NSF提案0802803“具有孤立奇点的复杂多样性的全局不变量及其应用”摘要 丘的研究计划包括七个领域的项目：（1）复几何中关于复簇和复几何中的基本问题。 C^N和它们的关系。(2)高阶Bergman函数及其显式计算(3)完备Reinhardt间双全纯映射的显式计算 域在复杂的品种只有商奇点。(4)无穷多连续数值不变量的构造 复簇中的完备Reinhardt域 奇点(5)完备Reinhardt整环模空间的构造 复簇中的严格伪凸域(6)中完备Reinhardt域的模空间的几何 复杂的品种。(7)紧强伪凸CR之间CR态射的刚性 丘引入高阶Bergman函数域的复杂品种，只有孤立的正常奇点。这些Bergman函数在双全纯映射下是不变的。他打算利用这些伯格曼函数来研究问题的双全纯等价域的复杂品种。Yau观察到他的高阶Bergman函数对两个完备Reinhardt域之间的双全纯映射施加了许多限制，由此可以显式地确定这些双全纯映射。他开发了一种新的技术，以建立一个家庭的无限数量的连续数值不变量的完整Reinhardt域躺在同一品种。他表明，这个家庭的无限数量的连续数值不变量实际上是一个完整的不变量集的所有严格pseudoconvular完整的莱因哈特域的品种或集的所有pseudoconvular完整的莱因哈特域与真实的解析边界的品种。特别是，这些域的模空间的各种明确的构造作为图像的这个完整的家庭的数值不变量。他用一个具体的A_1-variety的例子来说明这是如何工作的。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。由于他的完整家庭的数值不变量是可计算的，他解决了biholomorphically等价问题的一个大家庭的领域在C2。他建议继续他的工作，任何商奇点。他还建议研究刚性问题的CR态射之间的两个强伪凸紧凑CR嵌入流形的相同尺寸。如果维度至少是5和余维的目标流形是相对较小的，他表明，非常数CR态射必然是CR biholomorphisms。他计划证明最一般的刚性定理CR态射之间的两个强伪凸紧凑CR嵌入CR流形的相同尺寸。 众所周知，奇点理论在数学的主流以及科学的许多分支中起着至关重要的作用。例如，通过在射影流形上取一个锥，可以在原点得到一个孤立的奇点。射影流形的分类可以通过孤立奇点的分类来实现，因此在某种意义上代数几何也包含在奇点理论中。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（例如桌子角）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。该项目提出了一种新的方法来理解奇点的整体结构。丘成桐的国家科学基金会赠款用于部分支持中西部复杂分析和复杂几何研讨会，2007年4月13日至15日在伊利诺伊大学芝加哥。有11名发言者，其中两名是女性。丘和李松英为2007年10月5日至6日在德保罗大学举行的AMS会议组织了一个特别会议"分析和CR几何“。有23位发言人。私家侦探是高中生张乐天的导师，他在英特尔科学竞赛中被选为40强。Yau和Zhang的论文发表在Math Research Letter上。目前，P.I.正在为两名高中生的英特尔科学竞赛提供建议。其中一个是女学生。丘有一个女学生完成了博士学位。今年，他还有7名研究生在攻读博士学位，其中一个是非裔美国人丘先生与芝加哥州立大学（非博士）建立了研究和教育合作关系。以非洲裔美国学生为多数的授予机构）和弗吉尼亚州的约翰泰勒社区学院。

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.