Global Invariants for CR Geometry and Isolated Singularities

CR 几何和孤立奇点的全局不变量

• 批准号: 0503868
• 负责人: Stephen S. Yau
• 金额: --
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503868&HistoricalAwards=false
• 关键词: Global; Invariants; CR; Geometry; Isolated

AbstractAward: DMS-0503868Principal Investigator: Stephen S.T. YauThis research proposal contains projects in five areas: (1)Fundamental problem in CR geometry, (2) Explicit computation ofCR automorphism groups, (3) Simultaneous embedding problem for aCR family of compact CR manifolds, (4) Deformation of CRmanifolds and deformation of isolated singularities, (5)Holomorphic De Rham cohomology and complex Plateau problem. Yauintroduces a new Bergman function for any strongly pseudoconvexcomplex manifold, which is invariant under biholomorphic maps. Heintends to use this Bergman function to study the CR equivalentproblem of strongly pseudoconvex CR manifolds lying on the samevariety with isolated normal singularities. He developed a newtechnique to define a continuous numerical invariant on stronglypseudoconvex CR manifolds lying on the same variety. He showsthat his invariant varies continuously in R when the CR structureof strongly pseudoconvex CR manifold changes in the variety. Yauobserves that his new Bergman functions put a lot of restrictionon biholomorphic maps between strongly pseudoconvex CR manifold,from which the automorphism groups of the CR manifolds can bedetermined explicitly. He illustrates how this works in aconcrete example. The proposed work in (3) builds on his jointwork with Xiaojun Huang and Hing Sun Luk on simultaneousembedding of CR manifolds. It will be done in collaboration withXiaojun Huang. They intend to use their prior work on furtherextensions to study the most subtle problem on simultaneousembedding of 3-dimensional CR manifolds. Yau plans to use thetechniques developed in (1) and (3) above to understand therelation between the Kuranishi family of CR manifolds and versaldeformation of isolated normal singularities in a more concretemanner. Finally the work in (5) is part of Yau's ongoing projecton holomorphic De Rham cohomology and complex Plateau problem. Hehas shown that the vanishing of holomorphic De Rham cohomology ofthe CR manifold X gives a lot of restriction on the singularitiesof the variety which X bounds. He plans to use his result tosolve the clasical complex Plateau problem for 3-dimensionalstrongly pseudoconvex CR manifolds in 3-dimensional complexEuclidean space.This research proposal has an important theme of unifyingdifferent fields such as CR Geometry, Complex Analysis,Singularities Theory and Algebraic Geometry together. Thephilosophy and technique of the proposal will also be useful inbiotechnology. With the completion of human genomic sequence, itis an urgent task to accurate identify protein coding regions(exons) from genomic sequence. Similar to Yau's proposal, one cantry to find numerical characters of exons which introns do nothave. In this way, one may be able to identify all possible humanproteins. The next important task is to predict the properties ofthese newly discovered proteins. Using the similar techniquedeveloped in Yau's proposal, one may able to represent eachprotein as a point in certain n-dimensional space. Proteinswhich are closed to each other in this space should have similarproperties. In this way, one can predict some properties of newlyfound proteins. Biologists can do experiment to verify theseproperties.

摘要奖：DMS-0503868主要研究者：Stephen S.T. Yau本研究计划包含五个方面的项目：（1）CR几何的基本问题，（2）CR自同构群的显式计算，（3）CR族紧CR流形的同时嵌入问题，（4）CR流形的变形和孤立奇点的变形，（5）全纯De Rham上同调和复Plateau问题。Yau在强伪凸复流形上引入了一个新的Bergman函数，它在双全纯映射下是不变的.他打算利用这个Bergman函数来研究位于同一簇上的具有孤立法奇点的强伪凸CR流形的CR等价问题。他开发了一种新的技术来定义一个连续的数值不变量的stronglyprodoconvex CR流形躺在同一品种。证明了当强伪凸CR流形的CR结构在簇中变化时，他的不变量在R中连续变化。Yau指出，他的新Bergman函数对强伪凸CR流形之间的双全纯映射作了许多限制，由此可以明确地确定CR流形的自同构群。他用一个具体的例子说明了这是如何工作的。在（3）中提出的工作建立在他与Xiaojun Huang和Hing Sun Luk关于CR流形的错误嵌入的联合工作的基础上。它将与黄晓军合作完成。他们打算利用他们以前的工作进一步扩展，以研究最微妙的问题上的错误嵌入的三维CR流形。Yau计划使用上面（1）和（3）中开发的技术，以更具体的方式理解CR流形的Kuranishi族与孤立法向奇点的普遍变形之间的关系。最后，（5）中的工作是丘氏正在进行的全纯De Rham上同调和复Plateau问题项目的一部分。他证明了CR流形X的全纯De Rham上同调的消失对X所界定的簇的奇异性有许多限制。他计划利用他的结果解决三维复欧氏空间中三维强伪凸CR流形的经典复Plateau问题，这一研究计划的重要主题是将CR几何、复分析、奇点理论和代数几何等不同领域统一起来。该提案的理念和技术也将在生物技术方面有所帮助。随着人类基因组测序的完成，从基因组序列中准确识别蛋白质编码区（外显子）成为一项紧迫的任务。类似于Yau的建议，人们可以尝试找到内含子没有的外显子的数字字符。用这种方法，人们就可以鉴定出所有可能的人类蛋白质。下一个重要的任务是预测这些新发现的蛋白质的性质。使用丘的建议中开发的类似技术，人们可以将每个蛋白质表示为某个n维空间中的一个点。 在这个空间中相互接近的蛋白质应该具有相似的性质。通过这种方法，人们可以预测新发现的蛋白质的某些性质。生物学家可以通过实验来验证这些特性。