Mathematical Sciences: Complex Manifolds and Meromorphic Mappings

数学科学：复流形和亚纯映射

• 批准号: 9204037
• 负责人: Bernard Shiffman
• 金额: $ 10.5万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-01 至 1995-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204037&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Complex; Manifolds; Meromorphic

This project continues mathematical research into problems associated with the geometry of several complex variables and mappings defined on regions within such spaces. The work focuses on complex varieties, meromorphic mappings and holomorphic vector bundles. Work on meromorphic mappings seeks generalizations of Hartogs' theorems on separate analyticity and on holomorphic extendibility. Also to be investigated are Hartogs-type results for analytic subvarieties with possible application to the characterization of holomorphic chains of codimension greater than unity. Studies of holomorphic vector bundles concerns singular hermitian metrics on numerically effective line bundles on projective-algebraic manifolds. A continuation of work on cohomology vanishing theorems and sectional curvatures of indecomposable Hermitian-Einstein vector bundles on complex projective space will be carried out. Finally, the geometry of affine algebraic varieties and degree bounds for the division problem for special polynomial ideals will be studied. Complex function theory encompasses the study of differentiable functions of a several complex variables and related classes of functions such as boundary values of holomorphic functions. The subject is highly geometric; many of the problems concern the properties of various sets and how they transform by mappings composed of holomorphic functions. Applications of the theory to potential theory and fluid dynamics is now standard in engineering circles.

该项目继续数学研究与几个复杂变量的几何相关的问题，并在这些空间内的区域上定义映射。研究的重点是复变异、亚纯映射和全纯向量束。亚纯映射的工作寻求Hartogs关于分离解析性和全纯可扩展性的定理的推广。还研究了解析子变种的hartogs型结果，这些结果可能应用于协维数大于1的全纯链的表征。全纯向量束的研究涉及射影代数流形上数值有效线束上的奇异厄密度量。本文对复射影空间上不可分解hermite - einstein向量束的上同调消失定理和截面曲率作了进一步的研究。最后，研究了特殊多项式理想除法问题的仿射代数变异的几何性质和阶界。复变函数理论包括研究若干复变量的可微函数和相关的函数类，如全纯函数的边值。这个主题是高度几何化的；许多问题涉及各种集合的性质以及它们如何通过由全纯函数组成的映射进行变换。该理论在位势理论和流体力学中的应用现已成为工程界的标准。