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.

这个项目继续对与定义在这样的空间内的区域上的几个复变量和映射的几何相关的问题进行数学研究。主要研究复簇、亚纯映射和全纯向量丛。关于亚纯映射的工作寻求Hartogs关于分离解析性和关于全纯可扩性的定理的推广。还将研究解析亚簇的Hartogs型结果，它可能应用于余维大于1的全纯链的刻画。全纯向量丛的研究涉及射影-代数流形上数值有效线丛上的奇异Hermite度量。我们将继续研究复射影空间上不可分解的厄米-爱因斯坦向量丛的上同调、零化定理和截曲率。最后，我们将研究特殊多项式理想除法问题的仿射代数簇的几何和度界。复变函数论包括研究多个复变数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。将该理论应用于位势理论和流体力学现在是工程界的标准。