Mathematical Sciences: Complex Manifolds and Meromorphic Mappings

这个项目继续数学研究的问题 与几个复变量的几何学有关， 在这些空间内的区域上定义的映射。 工作重点 关于复簇、亚纯映射和全纯向量 捆起来。 关于亚纯映射的工作寻求 关于分离解析性和全纯性的Hartogs定理 可扩展性 也要研究Hartogs型结果 对于解析子簇，可能应用于 余维数大于1的全纯链的刻划 比团结。 全纯向量丛的研究 数值有效线丛上的奇异厄米度量 关于射影代数流形 继续开展关于 上同调消失定理和截曲率 复形上不可分解的Hermitian-Einstein向量丛 将进行空间投影。 最后， 仿射代数簇与除的度界 问题的特殊多项式理想将被研究。 复变函数理论包括研究 多复变量的可微函数和 相关的函数类，例如 全纯函数这个主题是高度几何的;许多 这些问题涉及各种集合的性质以及它们如何 由全纯函数构成的映射变换。 该理论在势理论和流体力学中的应用 现在是工程界的标准。

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.