Pseudoconcave Sets and Positive Closed Currents

摘要：本课题的总体目标是研究拓扑薄伪凹集支撑的正闭合电流的结构，主要关注电流性质与其支撑之间的关系。两个复维空间的拓扑薄集是由一组复线沿无密度集编织而成的拓扑薄集；它们最重要的子类是由多极伪凸类构成的。要考虑的具体问题是围绕着正闭流支点的表征问题，正闭流的拓扑表征问题，正闭流的唯一性现象，以及细假凹集分支结构的相关问题。分支结构问题涉及到复射影空间中薄伪凹集的单项式群的类似问题以及该集合的补的约同调群上的阶结构性质。在一般情况下，本项目解决的问题涉及具有单一支撑的电流结构。电流是几何分布的概括，因此，对于分析现代科学中越来越频繁研究的低规律性现象是有用的。本计画更广泛的目标是促进对电流概念的理解，并增加其适用性。

ABSTRACT: The general aim of this project is to study the structure of positive closed currents supported by topologically thin pseudoconcave sets, focusing primarilly on the relationship between properties of currents and their supports. Topologically thin sets in two complex dimensions are those fibered along nowhere dense sets by a family of complex lines; their most important subclass is formed by pluripolar pseudoconvex ones. The specific questions to be considered group around the problems of characterization of supports of positive closed currents, characterization of positive closed currents in topological terms, uniqueness phenomena for positive closed currents, and related problems of branching structure of thin pseudoconcave sets. Problems involving the branching structure concern analogs of the mondromy group for thin pseudoconcave sets and properties of the order structure on the reduced homology group of the complement of the set in the ambient complex projective space. In the general context the problems addressed in this project concern the the structure of currents with the singular supports. Currents are generalizations of distributions to the geometrical setting and, as such, areuseful to analize low regularity phenomena that are studied with increasingfrequency in modern science. The broader aim of this project is to contribute to the understanding of the notion of current and to increase its applica- bility.