Pseudoconcave Sets and Positive Closed Currents

文摘：本项目的总体目标是研究拓扑稀疏伪凹集支撑的正闭合电流的结构，主要集中在电流性质与其支撑体之间的关系。二维复数维拓扑稀疏集是指由一族复数直线沿着无处密集的集合形成的集合；它们最重要的子类是由多极伪凸集形成的。要考虑的具体问题围绕着正闭电流的支集的刻画、正闭电流的拓扑刻画、正闭电流的唯一性现象以及薄伪凹集的分支结构的相关问题。涉及分枝结构的问题涉及薄伪凹集的随机群的类比和环境复射影空间中该集合的补的约化同调群上的序结构的性质。在一般情况下，本项目所涉及的问题涉及具有单一支撑的水流结构。电流是几何背景下分布的推广，因此有助于分析在现代科学中随着频率的增加而研究的低规律性现象。该项目的更广泛目标是促进对当前概念的理解，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.