Pseudoconcave Sets and Positive Closed Currents

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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, are useful to analize low regularity phenomena that are studied with increasing frequency 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.