Some applications of metric currents to complex analysis

TL;DR

This paper explores the application of metric currents in complex analysis, establishing foundational results, introducing bidimension concepts, and solving the ar equation in various complex space settings.

Contribution

It introduces bidimension for metric currents, develops a structure theorem for currents on complex spaces, and defines quasi-local currents to solve ar equations with bounded support.

Findings

Established comparison theorem between metric and classical currents.

Defined bidimension for metric currents and analyzed its properties.

Solved ar equation for currents with bounded support.

Abstract

The aim of this paper is to show two applications of metric currents to complex analysis. After recalling the basic definitions, we give a detailed proof of the comparison theorem between metric currents and classical ones on a manifold. In Section 3 we introduce the concept of bidimension for a metric current on a finite dimensional space, showing that the usual properties of $(p,q)-$currents still hold, except for the existence of a Dolbeault decomposition. Section 4 is devoted to the analysis of a particular class of complex spaces, whose structure allows us to give a structure theorem for currents, solve the Cauchy-Riemann equation and characterize holomorphic currents. In Section 5, we introduce the concept of bidimension of (global) metric currents on a Banach space and relate it to the behaviour of the finite dimensional projections of the currents. In section 6 we define a new class of currents, the \emph{quasi-local} metric currents, which are usual metric currents when restricted to bounded sets, and we give a definition of $(p,q)-$current in this new class. The last Section shows how to employ these newly defined quasi-local currents in order to obtain a solution to the equation $\debar U=T$, when $T$ is of bidimension $(0,q)$ and its support is bounded; finally, we extend the result to a current $T$ with generic bidimension, $\debar-$closed, with bounded support.