Limit currents and value distribution of holomorphic maps

TL;DR

This paper introduces a new method for constructing positive currents associated with holomorphic maps using higher-dimensional analogues of the Ahlfors inequality, with applications to value distribution and equidistribution problems.

Contribution

It develops a novel approach to creating Ahlfors currents in higher dimensions, expanding the tools available for value distribution theory.

Findings

Construction of $d$-closed and $dd^c$-closed positive currents

Application to equidistribution problems in value distribution theory

Extension of Ahlfors inequality to higher dimensions

Abstract

We construct $d$-closed and $dd^c$-closed positive currents associated to a holomorphic map $\phi$ via cluster points of normalized weighted truncated image currents. They are constructed using analogues of the Ahlfors length-area inequality in higher dimensions. Such classes of currents are referred to as Ahlfors currents. We give some applications to equidistribution problems in value distribution theory.