Directed harmonic currents near hyperbolic singularities

TL;DR

This paper investigates the behavior of harmonic currents near hyperbolic singularities in holomorphic foliations, showing the vanishing of the Lelong number at the singularity and exploring implications for global mass distribution and leaf recurrence.

Contribution

It establishes the vanishing of the Lelong number for harmonic currents near hyperbolic singularities and applies this to global mass distribution and recurrence in complex surfaces.

Findings

Lelong number of harmonic current at hyperbolic singularity is zero.

Global mass distribution of harmonic currents is characterized.

Recurrence properties of generic leaves are analyzed.

Abstract

Let \Fc be a holomorphic foliation by curves defined in a neighborhood of 0 in \C^2 having 0 as a hyperbolic singularity. Let T be a harmonic current directed by \Fc which does not give mass to any of the two separatrices. Then we show that the Lelong number of T at 0 vanishes. Next, we apply this local result to investigate the global mass-distribution for directed harmonic currents on singular holomorphic foliations living on compact complex surfaces. Finally, we apply this global result to study the recurrence phenomenon of a generic leaf.