Singular holomorphic foliations by curves. III: Zero Lelong numbers

TL;DR

This paper proves that positive harmonic currents directed by certain singular holomorphic foliations in complex space have zero Lelong number at weakly hyperbolic singularities, with implications for understanding the structure of such currents.

Contribution

It establishes the vanishing of Lelong numbers for directed harmonic currents near weakly hyperbolic singularities, linking local and global properties of singular holomorphic foliations.

Findings

Lelong number of directed harmonic current at the singularity is zero.

Application of local result to global foliation context.

Discussion of relations between harmonic currents, ddc-closed currents, and Lelong numbers.

Abstract

Let $\mathcal{F}$ be a holomorphic foliation by curves defined in a neighborhood of $0$ in $\mathbb{C}^n$ ($n\geq 2$) having $0$ as a weakly hyperbolic singularity. Let $T$ be a positive harmonic current directed by $\mathcal{F}$ which does not give mass to any of the $n$ coordinate invariant hyperplanes $\{z_j=0\}$ for $1\leq j\leq n.$ Then we show that the Lelong number of $T$ at $0$ vanishes. Moreover, an application of this local result in the global context is given. We discuss also the relation between several basic notions such as directed positive harmonic currents, directed positive ddc-closed currents, Lelong numbers etc. in the framework of singular holomorphic foliations.