Directed harmonic currents near non-hyperbolic linearized singularities

TL;DR

This paper investigates the behavior of harmonic currents near non-hyperbolic linearized singularities in holomorphic foliations, revealing conditions under which the Lelong number is positive or zero based on the parameter bb.

Contribution

It extends Nguyen's 2014 results by analyzing the Lelong number for non-hyperbolic cases where bbbb; it characterizes when the Lelong number is positive or vanishes.

Findings

Lelong number is positive if bb>0

Lelong number vanishes if bbbb<0 and T is monodromy-invariant

Lelong number vanishes if bb<0 and bbbb is rational

Abstract

Let $(\mathbb{D}^2,\mathcal{F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb{D}^2$ defined by the linear vector field \[ z \,\frac{\partial}{\partial z}+ \lambda \,w \,\frac{\partial}{\partial w}, \] where $\lambda\in\mathbb{C}^*$. Such a foliation has a non-degenerate linearized singularity at $0$. Let $T$ be a harmonic current directed by $\mathcal{F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$ and whose the trivial extension $\tilde{T}$ across $0$ is $dd^c$-closed. The Lelong number of $T$ at $0$ describes the mass distribution on the foliated space. In 2014 Nguyen proved that when $\lambda\notin\mathbb{R}$, i.e. $0$ is a hyperbolic singularity, the Lelong number at $0$ vanishes. For the non-hyperbolic case $\lambda\in\mathbb{R}^*$ the article proves the following results. The Lelong number at $0$: 1) is strictly positive if $\lambda>0$; 2) vanishes if $\lambda\in\mathbb{Q}_{<0}$; 3) vanishes if $\lambda<0$ and $T$ is invariant under the action of some cofinite subgroup of the monodromy group.