Directed harmonic currents near non-hyperbolic linearized singularities

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.