Directed harmonic currents near non-hyperbolic linearizable singularities

Abstract

Abstract Let $(\mathbb {D}^2,\mathscr {F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb {D}^2$ defined by the linear vector field $$ \begin{align*} z \frac{\partial}{\partial z}+ \unicode{x3bb} w \frac{\partial}{\partial w}, \end{align*} $$ where $\unicode{x3bb} \in \mathbb {C}^*$ . Such a foliation has a non-degenerate singularity at the origin ${0:=(0,0) \in \mathbb {C}^2}$ . Let T be a harmonic current directed by $\mathscr {F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$ . Assume $T\neq 0$ . The Lelong number of T at $0$ describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when $\unicode{x3bb} \notin \mathbb {R}$ , that is, when $0$ is a hyperbolic singularity, the Lelong number at $0$ vanishes. Suppose the trivial extension $\tilde {T}$ across $0$ is $dd^c$ -closed. For the non-hyperbolic case $\unicode{x3bb} \in \mathbb {R}^*$ , we prove that the Lelong number at $0$ : (1) is strictly positive if $\unicode{x3bb}>0$ ; (2) vanishes if $\unicode{x3bb} \in \mathbb {Q}_{<0}$ ; (3) vanishes if $\unicode{x3bb} <0$ and T is invariant under the action of some cofinite subgroup of the monodromy group.