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 \[ z \,\frac{\partial}{\partial z}+ λ\,w \,\frac{\partial}{\partial w}, \] where $\lambda \in {\mathbb{C}}^{\ast }$. 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 $d{d}^c$-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 $\lambda \in {\mathbb{R}}^{\ast }$ 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.

arXiv comment24 pages, 15 figures

References in corpus (2)

• Ergodic theorems for laminations and foliations: recent results and perspectives
• Harmonic currents directed by foliations by Riemann surfaces