Directed harmonic currents near non-hyperbolic linearized singularities

• Published in: arXiv.org 2020-11
• Main Author: Chen, Zhangchi
• Format: Article
• Language: English
• Subjects: Fields (mathematics); Linearization; Mass distribution; Singularities; Subgroups
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2011.05909

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}_{