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Directed harmonic currents near non-hyperbolic linearizable singularities

Part of: Global differential geometry Singularities Analytic spaces Complex spaces with a group of automorphisms Automorphic functions

Published online by Cambridge University Press:  07 July 2022

ZHANGCHI CHEN Open the ORCID record for ZHANGCHI CHEN [Opens in a new window]
ZHANGCHI CHEN*
Affiliation:
Morningside Center of Mathematics, Chinese Academy of Science, Beijing, China, http://www.mcm.ac.cn/people/postdocs/202110/t20211022_666685.html
*
e-mail: [email protected]
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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. (1) is strictly positive if $\unicode{x3bb}>0$ ;

  2. (2) vanishes if $\unicode{x3bb} \in \mathbb {Q}_{<0}$ ;

  3. (3) vanishes if $\unicode{x3bb} <0$ and T is invariant under the action of some cofinite subgroup of the monodromy group.

Keywords

holomorphic foliationharmonic currentnon-hyperbolic linearizable singularityLelong number

MSC classification

Primary: 32M25: Complex vector fields
Secondary: 32S65: Singularities of holomorphic vector fields and foliations 32C30: Integration on analytic sets and spaces, currents 53C65: Integral geometry
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 7 , July 2023 , pp. 2228 - 2257
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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