非双曲线性化奇点附近的有向调和流
Directed harmonic currents near non-hyperbolic linearized singularities
AI总结：
本文研究非双曲线性化奇点附近的有向调和流，证明了实参数λ不同情形下0点Lelong数的取值结论，补充了Nguyen关于双曲情形的已有结果。
摘要：
设$(\n\nmathbb{D}^2,\n\nmathcal{F},\{0\})$是单位双圆盘$\n\nmathbb{D}^2$上由线性向量场$z \frac{\partial}{\partial z}+ \lambda w \frac{\partial}{\partial w}$定义的奇异全纯叶状结构，其中$\lambda\in\n\nmathbb{C}^*$。这类叶状结构在0点处具有非退化线性化奇点。设$T$是由$\n\nmathcal{F}$引导的调和流，它不对两条分界线$(z=0)$和$(w=0)$中的任何一条赋予质量，且其穿过0点的平凡扩张$\n\tilde{T}$是$dd^c$-闭的。$T$在0点的Lelong数描述了叶状空间上的质量分布。2014年Nguyen证明了当$\lambda\notin\n\nmathbb{R}$（即0点为双曲奇点）时，0点处的Lelong数为零。对于非双曲情形$\lambda\in\n\nmathbb{R}^*$，本文证明了以下结果：0点处的Lelong数：1) 若$\lambda>0$则严格为正；2) 若$\lambda\in\n\nmathbb{Q}_{<0}$则为零；3) 若$\lambda<0$且$T$在单值群的某个余有限子群作用下不变，则为零。
英文摘要：
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 $λ\in\mathbb{C}^*$. 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 $λ\notin\mathbb{R}$, i.e. $0$ is a hyperbolic singularity, the Lelong number at $0$ vanishes. For the non-hyperbolic case $λ\in\mathbb{R}^*$ the article proves the following results. The Lelong number at $0$: 1) is strictly positive if $λ>0$; 2) vanishes if $λ\in\mathbb{Q}_{<0}$; 3) vanishes if $λ<0$ and $T$ is invariant under the action of some cofinite subgroup of the monodromy group.
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