论文标题
非纤维线性奇异性附近的定向谐波电流
Directed harmonic currents near non-hyperbolic linearized singularities
论文作者
论文摘要
令$（\ Mathbb {d}^2，\ Mathcal {f}，\ {0 \}）$是单位bidisc $ \ mathbb {d}^2 $在线性bidisc $ \ mathbb {d}^2 $上定义的单数全态叶子\，\ frac {\ partial} {\ partial w}，\]其中$λ\ in \ mathbb {c}^*$。这样的叶面的非分类线性奇异性为$ 0 $。令$ t $为$ \ Mathcal {f} $指导的谐波电流，它不会给两个分隔$（z = 0）$和$（W = 0）$中的任何一个，并且其琐碎的$ \ tilde $ \ tilde {t} $ in $ 0 $ 0 $ is $ nes $ is $ is $ dd $ dd^c $。 $ t $ $ 0 $的Lelong数量描述了Foliated空间上的质量分布。 Nguyen在2014年证明，当$λ\ notin \ notin \ mathbb {r} $时，即$ 0 $是一种双曲线奇异性，lelong的数字为$ 0 $ $ nishes。对于非hyperbolic情况，$λ\ in \ mathbb {r}^*$中的文章证明了以下结果。 $ 0 $的Lelong号码： 1）如果$λ> 0 $，则严格为正； 2）如果$λ\ in \ mathbb {q} _ {<0} $; 3）如果$λ<0 $和$ t $在单一组的某些辅助亚组的作用下消失。
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.
