标题： 微分层边界的连通性 显示英文标题

标题： Connectedness of the boundaries of the strata of differentials

摘要： 设$\mathcal{P}(\mu)^{\circ}$为定义了微分零点和极点阶数的光滑复曲线上的微分层射影化分支，记作$\mu$。 当$\mathcal{P}(\mu)^{\circ}$的复维度至少为二时，Dozier--Grushevsky--Lee 通过显式退化技术证明了，在由Bainbridge--Chen--Gendron--Grushevsky--Möller构造的多尺度紧化中，$\mathcal{P}(\mu)^{\circ}$的边界是连通的。 一个自然的问题是，$\mathcal{P}(\mu)^{\circ}$的边界的连通性是否由其内在性质决定。 对于亚纯微分的情形，我们基于亚纯微分层是仿射簇这一事实，给出了一个简洁的解释：在任何完备代数紧化中，$\mathcal{P}(\mu)^{\circ}$的边界总是连通的。 我们还观察到，相同的结果对于亚纯微分的线性子簇以及具有至少为$k$阶极点的$k$-微分的层也成立。在全纯微分的情况下，利用Teichmüller曲线的性质，我们提供了另一种论证，表明$\mathcal{P}(\mu)^{\circ}$的水平边界和其垂直边界的每个不可约分支在多尺度紧化中非平凡地相交。 显示英文摘要

摘要： Let $\mathcal{P}(\mu)^{\circ}$ be a connected component of the projectivized stratum of differentials on smooth complex curves, where the zero and pole orders of the differentials are specified by $\mu$. When the complex dimension of $\mathcal{P}(\mu)^{\circ}$ is at least two, Dozier--Grushevsky--Lee, through explicit degeneration techniques, showed that the boundary of $\mathcal{P}(\mu)^{\circ}$ is connected in the multi-scale compactification constructed by Bainbridge--Chen--Gendron--Grushevsky--M\"oller. A natural question is whether the connectedness of the boundary of $\mathcal{P}(\mu)^{\circ}$ is determined by its intrinsic properties. In the case of meromorphic differentials, we provide a concise explanation that the boundary of $\mathcal{P}(\mu)^{\circ}$ is always connected in any complete algebraic compactification, based on the fact that the strata of meromorphic differentials are affine varieties. We also observe that the same result holds for linear subvarieties of meromorphic differentials, as well as for the strata of $k$-differentials with a pole of order at least $k$. In the case of holomorphic differentials, using properties of Teichm\"uller curves, we provide an alternative argument showing that the horizontal boundary of $\mathcal{P}(\mu)^{\circ}$ and every irreducible component of its vertical boundary intersect non-trivially in the multi-scale compactification.