Connectedness of the boundaries of the strata of differentials
Dawei Chen
TL;DR
The paper investigates the boundary topology of the projectivized stratum $\mathcal{P}(\mu)$ of (possibly meromorphic) differentials and its connected component $\mathcal{P}(\mu)^{\circ}$, focusing on when the boundary is connected in the multi-scale compactification $\mathcal{MS}(\mu)$. Central to the approach is the affineness of meromorphic strata, which implies that for $\dim_{\mathbb C}\mathcal{P}(\mu)^{\circ}\ge 2$ the total boundary $\Delta_{\rm H}\cup \Delta_{\rm V}$ and the vertical boundary $\Delta_{\rm V}$ are connected, with extensions to irreducible subvarieties $\mathcal N$ of dimension $\ge 2$ (including linear subvarieties and $k$-differentials with pole order at least $k$). For holomorphic (i.e., non-meromorphic) signatures, an alternative Teichmüller-curve-based argument shows $\Delta_{\rm H}$ is non-empty and every vertical boundary component meets $\Delta_{\rm H}$, leading to boundary connectedness in the multi-scale compactification. These results indicate that boundary connectedness is governed by intrinsic geometric properties of the strata, and they provide a unified framework applicable to a range of strata and subvarieties, including those lifted via canonical coverings.
Abstract
Let $\mathcal{P}(μ)^{\circ}$ 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 $μ$. When the complex dimension of $\mathcal{P}(μ)^{\circ}$ is at least two, Dozier--Grushevsky--Lee, through explicit degeneration techniques, showed that the boundary of $\mathcal{P}(μ)^{\circ}$ is connected in the multi-scale compactification constructed by Bainbridge--Chen--Gendron--Grushevsky--Möller. A natural question is whether the connectedness of the boundary of $\mathcal{P}(μ)^{\circ}$ is determined by its intrinsic properties. In the case of meromorphic differentials, we provide a concise explanation that the boundary of $\mathcal{P}(μ)^{\circ}$ 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 Teichmüller curves, we provide an alternative argument showing that the horizontal boundary of $\mathcal{P}(μ)^{\circ}$ and every irreducible component of its vertical boundary intersect non-trivially in the multi-scale compactification.