Connectedness of the boundaries of the strata of differentials

TL;DR

This paper investigates the connectedness of the boundaries of strata of differentials on complex curves, showing that for meromorphic differentials the boundary is always connected in any algebraic compactification, and providing new insights for holomorphic differentials.

Contribution

It demonstrates that the boundary of the stratum of meromorphic differentials is always connected in any algebraic compactification, and offers alternative proofs for holomorphic differentials using Teichmüller curve properties.

Findings

Boundary of meromorphic differential strata is always connected in algebraic compactifications.

For holomorphic differentials, the boundary components intersect non-trivially in the multi-scale compactification.

The results extend to linear subvarieties and certain k-differentials with poles.

Abstract

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.