For meromorphic differentials, the boundary of a stratum is connected in every complete algebraic compactification; for holomorphic differentials, the horizontal boundary meets every vertical boundary component.
Compactification of strata of Abelian differentials
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Connectedness of the boundaries of the strata of differentials
For meromorphic differentials, the boundary of a stratum is connected in every complete algebraic compactification; for holomorphic differentials, the horizontal boundary meets every vertical boundary component.