Components of strata of k-differentials and their orbit closures

Abstract

We obtain a complete classification of components of strata of holomorphic and meromorphic k-differentials. We show that, when genus is at least two and outside of explicit exceptions when k < 4, there is one primitive nonhyperelliptic component unless k is odd and all singularities have even order, in which case there are two distinguished by their Arf invariant. The exceptions include new sporadic components of strata of cubic differentials. Our work provides a new proof of earlier results of Kontsevich-Zorich, Boissy, Lanneau, and Chen-Gendron when k = 1, 2. The proofs are almost purely algebraic, relying on the multiscale compactification of Bainbridge-Chen-Gendron-Grushevsky-Moller. This answers a question of Chen-Yu. We also show that for any component of a stratum of finite area $k$-differentials of positive genus, the smallest GL(2,R)-orbit closure containing all of its holonomy covers is as big as possible. This answers the positive genus analogue of a "long term goal" of Mirzakhani-Wright. The result also holds in genus zero strata that contain surfaces with Euclidean cylinders, thus addressing infinitely many cases of the original question of Mirzakhani-Wright.