Components of strata of k-differentials and their orbit closures

TL;DR

This paper classifies the components of strata of holomorphic and meromorphic k-differentials, revealing their structure and orbit closures, with algebraic proofs and implications for dynamics and geometry.

Contribution

It provides a complete classification of strata components for k-differentials, including new sporadic cases, and proves maximal orbit closure properties in positive genus.

Findings

Classified components of k-differential strata for genus ≥ 2.

Identified conditions for the number of distinguished components.

Proved maximal orbit closure for holonomy covers in positive genus.

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.