Components of strata of k-differentials and their orbit closures

Juliet Aygun; Paul Apisa

arxiv: 2604.26177 · v1 · submitted 2026-04-28 · 🧮 math.AG · math.DS· math.GT

Components of strata of k-differentials and their orbit closures

Paul Apisa , Juliet Aygun This is my paper

Pith reviewed 2026-05-07 12:21 UTC · model grok-4.3

classification 🧮 math.AG math.DSmath.GT

keywords genusstratacomponentswhenanswerscomponentdifferentialsexceptions

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The pith

Strata of k-differentials have one or two components except for explicit exceptions, with new sporadic cubic examples, and their holonomy covers have maximal GL(2,R) orbit closures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Riemann surfaces can be equipped with k-differentials, which are like holomorphic 1-forms but raised to the k-th power and allowed to have poles. These objects are grouped into strata according to the orders of their zeros and poles. The paper determines how many separate connected pieces each stratum has. In most cases there is a single main piece, but when k is odd and every singularity has even order there are two pieces separated by a topological invariant called the Arf invariant. A few extra unexpected pieces appear for cubic differentials. The authors also study how these surfaces can be continuously deformed by linear transformations coming from GL(2,R) and show that the smallest space containing all such deformations from a given component is the largest possible one. The arguments are mostly algebraic and rest on a compactification of the space of differentials that was constructed earlier.

Core claim

We obtain a complete classification of components of strata of holomorphic and meromorphic k-differentials. ... 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. ... 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.

Load-bearing premise

The algebraic arguments rely on the multiscale compactification of Bainbridge-Chen-Gendron-Grushevsky-Moller possessing the precise properties needed to detect components and orbit closures, particularly outside the listed low-k exceptions and for genus at least two.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Apisa and Aygun classify components of k-differential strata for general k and prove maximal orbit closures via algebraic arguments on the multiscale compactification.

read the letter

This paper gives the first complete classification of connected components in strata of holomorphic and meromorphic k-differentials. For genus at least two, outside listed exceptions for k under 4, there is one primitive nonhyperelliptic component except when k is odd and all singularities have even order, in which case there are two separated by the Arf invariant. They also turn up new sporadic components for cubic differentials and supply algebraic proofs for the k=1 and k=2 cases that were previously known by other routes. The second main result is that the smallest GL(2,R)-orbit closure containing the holonomy covers of any positive-genus finite-area component is as large as possible, which settles the positive-genus part of the Mirzakhani-Wright question and many genus-zero cases with cylinders.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard algebraic-geometry tools and one key prior construction; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption The multiscale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-Moller has the stated topological and algebraic properties used to detect components.
Invoked throughout the algebraic proofs of the classification and orbit-closure statements.

pith-pipeline@v0.9.0 · 5534 in / 1488 out tokens · 104442 ms · 2026-05-07T12:21:10.810408+00:00 · methodology

Components of strata of k-differentials and their orbit closures

Core claim

Load-bearing premise

discussion (0)