The field of definition of affine invariant submanifolds of the moduli space of abelian differentials

TL;DR

This paper establishes that the field of definition of affine invariant submanifolds in the moduli space of abelian differentials equals the intersection of holonomy fields, with implications for orbit closures and Teichmüller curves.

Contribution

It proves the equality of the field of definition with the intersection of holonomy fields and analyzes the tangent bundle structure, advancing understanding of affine invariant submanifolds.

Findings

Field of definition equals intersection of holonomy fields.

Projection of tangent bundle to absolute cohomology is simple.

Provides explicit sets with large orbit closures and supports finiteness conjectures.

Abstract

The field of definition of an affine invariant submanifold M is the smallest subfield of the reals such that M can be defined in local period coordinates by linear equations with coefficients in this field. We show that the field of definition is equal to the intersection of the holonomy fields of translation surfaces in M, and is a real number field of degree at most the genus. We show that the projection of the tangent bundle of M to absolute cohomology H^1 is simple, and give a direct sum decomposition of H^1. Applications include explicit full measure sets of translation surfaces whose orbit closures are as large as possible, and evidence for finiteness of algebraically primitive Teichm\"uller curves. The proofs use recent results of Artur Avila, Alex Eskin, Maryam Mirzakhani, Amir Mohammadi, and Martin M\"oller.