Geometry & Topology

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

Alex Wright

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The field of definition of an affine invariant submanifold is the smallest subfield of such that 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 , and is a real number field of degree at most the genus.

We show that the projection of the tangent bundle of to absolute cohomology H1 is simple, and give a direct sum decomposition of H1 analogous to that given by Möller in the case of Teichmüller curves.

Applications include explicit full measure sets of translation surfaces whose orbit closures are as large as possible, and evidence for finiteness of algebraically primitive Teichmüller curves.

The proofs use recent results of Avila, Eskin, Mirzakhani, Mohammadi and Möller.

Article information

Geom. Topol., Volume 18, Number 3 (2014), 1323-1341.

Received: 13 June 2013
Revised: 29 November 2013
Accepted: 12 January 2014
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

translation surface abelian differential $\mathrm{SL}(2,\mathbb{R})$–action Teichmuller dynamics


Wright, Alex. The field of definition of affine invariant submanifolds of the moduli space of abelian differentials. Geom. Topol. 18 (2014), no. 3, 1323--1341. doi:10.2140/gt.2014.18.1323.

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