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

Alex Wright

Geometry & Topology 18 (2014) 1323–1341

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.

translation surface, abelian differential, $\mathrm{SL}(2,\mathbb{R})$–action, Teichmuller dynamics
Mathematical Subject Classification 2010
Primary: 32G15, 37D40
Received: 13 June 2013
Revised: 29 November 2013
Accepted: 12 January 2014
Published: 7 July 2014
Proposed: Benson Farb
Seconded: Danny Calegari, Leonid Polterovich
Alex Wright
Math Department
University of Chicago
5734 South University Avenue
Chicago, IL 60637