The field of definition of an affine invariant submanifold
is the smallest
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
is a real number field of degree at most the genus.
We show that the projection of the tangent bundle of
is simple, and give a direct sum decomposition of
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
The proofs use recent results of Avila, Eskin, Mirzakhani, Mohammadi and