Mirzakhani's arithmeticity conjecture for higher-rank affine invariant submanifolds

An affine invariant submanifold is a submanifold of a stratum of abelian differentials cut out by real linear homogeneous equations in period coordinates. Its rank is the complex dimension of the projection of its tangent space to absolute cohomology, and it has higher rank when this rank is greater than one. An affine invariant submanifold is arithmetic when it arises from arithmetic data in the sense of the source.

Mirzakhani's arithmeticity conjecture. Higher-rank affine invariant submanifolds are arithmetic.

This conjecture is one of two conjectures articulated by Mirzakhani to explain the scarcity of known geometrically primitive orbit closures. The source applies these conjectures to classify orbit closures in hyperelliptic components, but does not provide a general resolution here.