Flat affine subvarieties in Oeljeklaus-Toma manifolds

The Oeljeklaus-Toma (OT-)manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field K and a torsion-free subgroup U in the group of units of the ring of integers of K, with rank of U equal to the number of real embeddings of K. OT-manifolds are equipped with a torsion-free flat affine connection preserving the complex structure (this structure is known as flat affine structure). We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all elements in U\{1} are primitive in K, then X contains no proper analytic subvarieties.