Flat affine subvarieties in Oeljeklaus-Toma manifolds

TL;DR

This paper proves that in Oeljeklaus-Toma manifolds, the smallest positive-dimensional complex subvarieties are flat affine, and under certain conditions, these manifolds contain no proper complex subvarieties, highlighting their geometric rigidity.

Contribution

It establishes that minimal positive-dimensional subvarieties in OT-manifolds are flat affine and identifies conditions for the absence of proper subvarieties.

Findings

Smallest positive-dimensional subvarieties are flat affine.

Under certain algebraic conditions, OT-manifolds have no proper complex subvarieties.

Abstract

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 $K$. 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 non-trivial elements in $U$ are primitive in $K$, then $X$ contains no proper complex subvarieties.