Oeljeklaus-Toma manifolds admitting no complex subvarieties

TL;DR

This paper proves that certain Oeljeklaus-Toma manifolds, constructed from number fields, do not contain non-trivial complex subvarieties if they admit a locally conformally Kahler structure, using number theory and complex geometry tools.

Contribution

The authors demonstrate that OT-manifolds with a locally conformally Kahler structure have no non-trivial complex subvarieties, employing a novel combination of geometric and number-theoretic methods.

Findings

OT-manifolds with LCK structure lack non-trivial subvarieties

Construction of a semipositive holomorphic line bundle with trivial Chern class

Application of Strong Approximation theorem to Zariski density

Abstract

The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally Kahler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of w is Zariski dense.