Several complex structures on the Oeljeklaus-Toma manifolds

TL;DR

This paper studies complex structures on Oeljeklaus-Toma manifolds, providing algebraic conditions for biholomorphism and constructing examples of non-Kähler manifolds with multiple rigid complex structures.

Contribution

It offers an algebraic characterization of when these manifolds are biholomorphic and constructs new examples with multiple distinct complex structures.

Findings

Algebraic criteria for biholomorphism of Oeljeklaus-Toma manifolds

Existence of compact non-Kähler manifolds with multiple rigid complex structures

Construction of manifolds with exponentially many complex structures

Abstract

We investigate complex structures on the Oeljeklaus-Toma manifolds. The Oeljeklaus-Toma manifolds are defined using complex embeddings of number fields. By replacing these embeddings with their conjugates, one obtains other manifolds that share the same underlying differential structure. In this paper, we give an algebraic description of when such manifolds are biholomorphic. As a simple application, we obtain compact non-K\"{a}hler manifolds of dimension $2t+1$ that admit $2^t$ different rigid complex structures.