Several complex structures on the Oeljeklaus-Toma manifolds

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.