Stud. Univ. Babe¸ s-Bolyai Math. 58(2013), No. 4, 459–468 Oeljeklaus-Toma manifolds and locally conformally K¨ ahler metrics. A state of the art Liviu Ornea and Victor Vuletescu To the memory of Professor Mircea-Eugen Craioveanu (1942-2012) Abstract. We review several properties about Oeljeklaus-Toma manifolds, espe- cially the locally conformally K¨ahler ones, with focus on the non-existence of certain complex submanifolds. Mathematics Subject Classification (2010): 53C55, 32J18. Keywords: Compact complex manifolds, algebraic number fields, algebraic units, locally conformally K¨ahler metrics, complex submanifold. 1. Introduction The idea of associating compact complex manifolds to number fields is present since the very beginnings of complex geometry. If one was to write a history of this ideas, he would probably start from elliptic curves, which subtle links to number theory were felt by L. Kronecker and K. Weierstrass, would then include H. Weyl, whose research on complex tori have roots in the study of number fields units, and would then arrive to A. Weil who extended this line of research to K¨ ahler manifolds. The goal of the present paper is to give an account on the recent progress in a highly interesting class of compact complex manifolds associated to certain number fields introduced by K. Oeljeklaus & M. Toma in 2005. Despite being a relatively new topic, this kind of manifolds already provided a number of surprising results in the non-K¨ ahler geometry, as we shall see below. 2. Basic facts from algebraic number theory We recall (cf. e.g. [7]) that an (abstract) number field is a finite extension K of Q; it follows that K is isomorphic (as Q−algebras) to Q[X]/(f ) where f ∈ Z[X] is a (monic) irreducibile polynomial. An abstract number field K can be embedded into C Partially supported by CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0118.