A class of compact complex manifolds without complex submanifolds

This paper provides a comprehensive survey of the study of complex submanifolds in compact non-Kähler complex manifolds, focusing on the existence and non-existence of such submanifolds, particularly curves and surfaces. It explores classical constructions, including Inoue surfaces, as well as their higher-dimensional generalizations by Oeljeklaus and Toma (OT-manifolds), and more recent families introduced by Endo and Pajitnov (EP-manifolds). These manifolds exhibit a variety of geometric structures and present distinct behaviors concerning the presence of complex submanifolds. The paper revisits key constructions, examining their algebraic, and topological properties, and provides insights into how these properties influence the existence of complex subvarieties. In particular, this study highlights the interplay between number-theoretic data and geometric properties, such as the lack of complex curves in certain OT-manifolds and the nuanced behavior of Endo–Pajitnov manifolds, where the presence of complex submanifolds is sensitive to algebraic parameters. The aim is to offer a unified perspective on the rigidity phenomena characterizing these manifolds, with an emphasis on the interplay between algebraic structures and complex geometry, while also suggesting avenues for future research in this fascinating area of non-Kähler geometry.