Manifolds Which Are Complex and Symplectic But Not Kähler

Abstract

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a Kahler structure is the renowned Kodaira–Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply connected eight-dimensional compact manifold of Fernandez and Munoz (Ann Math (2), 167(3):1045–1054, 2008) admits both symplectic and complex structures but does not carry Kahler metrics.