Manifolds which are complex and symplectic but not K\"ahler

TL;DR

This paper discusses the existence of complex and symplectic structures on certain manifolds that do not admit K"ahler metrics, expanding understanding beyond classical examples like the Kodaira-Thurston manifold.

Contribution

It generalizes the paradigm of non-K"ahler manifolds with complex and symplectic structures to a simply-connected 8-dimensional case, showing this phenomenon is not limited by fundamental group.

Findings

The 8-dimensional manifold admits both structures but is not K"ahler.

The phenomenon is not related to the fundamental group.

Provides a broader class of examples beyond the Kodaira-Thurston manifold.

Abstract

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a K\"ahler 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 $8$-dimensional compact manifold of [\textsc{M. Fern\'{a}ndez and V. Mu\~{n}oz}, \emph{An 8-dimensional non-formal simply connected symplectic manifold}, Ann. of Math. (2) \textbf{167}, no. 3, 1045--1054, 2008.] admits both symplectic and complex structures but does not carry K\"ahler metrics.