Manifolds which are complex and symplectic but not Kähler

2014/04/30 by Bazzoni, Giovanni, Muñoz, Vicente
#53D05 #55P62 #Algebraic Topology (math.AT) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1404.7662

Abstract

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a Kähler 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 [M. Fernández and V. Muñoz, An 8-dimensional non-formal simply connected symplectic manifold, Ann. of Math. (2) 167, no. 3, 1045--1054, 2008.] admits both symplectic and complex structures but does not carry Kähler metrics.

Related

• A 6-dimensional simply connected complex and symplectic manifold with no Kähler metric
• An 8-dimensional non-formal simply connected symplectic manifold
• Compact complex manifolds whose complement of an analytic subset is Kahler
• Complex non-Kähler manifolds that are cohomologically close to, or far from, being Kähler
• Fundamental group and pluridifferentials on compact Kähler manifolds