A $6$-dimensional simply connected complex and symplectic manifold with no K\"ahler metric

TL;DR

This paper constructs a 6-dimensional simply connected compact manifold that admits complex and symplectic structures but does not support any K"ahler metric, highlighting a unique example in the lowest such dimension.

Contribution

It provides the first known example of a simply connected 6-dimensional manifold with complex and symplectic structures lacking a K"ahler metric, expanding understanding of geometric structures.

Findings

The manifold is formal and has even odd-degree Betti numbers.

It does not satisfy the Lefschetz property for any symplectic form.

Abstract

We construct a simply connected compact manifold which has complex and symplectic structures but does not admit K\"ahler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form.