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.
A 6-dimensional simply connected complex and symplectic manifold with no Kähler metric
Bazzoni G;
2018-01-01
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.
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