Journal of Symplectic Geometry

Volume 16 (2018)

Number 4

A $6$-dimensional simply connected complex and symplectic manifold with no Kähler metric

Pages: 1001 – 1020



Giovanni Bazzoni (Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany)

Marisa Fernández (Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Bilbao, Spain)

Vicente Muñoz (Facultad de Matemáticas, Universidad Complutense de Madrid, Spain)


We construct a simply connected compact manifold which has complex and symplectic structures but does not admit Kähler metric, 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.

Received 16 August 2015

Accepted 9 January 2018

Published 11 February 2019