A family of compact complex and symplectic Calabi–Yau manifolds that are non-Kähler

Abstract

We construct a family of $6$–dimensional compact manifolds $M\left(A\right)$ which are simultaneously diffeomorphic to complex Calabi–Yau manifolds and symplectic Calabi–Yau manifolds. They have fundamental groups $\mathbb{Z}\oplus \mathbb{Z}$, their odd-degree Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, $M\left(A\right)\times Y$ is never homotopy equivalent to a compact Kähler manifold for any topological space $Y$. The main ingredient to show the non-Kählerness is a structure theorem of cohomology jump loci due to the second author.