Volume 22, issue 4 (2018)

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

Lizhen Qin and Botong Wang

Geometry & Topology 22 (2018) 2115–2144
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 $ℤ\oplus ℤ$, their odd-degree Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, $M\left(A\right)×Y$ 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.

Keywords
Kähler manifolds, Calabi-Yau manifolds
Mathematical Subject Classification 2010
Primary: 32J27, 53D05