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

Qin, Lizhen¹ ; Wang, Botong²

¹ Department of Mathematics, Nanjing University, Nanjing, Jiangsu, China
² Department of Mathematics, University of Wisconsin, Madison, WI, United States

Geometry & topology, Tome 22 (2018) no. 4, pp. 2115-2144.

Résumé

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.

Détail

DOI : 10.2140/gt.2018.22.2115

Classification : 32J27, 53D05
Keywords: Kähler manifolds, Calabi-Yau manifolds

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