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

Qin, Lizhen; Wang, Botong

doi:10.2140/gt.2018.22.2115

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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

DOI: 10.2140/gt.2018.22.2115

Abstract

We construct a family of $6$ –dimensional compact manifolds $M (A)$ 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, $M (A) \times Y$ 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.

Keywords

Kähler manifolds, Calabi-Yau manifolds

Mathematical Subject Classification 2010

Primary: 32J27, 53D05

References

Publication

Received: 17 June 2016

Revised: 15 April 2017

Accepted: 15 June 2017

Published: 5 April 2018

Proposed: Yasha Eliashberg

Seconded: Richard Thomas, Dan Abramovich

Authors

Lizhen Qin
Department of Mathematics Nanjing University Nanjing, Jiangsu China

Botong Wang
Department of Mathematics University of Wisconsin Madison, WI United States

Volume 22, issue 4 (2018)