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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(A) which are simultaneously diffeomorphic to complex Calabi–Yau manifolds and symplectic Calabi–Yau manifolds. They have fundamental groups , their odd-degree Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, M(A) × 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