A Family of Compact Complex-Symplectic Calabi-Yau Manifolds that are Nonk\"ahler

TL;DR

This paper constructs a family of 6-dimensional compact manifolds that are both complex and symplectic Calabi-Yau, yet are non-Kähler, highlighting new examples in complex geometry.

Contribution

It introduces a novel family of manifolds that are simultaneously complex and symplectic Calabi-Yau but do not admit Kähler structures, expanding understanding of non-Kähler Calabi-Yau manifolds.

Findings

Manifolds have fundamental group Z⊕Z

They satisfy the hard Lefschetz property

Their real homotopy types are formal

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 $\mathbb{Z} \oplus \mathbb{Z}$, their odd-dimensional Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, $M(A) \times Y$ are not homotopy equivalent to any compact K\"ahler manifold for any topological space $Y$.