Non-K\"ahler Calabi-Yau manifolds

TL;DR

This paper investigates a broad class of non-Kähler Calabi-Yau manifolds characterized by vanishing first Bott-Chern class, exploring their properties, examples, and the challenges in defining canonical metrics.

Contribution

It extends the understanding of Calabi-Yau manifolds beyond the Kähler case by analyzing non-Kähler examples and their geometric properties.

Findings

Manifolds with vanishing first Bott-Chern class include all with torsion canonical bundle.

A manifold in Fujiki's class C with this property has a torsion canonical bundle.

Provided examples of non-Kähler Calabi-Yau manifolds and discussed canonical metrics.

Abstract

We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kahler Calabi-Yau manifolds, and discuss the problem of defining and constructing canonical metrics on them.