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2015•Contemporary mathematics - American Mathematical SocietyOpen access

Non-Kähler Calabi-Yau manifolds

Valentino Tosatti

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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 \mathcal {C} with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kähler Calabi-Yau manifolds, and discuss the problem of defining and constructing canonical metrics on them.

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What this paper is about

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 \mathcal {C} with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kähler Calabi-Yau manifolds, and discuss the problem of defining and constructing canonical metrics on them.

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Available 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 \mathcal {C} with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kähler Calabi-Yau manifolds, and discuss the problem of defining and constructing canonical metrics on them.

Key concepts: Calabi–Yau manifold, Chern class, Canonical bundle, Mathematics, Pure mathematics, Hyperkähler manifold, Torsion (gastropod), Manifold (fluid mechanics)

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