Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

By Joel Fine, Dmitri Panov. 2009

70 citations recorded by the source corpus.

We use hyperbolic geometry to construct simply connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C 4 : the smoothing is a natural S 3 ‐bundle over H 3 , its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S 2 ‐bundle over H 4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions. In particular, we find the first example of a simply connected symplectic 6‐manifold with c1D 0 that does not admit a compatible Kahler structure. We also find infinitely many distinct complex structures on 2.S 3 S 3 / #.S 2 S 4 / with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kahler “Fano” manifolds of dimension 12 and higher. 53D35, 32Q55; 51M10, 57M25

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/39687ea5ed20f212f1abd890ee9926d6c6be072c

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Non-Kähler Calabi-Yau manifolds (2014, background): “On the other hand, much interest has been devoted to studying non-Kähler compact complex manifolds with holomorphically trivial (or more generally torsion) canonical bundle, and many examples can be found in [6, 15, 17, 20, 23, 33, 35, 36, 37, 40, 43, 53, 60, 78, 83, 86] and references therein.”
Non-K\"ahler Calabi-Yau manifolds (2014, background): “On the other hand, much interest has been devoted to studying non-Kähler compact complex manifolds with holomorphically trivial (or more generally torsion) canonical bundle, and many examples can be found in [6, 15, 17, 20, 23, 33, 35, 36, 37, 40, 43, 53, 60, 78, 83, 86] and references therein.”