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Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

Joel Fine and Dmitri Panov

Geometry & Topology 14 (2010) 1723–1763

We use hyperbolic geometry to construct simply connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kähler structure. We start with the desingularisations of the quadric cone in 4: the smoothing is a natural S3–bundle over H3, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S2–bundle over H4 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 c1 = 0 that does not admit a compatible Kähler structure. We also find infinitely many distinct complex structures on 2(S3 × S3) # (S2 × S4) with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kähler “Fano” manifolds of dimension 12 and higher.

symplectic manifold, complex manifold, trivial canonical bundle, hyperbolic geometry
Mathematical Subject Classification 2000
Primary: 53D35, 32Q55
Secondary: 51M10, 57M25
Received: 26 October 2009
Revised: 16 March 2010
Accepted: 3 June 2010
Published: 13 July 2010
Proposed: Simon Donaldson
Seconded: Ron Stern, Gang Tian
Joel Fine
Départment de Mathématique
Université Libre de Bruxelles
Boulevard du Triomphe
Bruxelles 1050
Dmitri Panov
Department of Mathematics
Imperial College London
South Kensington Campus
London SW7 2AZ
United Kingdom