Geometry & Topology 14 (2010) 1723–1763

DOI: 10.2140/gt.2010.14.1723

Abstract

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 C⁴: 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.

Keywords

symplectic manifold, complex manifold, trivial canonical bundle, hyperbolic geometry

Mathematical Subject Classification

Primary: 32Q55, 53D35

Secondary: 51M10, 57M25

References

Publication

Received: 26 October 2009
Revised: 16 March 2010
Accepted: 3 June 2010
Published: 13 July 2010
Proposed: Simon Donaldson
Seconded: Ron Stern, Gang Tian

Authors