Geometry & Topology

Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

Joel Fine and Dmitri Panov

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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.

Article information

Geom. Topol., Volume 14, Number 3 (2010), 1723-1763.

Received: 26 October 2009
Revised: 16 March 2010
Accepted: 3 June 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 32Q55: Topological aspects of complex manifolds
Secondary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

symplectic manifold complex manifold trivial canonical bundle hyperbolic geometry


Fine, Joel; Panov, Dmitri. Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle. Geom. Topol. 14 (2010), no. 3, 1723--1763. doi:10.2140/gt.2010.14.1723.

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