2009/05/31 by Joël Fine, Joel Fine, Dmitri Panov
Mathematics · #Bundle #Canonical bundle #Cotangent bundle #Differential geometry #Frame bundle #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Hyperbolic 3-manifold #Hyperbolic function #Hyperbolic geometry #Hyperbolic group #Hyperbolic manifold #Hyperbolic set #Mathematical analysis #Mathematics #Metric (unit) #Normal bundle #Orbifold #Pure mathematics #Symplectic geometry #Tangent bundle #Tangent space #Vector bundle #math.DG #math.SG
paper · pdf · doi:10.2140/gt.2010.14.1723
published as Geometry and Topology 14 (2010) 1723-1763 · 27 pages. v4 corrected error in discussion of topology of symplectic example
arxiv created 2009/12/22 · openalex publication_date 2010/07/13 · arxiv updated 2017/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We use hyperbolic geometry to construct simply connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Khler 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 c 1 D 0 that does not admit a compatible Khler 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-Khler "Fano" manifolds of dimension 12 and higher.