Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

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 C: the smoothing is a natural S-bundle over H, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S-bundle over H 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 non-Kähler symplectic manifold with c1 = 0 as well as infinitely many distinct complex structures on 2(S × S)#(S × S) with trivial canonical bundle. We also explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kähler “Fano” manifolds of dimension 12 and higher.