Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

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 ${ℂ}^4$: 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 $c_1=0$ that does not admit a compatible Kähler structure. We also find infinitely many distinct complex structures on $2\left(S^3\times S^3\right)\#\left(S^2\times S^4\right)$ 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.