Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle

TL;DR

This paper constructs new simply-connected symplectic and complex manifolds with trivial canonical bundle using hyperbolic geometry, including the first known example of a non-Kahler symplectic 6-manifold with c_1=0.

Contribution

It introduces a novel hyperbolic geometric approach to produce non-Kahler manifolds with trivial canonical bundle, expanding the landscape of known examples.

Findings

First example of a simply-connected symplectic 6-manifold with c_1=0 that is non-Kahler

Infinite complex structures on a specific 6-manifold with trivial canonical bundle

Higher-dimensional hyperbolic constructions yield non-Kahler symplectic 'Fano' manifolds

Abstract

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler 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 Kahler structure. We also find infinitely many distinct complex structures on 2(S^3xS^3)#(S^2xS^4) with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kahler "Fano" manifolds of dimension 12 and higher.