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 D 0 that does not admit a compatible Kähler 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-Kähler "Fano" manifolds of dimension 12 and higher.53D35, 32Q55; 51M10, 57M25

Non-Kähler SYZ Mirror Symmetry

“…Construction of non-Kähler Calabi-Yau manifolds has been studied in several recent works [Cal58,STY02,GP04,GGP08,Wu06,FP10,FLY12,FP13]. The survey by Fu [Fu10] gave an excellent introduction to this topic.…”

S1‐invariant symplectic hypersurfaces in dimension 6 and the Fano condition

“…The corresponding question in dimensions 6, 8 and 10 is open. As was noted in [8], starting from dimension 12, symplectic twistor spaces of hyperbolic manifolds studied by Reznikov in [36] give rise to infinitely many symplectic Fano manifolds which are non-Kähler.…”

The diversity of symplectic Calabi-Yau 6-manifolds

“…From here, the main work required to prove Theorem 1 is to show that all symplectic six-dimensional orbifolds generated by Theorem 2 admit crepant symplectic resolutions (that do not change the fundamental group). A similar strategy was already applied in [7] to a particular simply connected, hyperbolic orbifold; this led to the first-known example of a simply connected, non-Kähler, symplectic Calabi-Yau. In the simply connected case of Theorem 1, where J 0 ∈ so(4) is a choice of almost complex structure on R 4 (that is, J 2 0 = −1).…”