Non-Kähler Calabi-Yau manifolds and holomorphic geometric structures
Indranil Biswas, Sorin Dumitrescu
TL;DR
The work analyzes holomorphic geometric structures on non-Kähler Calabi–Yau manifolds, especially Vaisman Calabi–Yau spaces, and extends Kähler Calabi–Yau methods via a Beauville–Bogomolov-type decomposition and a weak Bochner principle. It shows that affine-type holomorphic structures on Vaisman CY manifolds are locally homogeneous, and that rigidity forces Kodaira-type orbifold bases, with further implications for fundamental groups under semistability or equal-tangent-bundle section conditions. It also provides explicit constructions of simply connected non-Kähler Calabi–Yau manifolds bearing nonclosed holomorphic 1-forms, and studies principal torus bundles over Kähler Calabi–Yau bases to illustrate limits and generality of the rigidity phenomena. Overall, the results highlight how deformation, stability, and decomposition interact to constrain holomorphic geometric structures beyond the Kähler setting, and they connect to Kodaira-type geometries and elliptic fibrations in the non-Kähler realm.
Abstract
We study holomorphic geometric structures on non-Kähler compact complex manifolds with trivial canonical line bundle. For Vaisman Calabi-Yau manifolds we prove that all holomorphic geometric structures of affine type on them are locally homogeneous. Moreover, if the geometric structure is rigid, then the Vaisman manifold must be a Kodaira manifold. The proof uses a Beauville-Bogomolov type decomposition from [Is] together with a weak form of Bochner principle for Vaisman Calabi-Yau manifolds that we prove here. Other results show that a compact complex manifold with self-dual holomorphic tangent bundle bearing a rigid holomorphic geometric structure of affine type have infinite fundamental group. We prove the same result for compact complex manifolds with trivial canonical line bundle having semistable holomorphic tangent bundle, with respect to some Gauduchon metric. We exhibit (non-Kähler) compact complex simply connected manifolds with trivial canonical line bundle that admit non-closed holomorphic one-forms.