Fujiki class $\mathcal{C}$ and holomorphic geometric structures

Abstract

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class $\mathcal{C}$, whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class $\mathcal{C}$ and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.

This is the revised version accepted for publication in International Journal of Mathematics

References in corpus (1)

• Non-Kähler Calabi-Yau manifolds

Cited by in corpus (1)

• A Bochner principle and its applications to Fujiki class $\mathcal{C}$ manifolds with vanishing first Chern class