A Bochner principle and its applications to Fujiki class 𝒞 manifolds with vanishing first Chern class

Abstract

We prove a Bochner type vanishing theorem for compact complex manifolds Y in Fujiki class C, with vanishing first Chern class, that admit a cohomology class [α] ∈ H 1,1 (Y, R) which is numerically effective (nef) and has positive self-intersection (meaning Y α n > 0, where n = dim C Y ). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold Y are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of Y must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold Y admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant.2010 Mathematics Subject Classification. 53B35, 53C55, 53A55.

Holomorphic G-Structures and Foliated Cartan Geometries on Compact Complex Manifolds

“…Earlier works, [BD1,BD2,BD3,Du1,Du2,BDG], which were inspired by [Gro, DG], aimed to adapt Gromov's ideas and arguments to holomorphic geometric structures on compact complex manifolds. In that vein, the third-named author proved the following theorem:…”