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

International audienceWe prove a Bochner type vanishing theorem for compact complex manifolds Y in Fujiki class C, with vanishing first Chern class, that admit a co-homology class [α] ∈ H 1,1 (Y, R) which is numerically effective (nef) and has positive self-intersection. 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