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

Abstract

We prove a Bochner type vanishing theorem for compact complex manifolds $Y$ in Fujiki class $\mathcal{C}$, with vanishing first Chern class, that admit a cohomology class $[\alpha ]\in H^{1,1}(Y,\mathbb{R})$ which is numerically effective (nef) and has positive self-intersection (meaning ${\int }_Y{\alpha }^n>0$, where $n={\dim }_{\mathbb{C}}Y$). 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.

21 pages

References in corpus (1)

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