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

## TL;DR

This paper establishes a Bochner vanishing theorem for certain Fujiki class C manifolds with vanishing first Chern class, leading to results on the local homogeneity of affine geometric structures and properties of holomorphic Riemannian metrics.

## Contribution

It introduces a Bochner type vanishing theorem for Fujiki class C manifolds with nef classes and applies it to analyze geometric structures and Riemannian metrics on these manifolds.

## Key findings

- Holomorphic geometric structures are locally homogeneous on a Zariski open subset.
- Rigid geometric structures imply an infinite fundamental group.
- Manifolds with holomorphic Riemannian metrics have finite covers by complex tori.

## 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.