Stability and holomorphic connections on vector bundles over LVMB manifolds

TL;DR

This paper characterizes LVMB manifolds with specific holomorphic tangent bundle properties and shows that certain holomorphic vector bundles over these manifolds admit only flat connections.

Contribution

It provides a complete characterization of LVMB manifolds where the tangent bundle is spanned by global holomorphic vector fields with zeros, and proves flatness of holomorphic connections under these conditions.

Findings

Characterization of LVMB manifolds with tangent bundle spanned by global vector fields

Holomorphic connections on semi-stable bundles are always flat in this setting

Identification of conditions for the existence of holomorphic connections with specific properties

Abstract

We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previous property are always flat.