Complex manifolds with generating tangent bundles

TL;DR

This paper proves that a compact complex manifold with a tangent bundle generated by global sections is necessarily homogeneous, using a complex version of the Chow-Rashevskii theorem in Carnot-Caratheodory spaces.

Contribution

It establishes a new criterion linking the generation of tangent bundle sections to the homogeneity of complex manifolds, extending classical results.

Findings

Manifolds with globally generated tangent bundles are homogeneous.

The proof utilizes the complex Chow-Rashevskii theorem.

Provides a new geometric characterization of homogeneous complex manifolds.

Abstract

Let $M$ be a close complex manifold and $TM$ its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then $M$ is a complex homogeneous manifold. Our proof depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces.