Triviality and split of vector bundles on rationally connected varieties

Triviality and split of vector bundles on rationally connected varieties Xuanyu PanIn this paper, we use the existence of a family of rational curves on a separably connected variety, which satisfies the Lefschetz condition, to give a simple proof of a triviality criterion due to I. Biswas and J. Pedro and P. Dos Santos.We also prove that a vector bundle on a homogenous space is trivial if and only if the restriction of the vector bundle to every Schubert line is trivial.Using this result and the theory of Chern classes of vector bundles, we give a general criterion for a uniform vector bundle on a homogenous space to be splitting.As an application, we prove that a uniform vector bundle on a classical Grassmannian of low rank is splitting.