Holomorphic Lie algebroid connections over rationally connected varieties

Abstract

Take a holomorphic Lie algebroid $(V,ϕ)$ over a rationally connected smooth complex projective variety $X$. We show that, under certain conditions, a vector bundle $E$ over $X$ admits a $(V,ϕ)$-connection if and only if $E$ is trivial. Moreover, we prove that under the same conditions, any $(V,ϕ)$-connection over $X$ is flat.