On the infinitesimal automorphisms of principal bundles

Abstract

We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$ the extended Lie algebra of infinitesimal automorphisms of $L$. If the representation of $\mathfrak{g}_L$ on the space of holomorphic sections of $L$ is irreducible then $Z$ is rational; (2) Let $P$ be a holomorphic principal bundle over the Riemann sphere, with structural group $G$ whose Lie algebra is not equal to its nilpotent radical. Then there exists a Lie subgroup $H$ of $G$ which is a quotient of a Borel subgroup of ${\rm SL}(2)$ and such that $P$ admits a reduction to $H$.