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:

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.

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 complex Lie subgroup $H$ of $G$ with the following properties:

1. $H$ is a quotient of a Borel subgroup of ${\mathrm {SL}}(2)$ .

2. $P$ admits a holomorphic reduction to $H$.