On the infinitesimal automorphisms of principal bundles

TL;DR

This paper explores the structure of infinitesimal automorphisms of principal bundles and line bundles over complex manifolds, establishing rationality results and reduction properties in specific geometric contexts.

Contribution

It extends the Birkhoff-Grothendieck theorem and provides new insights into the automorphism Lie algebras of line bundles and principal bundles over complex manifolds.

Findings

Rationality of complex manifolds with irreducible automorphism representations

Existence of reductions of principal bundles over the Riemann sphere to specific subgroups

Extension of classical theorems to broader complex-analytic settings

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 $H$.