On the infinitesimal automorphisms of principal bundles
Radu Pantilie

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 be a compact complex manifold endowed with a very ample line bundle . Denote by the extended Lie algebra of infinitesimal automorphisms of . If the representation of on the space of holomorphic sections of is irreducible then is rational; (2) Let be a holomorphic principal bundle over the Riemann sphere, with structural group whose Lie algebra is not equal to its nilpotent radical. Then there exists a Lie subgroup of which is a quotient of a Borel subgroup of and such that admits a reduction to .
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