On the equivariant vector bundles on $\mathbb{CP}^1$

TL;DR

This paper classifies holomorphic vector bundles on the complex projective line that are equivariant under certain subgroups of PGL(2,C) or SL(2,C), extending previous results to broader classes of groups.

Contribution

It generalizes prior classifications by considering more general subgroups H, beyond finite abelian groups, and describes the structure of H-equivariant vector bundles on rac{1}{1}^1.

Findings

Classification of H-equivariant holomorphic vector bundles on rac{1}{1}^1.

Extension of previous results to broader subgroup classes.

Provides a framework for understanding equivariant bundles under complex algebraic groups.

Abstract

Let $H$ be a subgroup of ${\rm PGL}(2,\mathbb C)$ (respectively, ${\rm SL}(2,\mathbb C)$) such that the Zariski closure in ${\rm PGL}(2,\mathbb C)$ (respectively, ${\rm SL}(2,\mathbb C)$) of some compact subgroup of $H$ contains $H$. We classify the $H$--equivariant holomorphic vector bundles on $\mathbb{CP}^1$. This generalizes \cite{BM} where $H$ was assumed to be a finite abelian group.