Equivariant vector bundles over the complex projective line

TL;DR

This paper classifies all algebraic vector bundles over the complex projective line that are compatible with a finite abelian group action, extending previous classifications on invariant affine open subsets.

Contribution

It provides a complete classification of G-equivariant algebraic vector bundles over the entire complex projective line, building on prior work limited to affine subsets.

Findings

Complete classification of G-equivariant vector bundles on ${f C}P^1$

Extension of previous affine subset classifications

Results applicable to finite abelian group actions

Abstract

Let $G$ be a finite abelian group acting faithfully on ${\mathbb C}{\mathbb P}^1$ via holomorphic automorphisms. In \cite{DF2} the $G$--equivariant algebraic vector bundles on $G$--invariant affine open subsets of ${\mathbb C}{\mathbb P}^1$ were classified. We classify the $G$--equivariant algebraic vector bundles on ${\mathbb C}{\mathbb P}^1$.