Grothendieck's Classification of Line Bundles over the Riemann Sphere

TL;DR

This paper provides a comprehensive, self-contained proof of Grothendieck's classification of holomorphic vector bundles over the Riemann sphere, including background, theory, and generalization to principal G-bundles.

Contribution

It offers a detailed, accessible proof of Grothendieck's theorem, extending the classification to principal G-bundles and consolidating foundational concepts.

Findings

Classification of holomorphic vector bundles over the Riemann sphere

Proof of Grothendieck's theorem in full generality

Extension to principal G-bundles

Abstract

In this paper we look at Grothendieck's work on classifying holomorphic bundles over the complex projective line. The paper is divided into $4$ parts. The first and second part we build up the necessary background to talk about vector bundles, sheaves, cohomology, etc. The main result of the $3^{rd}$ chapter is the classification of holomorphic vector bundles over the complex projective line. In the $4^{th}$ chapter we introduce principal $G$-bundles and some of the theory behind them and finish off by proving Grothendieck's theorem in full generality. The goal is a (mostly) self-contained proof of Grothendieck's result accessible to someone who has taken differential geometry.