1 - Lectures on Principal Bundles

Summary

To Peter Newstead on his 65th birthday

Introduction

The aim of these lectures is to give a brief introduction to principal bundles on algebraic curves towards the construction of the moduli spaces of semistable principal bundles. The second section develops the basic machinery on principal bundles and their automorphisms. At the end of the second section, we give a proof of theorem of Grothendieck on orthogonal bundles. The third section, after developing the notions of semistability and stability, gives a modern proof of the main part of Grothendieck's theorem on classification of principal bundles on the projective line. The last section gives an outline of the construction of the moduli space of principal bundles on curves. The moduli space was constructed by A.Ramanthan in 1976. The method outlined here is from a construction in [BS]. These notes are a transcription of the lectures given in Mexico and therefore have an air of informality about them. I have consciously retained this informality despite criticism from a learned referee on the lack of rigor in some places. Indeed “these notes are almost exactly in the form in which they were first written and distributed: as class notes, supplementing and working out my oral lectures. As such, they are far from polished and ask a lot of the reader. …Be that as it may, my hope is that a well-intentioned reader will still be able to penetrate these notes and learn something of the subject”