12 - Moduli Spaces for Principal Bundles

Summary

Dedicated to Peter Newstead on occasion of his 65th birthday

Abstract

In this note, we will survey recent developments in the construction of moduli spaces of principal bundles with reductive structure group on algebraic varieties with special emphasis on smooth projective curves. An application of the theory of moduli spaces to the theory of moduli stacks will also be mentioned.

Introduction

Let k be an algebraically closed field, X a smooth, connected, and projective curve over k, and G a connected reductive linear algebraic group over k. This survey article is devoted to various aspects of the classification of principal G-bundles on X.

To begin with, let us look at the topological side. If k = ℂ, then any principal G-bundle may be viewed as a holomorphic principal G-bundle over X (viewed as a compact Riemann surface). As such, it defines a topological principal G-bundle on X. It is not hard to see that there is a bijection between the set of isomorphy classes of topological principal G-bundles on X and the fundamental group π₁(G) ([Ram01], Proposition 5.1 see also Section 5.1). The fundamental group of G may be defined in terms of the root data of G. In particular, it is defined for reductive linear algebraic groups over any field k. Then, one can show that, over any field K, π₁(G) parameterizes the algebraic equivalence classes of principal G-bundles on X ([DS], Proposition 5, [Holla], Proposition 3.15).