Stable augmented bundles over Riemann surfaces

Summary

Introduction

At the symposium in Durham, the proceedings of which are reflected in this volume, there was a significant number of talks on what can generally be called “augmented holomorphic bundles”. What we mean by this term is a holomorphic object which consists of one or more holomorphic bundles together with certain extra holomorphic data, typically in the form of prescribed holomorphic sections. We were ourselves responsible for discussions of so-called holomorphic pairs (i.e. a single bundle with one prescribed section), holomorphic k-pairs (i.e. a single bundle with k prescribed sections), and holomorphic triples (i.e. two bundles plus a holomorphic map between them). There were also discussions of Higgs bundles (i.e. bundles together with a section of a specific associated bundle), and of objects consisting of a bundle plus a k-dimensional linear subspace of its space of holomorphic sections (called “coherent systems” by Le Potier, and “Brill-Noether pairs” by King and Newstead).

While each variant has special features, there are important aspects common to all these types of augmented bundles. Perhaps the most significant is the fact that all admit definitions of stability which extend the usual notion of stability for a holomorphic bundle, and which allow the construction of moduli spaces. Furthermore, except for the case of Higgs bundles, the definitions each involve a real parameter. By varying the parameter, this leads to a chain of birationally equivalent moduli spaces.