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Vector bundles and their associated moduli spaces are of fundamental importance in algebraic geometry. In recent decades the richness of the subject has been enhanced by its relationship to other areas of mathematics, including differential geometry (which has always played a fundamental role), topology (where the moduli spaces not only present interesting topological features but turn out to be useful in settling topological problems) and, perhaps most surprising of all, theoretical physics, in particular gauge theory, quantum field theory and string theory.
The concept of stability for vector bundles, and the ensuing construction of the moduli spaces which classify these bundles, goes back more than 40 years to the fundamental work of Mumford and of Narasimhan and Seshadri. Since then, there has been much progress in describing the detailed structure of these spaces using methods from pure algebraic geometry, topology, differential geometry, number theory, representation theory and theoretical physics.
Peter E. Newstead has been a leading figure in this field almost from its inception and has made many seminal contributions to our understanding of moduli spaces of stable bundles. His “Tata lecture notes” have helped an entire generation of algebraic geometers learn the foundations of moduli theory and vector bundles. He is Chair of the international research group “Vector Bundles on Algebraic Curves” (VBAC), which he initiated in 1993 and whose purpose was to encourage the development of this area of research, particularly among young mathematicians.
Dedicated to Peter Newstead on the occasion of his 65th birthday.
A coherent system is a pair (E, V) where E is a holomorphic bundle and V is a linear subspace of its space of holomorphic sections. If E is a semistable bundle, then the existence of such objects is equivalent to the non-emptiness of a higher rank Brill-Noether locus. This connection to higher rank Brill-Noether theory provides one of the motivations for studying coherent systems. It certainly motivated Peter Newstead's guiding role in the development of the subject, and thus makes a volume in honor of his 65th birthday a fitting place for a survey.
Interest in coherent systems extends beyond Brill-Noether theory, mainly because there is a stability notion for a pair (E, V), distinct from the stability of the bundle E. The natural definition of such stability depends on a real parameter (denoted by α) and leads to a finite family of moduli spaces of α-stable coherent systems. These moduli spaces present a rich display of topological and geometric phenomena, most of which have yet to be fully explored.
This survey will be limited in scope because of space constraints and because of a survey in preparation by Peter Newstead based on his lectures at a Clay Institute workshop held in October 2006 ([N]).
Vector bundles and their associated moduli spaces are of fundamental importance in algebraic geometry. In recent decades this subject has been greatly enhanced by its relationships with other areas of mathematics, including differential geometry, topology and even theoretical physics, specifically gauge theory, quantum field theory and string theory. Peter E. Newstead has been a leading figure in this field almost from its inception and has made many seminal contributions to our understanding of moduli spaces of stable bundles. This volume has been assembled in tribute to Professor Newstead and his contribution to algebraic geometry. Some of the subject's leading experts cover foundational material, while the survey and research papers focus on topics at the forefront of the field. This volume is suitable for both graduate students and more experienced researchers.