Coherent Systems: a Brief Survey

Abstract

Dedicated to Peter Newstead on the occasion of his 65th birthday . Introduction 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]).