This is a survey article on the Horrocks-Mumford bundle and its geometry. Since its discovery in 1972 this bundle has attracted the interest of several algebraic geometers (see e.g. [BHM1], [BHM2], [BM], [DS1], [DS2], [HL2], [HV], [HKW2], [K], [Sa1], [Sch], [Su2]). It is my point of view that we now understand the geometry of this bundle quite well (of course this does not mean that other interesting and surprising facts might not be discovered in the future). In any case the knowledge of the geometry of the Horrocks- Mumford bundle has already proved useful in other contexts, see e.g. the article by Decker and Popescu in this volume.

I would like to thank the organizers of the Durham symposium on Vector Bundles in Algebraic Geometry for organizing this meeting and for giving me the opportunity to speak about the Horrocks-Mumford bundle. I also lectured on the Horrocks-Mumford bundle during a stay at Tokyo Metropolitain University in March 1993. I would like to take this opportunity to thank Professor N. Sasakura for inviting me to Tokyo. This article is based on both my lectures in Tokyo and my talks in Durham.

I am grateful to the Deutsche Forschungsgemeinschaft for partial support over the last few years in the framework of the Schwerpunktprogramm “Komplexe Mannigfaltigkeiten”.

Throughout this paper I shall work over the complex numbers ℂ.

Vector bundles on ℙn

In order to put the Horrocks-Mumford bundle into perspective, I first want to recall some known results about the existence of vector bundles on ℙn.