Vector Bundles on Complex Projective Spaces (Progress in Mathematics; V. 3): M. Schneider, Heinz Spindler, Christian Okonek

These lecture notes are intended as. an introduction to the methods. of classification of holomorphic vector bundles over projectire algebraic manifolds X. To be as concrete as possible we have mostly restricted ourselves to the case X = P_n. According to Serre (GAGA) the classification of hol?morphic vector bundles is equivalent to the classification of algebraic vector bundles. Here we have used almost exclusively the langUage of analytic geometry. The book is intended for students who have a basic  knowledge of analytic and (or) algebraic geometry. Some fundamental results fr ?m these fields are summarized at the beginning. One of the authors gave a Survey in the Seminaire Bourbaki i978 on the current state of the classification of holomorphic vector bundles over P_n This lecture then served a the basis for a course of lectures in GSttingen in the Winter Semester 78/79. The present work is an extended and up-dated exposition of that course. Because of the introductory nature of this book we have had to leave ou some difficult topics such as the restriction theorem of Barth. As compensation we have appended to each section a paragraph in which'historical remarks are made, further results indicated 'and unsolved problems presented.

The book is divided into two chapters. Each chapter is subdivided into several sections which in tqrn are made up of a number of paragraphs. Each section'is preceeded by a short description of its contents.

In assembling the list of literature we have done our best tO. include all the articles about vector bundles over  which are known to us. On the other hand .we have not thought it necessary to include works about the classification of holomorphic vector bundles over curves. The reader interested in this highly developed theory is recommended'to read' an article-bY Tjurin (Russian Math. Surveys'1974) or the lecture notes of a course held at Tara Institute by Newstead.

Part of the present interest in holomorphic vector bundles comes from the connection to physics. The mathematician who is interested in this connection is recommended to see the ENS-Seminaire of Douady and Verdier. In the final paragraph of the present lecture notes he will also find remarks about that topic and some literature citations.

R. M. Switzer has not only translated the manuscript of these notes into english but has also aided us in answering many mathematical.questions. For this assitance we wish to thank him heartily. Furthermore we wish to thank. Mrs. M. Schneider for doing such a good job with the unpleasant task of typing these  notes and H. Hoppe for assembling the index and doing the difficult job of inserting the mathematical symbols.