These lecture notes are intended as an introduction to the methods
of classification of holomorphic vector bundles over projective
algebraic manifolds X. To be as concrete as possible we have mostly
restricted ourselves to the case X = Fn. According to Serre (GAGA)
the classification of holomorphic 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 funda mental results from these
fields are summarized at the beginning. One of the authors gave a
survey in the Seminaire Bourbaki 1978 on the current state of the
classification of holomorphic vector bundles overFn. This lecture
then served as the basis for a course of lectures in Gottingen 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 out some difficult topics
such as the restriction theorem of Barth. As compensation we have
appended to each sec tion 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 turn are made up of a
number of paragraphs. Each section is preceeded by a short
description of iv its contents."
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