These lecture notes are intended as an introduction to the methods
of classi?cation of holomorphic vector bundles over projective
algebraic manifolds X. To be as concrete as possible we have mostly
restricted ourselves to the case X = P . According to Serre (GAGA)
the class- n cation of holomorphic vector bundles is equivalent to
the classi?cation 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 from these ?elds
are summarized at the beginning. One of the authors gave a survey
in the Seminaire Bourbaki 1978 on the current state of the
classi?cation of holomorphic vector bundles over P . This lecture
then served as the basis for a course of lectures n in G]ottingen
in the Winter Semester 78/79. The present work is an extended and
up-dated exposition of that course. Because of the - troductory
nature of this book we have had to leave out some di?cult 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 turn are made up of a
number of pa- graphs. Each section is preceded by a short
description of its contents."
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