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 S´eminaire 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.
