XIV

Preface

There were instances where the course continued through the summer

as a reading course for students in group representations. One summer, the

objective was to prove the Borel-Weil-Bott theorem; another time, it was to

explore a complex analysis approach to the study of representations of real

semisimple Lie groups. Material from these summer courses was expanded

and then included in the text as the final three chapters.

The material on several complex variables in the text owes a great debt

to the text of Gunning and Rossi [GR], and the recent rewriting of that text

by Gunning [Guj. It was from Gunning and Rossi that I learned the subject,

and the approach to the material that is used in Gunning and Rossi is also

the approach used in this text. This means a thorough treatment of the local

theory using the tools of commutative algebra, an extensive development of

sheaf theory and the theory of coherent analytic sheaves, proofs of the main

vanishing theorem for such sheaves (Cartan's Theorem B) in full general-

ity, and a complete proof of the finite dimensionality of the cohomologies

of coherent sheaves on compact varieties (the Cartan-Serre theorem). This

does not mean that I have included treatments of all the topics covered in

Gunning and Rossi. There is no discussion of pseudoconvexity, for example,

or global embeddings, or the proper mapping theorem, or envelopes of holo-

morphy. I have included, however, a more extensive list of applications of

the main results of the subject - particularly if one includes in this category

Serre's GAGA theorems and the material on complex semisimple Lie groups

and the proof of the Borel-Weil-Bott theorem.

Several complex variables is a very rich subject, wThich can be approached

from a variety of points of view. The serious student of several complex

variables should consult, not only Gunning's rewriting of Gunning and Rossi,

but also the many excellent texts which approach the subject from other

points of view. These include [D], [Fi], [GRe], [GRe2], [Ho], [K], and [N],

to name just a few.

Interwoven with the material on several complex variables in this text is

a simultaneous treatment of basic complex algebraic geometry. This includes

the structure theory of local rings of regular functions and germs of varieties,

dimension theory, the vanishing theorems for coherent and quasi-coherent

algebraic sheaves, structure of regular maps between varieties, and the main

theorems on the cohomology of coherent sheaves on projective spaces.

There are real advantages to this simultaneous development of algebraic

and analytic geometry. Results in the two subjects often have essentially

the same proofs; they both rely heavily on the same background material -

commutative algebra for the local theory and homological algebra and sheaf

theory for the global theory; and often a difficult proof in several complex

variables can be motivated and clarified by an understanding of the often