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
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