Preface
xv
similar but technically simpler proof of the analogous result in algebraic
geometry.
Several complex variables and complex algebraic geometry are not just
similar; they are equivalent when done in the context of projective varieties.
This is the content of Serre's GAGA theorems. We give complete proofs of
these results in Chapter 13, after first studying the cohomology of coherent
sheaves on projective spaces in Chapter 12.
The text could easily have ended with Chapter 13. This is where the
course typically ends. The material in Chapters 14 through 16 is on quite
a different subject - Lie groups and their representations - albeit one that
involves the extensive use of several complex variables and algebraic geome-
try. Chapter 16 is devoted to a proof of the Borel-Weil-Bott theorem. This
is the theorem which pinpoints the relationship between finite dimensional
holomorphic representations of a complex semisimple Lie group G and the
cohomologies of G-equivariant holomorphic line bundles on a projective va-
riety, called the flag variety, constructed from G. Chapter 15 is a brief
treatment of the subject of complex algebraic groups. This is included in
order to provide proofs of some of the basic structure results for complex
semisimple Lie groups that are needed in the formulation and proof of the
Borel-Weil-Bott theorem. Chapter 14 is a survey of the background mate-
rial needed if one is to understand Chapters 15 and 16. It includes material
on topological groups and their representations, compact groups, Lie groups
and Lie algebras, and finite dimensional representations of semisimple Lie
algebras. These last three chapters are included primarily for the benefit
of the student of Lie theory and group representations. This material illus-
trates that both several complex variables and complex algebraic geometry
are essential tools in the modern study of group representations. The chapter
on algebraic groups (Chapter 15) provides particularly compelling examples
of the utility of algebraic geometry applied in the context of the structure
theory of Lie groups. The proof of the Borel-Weil-Bott theorem in Chapter
16 involves applications of a wide range of material from several complex
variables and algebraic geometry. In particular, it provides nice applications
of the sheaf theory of Chapter 7, the Cartan-Serre theorem from Chapter
11, the material on projective varieties in Chapter 12, Serre's theorems in
Chapter 13, and of course, the background material on algebraic groups and
general Lie theory from Chapters 14 and 15.
I have tried to make the text as self-contained as possible. However,
students who attempt to use it will need some background. This should
include knowledge of the material from typical first year graduate courses
in real and complex analysis, modern algebra, and topology. Also, students
who wishes to confront the material in Chapters 14 through 16 will be
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