This text evolved from notes I developed for use in a course on several com-
plex variables at the University of Utah. The eclectic nature of the topics
presented in the text reflects the interests and motivation of the graduate
students who tended to enroll for this course. These students were almost all
planning to specialize in either algebraic geometry or representation theory
of semisimple Lie groups. The algebraic geometry students were primarily
interested in several complex variables because of its connections with al-
gebraic geometry, while the group representations students were primarily
interested in applications of complex analysis - both algebraic and analytic
- to group representations.
The course I designed to serve this mix of students involved a simulta-
neous development of basic complex algebraic geometry and basic several
complex variables, which emphasized and capitalized on the similarities in
technique of much of the foundational material in the two subjects. The
course began with an exposition of the algebraic properties of the local
rings of regular and holomorphic functions, first on C
and then on vari-
eties. This was followed by a development of abstract sheaf theory and sheaf
cohomology and then by the introduction of coherent sheaves in both the
algebraic and analytic settings. The fundamental vanishing theorems for
both kinds of coherent sheaves were proved and then exploited. Typically
the course ended with a proof and applications of Serre's GAGA theorems,
which show the equivalence of the algebraic and analytic theories in the case
of projective varieties. The notes for this course were corrected and refined,
with the help of the students, each time the course was taught. This text is
the result of that process.
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