Preface

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

n

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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