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. xin
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