By Joseph L. Taylor

ISBN-10: 082183178X

ISBN-13: 9780821831786

This article offers an built-in improvement of the idea of a number of advanced variables and intricate algebraic geometry, resulting in proofs of Serre's celebrated GAGA theorems concerning the 2 matters, and together with purposes to the illustration concept of complicated semisimple Lie teams. It features a thorough therapy of the neighborhood concept utilizing the instruments of commutative algebra, an in depth improvement of sheaf conception and the speculation of coherent analytic and algebraic sheaves, proofs of the most vanishing theorems for those different types of sheaves, and an entire facts of the finite dimensionality of the cohomology of coherent sheaves on compact types. The vanishing theorems have a large choice of purposes and those are coated in detail.

Of specific curiosity are the final 3 chapters, that are dedicated to purposes of the previous fabric to the research of the constitution and representations of advanced semisimple Lie teams. integrated are introductions to harmonic research, the Peter-Weyl theorem, Lie idea and the constitution of Lie algebras, semisimple Lie algebras and their representations, algebraic teams and the constitution of advanced semisimple Lie teams. All of this culminates in Miličić's facts of the Borel-Weil-Bott theorem, which makes wide use of the fabric built previous within the text.

There are quite a few examples and workouts in every one bankruptcy. this contemporary remedy of a vintage perspective will be a good textual content for a graduate direction on a number of complicated variables, in addition to an invaluable reference for the expert.

Readership: Graduate scholars and examine mathematicians drawn to ODEs, algebraic geometry, workforce idea, generalizations, and summary harmonic research.