Algebraic and Analytic Geometry: (Series Number 345 London Mathematical Society Lecture Note Series)

This textbook, for an undergraduate course in modern algebraic geometry, recognizes that the typical undergraduate curriculum contains a great deal of analysis and, by contrast, little algebra. Because of this imbalance, it seems most natural to present algebraic geometry by highlighting the way it connects algebra and analysis; the average student will probably be more familiar and more comfortable with the analytic component. The book therefore focuses on Serre's GAGA theorem, which perhaps best encapsulates the link between algebra and analysis. GAGA provides the unifying theme of the book: we develop enough of the modern machinery of algebraic geometry to be able to give an essentially complete proof, at a level accessible to undergraduates throughout. The book is based on a course which the author has taught, twice, at the Australian National University.

Table of Contents:
Foreword; 1. Introduction; 2. Manifolds; 3. Schemes; 4. The complex topology; 5. The analytification of a scheme; 6. The high road to analytification; 7. Coherent sheaves; 8. Projective space - the statements; 9. Projective space - the proofs; 10. The proof of GAGA; Appendix. The proofs concerning analytification; Bibliography; Glossary; Index.

About the Author :
Amnon Neeman is professor of mathematics at Australian National University. He is the author of the book Triangulated Categories.

Review :
All in all, the book under review is a masterpiece of expository writing in modern algebraic geometry. It is exactly what the author promised: no comprehensive text to train future algebraic geometers, but rather an attempt to convince students of the fascinating beauty, the tremendous power, and the high value of the methods of algebraic and analytic geometry ... The author has reached his declared goal in an admirable, truly brilliant manner. His lucid exposition of these modern, fairly advanced topics in the field for beginners breathes his passion for the subject and for grippingly teaching it, and his style of writing bespeaks a good sympathetic understanding of students and their needs. Numerous synoptic sections, asides for the expert reader, and additional remarks help the student grasp the vast material thoroughly, and both the careful glossary and the detailed index facilitate active work with this highly inspiring book. Also, the author makes concrete, instructive proposals for how to use his text for different variants of self-study or teaching, which is particularly useful as its current extended contents now offer far more than can be covered in a one-semester course ... It remains to wish that this very special introduction to the realm of(complex)algebraic geometry finds deserved wide-spread interest within the mathematical community. Werner Kleinert, Zentralblatt MATH