Abelian Varieties (TATA INSTITUTE OF FUNDAMENTAL RESEARCH, BOMBAY// STUDIES IN MATHEMATICS)

Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. Mumford describes the analytic methods and results applicable when the ground field k is the complex field C and discusses the scheme-theoretic methods and results used to deal with inseparable isogenies when the ground field k has characteristic p. The author also provides a self-contained proof of the existence of a dual abeilan variety, reviews the structure of the ring of endormorphisms, and includes in appendices "The Theorem of Tate" and the "Mordell-Weil Thorem." This is an established work by an eminent mathematician and the only book on this subject. Titrle Page......Page 2 Copyright Page......Page 3 Introduction......Page 4 Preface to Second Edition......Page 8 Contents......Page 10 1. Complex Tori......Page 12 2. Line bundles on a complex torus......Page 24 3. Algebraizability of tori......Page 35 4. Definition of abelian varieties......Page 50 5. Cohomology and base change......Page 57 6. The theorem of the cube: I......Page 66 7. Dividing varieties by finite groups......Page 76 8. The dual abelian variety: char 0......Page 85 9. The case k=C......Page 93 10. The theorem of the cube: II......Page 100 11. Basic theory of group schemes......Page 104 12. Quotients by finite group schemes......Page 119 13. The dual abelian variety in any characteristic......Page 134 14. Duality theory of finite commutative group schemes.......Page 143 15. Applications to abelian varieties......Page 154 16. Cohomology of line bundles......Page 161 17. Very ample line bundles......Page 174 18. Etale coverings......Page 178 19. Structure of Hom(X,X)......Page 183 20. Riemann forms......Page 194 21. Positivity of the Rosati involution......Page 203 22. Examples.......Page 221 23. The group #(L).......Page 232 24. The case k = C......Page 246 Appendix I : The Theorem of Tate by C.P.Ramanujam......Page 251 Appendix II: Mordell-Weil Theorem by Yuri Manin......Page 272 Bibliography......Page 287 Index......Page 289