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Complex Abelian Varieties : Grundlehren der mathematischen Wissenschaften - Christina Birkenhake

Complex Abelian Varieties

Grundlehren der mathematischen Wissenschaften


Published: 22nd April 2004
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The subject of this book is the theory of abelian varieties over the field of complex numbers. It covers the main results of the theory, both classic and recent, in modern language. It is intended to give a comprehensive introduction to the field, but also to serve as a reference. The focal topics are the projective embeddings of an abelian variety, their equations and geometric properties. Moreover several moduli spaces of abelian varieties with additional structure are constructed. Some special results on Jacobians and Prym varieties allow applications to the theory of algebraic curves. The second edition contains five new chapters which present some of the most important recent result on the subject. Among them are results on automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture.

From the reviews:

"... the authors have somehow managed to make it unique in several respects: not only is it far more readable than most of other book on the subject, but it is also much more complete. It is, in my opinion, a very valuable reference book ... . [...] ...prerequisites are kept to a minimum. Very little background in algebraic geometry is necessary, almost all proofs are complete and accessible references are provided whenever they are not. ...Olivier Debarre in Mathematical Reviews, 1994

"It is a great reference and textbook, detailed, very up-to-date, thorough, clearly written and perfectly arranged."
W. Kleinert in Zentralblatt MATH, 1993

"... written in an understandable and systematical way and can be recommended to all mathematicians and physicists interested in the subject."
Newsletter of the European Mathematical Society, 1993

From the reviews of the second edition:

"This book aims to be a course in Lie groups that can be covered in one year with a group of seasoned graduate students. ... offers a wealth of complementary, partly quite recent material that is not found in any other textbook on Lie groups. ... this book covers an unusually wide spectrum of topics ... . the entire presentation is utmost thorough, comprehensive, lucid and absolutely user-friendly. ... All together, this graduate text his a highly interesting, valuable and welcome addition ... . (Werner Kleinert, Zentralblatt MATH, Vol. 1053, 2005)

From the reviews of the second edition:

"This is the second, substantially extended edition of the book ... . The book well deserves to become a standard reference for more researchers working or interested in the theory of abelian varieties." (Fumio Hazama, Mathematical Reviews, 2005c)

"The book under review is the second, essentially augmented edition of the original standard text, which now also reflects some of the very recent developments. ... the bibliography has been accordingly up-dated and enhanced. ... Summing up, the second, amply enlarged and up-dated edition of this outstanding standard monograph on complex abelian varieties has increased its utility in a significant degree, and its leading position among the existing books on the subject has been evidently strengthened." (Werner Kleinert, Zentralblatt MATH, Vol. 1056, 2005)

Complex torip. 7
Line bundles on complex torip. 23
Cohomology of line bundlesp. 45
Abelian varietiesp. 69
Endomorphisms of Abelian varietiesp. 113
Theta and Heisenberg groupsp. 145
Equations for Abelian varietiesp. 179
Modulip. 209
Moduli spaces of Abelian varieties with endomorphism structurep. 243
Abelian surfacesp. 281
Jacobian varietiesp. 315
Prym varietiesp. 363
Automorphismsp. 411
Vector bundles on Abelian varietiesp. 439
Further results on line bundles and the theta divisorp. 479
Cycles on Abelian varietiesp. 521
The Hodge conjecture for general Abelian and Jacobian varietiesp. 549
Algebraic varieties and complex analysis spacesp. 567
Line bundles and factors of automorphyp. 571
Some algebraic geometric resultsp. 577
Derived categoriesp. 583
Moduli spaces of sheavesp. 591
Abelian schemesp. 597
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540204886
ISBN-10: 3540204881
Series: Grundlehren der mathematischen Wissenschaften
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 638
Published: 22nd April 2004
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.8 x 16.2  x 4.0
Weight (kg): 1.14
Edition Number: 2
Edition Type: Revised