This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language. The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. ". . . far more readable than most . . . it is also much more complete." Olivier Debarre in Mathematical Reviews, 1994.
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Book Description
--This text refers to an alternate
Hardcover
edition.
Editorial Reviews
Review
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)
--This text refers to an alternate Hardcover edition.From the Back Cover
Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and 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 onJacobians and Prym varieties allow applications to the theory of algebraic curves. The main tools for the proofs are the theta group of a line bundle, introduced by Mumford, and the characteristics, to be associated to any nondegenerate line bundle. They are a direct generalization of the classical notion of characteristics of theta functions.
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.
--This text refers to an alternate Hardcover edition.
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First Sentence:
A hyperelliptic integral is by definition an integral of the form where is a path in the complex plane C with coordinate z and f(z) = (z - ai) . . . (z - ad) with pairwise different constants ai. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
canonical theta functions, definite line bundle, nondegenerate line bundle, complementary abelian subvarieties, irreducible principal polarization, complementary abelian subvariety, endomorphism structure, ample symmetric line bundle, totally indefinite quaternion algebra, symmetric theta structures, symmetric line bundles, polarized abelian variety, normal complex analytic space, polarized abelian varieties, polarized abelian threefold, tetragonal curve, symmetric divisor, dual complex torus, complex subtorus, normalized isomorphism, totally complex quadratic extension, theta transformation formula, trisecant identity, real coordinate functions, totally definite quaternion algebra Key Phrases - Capitalized Phrases (CAPs): (learn more)
Isogeny Theorem, Riemann-Roch Theorem, Appell-Humbert Theorem, Riemann Relations, Seesaw Principle, Decomposition Theorem, Riemann's Singularity Theorem, Proof According, Riemann's Theorem, Theorem of the Square, Moving Lemma, Torelli's Theorem, Inverse Formula, Kawamata-Viehweg Vanishing Theorem, Definition of the Jacobian Variety, Hence the Leray, Proof Step, Base Change Formula, Holomorphic Lefschetz Fixed Point Formula, Poincaré's Complete Reducibility Theorem, Proof First, Rigidity Lemma, Using Exercise New!
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A hyperelliptic integral is by definition an integral of the form where is a path in the complex plane C with coordinate z and f(z) = (z - ai) . . . (z - ad) with pairwise different constants ai. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
canonical theta functions, definite line bundle, nondegenerate line bundle, complementary abelian subvarieties, irreducible principal polarization, complementary abelian subvariety, endomorphism structure, ample symmetric line bundle, totally indefinite quaternion algebra, symmetric theta structures, symmetric line bundles, polarized abelian variety, normal complex analytic space, polarized abelian varieties, polarized abelian threefold, tetragonal curve, symmetric divisor, dual complex torus, complex subtorus, normalized isomorphism, totally complex quadratic extension, theta transformation formula, trisecant identity, real coordinate functions, totally definite quaternion algebra Key Phrases - Capitalized Phrases (CAPs): (learn more)
Isogeny Theorem, Riemann-Roch Theorem, Appell-Humbert Theorem, Riemann Relations, Seesaw Principle, Decomposition Theorem, Riemann's Singularity Theorem, Proof According, Riemann's Theorem, Theorem of the Square, Moving Lemma, Torelli's Theorem, Inverse Formula, Kawamata-Viehweg Vanishing Theorem, Definition of the Jacobian Variety, Hence the Leray, Proof Step, Base Change Formula, Holomorphic Lefschetz Fixed Point Formula, Poincaré's Complete Reducibility Theorem, Proof First, Rigidity Lemma, Using Exercise New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
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47
books
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- New Trends in Algebraic Geometry (London Mathematical Society Lecture Note Series) by K. Hulek in Back Matter (1), Back Matter (2), and Back Matter (3)
- Intersection Theory, 2nd Edition by William Fulton in Back Matter (1), and Back Matter (2)
- Abelian Varieties (Tata Institute of Fundamental Research, Bombay// Studies in Mathematics) by David Mumford in Back Matter (1), and Back Matter (2)
- Geometry of Algebraic Curves: Volume I (Grundlehren der mathematischen Wissenschaften) by Enrico Arbarello in Back Matter (1), and Back Matter (2)
- Cohomology of Finite Groups (Grundlehren der mathematischen Wissenschaften) by Alejandro Adem in Back Matter
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- Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993 ([De Gruyter Proceedings in Mathematics]) by Wolf Barth on page 18, and page 23
- Recent Progress in Intersection Theory by Geir Ellingsrud on page 196, and page 297
- Cohomology of Finite Groups (Grundlehren der mathematischen Wissenschaften) by Alejandro Adem in Back Matter
- Elliptic Cohomology (University Series in Mathematics) by Charles B. Thomas in Back Matter
- Metric Spaces of Non-Positive Curvature (Grundlehren der mathematischen Wissenschaften) by Martin R. Bridson in Back Matter
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