Overview

"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."

Full Product Details

Author:   Christina Birkenhake ,  Herbert Lange
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   Softcover reprint of hardcover 2nd ed. 2004
Volume:   302
Dimensions:   Width: 15.50cm , Height: 3.30cm , Length: 23.50cm
Weight:   0.985kg
ISBN:  

9783642058073

ISBN 10:   3642058078
Pages:   638
Publication Date:   01 December 2010
Audience:   Professional and scholarly ,  Professional and scholarly ,  Professional & Vocational ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Notation.- 1. Complex Tori.- 2. Line Bundles on Complex Tori.- 3. Cohomology of Line Bundles.- 4. Abelian Varieties.- 5. Endomorphisms of Abelian Varieties.- 6. Theta and Heisenberg Groups.- 7. Equations for Abelian Varieties.- 8. Moduli.- 9. Moduli Spaces of Abelian Varieties with Endomorphism Structure.- 10. Abelian Surfaces.- 11. Jacobian Varieties.- 12. Prym Varieties.- 13. Automorphisms.- 14. Vector bundles on Abelian Varieties.- 15. Further Results on Line Bundles an the Theta Divisor.- 16. Cycles on Abelian varieties.- 17. The Hodge Conjecture for General Abelian and Jacobian Varieties.- A. Algebraic Varieties and Complex Analytic Spaces.- B. Line Bundles and Factors of Automorphy.- C. Some Algebraic Geometric Results.- C.1 Some Properties of ?-Divisors.- C.2 The Kodaira Dimension.- C.3 Vanishing Theorems.- C.6 Some Results from Intersection Theory.- C.7 Adjoint Ideals.- D. Derived Categories.- D.1 Definition and First Properties.- D.2 Derived Functors.- D.3 The Grothendieck-Riemann-Roch Theorem.- E. Moduli Spaces of Sheaves.- F. Abelian Schemes.- F.1 Abelian Schemes and the Poincar´e Bundle.- F.2 Relative Fourier Functor.- F.3 The Relative Jacobian.- Glossary of Notation.

Reviews

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)

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)

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)

"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 standardtext, 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)"

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