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Abelian Varieties, Theta Functions and the Fourier Transform

Abelian Varieties, Theta Functions and the Fourier Transform

$118.00

Part of Cambridge Tracts in Mathematics

  • Date Published: April 2003
  • availability: Available
  • format: Hardback
  • isbn: 9780521808040

$118.00
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About the Authors
  • This book is a modern treatment of the theory of theta functions in the context of algebraic geometry. The novelty of its approach lies in the systematic use of the Fourier-Mukai transform. Alexander Polishchuk starts by discussing the classical theory of theta functions from the viewpoint of the representation theory of the Heisenberg group (in which the usual Fourier transform plays the prominent role). He then shows that in the algebraic approach to this theory (originally due to Mumford) the Fourier-Mukai transform can often be used to simplify the existing proofs or to provide completely new proofs of many important theorems. This incisive volume is for graduate students and researchers with strong interest in algebraic geometry.

    • A modern treatment of the theory of theta functions in the context of algebraic geometry
    • Discusses the classical theory of theta functions from the view of representation theory of the Heisenberg group
    • Ideal for graduate students and researchers with interest in algebraic geometry
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    Reviews & endorsements

    "I would definitely recommend this book to a reader already acquainted with abelian varieties wishing to go beyond the basics of the subject. It is stimulatig and provocative and at the same time well-organized. Even the expert will learn a lot from reading it." Bulletin of the AMS

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    Product details

    • Date Published: April 2003
    • format: Hardback
    • isbn: 9780521808040
    • length: 308 pages
    • dimensions: 229 x 152 x 21 mm
    • weight: 0.62kg
    • contains: 88 exercises
    • availability: Available
  • Table of Contents

    Part I. Analytic Theory:
    1. Line bundles on complex tori
    2. Representations of Heisenberg groups I
    3. Theta functions
    4. Representations of Heisenberg groups II: intertwining operators
    5. Theta functions II: functional equation
    6. Mirror symmetry for tori
    7. Cohomology of a line bundle on a complex torus: mirror symmetry approach
    Part II. Algebraic Theory:
    8. Abelian varieties and theorem of the cube
    9. Dual Abelian variety
    10. Extensions, biextensions and duality
    11. Fourier–Mukai transform
    12. Mumford group and Riemann's quartic theta relation
    13. More on line bundles
    14. Vector bundles on elliptic curves
    15. Equivalences between derived categories of coherent sheaves on Abelian varieties
    Part III. Jacobians:
    16. Construction of the Jacobian
    17. Determinant bundles and the principle polarization of the Jacobian
    18. Fay's trisecant identity
    19. More on symmetric powers of a curve
    20. Varieties of special divisors
    21. Torelli theorem
    22. Deligne's symbol, determinant bundles and strange duality
    Bibliographical notes and further reading
    References.

  • Author

    Alexander Polishchuk, Boston University
    fm.author_biographical_note1

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