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Hodge Theory and Complex Algebraic Geometry I

Hodge Theory and Complex Algebraic Geometry I

Volume 1

$196.00 (P)

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: January 2003
  • availability: Available
  • format: Hardback
  • isbn: 9780521802604

$ 196.00 (P)
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About the Authors
  • This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions.

    • Self-contained with full proofs, making it understandable to graduate students
    • A modern treatment of the subject, now in paperback
    • Exercises complement the main text, and give useful extra results
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    Reviews & endorsements

    "...this book is going to become a very common reference in this field. ...useful for both a student trying to learn the subject as well as the researcher that can find a wealth of results in a clear and compact format. The exposition is very precise and the introduction that precedes each chapter helps the reader to focus on the main ideas in the text." Mathematical Reviews

    "Mathematical rewards [await] those who invest their mathematical energies in this beautiful pair of volumes." Bulletin of the AMS

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    Product details

    • Date Published: January 2003
    • format: Hardback
    • isbn: 9780521802604
    • length: 336 pages
    • dimensions: 236 x 158 x 23 mm
    • weight: 0.57kg
    • contains: 30 exercises
    • availability: Available
  • Table of Contents

    Introduction
    Part I. Preliminaries:
    1. Holomorphic functions of many variables
    2. Complex manifolds
    3. Kähler metrics
    4. Sheaves and cohomology
    Part II. The Hodge Decomposition:
    5. Harmonic forms and cohomology
    6. The case of Kähler manifolds
    7. Hodge structures and polarisations
    8. Holomorphic de Rham complexes and spectral sequences
    Part III. Variations of Hodge Structure:
    9. Families and deformations
    10. Variations of Hodge structure
    Part IV. Cycles and Cycle Classes:
    11. Hodge classes
    12. Deligne-Beilinson cohomology and the Abel-Jacobi map
    Bibliography
    Index.

  • Instructors have used or reviewed this title for the following courses

    • Introduction to Algebraic Geometry ll
  • Author

    Claire Voisin, Institut des Hautes Études Scientifiques, Paris
    Claire Voisin is a Professor at the Institut des Hautes Études Scientifiques, France

    Translator

    Leila Schneps

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