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Hodge Theory and Complex Algebraic Geometry I

Volume 1


Part of Cambridge Studies in Advanced Mathematics

  • Date Published: December 2007
  • availability: Available
  • format: Paperback
  • isbn: 9780521718011

£ 44.99

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About the Authors
  • The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

    • 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 introductory text to Hodge theory and Kahlerian geometry is an excellent and modern introduction to the subject, shining with comprehensiveness, strictness, clarity, rigor, thematic steadfastness of purpose, and catching enthusiasm for this fascinating field of contemporary mathematical research. This book is exceedingly instructive, inspiring, challenging and user-friendly, which makes it truly outstanding and extremely valuable for students, teachers, and researchers in complex geometry.' Zentralblatt MATH

    'I would recommend anyone interested in learning about a topic in complex differential or algebraic geometry to read Voisin's volumes. She has done a remarkably good job.' Proceedings of the Edinburgh Mathematical Society

    '… 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

    'The book provides a very satisfying exposition of all the methods of studying algebraic cycles that have come out of Hodge theory.' Bulletin of the London Mathematical Society

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

    Prize Winner Cambridge University Press congratulates Claire Voisin, winner of the 2007 Ruth Lyttle Satter Prize in Mathematics!

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    Product details

    • Date Published: December 2007
    • format: Paperback
    • isbn: 9780521718011
    • length: 334 pages
    • dimensions: 227 x 153 x 18 mm
    • weight: 0.53kg
    • contains: 30 exercises
    • availability: Available
  • Table of Contents

    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

  • 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


    Leila Schneps

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