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Period Mappings and Period Domains

2nd Edition

$89.99 (P)

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: August 2017
  • availability: Available
  • format: Hardback
  • isbn: 9781108422628

$ 89.99 (P)
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About the Authors
  • This up-to-date introduction to Griffiths' theory of period maps and period domains focusses on algebraic, group-theoretic and differential geometric aspects. Starting with an explanation of Griffiths' basic theory, the authors go on to introduce spectral sequences and Koszul complexes that are used to derive results about cycles on higher-dimensional algebraic varieties such as the Noether–Lefschetz theorem and Nori's theorem. They explain differential geometric methods, leading up to proofs of Arakelov-type theorems, the theorem of the fixed part and the rigidity theorem. They also use Higgs bundles and harmonic maps to prove the striking result that not all compact quotients of period domains are Kähler. This thoroughly revised second edition includes a new third part covering important recent developments, in which the group-theoretic approach to Hodge structures is explained, leading to Mumford–Tate groups and their associated domains, the Mumford–Tate varieties and generalizations of Shimura varieties.

    • A completely revised and up-to-date new edition, covering all major new developments in the field
    • Accessible to graduate students with modest backgrounds in algebraic topology and algebra
    • Begins by providing a comprehensive introduction to the basic theory as developed by Griffiths
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    Reviews & endorsements

    Review of previous edition: 'This book, dedicated to Philip Griffiths, provides an excellent introduction to the study of periods of algebraic integrals and their applications to complex algebraic geometry. In addition to the clarity of the presentation and the wealth of information, this book also contains numerous problems which render it ideal for use in a graduate course in Hodge theory.' Mathematical Reviews

    Review of previous edition: '… generally more informal and differential-geometric in its approach, which will appeal to many readers … the book is a useful introduction to Carlos Simpson's deep analysis of the fundamental groups of compact Kähler manifolds using harmonic maps and Higgs bundles.' Burt Totaro, University of Cambridge

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    Product details

    • Edition: 2nd Edition
    • Date Published: August 2017
    • format: Hardback
    • isbn: 9781108422628
    • dimensions: 238 x 160 x 42 mm
    • weight: 1.03kg
    • contains: 35 b/w illus. 3 tables 180 exercises
    • availability: Available
  • Table of Contents

    Part I. Basic Theory:
    1. Introductory examples
    2. Cohomology of compact Kähler manifolds
    3. Holomorphic invariants and cohomology
    4. Cohomology of manifolds varying in a family
    5. Period maps looked at infinitesimally
    Part II. Algebraic Methods:
    6. Spectral sequences
    7. Koszul complexes and some applications
    8. Torelli theorems
    9. Normal functions and their applications
    10. Applications to algebraic cycles: Nori's theorem
    Part III. Differential Geometric Aspects:
    11. Further differential geometric tools
    12. Structure of period domains
    13. Curvature estimates and applications
    14. Harmonic maps and Hodge theory
    Part IV. Additional Topics:
    15. Hodge structures and algebraic groups
    16. Mumford–Tate domains
    17. Hodge loci and special subvarieties
    Appendix A. Projective varieties and complex manifolds
    Appendix B. Homology and cohomology
    Appendix C. Vector bundles and Chern classes
    Appendix D. Lie groups and algebraic groups
    References
    Index.

  • Authors

    James Carlson, University of Utah
    James Carlson is Professor Emeritus at the University of Utah. From 2003 to 2012, he was president of the Clay Mathematics Institute, New Hampshire. Most of Carlson's research is in the area of Hodge theory.

    Stefan Müller-Stach, Johannes Gutenberg Universität Mainz, Germany
    Stefan Müller-Stach is Professor of number theory at Johannes Gutenberg Universität Mainz, Germany. He works in arithmetic and algebraic geometry, focussing on algebraic cycles and Hodge theory, and his recent research interests include period integrals and the history and foundations of mathematics. Recently, he has published monographs on number theory (with J. Piontkowski) and period numbers (with A. Huber), as well as an edition of some works of Richard Dedekind.

    Chris Peters, Université Grenoble Alpes, France
    Chris Peters is a retired professor from the Université Grenoble Alpes, France and has a research position at the Eindhoven University of Technology, The Netherlands. He is widely known for the monographs Compact Complex Surfaces (with W. Barth, K. Hulek and A. van de Ven, 1984), as well as Mixed Hodge Structures, (with J. Steenbrink, 2008). He has also written shorter treatises on the motivic aspects of Hodge theory, on motives (with J. P. Murre and J. Nagel) and on applications of Hodge theory in mirror symmetry (with Bertin).

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