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Lectures on Kähler Geometry

Part of London Mathematical Society Student Texts

  • Date Published: March 2007
  • availability: Available
  • format: Paperback
  • isbn: 9780521688970


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About the Authors
  • Kähler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kähler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kähler identities. The final part of the text studies several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböck techniques, Calabi–Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory.

    • The first graduate-level text on Kähler geometry, providing a concise introduction for both mathematicians and physicists with a basic knowledge of calculus in several variables and linear algebra
    • Over 130 exercises and worked examples
    • Self-contained and presents varying viewpoints including Riemannian, complex and algebraic
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    Product details

    • Date Published: March 2007
    • format: Paperback
    • isbn: 9780521688970
    • length: 182 pages
    • dimensions: 229 x 153 x 12 mm
    • weight: 0.266kg
    • contains: 131 exercises
    • availability: Available
  • Table of Contents

    Part I. Basics on Differential Geometry:
    1. Smooth manifolds
    2. Tensor fields on smooth manifolds
    3. The exterior derivative
    4. Principal and vector bundles
    5. Connections
    6. Riemannian manifolds
    Part II. Complex and Hermitian Geometry:
    7. Complex structures and holomorphic maps
    8. Holomorphic forms and vector fields
    9. Complex and holomorphic vector bundles
    10. Hermitian bundles
    11. Hermitian and Kähler metrics
    12. The curvature tensor of Kähler manifolds
    13. Examples of Kähler metrics
    14. Natural operators on Riemannian and Kähler manifolds
    15. Hodge and Dolbeault theory
    Part III. Topics on Compact Kähler Manifolds:
    16. Chern classes
    17. The Ricci form of Kähler manifolds
    18. The Calabi–Yau theorem
    19. Kähler–Einstein metrics
    20. Weitzenböck techniques
    21. The Hirzebruch–Riemann–Roch formula
    22. Further vanishing results
    23. Ricci–flat Kähler metrics
    24. Explicit examples of Calabi–Yau manifolds

  • Author

    Andrei Moroianu, Ecole Polytechnique, Paris
    Andrei Moroianu is a Researcher at CNRS and a Professor of Mathematics at Ecole Polytechnique.

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