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Lectures on Kahler Geometry

Lectures on Kahler Geometry

Andrei Moroianu - Collection London Mathematical Society Student Texts

172 pages, parution le 19/04/2007

Résumé

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

Sommaire

  • Basics on Differential Geometry
    • Smooth manifolds;
    • Tensor fields on smooth manifolds;
    • The exterior derivative;
    • Principal and vector bundles;
    • Connections;
    • Riemannian manifolds;
  • Complex and Hermitian Geometry
    • Complex structures and holomorphic maps;
    • Holomorphic forms and vector fields;
    • Complex and holomorphic vector bundles;
    • Hermitian bundles;
    • Hermitian and Kähler metrics;
    • The curvature tensor of Kähler manifolds;
    • Examples of Kähler metrics;
    • Natural operators on Riemannian and Kähler manifolds;
    • Hodge and Dolbeault theory;
  • Topics on Compact Kähler Manifolds
    • Chern classes;
    • The Ricci form of Kähler manifolds;
    • The Calabi-Yau theorem;
    • Kähler-Einstein metrics;
    • Weitzenböck techniques;
    • The Hirzebruch-Riemann-Roch formula;
    • Further vanishing results;
    • Ricci-flat Kähler metrics;
    • Explicit examples of Calabi-Yau manifolds;
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Caractéristiques techniques

  PAPIER
Éditeur(s) Cambridge University Press
Auteur(s) Andrei Moroianu
Collection London Mathematical Society Student Texts
Parution 19/04/2007
Nb. de pages 172
Format 15,5 x 22,5
Couverture Broché
Poids 266g
Intérieur Noir et Blanc
EAN13 9780521688970
ISBN13 978-0-521-68897-0

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