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Introduction to Characteristic Classes and Index Theory


Introduction to Characteristic Classes and Index Theory (2000) This book is based on a course given by the author at the university of Lisbon during the academic year 1997-1998. Its aim is to give the reader an idea of how the theory of characteristic classes can be applied to solve index problems. Starting from the Lefschetz fixed point theorem and its application to the computation of the Euler-Poincaré characteristic of a compact orientable manifold, we first develop the theory of Euler classes of orientable manifolds and real vector bundles. Then, we study the Stiefel-Whitney classes and the general modulo 2 characteristic classes of real vector bundles. Similar considerations for complex vector bundles lead us to the Chern classes. We conclude the part devoted to characteristic classes by a study of global and local Chern characters. The rest of the book is then centered around the Riemann-Roch theorem. We present first a very simple proof which works for compact complex curves and allows us to make links with the original results of Riemann and Roch. Then, we treat in details the case of compact complex projective manifolds by more advanced methods.
Categorias:
  • Textos de Matemática da FCUL

  • Autor: Jean-Pierre Schneiders