文档介绍：. Raynor
The Riemann-Roch theorem is a special case of
the Atiyah-Singer index formula
Master thesis defended on 5 March, 2010
Thesis supervisor: dr. M. L¨ubke
Mathematisch Instituut, Universiteit Leiden
Contents
Introduction 5
Chapter 1. Review of Basic Material 9
1. Vector bundles 9
2. Sheaves 18
Chapter 2. The Analytic Index of an plex 27
1. Elliptic differential operators 27
2. plexes 30
Chapter 3. The Riemann-Roch Theorem 35
1. Divisors 35
2. The Riemann-Roch Theorem and the analytic index of a divisor 40
3. The Euler characteristic and Hirzebruch-Riemann-Roch 42
Chapter 4. The Topological Index of a Divisor 45
1. De Rham Cohomology 45
2. The genus of a Riemann surface 46
3. The degree of a divisor 48
Chapter 5. Some aspects of algebraic topology and the T-characteristic 57
1. Chern classes 57
2. Multiplicative sequences and the Todd polynomials 62
3. The Todd class and the Chern Character 63
4. The T-characteristic 65
Chapter 6. The Topological Index of the Dolbeault operator 67
1. Elements of topological K-theory 67
2. The difference bundle associated to an elliptic operator 68
3. The Thom Isomorphism 71
4. The Todd genus is a special case of the topological index 76
Appendix: plexes and the topological index 81
Bibliography 85
3
Introduction
The Atiyah-Singer index formula equates a purely analytical property of an
elliptic differential operator P (resp. plex E) on pact manifold
called the analytic index inda(P ) (resp. inda(E)) with a purely topological prop-
erty, the topological index indt(P )(resp. indt(E)) and has been one of the most
significant single results in late twentieth century pure mathematics. It was an-
nounced by Michael Atiyah and Isadore Singer in 1963, with a sketch of a proof
using cohomological methods. Between 1968 and 1971, they published a series of
papers1 in which they proved the formula using topological K-theory, as well as
filling in the details of the original p