Synopses & Reviews
The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.
Review
"This treatise provides a new approach to the arithmetic Riemann-Roch problem, and a widely algebraic-geometric method to solve it."--Zentralblatt für Mathematik
Review
This treatise provides a new approach to the arithmetic Riemann-Roch problem, and a widely algebraic-geometric method to solve it. Zentralblatt für Mathematik
Synopsis
The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.
Table of Contents
| Introduction | |
| List of Symbols | |
| Lecture 1 | Classical Riemann-Roch Theorem | 3 |
| Lecture 2 | Chern Classes of Arithmetic Vector Bundles | 15 |
| Lecture 3 | Laplacians and Heat Kernels | 29 |
| Lecture 4 | The Local Index Theorem for Dirac Operators | 44 |
| Lecture 5 | Number Operators and Direct Images | 62 |
| Lecture 6 | Arithmetic Riemann-Roch Theorem | 77 |
| Lecture 7 | The Theorem of Bismut-Vasserot | 93 |
| References | 99 |