## Lectures on the Arithmetic Riemann-Roch TheoremThe 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

CLASSICAL RIEMANNROCH THEOREM | 3 |

CHERN CLASSES OF ARITHMETIC VECTOR BUNDLES | 15 |

LAPLACIANS AND HEAT KERNELS | 29 |

THE LOCAL INDEX THEOREM FOR DIRAC OPERATORS | 44 |

NUMBER OPERATORS AND DIRECT IMAGES | 62 |

ARITHMETIC RIEMANNROCH THEOREM | 77 |

THE THEOREM OF BISMUTVASSEROT | 93 |

### Other editions - View all

### Common terms and phrases

algebra argument arithmetic assertion assume asymptotic expansion basis bounded ch-class ch(E checks Chern Chern-classes choose closed coefficients coincides compact complex compute connection consider constant converges coordinates corresponding curvature cycle define definition deformation denote depend derivatives difference differential direct image divisor easy elements equal exact sequence example exists factor Finally finite fixed follows formally formula function Furthermore given gives heat kernel hermitian hermitian metric holds holomorphic implies independent induces integral isomorphic Kahler Laplacian lecture Lemma limit line bundle manifold metric morphism norm obtain operator perturbation positive projective bundle Proof prove rank regular relative replace represented respectively result Riemann-Roch scaling scheme secondary class sheaf side smooth split superconnection Suppose term theorem uniformly unique usual vanish vector bundle volume zero