文档介绍：LECTURES ON
THE ARITHMETIC
. RIEMANN-ROCH THEOREM
BY
GERD FALTINGS
ANNALS OF MATHEMATICS STUDIES
PRINCETON UNIVERSITY PRESS
Table of Contents
INTRODUCTION ............................ vii
LIST OF SYMBOLS ........................... ix
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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
INTRODUCTION
In the spring of 1990 I gave a graduate course at Princeton Univer-
sity on the arithmetic Riemann-Roth theorem, which had just been shown
by Bismut-Lebeau and Gillet-Soul& The main purpose was to under-
stand the techniques involved, and to simplify the presentation if possible.
These notes arise from the course, however with some later improvements.
Arakelov theory was invented with the purpose of applying techniques from
algebraic geometry to arithmetic problems, especially to obtain a proof of
the Mordell-conjecture. The main idea is to formulate algebraic properties
at finite places in terms of metrics, and then to try to find analogues at
infinity. In short one has to endow everyting with metrics. Around 1982
some progress was made. I showed that for arithmetic surfaces there is a
Riemann-Roth theorem, and that one can use it to derive various analogues
of properties plex surfaces, for example the Hodge-Index theorem
and the positivity of w2. The key to all this was the construction of volume
forms on the cohomology of hermitian bundles. At the same time Quillen
proposed a similar construction, using Ra