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COHOMOLOGICAL ARITHMETIC CHOW RINGS

Published online by Cambridge University Press:  21 September 2006

J. I. Burgos Gil ,
J. Kramer  and
U. Kühn
J. I. Burgos Gil
Affiliation:
Facultad de Matemáticas, Universidad de Barcelona, Gran Vía 318 4o 1a, 08007 Barcelona, Spain ([email protected])
J. Kramer
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
U. Kühn
Affiliation:
Department Mathematik, Universität Hamburg, Bundesstraβe 55, D-20146 Hamburg, Germany
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Abstract

We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.

Keywords

Arakelov geometrysheaf cohomologyDeligne–Beilinson cohomologygood hermitian metricsPrimary 14G4014G3514C1714C3011G18
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 6 , Issue 1 , January 2007 , pp. 1 - 172
Copyright
2006 Cambridge University Press

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