Cohomological arithmetic Chow rings

TL;DR

This paper develops a flexible cohomological framework for arithmetic Chow rings, enabling various variants and applications, including automorphic line bundles and heights of Hecke correspondences.

Contribution

It introduces a generalized theory of arithmetic Chow rings using complexes of abelian groups, unifying and extending existing intersection theories with new applications.

Findings

Recovered Gillet-Soulé intersection theory for projective varieties

Developed covariant arithmetic Chow groups for proper morphisms

Computed Faltings heights of Hecke correspondences on modular curves

Abstract

We develop a theory of abstract arithmetic Chow rings where the role of the fibers at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. This theory allows the construction of many variants of the arithmetic Chow groups with different properties. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soul\'e 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 crossings divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to a logarithmically singular hermitian line bundle to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.