Cohomological arithmetic Chow rings

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.