Chapter I: Arithmetic Intersection

Abstract

The algebraic intersection theory on a smooth projective variety X over a field is a classical topic. Given two subvarieties Y and Z in X, with complementary dimensions, one defines an integer Y.Z, their algebraic intersection number.

At the beginning of the seventies, Arakelov defined an arithmetic intersection theory on a regular integral model X of a curve over a number field. Given two divisors Y and Z in X, he defines a real number, the arithmetic intersection of Y and Z.

The goal of this chapter is to extend Arakelov theory to higher dimensions. More precisely, given a regular projective scheme X over the integers and an hermitian line bundle $\bar {L}$ on X, to any closed subset Y ⊂ X is attached a real number $h_{\bar {L}}(Y)$, the Faltings height of Y . Our definition is given by a list of axioms.

In a last paragraph we develop, without proofs, an arithmetic intersection theory for X, in terms of arithmetic Chow groups. We also explain how to view $h_{\bar {L}}(Y)$ as an intersection number.