Higher arithmetic intersection theory

Abstract

We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over an arithmetic field, which is similar to Gillet and Soulé's definition of arithmetic Chow groups. We also give a compact description of the intersection theory of such groups. A consequence of this theory is the definition of a heigh pairing between two higher algebraic cycles of complementary dimensions, whose real regulator class is zero. This description agrees with Beilinson's height pairing for the classical arithmetic Chow groups. We also give examples of higher arithmetic intersection pairing in dimension zero.