Higher Arithmetic Intersection Theory

TL;DR

This paper introduces a new framework for higher arithmetic Chow groups on smooth projective varieties over number fields, extending classical intersection theory and defining a height pairing for higher cycles.

Contribution

It provides a novel definition of higher arithmetic Chow groups, a compact intersection theory, and a height pairing consistent with Beilinson's pairing, with new examples under a conjecture.

Findings

New definition of higher arithmetic Chow groups

A compact description of their intersection theory

Examples of non-zero higher intersection pairings under a conjecture

Abstract

We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over a number field, which is similar to Gillet and Soul\'e'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 height 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 the higher arithmetic intersection pairing in dimension zero that, assuming a conjecture by Milnor on the independence of the values of the dilogarithm, are non zero.