A generalization of Beilinson's geometric height pairing

TL;DR

This paper generalizes Beilinson's height pairing from curves to higher-dimensional bases, constructing a new pairing for varieties over function fields of higher-dimensional varieties, with potential geometric significance.

Contribution

It extends Beilinson's height pairing to varieties over the function field of higher-dimensional bases, providing a new pairing valued in the second $ ext{\ell}$-adic cohomology group.

Findings

Constructed a pairing for varieties over higher-dimensional bases.

The pairing is $ ext{\mathbb{Q}}$-valued over $ ext{\mathbb{C}}$.

Speculates on the geometric origin of the pairing.

Abstract

In the first section of his seminal paper on height pairings, Beilinson constructed an $\ell$-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the function field of a curve over an algebraically closed field, and asked about an generalization to higher dimensional bases. In this paper we answer Beilinson's question by constructing a pairing for varieties defined over the function field of a smooth variety $B$ over an algebraically closed field, with values in the second $\ell$-adic cohomology group of $B$. Over $\Bbb C$ our pairing is in fact $\Bbb Q$-valued, and in general we speculate about its geometric origin.