Refined height pairing

Kahn, Bruno

doi:10.2140/ant.2024.18.1039

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Refined height pairing

Bruno Kahn

Appendix: Qing Liu

Vol. 18 (2024), No. 6, 1039–1079

DOI: 10.2140/ant.2024.18.1039

Abstract

For a $d$ -dimensional regular proper variety $X$ over the function field of a smooth variety $B$ over a field $k$ and for $i \geq 0$ , we define a subgroup ${CH}^{i} {(X)}^{(0)}$ of ${CH}^{i} (X)$ and construct a “refined height pairing”

{CH}^{i} {(X)}^{(0)} \times {CH}^{d + 1 - i} {(X)}^{(0)} \to {CH}^{1} (B)

in the category of abelian groups up to isogeny. For $i = 1, d$ , ${CH}^{i} {(X)}^{(0)}$ is the group of cycles numerically equivalent to $0$ . This pairing relates to pairings defined by P. Schneider and A. Beilinson if $B$ is a curve, to a refined height defined by L. Moret-Bailly when $X$ is an abelian variety, and to a pairing with values in $H^{2} (B_{\bar{k}}, ℚ_{l} (1))$ defined by D. Rössler and T. Szamuely in general. We study it in detail when $i = 1$ .

Keywords

intersection theory, alteration theory, category theory

Mathematical Subject Classification

Primary: 14C17

Secondary: 14E15

Milestones

Received: 25 March 2021

Revised: 11 June 2023

Accepted: 3 September 2023

Published: 30 April 2024

Authors

Bruno Kahn
CNRS, IMJ-PRG Sorbonne Université and Université Paris Cité Paris France

Qing Liu
Institut de Mathématiques de Bordeaux Université de Bordeaux, CNRS Talence France

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Vol. 18, No. 6, 2024