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Refined height pairing

Bruno Kahn

Appendix: Qing Liu

Vol. 18 (2024), No. 6, 1039–1079

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

CH i(X)(0) × CH d+1i(X)(0) 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 H2(Bk¯, l(1)) defined by D. Rössler and T. Szamuely in general. We study it in detail when i = 1.

intersection theory, alteration theory, category theory
Mathematical Subject Classification
Primary: 14C17
Secondary: 14E15
Received: 25 March 2021
Revised: 11 June 2023
Accepted: 3 September 2023
Published: 30 April 2024
Bruno Kahn
Sorbonne Université and Université Paris Cité
Qing Liu
Institut de Mathématiques de Bordeaux
Université de Bordeaux, CNRS

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