Refined height pairing

Abstract

For a $d$-dimensional smooth projective variety $X$ over the function field of a smooth variety $B$ over a field $k$ and for $i\ge 0$, we define a subgroup $C{H}^i(X{)}^{(0)}$ of $C{H}^i(X)$ 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 modulo isogeny. For $i=1,d$, $C{H}^i(X{)}^{(0)}$ 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 $H^2(B_{\bar{k}},{\mathbf{Q}}_l(1))$ defined by D. Rössler and T. Szamuely in general. We study it in detail when $i=1$.

To appear in Alg. & Number theory. Added after Def. 2.2: Even if it is not apparent anymore, this definition was inspired by [8, Assumption 2] and [5, 1.2]

References in corpus (3)

• Néron models of algebraic curves
• Parameter spaces for algebraic equivalence
• Albanese kernels and Griffiths groups