Refined height pairing

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

$${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^i{(X)}^{(0)}\times {\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^{d+1-i}{(X)}^{(0)}\to {\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^1(B)$$

in the category of abelian groups up to isogeny. For $i=1,d$, ${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^i{(X)}^{(0)}$ 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 $H^2(B_{\overline{k}},{\mathbb{Q}}_l(1))$ defined by D. Rössler and T. Szamuely in general. We study it in detail when $i=1$.