Refined height pairing

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 $CH^i(X)^{(0)}$ 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 modulo 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 $H^2(B_{\bar k},\mathbf{Q}_l(1))$ defined by D. R\"ossler and T. Szamuely in general. We study it in detail when $i=1$.