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 ≥ 0, we define a subgroup CH i (X) (0) of CH i (X) and construct a "refined height pairing"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 (Bk, Q l (1)) defined by D. Rössler and T. Szamuely in general.We study it in detail when i = 1.