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”

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 $i=1$.

为一个$d$维正则簇$X$在光滑簇的函数域上$乙$在一片田野上$k$并为$我\ge 0$，我们定义一个子群${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^{我}{(X）}^{(0）}$的${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^{我}(X）$并构建“精致的身高配对”

在阿贝尔群的范畴中直至同系。为了$我=1,d$,${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^{我}{(X）}^{(0）}$是循环组在数值上等价于$0$。该配对与 P. Schneider 和 A. Beilinson 定义的配对相关，如果$乙$是一条曲线，达到由 L. Moret-Bailly 定义的精确高度，当$X$是一个阿贝尔变体，并且与以下值配对$H^2({乙}_{\stackrel{￣}{k}},{\mathbb{Q}}_{我}(1））$一般由 D. Rössler 和 T. Szamuely 定义。我们详细研究一下$我=1$。