Let $f$ be an invertible meromorphic function on $X$. Let us view it as an invertible meromorphic section of the trivial metrized line bundle $\overline{\mathscr{O}_{X}}$. Let $\overline{L}$ be any line bundle on $X$ with an admissible adelic metric. Then,

$$({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{\mathscr{O}_{X}})|X)=0.$$

Moreover, according to Theorem 1.3 of [19] (see Section 1.3),

$$({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{\mathscr{O}_{X}})|X)=({\widehat{c}}_{1}(\overline{L})|\operatorname{div}(f))+\sum_{v\in M(F)}\int_{X_{v}}\log\left|{f}\right|_{v}^{-1}c_{1}(\overline{L})_{v}.$$

In other words, this furnishes an integral formula for the height (relative to $\overline{L}$) of any divisor which is rationally equivalent to $0$ :

$$({\widehat{c}}_{1}(\overline{L})|\operatorname{div}(f))=\sum_{v\in M(F)}\int_{X_{v}}\log\left|{f}\right|_{v}c_{1}(\overline{L})_{v}.$$