We want to apply this formula to a specific metrized line bundle on $X$. The Jacobian $J$ of $X$ is an Abelian variety of dimension $g$. We also choose a divisor $D$ of degree $1$ on $X$ and correspondingly fix an embedding $\iota$ of $X$ into $J$. (For this, we may need to enlarge the ground field $F$.) Finally, we let $\Theta$ be the theta divisor of $J$, defined as the image of $X^{g-1}$ by the map $(x_{1},\dots,x_{g-1})\mapsto\sum_{j=1}^{g-1}\iota(x_{j})$.

As described above, the line bundle $\mathscr{O}_{J}(\Theta)$ admits a canonical metrization ; this induces a metrization on its inverse image $L=\iota^{*}\mathscr{O}_{J}(\Theta)$ on $X$. The metrized line bundle $\overline{\mathscr{O}_{J}(\Theta)}$ gives rise to the (theta) Néron–Tate height on $J$. Consequently, decomposing $\operatorname{div}(f)=\sum n_{P}P$, we obtain

$$({\widehat{c}}_{1}(\overline{L})|\operatorname{div}(f))=\sum n_{P}[F(P):F]\widehat{h}_{\Theta}(\iota(P))=\sum n_{P}[F(P):F]\widehat{h}_{\Theta}([P-D]),$$

where $D$ is the fixed divisor of degree $1$ on $X$.