The local height on $X$ is the function that, to each $d$-dimensional cycle $Y$ and each family of integrable metrized line bundles with sections $({\overline{L}}_{i},s_{i})$, $i=0,\dots,d$, such that the sections meet $Y$ properly, associates a real number $\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})$ determined inductively by the properties:

1. (1)

   $\operatorname{h}(\emptyset)=0$;

2. (2)

   if $Y$ is a cycle of dimension $d\geq 0$, then

   $$\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})=\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1}}(Y\cdot\operatorname{div}s_{d};s_{0},\dots,s_{d-1})\\ -\int_{X^{{\text{\rm an}}}}\log\|s_{d}\|\operatorname{c}_{1}(\overline{L}_{0})\land\dots\wedge\operatorname{c}_{1}(\overline{L}_{d-1})\land\delta_{Y}.$$