The local height function satisfies the following properties.

1. (1)

   It is symmetric and multilinear with respect to $\otimes$ in the pairs $({\overline{L}}_{i},s_{i})$, $i=0,\dots,d$, provided that all terms are defined.

2. (2)

   Let $\varphi\colon X^{\prime}\to X$ be a morphism of proper varieties over $K$, $Y$ a $d$-dimensional cycle of $X^{\prime}$, and $({\overline{L}}_{i},s_{i})$ an integrable metrized line bundle on $X$ and a section, $i=0,\dots,d$. Then

   $$\operatorname{h}_{\varphi^{\ast}{\overline{L}}_{0},\dots,\varphi^{\ast}{\overline{L}}_{d}}(Y;\varphi^{\ast}s_{0},\dots,\varphi^{\ast}s_{d})=\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(\varphi_{\ast}Y;s_{0},\dots,s_{d}),$$

   provided that both terms are defined.

3. (3)

   Let $Z$ be the zero-cycle $Y\cdot\operatorname{div}(s_{0})\cdots\operatorname{div}(s_{d-1})$ and $f$ a rational function such that the section $fs_{d}$ meets $Z$ properly. Then

   $$\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})-\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,fs_{d})=\log|f(Z)|,$$

   where, if $Z=\sum_{l}m_{l}p_{l}$, then $f(Z)=\prod_{l}f(p_{l})^{m_{l}}$.

4. (4)

   Let ${\overline{L^{\prime}}}_{d}=(L_{d},\|\cdot\|^{\prime})$ be another choice of metric. Then

   $$\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})-\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}^{\prime}_{d}}(Y;s_{0},\dots,s_{d})=\\ -\int_{Y}\log(\|s_{d}(p)\|/\|s_{d}(p)\|^{\prime})\operatorname{c}_{1}(\overline{L}_{0})\land\dots\wedge\operatorname{c}_{1}(\overline{L}_{d-1})\land\delta_{Y}$$

   is independent of the choice of sections.