The global height of integrable cycles satisfies the following properties.

1. (1)

   It is symmetric and multilinear with respect to tensor products of integrable metrized line bundles.

2. (2)

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

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