Proposition 6.35 . [02WU]

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Proposition 6.35.

With notations as above, let $Y$ be either the closure of an orbit or a toric subvariety. Then $Y$ is integrable with respect to ${\overline{L}}_{0},\dots,{\overline{L}}_{d}$ in the sense of Definition 2.53. Moreover, its global height is given by

\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)=\left[\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)\right]\in\mathbb{R}/\operatorname{def}(\mathbb{K}^{\times}).