Proposition 6.4 . [02W6]

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Proposition 6.4.

The toric local height is symmetric and multilinear with respect to tensor product of metrized toric line bundles. In particular, let $\Sigma$ be a complete fan, ${\overline{L}}_{i}$ a family of $d+1$ toric line bundles with integrable toric metrics and $Y$ an algebraic cycle of $X_{\Sigma}$ of dimension $d$ . Then

(6.5)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)=\sum_{j=0}^{d}(-1)^{d-j}\sum_{1\leq i_{0}<\cdots<i_{j}\leq d}\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{i_{0}}\otimes\cdots\otimes{\overline{L}}_{i_{j}}}(Y).