6.1. Residually metrized models [017D]

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6.1. Residually metrized models

As explained above, to any model ${\mathcal{L}}$ of a line bundle $L$ on $X$ , defined on a proper dlt model ${\mathcal{X}}$ of $X$ , we can associate a skeleton $\operatorname{Sk}({\mathcal{L}})\subset\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}$ . To produce a measure on $\operatorname{Sk}({\mathcal{L}})$ we need additional data.

Definition 6.1.

Let $L$ be a line bundle on $X$ . A residually metrized model of $L$ is a pair ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ where ${\mathcal{L}}$ is a model of $L$ , determined on a proper dlt model ${\mathcal{X}}$ of $X$ , and $\psi_{0}$ is a continuous Hermitian metric on ${\mathcal{L}}_{0}:={\mathcal{L}}|_{{\mathcal{X}}_{0}}$ , viewed as a holomorphic line bundle over the complex space ${\mathcal{X}}_{0}$ . A residually metrized model metric $\psi^{\#}$ on $L$ is an equivalence class of such pairs, modulo pull-back to a higher model.

Example 6.2.

If $L$ is trivial, then any choice of trivialization $s\in H^{0}(X,L)$ defines a residually metrized model metric $\psi^{\#}$ on $L$ , determined on any model ${\mathcal{X}}$ by ${\mathcal{L}}={\mathcal{O}}_{{\mathcal{X}}}$ and $\psi_{0}$ the trivial metric on ${\mathcal{O}}_{{\mathcal{X}}_{0}}$ .