6.3. Measures on dual complexes [017I]

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6.3. Measures on dual complexes

We now define measures associated to residually metrized model metrics.

Definition 6.4.

Let ${\mathcal{L}}^{\#}$ be a residually metrized model of $K_{X}$ , determined on a proper dlt model ${\mathcal{X}}$ of $X$ . To ${\mathcal{L}}^{\#}$ we associate a positive measure $\mu_{{\mathcal{L}}^{\#}}$ on $\Delta({\mathcal{L}})\subset\Delta({\mathcal{X}})$ defined by

\mu_{{\mathcal{L}}^{\#}}=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}({\mathcal{L}}^{\#})\right)b_{\sigma}^{-1}\lambda_{\sigma},

where $\sigma$ runs over the top-dimensional faces of $\Delta({\mathcal{L}})$ .

By Lemma 1.2, we have

\mu_{{\mathcal{L}}^{\#}}(\sigma)=\frac{\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}({\mathcal{L}}^{\#})}{d!\prod_{i\in J}b_{i}}

for each face $\sigma$ corresponding to a component of some $E_{J}$ .