Proposition-Definition 2.18 . [019J]

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Proposition-Definition 2.18.

To any $n$ -tuple $(\theta_{1},\dots,\theta_{n})$ of closed $(1,1)$ -forms we can associated a signed atomic measure $\theta_{1}\wedge\dots\wedge\theta_{n}$ supported on $X^{\mathrm{div}}$ such that

(2.2)

\int_{X}f\,\theta_{1}\wedge\dots\wedge\theta_{n}=\sum_{i\in I}b_{i}f(x_{i})\,(\theta_{1,\mathcal{X}}|_{E_{i}}\cdot\ldots\cdot\theta_{n,\mathcal{X}}|_{E_{i}})

for any common determination $\mathcal{X}$ of the forms $\theta_{i}$ , and for any model function $f$ . Here we have written the special fiber as $\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$ and $x_{i}=x_{E_{i}}$ is the divisorial point associated to $E_{i}$ .

Further, $(\theta_{1},\dots,\theta_{n})\mapsto\theta_{1}\wedge\dots\wedge\theta_{n}$ is multilinear and symmetric.