Lemma 2.2.2 . [01JC]

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Lemma 2.2.2.

Let $\overline{L}=\overline{\mathscr{O}(D+f)}_{\mathfrak{X}}$ be a metrized line bundle on $\mathrm{X}$ associated to a divisor $D$ on $\mathrm{X}$ and a continuous function $f$ on the graph $R(\mathfrak{X})$ . If it is semi-positive, resp. admissible in the sense of [57] then it is semi-positive, resp. admissible in the sense of this article, and one has

c_{1}(\overline{L})=\iota_{*}{\operatorname{curv}}(\overline{\mathscr{O}(D+f)})\log\left|{\pi}\right|^{-1}.