Proposition 1.3.2 . [01J3]

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Proposition 1.3.2.

Let $\varphi$ be a smooth function on $\mathrm{X}$ and let $\overline{L}_{1},\dots,\overline{L}_{k}$ be admissible metrized line bundles ; let $\mathrm{Z}$ be a $k$ -dimensional subvariety of $\mathrm{X}$ and let $s$ be an invertible meromorphic sections of $\overline{L}_{1}$ . Then,

\int_{\mathrm{X}}\varphi c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}\\ =\int_{\mathrm{X}}\varphi c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\operatorname{div}(s|\mathrm{Z})}+\int_{X}\log\left\|{s}\right\|^{-1}\mathop{\mathrm{d}\mathrm{d}^{c}}\varphi c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}.