ScalingStacks

Lemma 4.6 . [05AK]

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Lemma 4.6.

Let 𝔛\mathfrak{X} be an admissible formal scheme over K∘K^{\circ} of dimension n+1n+1 with boundaryless generic fibre 𝔛an\mathfrak{X}^{\textup{an}} and L0,…,LnL_{0},...,L_{n} line bundles on 𝔛an\mathfrak{X}^{\textup{an}} endowed with formal metrics corresponding to the models 𝔏0,…,𝔏n\mathfrak{L}_{0},...,\mathfrak{L}_{n} on 𝔛\mathfrak{X}. Suppose that L0=L1=π’ͺ𝔛anL_{0}=L_{1}=\mathcal{O}_{\mathfrak{X}^{\textup{an}}}, denote by βˆ₯β‹…βˆ₯0\|\cdot\|_{0} and βˆ₯β‹…βˆ₯1\|\cdot\|_{1} the metrics on L0L_{0} respectively L1L_{1} and set f0:=βˆ’log⁑‖1β€–0f_{0}:=-\log\|1\|_{0}, f1:=βˆ’log⁑‖1β€–1f_{1}:=-\log\|1\|_{1}. Suppose that f0f_{0} and f1f_{1} have compact support. Then

βˆ«π”›anf0​c1​(𝔏1)βˆ§β€¦βˆ§c1​(𝔏n)=βˆ«π”›anf1​c1​(𝔏0)∧c1​(𝔏2)βˆ§β€¦βˆ§c1​(𝔏n).\int_{\mathfrak{X}^{\textup{an}}}f_{0}\;c_{1}(\mathfrak{L_{1}})\wedge...\wedge c_{1}(\mathfrak{L}_{n})=\int_{\mathfrak{X}^{\textup{an}}}f_{1}\;c_{1}(\mathfrak{L}_{0})\wedge c_{1}(\mathfrak{L}_{2})\wedge...\wedge c_{1}(\mathfrak{L}_{n}).

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