ScalingStacks

Remark 4.4 . [05AH]

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Remark 4.4.

There is a close connection of the Monge-AmpΓ¨re measure with the intersection product on formal schemes as defined in [Gub98]: Assume that 𝔛\mathfrak{X} has irreducible, reduced and boundaryless generic fibre and reduced special fibre. In addition to 𝔏1,…,𝔏n\mathfrak{L}_{1},...,\mathfrak{L}_{n} let 𝔏0\mathfrak{L}_{0} be a formal line bundle on 𝔛\mathfrak{X} which is trivial on the generic fibre and set f:=βˆ’log⁑‖1β€–f:=-\log\|1\| where βˆ₯β‹…βˆ₯\|\cdot\| is the formal metric induced by 𝔏0\mathfrak{L}_{0}. Suppose that ff has compact support and let D:=div⁑(1)D:=\Div(1) be the Cartier divisor on 𝔛\mathfrak{X} induced by 11 as in [Gub98, Remark 3.1]. We examine the Weil divisor cyc⁑(D)\cyc(D) associated to DD as defined in [Gub98, Β§3]. Since 𝔏0\mathfrak{L}_{0} is trivial on the generic fibre, the horizontal part of cyc⁑(D)\cyc(D) is zero while the vertical part is by definition ([Gub98, 3.8]) given by βˆ‘Y∈irr⁑(𝔛~)f⁑(ΞΆY)β‹…Y\sum_{Y\in\irr(\tilde{\mathfrak{X}})}f(\zeta_{Y})\cdot Y. Now since 𝔛an\mathfrak{X}^{\textup{an}} has no boundary, every irreducible component of 𝔛~\tilde{\mathfrak{X}} is proper by Corollary A.4 and together with the definition of the intersection product ([Gub98, Β§4]) we obtain

βˆ«π”›anf​c1​(𝔏1)βˆ§β€¦βˆ§c1​(𝔏n)=βˆ‘Y∈irr⁑(𝔛~)f⁑(ΞΆY)β‹…deg𝔏1,…,𝔏n⁑(Y)=deg𝔏1,…,𝔏n⁑(cyc⁑(D)).\int_{\mathfrak{X}^{\textup{an}}}fc_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})=\sum_{Y\in\irr(\tilde{\mathfrak{X}})}f(\zeta_{Y})\cdot\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(Y)=\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(\cyc(D)).

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