ScalingStacks

Theorem 1.1 . [0595]

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Theorem 1.1.

Let XX be an nn-dimensional proper algebraic variety over KK, L¯=(L,∥⋅∥)\overline{L}=(L,\|\cdot\|) a formally metrized line bundle on XanX^{\textup{an}} and τ\tau an open face of dimension nn of a skeleton corresponding to a strictly semistable formal model 𝔛\mathfrak{X} of XanX^{\textup{an}} on which L¯\overline{L} has a formal model 𝔏\mathfrak{L}. Let φ\varphi be a continuous function on XanX^{\textup{an}} such that ∥⋅∥e−φ\|\cdot\|e^{-\varphi} is a semipositive metric. Suppose that φ\varphi factorizes through the retraction p𝔛p_{\mathfrak{X}} onto the skeleton. Then

c1(L,∥⋅∥e−φ)n=[K~(S):K~]⋅n!⋅MA(φ|τ)c_{1}(L,\|\cdot\|e^{-\varphi})^{n}=[\tilde{K}(S):\tilde{K}]\cdot n!\cdot\MA\left(\varphi\Big|_{\tau}\right)

on p𝔛−1​(τ)p_{\mathfrak{X}}^{-1}(\tau) where MA\MA denotes the real Monge-Ampère operator on τ\tau which is considered to be a measure on p𝔛−1​(τ)p_{\mathfrak{X}}^{-1}(\tau) by pushforward via the inclusion and SS denotes the point in the special fibre of 𝔛\mathfrak{X} which is the image of τ\tau under the reduction map.

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