ScalingStacks

Proposition 5.9 . [05BF]

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Proposition 5.9.

Let KK be algebraically closed, 𝔛\mathfrak{X} a strongly nondegenerate polystable formal model of XanX^{\textup{an}} over K∘K^{\circ} with associated skeleton Ξ”\Delta, Ο„\tau an nn-dimensional open face of Ξ”\Delta and hh a rational piecewise affine linear convex function on Ο„\tau. Then the metric on π’ͺXan|pπ”›βˆ’1​(Ο„)\mathcal{O}_{X^{\textup{an}}}\Big|_{p_{\mathfrak{X}}^{-1}(\tau)} which is given by βˆ₯1βˆ₯=eβˆ’h∘p𝔛\|1\|=e^{-h\circ p_{\mathfrak{X}}} is a semipositive piecewise β„š\mathbb{Q}-linear metric.

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