ScalingStacks

Proposition 5.4 . [0038]

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Proposition 5.4.

(NA MA-real MA comparison) Let (𝒳,β„’)(\mathcal{X},\mathcal{L}) be a semistable snc model of (XK,L)(X_{K},L), and Int​(Ξ”J)\text{Int}(\Delta_{J}) be an nn-dimensional open face of Δ𝒳\Delta_{\mathcal{X}}. Recall the retraction map r𝒳:XKa​n→Δ𝒳r_{\mathcal{X}}:X_{K}^{an}\to\Delta_{\mathcal{X}}. Let Ο•βˆˆC0​(XKa​n)\phi\in C^{0}(X_{K}^{an}) be the potential of a semipositive metric ‖⋅‖ℒ​eβˆ’Ο•\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi}, and suppose Ο•=Ο•βˆ˜r𝒳\phi=\phi\circ r_{\mathcal{X}} on rπ’³βˆ’1​(Ξ”J)r_{\mathcal{X}}^{-1}(\Delta_{J}), then on Int​(Ξ”J)\text{Int}(\Delta_{J}) the pushforward of the NA MA measure

rπ’³βˆ—MA(β€–β‹…β€–eβˆ’Ο•)=n!MAℝ(Ο•|Int​(Ξ”J))r_{\mathcal{X}*}MA(\left\lVert\cdot\right\rVert e^{-\phi})=n!MA_{\mathbb{R}}(\phi|_{\text{Int}(\Delta_{J})})

equals the real MA measure of the convex function Ο•|Int​(Ξ”J)\phi|_{\text{Int}(\Delta_{J})} up to a factor n!n!.

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