ScalingStacks

Proposition 5.4. (NA MA-real MA comparison) Let $(\mathcal{X},\mathcal{L})$ be a semistable snc model of $(X_{K},L)$, and $\text{Int}(\Delta_{J})$ be an $n$-dimensional open face of $\Delta_{\mathcal{X}}$. Recall the retraction map $r_{\mathcal{X}}:X_{K}^{an}\to\Delta_{\mathcal{X}}$. Let $\phi\in C^{0}(X_{K}^{an})$ be the potential of a semipositive metric $\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi}$, and suppose $\phi=\phi\circ r_{\mathcal{X}}$ on $r_{\mathcal{X}}^{-1}(\Delta_{J})$, then on $\text{Int}(\Delta_{J})$ the pushforward of the NA MA measure

$$r_{\mathcal{X}*}MA(\left\lVert\cdot\right\rVert e^{-\phi})=n!MA_{\mathbb{R}}(\phi|_{\text{Int}(\Delta_{J})})$$

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