ScalingStacks

Proof. Let $\left\lVert\cdot\right\rVert_{FS}$ be a NA Fubini-Study approximation of $\left\lVert\cdot\right\rVert_{CY}$, with potential difference less than $\epsilon/2$. We construct the Fubini-Study metrics $\left\lVert\cdot\right\rVert_{FS,t}$ on $(X_{t},L)$ by the formula (10), so the curvature forms of $\left\lVert\cdot\right\rVert_{FS,t}^{1/|\log|t||}$ define the Kähler metrics $\omega_{FS,t}$ on $(X_{t},\frac{1}{|\log|t||}c_{1}(L))$. By construction, for sufficiently small $t$ the local potentials are $C^{0}$-close to that of $\left\lVert\cdot\right\rVert_{FS}$, which is $\epsilon$-close to the continuous metric $\left\lVert\cdot\right\rVert_{CY}$, so the uniform boundedness of the potentials can be guaranteed.

By our NA MA-real MA comparison assumption, over $\text{Int}(\Delta_{J})$ the potential $\phi_{0}$ of $\left\lVert\cdot\right\rVert_{CY}$ equals the pullback $\phi_{0}\circ r_{\mathcal{X}}$ via the retraction map $r_{\mathcal{X}}$. From our discussions on the hybrid topology in section 3.2, over $\text{Int}(\Delta_{J})$, for $\phi_{J,t}$ to be $C^{0}$-close to $\phi_{0}\circ r_{\mathcal{X}}$ means the same as saying $\phi_{J,t}$ is $C^{0}$-close to $\phi_{0}\circ\text{Log}_{\mathcal{X}}$.

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