ScalingStacks

Proof. (Sketch)

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  We first $C^{0}$ approximate the NA metric $\left\lVert\cdot\right\rVert_{CY}$ by some NA Fubini-Study metric, which arises naturally as a hybrid topology limit of usual Fubini-Study metrics on $X_{t}$ (cf. section 5.3). The Fubini-Study metrics are positive, and by construction their local potentials differ from $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ by an arbitrarily small amount in the $C^{0}$ sense.

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  We do not have direct control on the volume form of the Fubini-Study metrics; the degrees of the associated projective embeddings are gigantic. In contrast, the volume form of the local potential $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ has negligible difference from $\frac{(L^{n})}{|\log|t||^{n}}d\mu_{t}$, by the volume asymptote in section 3.1 and the real MA equation (17).

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  The idea is to perform a further regularization. We modify the Fubini-Study metric in the generic region of $X_{t}$, so that it essentially agrees with $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ in the generic region up to $C^{2}$-small error. In this step we appealed also to the regularity theory of real MA equation. The end result is $\omega_{\psi,t}$, which is Kähler by construction.

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  In the non-generic region, we do not perform regularization. Since the generic region already takes up $99.9\%$ of the $\omega_{\psi,t}^{n}$ measure for $|t|\ll 1$, the non-generic region has negligible total measure. We use this to argue for the total variation bound.

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