Proposition 6.4 (003K) · ScalingStacks

003K

Proposition 6.4. [53, Lemma 4.1, 4.2] Given any $\epsilon\ll 1$ , and let $t$ be small enough depending on $\epsilon$ . There is a Kähler metric $\omega_{\psi,t}$ , such that

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On $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$ , the local Kähler potentials $\phi_{J,t}$ of $\omega_{\psi,t}$ can be chosen to satisfy $|\phi_{J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|<\epsilon$ .
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The total variation $\int_{X_{t}}|\frac{|\log|t||^{n}\omega_{\psi,t}^{n}}{(L^{n})}-d\mu_{t}|<\epsilon.$
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The Kähler potential of $\omega_{\psi,t}$ relative to a fixed Fubini-Study reference metric, is uniformly bounded independent of $t,\epsilon$ .

003L

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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