4.1 Comparison Kähler metric I [00BC]

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4.1 Comparison Kähler metric I

We apply the Fubini-Study approximation to transfer the NA metric $\left\lVert\cdot\right\rVert_{CY}$ to the complex manifolds $X_{t}$ up to $C^{0}$ -small errors in the potential, while preserving the positivity of the metric.

Recall there is a logarithm map $\text{Log}_{\mathcal{X}}:X_{t}\to\Delta_{\mathcal{X}}$ , defined up to $O(\frac{1}{|\log|t||})$ coordinate ambiguity, so when we refer to $\text{Log}_{\mathcal{X}}$ over $\text{Int}(\Delta_{J})$ we will implicitly shrink $\text{Int}(\Delta_{J})$ by $O(\frac{1}{|\log|t||})$ to ensure the coordinate expression $x_{i}=\frac{\log|z_{i}|}{\log|t|}$ makes sense.

Lemma 4.1.

Given any $0<\epsilon\ll 1$ , then for sufficiently small $t$ depending on $\epsilon$ , there is a smooth Kähler metric $\omega_{FS,t}$ on $(X_{t},\frac{1}{|\log|t||}c_{1}(L))$ , such that

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The relative Kähler potential for any two choices of $\epsilon$ is bounded uniformly independent of small $t$ .
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On $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$ , the local Kähler potentials $\phi_{J,t}$ of $\omega_{FS,t}$ can be chosen to satisfy $|\phi_{J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|<\epsilon$ .

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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