4.4 Potential estimate II [00BF]

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4.4 Potential estimate II

Here we present a second strategy for the potential estimate, which aims to circumvent the uniform $L^{1}$ -stability estimate Theorem 2.6, and we explain why there is a difficulty with this second approach. Readers who wish to follow the main line of the proof may skip this section.

Lemma 4.9.

Given $0<\delta\ll 1$ , then for $0<\epsilon\ll 1$ depending on $\delta$ , and $t$ small enough depending on $\epsilon,\delta$ ,

\int_{\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n-1}\leq\frac{C\delta}{|\log|t||^{n}},

where $C$ is independent of $\delta,\epsilon,t$ .

Proof.

Pretending everything is smooth, a standard integration by part gives

\begin{split}&\int_{X_{t}}(\psi_{t}-\phi_{CY,t})(\omega_{CY,t}^{n}-\omega_{\psi,t}^{n})\\ =&\int_{X_{t}}(\psi_{t}-\phi_{CY,t})dd^{c}(-\psi_{t}+\phi_{CY,t})\wedge(\omega_{CY,t}^{n-1}+\ldots+\omega_{\psi,t}^{n-1})\\ =&\int_{X_{t}}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge(\omega_{CY,t}^{n-1}+\ldots+\omega_{\psi,t}^{n-1})\\ \geq&\int_{X_{t}}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge\omega_{\psi,t}^{n-1}.\end{split}

The same calculations work for continuous $\omega_{FS,t}$ -psh functions by standard pluripotential theory.

Combine $\left\lVert\phi_{CY,t}\right\rVert_{L^{\infty}}\leq C$ with the total variation bound in Lemma 4.2,

\int_{X_{t}}|\frac{|\log|t||^{n}\omega_{\psi,t}^{n}}{(L^{n})}-d\mu_{t}|<\delta,

we get

\int_{X_{t}}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge\omega_{\psi,t}^{n-1}\leq C\int_{X_{t}}|\omega_{CY,t}^{n}-\omega_{\psi,t}^{n}|\leq\frac{C\delta}{|\log|t||^{n}}.

Again by Lemma 4.2, the metric $\omega_{\psi,t}$ is uniformly controlled a.e. on $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ , so

\int_{\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n-1}\leq\frac{C\delta}{|\log|t||^{n}}.

∎

An outline of this strategy is

•

Choose some suitable integral normalisation on $\phi_{CY,t}$ . Apply Poincaré inequality to prove the average $L^{2}$ -integral of $\psi_{t}-\phi_{CY,t}$ is small in the generic region $W_{\delta}$ , which occupies most of the $d\mu_{t}$ -measure.
•

Deduce the measure is small on the set where $\phi_{CY,t}-\psi_{t}$ is perceptibly negative.
•

Apply the stability estimate Cor. 2.3 to show $\phi_{CY,t}-\psi_{t}$ cannot be perceptibly negative. Since $\left\lVert\psi_{t}\right\rVert_{C^{0}}$ is negligible, it shows the minimum of $\phi_{CY,t}$ is almost zero.
•

Using the small $L^{2}$ -average bound on $\phi_{CY,t}$ , one applies the mean value inequality to derive a small upper bound on $\phi_{CY,t}$ in the generic region $W_{\delta}$ .

The problem lies in the fact that the $n$ -dimensional open faces $\text{Int}(\Delta_{J})$ of $Sk(X)$ are disconnected, so the average values on each face are a priori unrelated. Thus the Poincaré inequality can only imply a small bound on $\phi_{CY,t}-\psi_{t}$ with a priori different normalisations associated to each open face, which is not good enough to get a small bound on average $L^{2}$ -integral of $\psi_{t}-\phi_{CY,t}$ .