4.3 Potential estimate I [00BE]

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4.3 Potential estimate I

We denote the CY metrics on $(X_{t},\frac{1}{|\log|t||}c_{1}(L))$ as

\omega_{CY,t}=\omega_{FS,t}+dd^{c}\phi_{CY,t}.

Using the local potentials $\phi_{J,t}$ of $\omega_{FS,t}$ , we can write $\omega_{CY,t}$ in terms of local absolute potentials on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$ :

\omega_{CY,t}=dd^{c}\phi_{CY,J,t},\quad\phi_{CY,J,t}=\phi_{J,t}+\phi_{CY,t}.

This depends on an implicit choice of $\omega_{FS,t}$ and $\phi_{J,t}$ in Lemma 4.1. Our goal is to find a suitable choice and show the smallness of $|\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|$ in the generic region.

Proposition 4.3.

For $0<|t|\ll 1$ , the CY potentials have a uniform bound $\left\lVert\phi_{CY,t}\right\rVert_{L^{\infty}}\leq C$ under the normalisation $\sup_{X_{t}}\phi_{CY,t}=0$ .

Proof.

Combine the uniform $C^{0}$ -estimate Theorem 2.2 and the uniform Skoda estimate Theorem 2.1, we know the CY potential with respect to any fixed choice of Fubini-Study background metric is uniformly bounded for small $t$ . Here the implicit choice of $\epsilon$ in $\omega_{FS,t}$ does not matter because of Lemma 4.1. ∎

Proposition 4.4.

Given small numbers $0<\lambda,\kappa\ll 1$ , then for $\delta,\epsilon,t$ sufficiently small depending on $\lambda$ and $\kappa$ , the function $\phi_{CY,t}$ is near its minimum with large probability:

d\mu_{t}(\{\phi_{CY,t}-\min_{X_{t}}\phi_{CY,t}\geq\kappa/5\})<\lambda.

Proof.

We wish to compare $\omega_{CY,t}$ with $\omega_{\psi,t}$ from Lemma 4.2 by an $L^{1}$ -stability estimate. Pick a parameter $c$ such that

d\mu_{t}(\{\psi_{t}-\phi_{CY,t}-c\leq 0\})\geq\lambda.

Since the potential $\phi_{CY,t}$ has a uniform bound, Theorem 2.1 implies another uniform Skoda estimate with modified constants

\int_{X_{t}}e^{-\alpha u}d\mu_{t}\leq A,\quad\forall u\in PSH(X_{t},\omega_{CY,t})\text{ with }\sup_{X_{t}}u=0.

We also have the $L^{1}$ -stability property for $\omega_{\psi,t}$ in Lemma 4.2:

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

We then apply the uniform $L^{1}$ -stability estimate Theorem 2.6, with $Y=X_{t}$ , $\omega=\omega_{CY,t}$ and $\phi=\psi_{t}-\phi_{CY,t}-c$ . In this construction $\delta,\epsilon,t$ are sufficiently small, chosen successively depending on $\lambda$ and $\kappa$ . We conclude

\sup_{X_{t}}(\psi_{t}-\phi_{CY,t}-c)\leq C(\lambda)\delta^{1/(2n+3)}\ll\kappa.

Now $|\psi_{t}|\leq 3\epsilon\ll\kappa$ , so $\inf_{X_{t}}\phi_{CY,t}>-c-\kappa/10.$

Taking the contrapositive, if we choose $c=-\inf_{X_{t}}\phi_{CY,t}-\kappa/10$ , then

d\mu_{t}(\{\psi_{t}-\phi_{CY,t}-c\leq 0\})<\lambda,

whence for $\epsilon\ll\kappa$ , using again $|\psi_{t}|\leq 3\epsilon$ ,

d\mu_{t}(\{\phi_{CY,t}-\inf\phi_{CY,t}\geq\kappa/5\})<\lambda.

∎

Remark 4.5.

The reason we use an asymmetric version of the $L^{1}$ -stability estimate, is that we have no control on the density of the comparison metric $\omega_{\psi,t}$ away from the the generic region except for a small bound on the measure contribution there.

We can reformulate this in terms of the local potentials of the CY metrics, and thereby eliminate auxiliary choices of Fubini-Study metric and regularisation.

Corollary 4.6.

Given small numbers $0<\lambda,\kappa\ll 1$ , then for $t$ small enough depending on $\lambda,\kappa$ , there exist appropriately chosen local potentials $\phi_{CY,J,t}$ on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$ , normalized to $\inf(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})=0$ , satisfying

d\mu_{t}(\{\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\geq\kappa/4\})<\lambda,

and $\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{L^{\infty}}\leq C$ independent of $\lambda,\kappa$ and small $t$ .

Proof.

By construction in Lemma 4.1, the local potential $\phi_{J,t}$ of $\omega_{FS,t}$ is $\epsilon$ -close to $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ , and since $\epsilon\ll\kappa$ these two are practically the same. Up to an overall normalisation constant, which is fixed by $\inf=0$ , we have $\phi_{CY,J,t}=\phi_{CY,t}+\phi_{J,t},$ so the measure bound follows from the previous result.

The uniform $L^{\infty}$ bound follows from Prop. 4.3 and Lemma 4.1 without any reference to $\lambda,\kappa$ . ∎

Given $0<\tau\ll 1$ , we consider the region obtained from shrinking the $n$ -dimensional faces near the boundary:

U_{J,t,\tau}=\text{Log}_{\mathcal{X}}^{-1}(\Delta_{J}\setminus\{|x_{i}|<\tau,|1-\sum_{1}^{n}x_{i}|<\tau\}).

The following theorem is a precise formulation for $C^{0}_{loc}$ -convergence of the local CY potentials to $\phi_{0}$ over the $n$ -dimensional open faces of $Sk(X)$ as $t\to 0$ .

Theorem 4.7.

( $C^{0}_{loc}$ -convergence estimate on the potential) Given $0<\tau,\kappa\ll 1$ , then for sufficiently small $t$ , on each $U_{J,t,\tau}$ there is a $C^{0}$ -bound

0\leq\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}<\kappa.

Proof.

We need to obtain upper bound on $\phi_{CY,J,t}$ . Consider $z\in U_{J,t,\tau}$ and $x=\text{Log}_{\mathcal{X}}(z)$ . Let $r\ll\tau$ be a parameter to be fixed, so $B(x,3r)\subset\text{Int}(\Delta_{J})$ . The function $\phi_{0}$ has an a priori Lipschitz estimate on $\Delta_{\mathcal{X}}$ , so the oscillation of $\phi_{0}$ on $B(x,3r)$ is less than $Cr\ll\kappa$ by choosing $r$ small enough.

Now we apply the mean value inequality to the psh function $\phi_{CY,J,t}$ on a ball in the local covering space of $U_{J,t,\tau}\subset(\mathbb{C}^{*})^{n}$ , which projects to $B(x,r)$ via $\text{Log}_{\mathcal{X}}$ . We have

\phi_{CY,J,t}(z)\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}\phi_{CY,J,t},

hence

\begin{split}(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})(z)&\leq\text{osc}_{B(r)}\phi_{0}+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})\\ &\leq\frac{\kappa}{10}+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}(\phi_{CY,J,t}-\text{Log}_{\mathcal{X}}).\end{split}

But on the ball $\phi_{CY,J,t}-\text{Log}_{\mathcal{X}}<\kappa/4$ , except on a subset of the ball with $d\mu_{t}$ -percentage $\leq C\lambda r^{-n}$ , on which we use the coarser bound $\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\leq C$ . Combining these,

(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})(z)<\kappa/2+C\lambda r^{-n}<\kappa,

by choosing $\lambda$ sufficiently small depending on $\kappa,\tau$ . ∎

Remark 4.8.

The above estimates do not use the full strength of the regularisation lemma 4.2. We only use the $L^{1}$ -stability of the volume density, not the metric information.