Subsection 4.7 (00TT) · ScalingStacks

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4.7 $L^{\infty}$ and stability estimates for CY potentials

We finally impose the Calabi-Yau condition, and consider the CY potential $\varphi=\varphi_{CY,s}$ normalised to $\sup_{X_{s}}\varphi=0$ , solving (20):

\omega_{CY,s}^{n}=(s^{-1}\omega_{FS}+\sqrt{-1}\partial\bar{\partial}\varphi)^{n}=a_{s}s^{-n}d\mu_{s}.

00SE

Theorem 4.23. ( $L^{\infty}$ -estimate) The Calabi-Yau potential $\varphi_{CY,s}$ satisfies the uniform $L^{\infty}$ -estimate $\left\lVert\varphi_{CY,s}\right\rVert_{L^{\infty}}\leq C$ .

00SF

Proof. We apply Kolodziej’s estimate in Thm 2.7. The Skoda type inequality (2) is verified in Cor. 4.20, hence the $L^{\infty}$ estimate. ∎

We now specialize to the Fermat case. Clearly $\varphi$ is invariant under the discrete symmetry of the hypersurface. Recall the regularisation is denoted as $\psi=\psi_{CY,s}$ , coming from the double Legendre transform construction $u=u_{CY,s}$ (cf. section 4.5). The local potentials of $\varphi_{CY,s}$ and $\psi_{CY,s}$ are denoted $\varphi_{m}=\varphi_{CY,s,m}$ and $\psi_{m}=\psi_{CY,s,m}$ according to the same convention as (19).

00SG

Theorem 4.24. In the Fermat case, there is a uniform stability estimate

\varphi_{CY,s}-\psi_{CY,s}\geq-Cs^{-1/2}\log s.

(30)

00SH

Proof. We apply Cor. 2.12. The Skoda estimate is verified in Cor. 4.20. The improved Skoda estimate Thm. 4.21 implies an exponential volume decay:

\frac{\int_{\varphi-\psi\leq-t}\omega_{\phi}^{n}}{\text{Vol}(X_{s})}\leq Ce^{-\alpha t\sqrt{s}},

hence there exists $c\gg 1$ , such that for $t_{0}=cs^{-1/2}\log s$ ,

\left(\frac{\int_{\varphi-\psi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(X_{s})}\right)^{1/2n}\leq Ce^{-\alpha t_{0}\sqrt{s}/2n}=Ce^{-\alpha c\log s/2n}\leq Cs^{-1/2}.

Thm 2.7 then implies $\varphi-\psi\geq-Cs^{-1/2}\log s$ as required. ∎

00SI

Remark 4.25. In the theorems above only an upper bound on the volume measure is actually needed. The intuition is that the Skoda inequality is already so close to an $L^{\infty}$ estimate, that a very tiny amount of extra assumptions are needed to conclude $L^{\infty}$ -estimate.

Combining this with the upper bound from Prop. 4.17,

00SJ

Corollary 4.26. In the Fermat case, there is a uniform $C^{0}$ -stability estimate:

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Inside $U^{s,*}_{w}\subset X_{s}$ , for $\langle m,w\rangle=1$ , the local potentials satisfy $|\varphi_{CY,s,m}-\psi_{CY,s,m}|\leq Cs^{-1/2}\log s,$ or equivalently $|\varphi_{CY,s}-\psi_{CY,s}|\leq Cs^{-1/2}\log s$ .
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Inside $U^{s,face}_{w}$ , the local potentials satisfy $|\varphi_{CY,s,0}-\psi_{CY,s,0}|\leq Cs^{-1/2}\log s$ , or equivalently $|\varphi_{CY,s}-\psi_{CY,s}|\leq Cs^{-1/2}\log s$ .

The point is that in the generic region of $X_{s}$ the Calabi-Yau local potentials are $C^{0}$ -approximated by their regularisations, which build in convexity by construction, and therefore have a priori Lipschitz bounds.

4.7 L∞L^{\infty} and stability estimates for CY potentials

4.7 $L^{\infty}$ and stability estimates for CY potentials