4.7 L ∞ and stability estimates for CY potentials [00TT]

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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}.

Theorem 4.23.

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

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).

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)

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. ∎

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,

Corollary 4.26.

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

•

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 $|\varphi_{CY,s}-\psi_{CY,s}|\leq Cs^{-1/2}\log s$ .
•

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 ∞ and stability estimates for CY potentials [00TT]

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

Theorem 4.23.

Proof.

Theorem 4.24.

Proof.

Remark 4.25.

Corollary 4.26.

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