Subsection 2.2 (00B0) · ScalingStacks

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2.2 Kolodziej’s estimate on pluripotentials

Given an $n$ -dimensional Kähler manifold $(Y,\omega)$ , for $\phi\in PSH(Y,\omega)\cap L^{\infty}$ , pluripotential theory allows one to make sense of the Monge-Ampère (MA) measure $\omega_{\phi}^{n}$ , generalising the notion of volume forms. A basic problem is to estimate $\phi$ from a priori bounds on $\omega_{\phi}^{n}$ . A prototypical result is (cf. [32, section 2.2] for an exposition based on [15][16]):

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Theorem 2.2. Let $(Y,\omega)$ be a compact Kähler manifold, and $\phi\in PSH(Y,\omega)\cap C^{0}$ , such that $\omega_{\phi}^{n}$ is an absolutely continuous measure. Assume there are positive constants $\alpha,A$ , such that the Skoda type estimate holds with respect to $\omega_{\phi}^{n}$ :

\int_{Y}e^{-\alpha u}\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.

(3)

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For fixed $n,\alpha,A$ , there is number $B(n,\alpha,A)$ , such that if $\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ for some $t_{0}$ , then $\min\phi\geq-t_{0}-4B(\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)})^{1/2n}$ .
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If $\sup_{Y}\phi=0$ , then $\left\lVert\phi\right\rVert_{C^{0}}\leq C(n,\alpha,A)$ .

The strength of this result is that it still applies when the complex/Kähler structures are highly degenerate, as it distills the dependence on $(Y,\omega)$ to only 3 constants $n,\alpha,A$ . A minor variant gives a criterion for two Kähler potentials to be close to each other.

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Corollary 2.3. (Stability estimate) Let $(Y,\omega)$ be a compact Kähler manifold, and $\phi,\psi\in PSH(Y,\omega)\cap C^{0}$ , such that $\omega_{\psi}^{n}$ is absolutely continuous. Assume $\left\lVert\phi\right\rVert_{C^{0}}\leq A^{\prime}$ and the Skoda type estimate (3). Then there is a number $B(n,A,A^{\prime},\alpha)$ , such that if $\frac{\int_{\psi-\phi\leq-t_{0}}\omega_{\psi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ for some $t_{0}$ , then

\min(\psi-\phi)\geq-t_{0}-4B\left(\frac{\int_{\psi-\phi\leq-t_{0}}\omega_{\psi}^{n}}{\text{Vol}(Y)}\right)^{1/2n}.