Subsection 2.2 (00T9) · ScalingStacks

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2.2 Kolodziej’s estimate on pluripotentials

Here we outline a method to estimate Kähler potentials, pioneered by Kolodziej, and further developed by [12] and [15][22]. Our exposition largely adapts [15][22][16], with special attention to the dependence of constants. Unlike in [15], we do not impose a volume normalisation.

Given an $n$ -dimensional Kähler manifold $(X,\omega)$ , for $\phi\in PSH(X,\omega)\cap L^{\infty}$ , pluripotential theory allows one to make sense of the Monge-Ampère (MA) measure $\omega_{\phi}^{n}$ , generalising the notion of volume forms. The basic problem is to estimate $\phi$ from a priori bounds on $\omega_{\phi}^{n}$ . A key concept is the capacity of subsets $K\subset X$ :

Cap_{\omega}(K)=\sup\{\int_{K}\omega_{u}^{n}|u\in PSH(X,\omega),0\leq u\leq 1\}.

We wish to sketch the main ideas behind a prototypical result:

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Theorem 2.7. Let $(X,\omega)$ be a compact Kähler manifold, and $\phi\in PSH(X,\omega)\cap C^{0}$ , 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_{X}e^{-\alpha u}\frac{\omega_{\phi}^{n}}{\text{Vol}(X)}\leq A,\quad\forall u\in PSH(X,\omega)\text{ with }\sup_{X}u=0.

(2)

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

The first ingredient is:

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Lemma 2.8. (cf. [15, Lemma 2.3]) The MA measure of sublevel sets controls the capacity of lower sublevel sets : for $\tau\geq 0$ and $0\leq t\leq 1$ ,

t^{n}Cap_{\omega}(\phi<-\tau-t)\leq\int_{\phi<-\tau}\omega_{\phi}^{n}.

The second ingredient below contains the most substance:

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Lemma 2.9. (Volume-capacity estimate) In the setting of Thm. 2.7, for any compact set $K\subset X$ ,

\frac{\int_{K}\omega_{\phi}^{n}}{\text{Vol}(X)}\leq Ae^{\alpha}\exp\left(-\alpha(\frac{\text{Vol}(X)}{\text{Cap}_{\omega}(K)})^{1/n}\right).

(3)

In particular there is a constant $B=B(n,\alpha,A)$ verifying the power law bound

\frac{\int_{K}\omega_{\phi}^{n}}{\text{Vol}(X)}\leq B^{2n}\left(\frac{\text{Cap}_{\omega}(K)}{\text{Vol}(X)}\right)^{2}.

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Proof. (Sketch) We may assume $K$ is not pluripolar, for otherwise $\int_{K}\omega_{\phi}^{n}=0$ and $\text{Cap}_{\omega}(K)=0$ . We introduce the Siciak extremal function

V_{K,\omega}=\sup\{u\in PSH(X,\omega)|u\leq 0\text{ on }K\},

whose upper semicontinuous regularisation $V_{K,\omega}^{*}\in PSH(X,\omega)$ . By the Alexander-Taylor comparison principle (cf. [22, Prop. 6.1]),

\exp(-\sup_{X}V_{K,\omega})\leq e\exp\left(-(\frac{\text{Vol}(X)}{\text{Cap}_{\omega}(K)})^{1/n}\right).

By the Skoda integrability assumption (2), and the fact that $V_{K,\omega}=V_{k,\omega}^{*}$ a.e with respect to $\omega^{n}$ (so by absolute continuity also for $\omega_{\phi}^{n}$ ),

\int_{X}e^{\alpha(\sup_{X}V_{K,\omega}-V_{K,\omega})}\omega_{\phi}^{n}=\int_{X}e^{\alpha(\sup_{X}V_{K,\omega}-V_{K,\omega}^{*})}\omega_{\phi}^{n}\leq A\text{Vol}(X),

hence

\int_{K}e^{-\alpha V_{K,\omega}}\omega_{\phi}^{n}\leq\int_{X}e^{-\alpha V_{K,\omega}}\omega_{\phi}^{n}\leq Ae^{\alpha}\text{Vol}(X)\exp\left(-\alpha(\frac{\text{Vol}(X)}{\text{Cap}_{\omega}(K)})^{1/n}\right).

The volume-capacity estimate (3) follows because $V_{K,\omega}\leq 0$ on $K$ . ∎

The third ingredient is an elementary decay lemma:

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Lemma 2.10. (cf. [15, Lemma 2.4 and Remark 2.5]) Let $f:[t_{0},\infty)\to[0,\infty)$ be a nonincreasing right-continuous function, such that

\begin{cases}f(t_{0})<\frac{1}{2B},\\ tf(\tau+t)\leq Bf(\tau)^{2},\quad\forall\tau\geq 0,\quad 0\leq t\leq 1,\\ \lim_{t\to\infty}f(t)=0.\end{cases}

Then $f(t)=0$ for $t\geq t_{0}+4Bf(t_{0})$ .

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Proof. (Thm 2.7) Combining the first two ingredients, the function $f(t)=(\frac{\int_{\phi\leq-t}\omega_{\phi}^{n}}{\text{Vol}(X)})^{1/2n}$ satisfies

tf(t+\tau)\leq Bf(\tau)^{2},\quad 0\leq t\leq 1,\quad\tau\geq 0,

We conclude that for $t>t_{0}+4Bf(t_{0})$ the sublevel set $\{\phi\leq-t\}$ has zero $\omega_{\phi}$ -measure, and therefore zero capacity by Lemma 2.8, so $\phi$ has the lower estimate as claimed in the first statement.

For the second statement, by (2) we have an a priori exponential decay

f(t)\leq A^{1/2n}e^{-\alpha t/2n},\quad t\geq 0,

which allows us to find an appropriate $t_{0}$ . ∎

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Remark 2.11. Thm. 2.7 implies a famous result of Kolodziej stating that if we fix $(X,\omega)$ and $p>1$ , then $\phi$ has a $C^{0}$ -bound depending only on $X,\omega,\left\lVert\frac{\omega_{\phi}^{n}}{\omega^{n}}\right\rVert_{L^{p}}$ . It is enough to check (2), which reduces by Hölder inequality to the standard Skoda inequality (cf. Thm 2.4), with modified constants. The strength of Thm. 2.7 is that it still applies when the complex/Kähler structures are highly degenerate, as it distills the dependence on $(X,\omega)$ to only 3 constants $n,\alpha,A$ .

Thm. 2.7 gives a criterion for two Kähler potentials to be close to each other.

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Corollary 2.12. (Stability estimate) Let $(X,\omega)$ be a compact Kähler manifold, and $\phi,\psi\in PSH(X,\omega)\cap C^{0}$ , such that $\omega_{\phi}^{n}$ is absolutely continuous. Assume $\left\lVert\psi\right\rVert_{C^{0}}\leq A^{\prime}$ and the Skoda type estimate (2). Then there is a number $B(n,A,A^{\prime},\alpha)$ , such that if $\frac{\int_{\phi-\psi\leq-t_{0}}\omega_{\phi}^{n}}{Vol(X)}<(2B)^{-2n}$ for some $t_{0}$ , then

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

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Proof. If $\psi$ is smooth, this follows from Thm. 2.7 by changing $\omega$ into $\omega_{\psi}$ , and changing $\phi$ into $\phi-\psi$ , and checking the Skoda type estimate holds with modified constants. In general, one can approximate $\psi\in PSH(X,\omega)$ by a decreasing sequence of functions in $PSH(X,\omega)\cap C^{\infty}$ [1], and since $\psi\in C^{0}$ the convergence is uniform by Dini’s theorem. ∎