ScalingStacks

A basic problem in pluripotential theory is to estimate Kähler potentials. Kolodziej pioneered a method to achieve the following effects. The basic versions of his theorems work on a fixed ambient Kähler manifold $(Y,\omega)$, and deal with Kähler potentials $\phi\in PSH(Y,\omega)$ normalized to $\sup_{Y}\phi=0$.

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  (‘Potential estimate’) If the volume density of $\omega_{\phi}^{n}$ has some integrability control such as an $L^{p}$-bound

  $$\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\leq C,\quad p>1,$$

  (10)

  then $\phi$ has an $L^{\infty}$ bound depending only on $(X,\omega),n,p,C$ [45][20][24]. In fact this can be improved to a $C^{\alpha}$ Hölder bound on $\phi$ [47]. (Notice $p=1$ would not suffice, as the $L^{1}$-bound $\int_{Y}\omega_{\phi}^{n}\leq\int_{Y}\omega^{n}$ is automatic).

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  (‘$L^{1}$-stability estimates’) Suppose $\phi,\psi\in PSH(Y,\omega)$ are both subject to the $L^{p}$ volume density integrability control (10), and the normalization $\sup_{Y}\phi=\sup_{Y}\psi=0$. If

  1. 1.

     Either $\phi,\psi$ are close together in the $L^{1}$-sense $\int_{Y}|\psi-\phi|\omega^{n}\ll 1$, (‘$L^{1}$-potential stability’)
     

  2. 2.

     Or the volume densities of $\omega_{\phi}^{n}$ and $\omega_{\psi}^{n}$ are close together in the $L^{1}$-sense, namely the total variation $\int_{Y}|\omega_{\phi}^{n}-\omega_{\psi}^{n}|\ll 1$, (‘$L^{1}$-volume stability’)

  Then $|\phi-\psi|$ is small in the $L^{\infty}$ sense with quantitative estimates [46].

In our applications, we need to work with a family of Kähler manifolds, and the estimates need to be uniform under very severe complex structure degenerations, and allowing the total volume to collapse to zero. Some subtleties are:

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  The Calabi-Yau volume measure is very different from the volume form of a Fubini-Study metric (cf. section 3.1), so the idea of volume density with respect to a Fubini-Study style ambient metric is no longer appropriate. The replacement of (10) turns out to be a Skoda type estimate (11).

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  Unlike $L^{\infty}$ bounds, the Hölder norm depends strongly on the choice of the ambient metric, which specifies a choice of distance function. We think it is highly non-obvious how to make a semi-explicit choice uniformly in the family, and therefore we do not attempt to generalize Hölder estimates.

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  A technical problem in our applications involving the comparison of two potentials, only one potential has volume density control. Thus unlike the $L^{1}$-stability estimates above, our version treats the two potentials asymmetrically.

Our analogue of the potential estimate is

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 $n,\alpha,A$. The relations to the more standard version above can be explained as follows:

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  For a fixed ambient Kähler metric, by the Hölder inequality and the standard Skoda inequality Theorem 4.3, under the density integrability assumption (10), then for $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, and some small enough $\beta>0$,

  $$\int_{Y}e^{-\beta u}\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}\leq\frac{1}{\text{Vol}(Y)}\left(\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\omega^{n}\right)^{1/p}\left(\int_{Y}e^{-p^{\prime}\beta u}\omega^{n}\right)^{1/p^{\prime}}\leq\text{const}.$$

  This means the Skoda type estimate (11) is a weaker assumption than (10), even in the standard setting.

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  The first part of the conclusion recovers Kolodziej’s potential estimate.

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  The second part of the conclusion is about comparing the two potentials $\phi$ versus $-t_{0}$. The condition $\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ means $\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}$ is small, which by the Hölder inequality and the density integrability (10) follows from the smallness of $\int_{\phi\leq-t_{0}}\omega^{n}$. This should be viewed as one half of the $L^{1}$-potential stability condition $\int_{Y}|\phi+t_{0}|\omega^{n}\ll 1.$ The conclusion for the lower bound on $\min\phi$, should be viewed as one half of a smallness bound on the $C^{0}$-norm of $\phi+t_{0}$, namely the two potentials $\phi$ and $-t_{0}$ are close together in $C^{0}$-norm.

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  To generalize to the cases with $\psi$ not necessarily constant, it suffices to replace $\omega$ by $\omega_{\psi}$, and $\phi$ by $\phi-\psi$ in Theorem 4.5.

Combining Thm. 4.4 with Thm. 4.5, we immediately obtain

Our adaption of the $L^{1}$-volume stability estimate is

Theorem 4.7 should be viewed as a one-sided version of $L^{1}$-volume stability estimate. The goal is to compare the two potentials $\phi$ and $0$, without loss of generality. The Skoda type estimate is the weakened version of the volume density integrability assumption (10) as before. Assume for the moment that this holds for both measures $d\mu$ and $d\nu$. After adjusting $\phi$ by an additive constant, we might as well assume $\mu(E_{0})\approx\nu(E_{0})\approx\frac{1}{2}$, noticing that the total variation between the two measures is small by assumption. Then we can reverse the role of $\omega_{\phi}$ and $\omega$, to deduce a two-sided smallness bound on $\phi$, which is the content of the $L^{1}$-volume stability estimate.