Subsection 6.3.1 (004X) · ScalingStacks

6.3.1 Producing convex functions

Recall the logarithm map $\text{Log}_{t}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$ .

\text{Log}_{t}(z_{1},\ldots z_{n})=\frac{1}{\log|t|}(\log|z_{1}|,\ldots\log|z_{n}|).

Consider an open convex subset $U\subset\mathbb{R}^{n}$ , and let $\phi$ be a psh function on $\text{Log}_{t}^{-1}(U)\subset(\mathbb{C}^{*})^{n}$ .

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Lemma 6.6. [52, Lemma 4.3] The fibrewise $T^{n}$ average function

\bar{\phi}(x_{1},\ldots x_{n})=\frac{1}{(2\pi)^{n}}\int_{T^{n}}\phi(e^{x_{1}\log|t|+i\theta_{1}},\ldots e^{x_{n}\log|t|+i\theta_{n}})d\theta_{1}\ldots d\theta_{n}

is a convex function in the variables $x_{1},\ldots x_{n}$ .

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Proof. Since the function $\bar{\phi}$ is an average of psh functions, it is psh as a $T^{n}$ -invariant function on $\text{Log}_{t}^{-1}(U)$ . Such functions correspond to convex functions downstairs. ∎

An important intuition is that on sufficiently collapsed toric regions inside $(\mathbb{C}^{*})^{n}$ , bounded Kähler potentials have a strong tendency to be approximated by convex functions.

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Proposition 6.7. Assume $\left\lVert\phi\right\rVert_{C^{0}}$ has a uniform bound independent of $t$ . Then after shrinking $U$ by a small amount independent of $t$ , we have

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The convex function $\bar{\phi}$ has a Lipschitz bound $|\bar{\phi}(x)-\bar{\phi}(x^{\prime})|\leq C|x-x^{\prime}|.$
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There is an upper bound $\phi-\bar{\phi}\leq\frac{C}{|\log|t||^{1/2}}$ .
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On each logarithmic dyadic scale $U_{a}=\{a_{i}\leq\log|z_{i}|\leq 2a_{i},\forall i\}\subset U$ , the $L^{1}$ -integral

$\int_{U_{a}}|\phi-\bar{\phi}|\prod\sqrt{-1}d\log z_{i}\wedge d\overline{\log z_{i}}\leq\frac{C}{|\log|t||^{1/2}}.$
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There is an improved Skoda inequality with uniform constants $\alpha,C$ independent of $t$ :

$\int_{U}e^{-\alpha|\log|t||^{1/2}(\phi-\bar{\phi})}d\mu_{t}\leq C.$

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Proof. (Sketch)

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Bounded convex functions automatically have Lipschitz bound on slightly shrinked convex domains.
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The second item follows from a slightly tricky application of mean value inequality for subharmonic functions, cf. [52, section 4.3].
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The third item is because the function $\phi-\bar{\phi}$ has mean value zero, so an upper bound implies an $L^{1}$ -bound, cf. [52, section 4.3].
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One first apply the basic Skoda estimate Thm. 4.2 to the function $\phi$ on each logarithmic dyadic scale, where $\bar{\phi}$ is almost constant by the Lipschitz bound. Then we sum over all the logarithmic dyadic scales (cf. [52, section 4.6]).

∎

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Remark 17. The improved Skoda estimate is one of the main discoveries in [52]. Intuitively, this means $\phi-\bar{\phi}$ can only be significantly below $-\frac{Const}{|\log|t||^{1/2}}$ on sets with exponentially small measure. Compounded with the upper bound $\phi-\bar{\phi}\leq\frac{C}{|\log|t||^{1/2}}$ , this means for sufficiently small $t$ , an arbitrary bounded psh function $\phi$ is very close to the convex function $\bar{\phi}$ except on exponentially small measure.

In our applications, the psh functions $\phi$ arise from the local potentials $\phi_{CY,J,t}$ of the Calabi-Yau metrics $\omega_{CY,t}$ on toric charts inside $X_{t}$ . Since $\omega_{CY,t}$ has uniformly bounded potential with respect to Fubini-Study reference metrics (cf. Thm. 4.6), it is easy to arrange the local potentials $\phi_{CY,J,t}$ on toric charts to be uniformly bounded, whence the convex functions $\bar{\phi}_{CY,J,t}$ are also uniformly bounded. Using the Lipschitz bound, by Arzela-Ascoli, we can extract a collection of subsequential limits as $t\to 0$ . By construction $\bar{\phi}_{CY,J,t}\to\bar{\phi}_{J,0}$ in the $C^{0}_{loc}$ sense on the interior of the $n$ -dimensional faces of $Sk(X)$ .