Lemma 5.3 (00SN) · ScalingStacks

00SN

Lemma 5.3. (Chern-Levine type estimate) Let $u$ be a psh function on the annulus region $U=\{|\log|z_{i}||<s,\forall i\}\subset(\mathbb{C}^{*})^{n}$ , with $\left\lVert u\right\rVert_{L^{\infty}}\lesssim 1$ . Then

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On the shrinked set $E=\{|\log|z_{i}||<s/2\}$ the measure

$\int_{E}(\sqrt{-1}\partial\bar{\partial}u)^{n}\leq Cs^{-n}.$

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Let $u+v$ is another psh function, with $\left\lVert v\right\rVert_{L^{\infty}}\ll 1$ . Let $f$ be any compactly supported function on the square $\{|x_{i}|<1\}\subset\mathbb{R}^{n}$ . Then

\int f\{(\sqrt{-1}\partial\bar{\partial}(u+v))^{n}-(\sqrt{-1}\partial\bar{\partial}u)^{n}\}\leq Cs^{-n}\left\lVert f\right\rVert_{C^{2}}\left\lVert v\right\rVert_{L^{\infty}}.

00SP

Proof. Let $\chi$ be a compactly supported nonnegative smooth function on the square $\{|x_{i}|<1\}\subset\mathbb{R}^{n}$ , equal to one on $\{|x_{i}|\leq 1/2\}$ . We identify $\chi$ with $\chi\circ s^{-1}\text{Log}$ , and denote $\omega_{std}=\sqrt{-1}\sum d\log z_{i}\wedge d\overline{\log z_{i}}$ . Then

-Cs^{-2}\omega_{std}\leq\sqrt{-1}\partial\bar{\partial}\chi\leq Cs^{-2}\omega_{std}.

The basic obervation is that if $T$ is a positive current of bidegree $(n-1,n-1)$ , then by integration by part,

\begin{split}&\int_{E}\sqrt{-1}\partial\bar{\partial}u\wedge T\leq\int_{\text{supp}(\chi)}\chi\sqrt{-1}\partial\bar{\partial}u\wedge T\\ &=\int_{\text{supp}(\chi)}u\sqrt{-1}\partial\bar{\partial}\chi\wedge T\leq Cs^{-2}\int_{\text{supp}(\chi)}\omega_{std}\wedge T.\end{split}

Iterating this argument to lower the power of $\sqrt{-1}\partial\bar{\partial}u$ ,

\int_{E}(\sqrt{-1}\partial\bar{\partial}u)^{n}\leq Cs^{-2n}\int_{U}\omega_{std}^{n}\leq Cs^{-n}.

The second statement is proved similarly by removing $\sqrt{-1}\partial\bar{\partial}v$ factors iteratively. ∎