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Proposition 17.13.35 (He–Testorf–Wang)E6915C7339CD

Proposition 17.13.35 (He–Testorf–Wang).E6915C7339CD Let $\ell \in \mathcal{R}(X,\theta ;\phi )$. Given any $\tau <\ell ^*_{\max }$ and $x\in X$, we have

DA238F3ECC6F 17.13.36
\begin{equation} \mathcal{I}\left(\ell ^*_{\tau }\right)_{x}=\left\{ f\in \mathcal{O}_{X,x}: |f|^2\int _0^{\infty } \exp (-\ell _ t+t\tau )\, \mathrm{d}t \textup{ is integrable near }x \right\} . \end{equation}

Proof. Fix $x\in X$, $\tau <\ell ^*_{\max }$ and $f\in \mathcal{O}_{X,x}$. Fix a Kähler form $\omega $ on $X$.

Step 1. We first assume that $f$ lies in the right-hand side of (17.13.36).

Given any $y\in X$, it follows from (17.13.14) that there is $t_0>0$ with

\[ \ell ^*_{\tau }(y)+1\geq \ell _{t_0}(y)-t_0\tau . \]

Observe that $t\mapsto \ell _ t-t\ell ^*_{\max }$ is decreasing in $t$ by corollary 17.13.31, it follows that for $t\in [t_0,t_0+1]$, we have

\[ \ell ^*_{\tau }(y)+1-t_0(\ell ^*_{\max }-\tau )\geq \ell _{t_0}(y)-t_0\ell ^*_{\max }\geq \ell _{t}(y)-t\ell ^*_{\max }. \]

Since $\tau < \ell ^*_{\max }$ and $t_0\geq t-1$, we deduce that

77E18B7AC795 17.13.37
\begin{equation} \ell ^*_{\tau }(y)+1+\ell ^*_{\max }-\tau \geq \ell _{t}(y)-t\tau ,\quad t\in [t_0,t_0+1]. \end{equation}

Take a sufficiently small open neighborhood $U$ of $x$ such that

\[ \int _ U |f|^2\int _0^{\infty } \exp (-\ell _ t+t\tau )\, \mathrm{d}t\, \omega ^ n<\infty . \]

Applying (17.13.37), we deduce that

\[ \int _ U |f|^2\exp \left(-\ell ^*_{\tau }\right)\, \omega ^ n<\infty . \]

Therefore, $f\in \mathcal{I}(\ell ^*_{\tau })_ x$.

Step 2. Assume that $f\in \mathcal{I}(\ell ^*_{\tau })_ x$.

It follows from the strong openness theorem theorem 5.16.21 (cf. (17.12.5)) that $f\in \mathcal{I}(\ell ^*_{\tau +\epsilon })_ x$ for some small enough $\epsilon >0$ with $\tau +\epsilon <\ell ^*_{\max }$. Take a sufficiently small open neighborhood $U$ of $x$ such that

\[ \int _ U |f|^2 \exp \left(-\ell ^*_{\tau +\epsilon }\right)\, \omega ^ n<\infty . \]

We compute

\[ \begin{aligned} \int _ U |f|^2\int _0^{\infty } \exp \left(-\ell _ t+t\tau \right)\, \mathrm{d}t\, \omega ^ n\leq & \int _ U |f|^2\int _0^{\infty } \exp \left(-\ell ^*_{\tau +\epsilon }-t\epsilon \right)\, \mathrm{d}t\, \omega ^ n\\ =& \frac{1}{\epsilon }\int _ U |f|^2\exp \left(-\ell ^*_{\tau +\epsilon }\right)\, \omega ^ n\\ <& \infty . \end{aligned} \]

Therefore, $f$ lies in the right-hand side of (17.13.36). $\square$

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