PomeCorollary 10.8.22.63196D2455BE Let $L$ and $T$ be holomorphic line bundles on $X$ with smooth quadratic metrics $h_0$ and $h_ T$, put $\theta =c_1(L,h_0)$, and choose $b_ T\geq 0$ with $c_1(T,h_ T)\geq -b_ T\omega $. Let $\varphi $ be quasi-plurisubharmonic with $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq \delta \omega $ for some $\delta >0$, and let $Y\subset X$ be a nonempty closed connected complex submanifold such that $\varphi |_ Y$ is not identically $-\infty $. Let $N$ and $C$ be the constants of theorem 10.8.18. Then for every integer $k\geq 1$ with $k\delta -b_ T\geq N$, every holomorphic section $f$ of $(T\otimes L^ k)|_ Y$ with finite norm for $h_ T\otimes h_0^ k\mathrm e^{-k\varphi |_ Y}$ extends to a holomorphic section $s$ of $T\otimes L^ k$ on $X$ with
Corollary 10.8.2263196D2455BE
\[ \int _ X|s|_{h_ T\otimes h_0^ k}^2\mathrm e^{-k\varphi }\, \mathrm dV \leq C\int _ Y|f|_{h_ T\otimes h_0^ k}^2 \mathrm e^{-k\varphi |_ Y}\, \mathrm dV_ Y. \]
Proof. The metric $h_ T\otimes h_0^ k\mathrm e^{-k\varphi }$ on $T\otimes L^ k$ is plurisubharmonic, its curvature current is $c_1(T,h_ T)+k(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )\geq (k\delta -b_ T)\omega \geq N\omega $, and its restriction to $Y$ is defined. Hence theorem 10.8.18 applies. Note that the finiteness of the norm on $Y$ means that $f$ lies locally in the multiplier ideal of $k\varphi |_ Y$, and that the constant $C$ is multiplied by $n!/(\dim Y)!$ if the volumes $\omega ^ n$ and $\omega |_ Y^{\dim Y}$ are used instead of $\mathrm dV$ and $\mathrm dV_ Y$. $\square$
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