Theorem 20.2.42.4F486DDB8BD0 Let $\varphi \in \mathrm{PSH}(X,\theta )$ such that $\nu (\varphi ,Y) =0$. Assume that $\theta _{\varphi }$ is a Kähler current. Then for any holomorphic line bundle $T$ on $X$ we have
\[ \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m = \lim _{k \to \infty } \frac{m!}{k^ m} \dim _{\mathbb C}\left\{ s|_ Y: s\in \mathrm{H}^0\left(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi )\right)\right\} . \]
Proof.
This is a consequence of theorem 18.1.46, theorem 20.2.28 and theorem 20.2.39:
\[ \begin{aligned} \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m & = \lim _{k \to \infty } \frac{m!}{k^ m} h^0\left(Y,T|_ Y\otimes L|_ Y^ k\otimes \mathcal{I}\left(k \operatorname{Tr}_ Y^{\theta }(\varphi )\right)\right)\\ & \leq \varliminf _{k \to \infty } \frac{m!}{k^ m} \dim _{\mathbb C}\left\{ s|_ Y: s\in \mathrm{H}^0\left(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi )\right)\right\} \\ & \leq \varlimsup _{k \to \infty } \frac{m!}{k^ m} \dim _{\mathbb C}\left\{ s|_ Y: s\in \mathrm{H}^0\left(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi )\right)\right\} \\ & \leq \lim _{k \to \infty } \frac{m!}{k^ m} h^0\left(Y ,T|_ Y\otimes L|_ Y^ k\otimes \operatorname{Res}_ Y\mathcal{I}(k\varphi )\right)\\ & = \int _ Y \left(\theta |_ Y+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^ m. \end{aligned} \]
This completes the proof.
$\square$
Pome
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