Corollary 18.1.52.DA9EAE05F341 Let $L$ be a pseudo-effective line bundle on $X$, and let $h$ be a smooth Hermitian metric on $L$ with $\theta =c_1(L,h)$. Then we have
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\begin{equation} \lim _{k\to \infty }\frac{n!}{k^ n}h^0\left(X,L^ k\right)=\int _ X \theta _{V_{\theta }}^ n. \end{equation}
Proof.
It suffices to observe that
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\begin{equation} \mathrm{H}^0\left(X,L^ k \otimes \mathcal{I}\left(kV_{\theta }\right) \right)=\mathrm{H}^0\left(X,L^ k\right) \end{equation}
for all $k>0$, since for any holomorphic section $s$ of $L^ k$, the function $\log |s|_{h^ k}^2$ is $\theta $-psh thanks to proposition 5.20.4 and therefore, we have
\[ \frac{1}{k}\log |s|_{h^ k}^2\preceq V_{\theta }. \]
In particular, $s\in \mathrm{H}^0\left(X,L^ k \otimes \mathcal{I}\left(kV_{\theta }\right) \right)$.
$\square$
Pome
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