Pome

Proposition 19.4.11AE88967EEE84

Proposition 19.4.11.AE88967EEE84 Assume that $\varphi $ has analytic singularities and $\theta _{\varphi }$ is a Kähler current. Then we have

\[ \Gamma ^{\infty }(\theta ,\varphi )\in \mathcal{S}'(\Delta _\nu (L)) \]

and

\[ \operatorname{vol}\Gamma ^{\infty }(\theta ,\varphi )=\frac{1}{n!}\int _ X \theta _{\varphi }^ n. \]

Proof. By theorem 5.18.8, we can choose a modification $\pi \colon Y\to X$ such that $\pi ^*\varphi $ has log singularities along an effective simple normal crossing $\mathbb Q$-divisor $D$. We regard $\nu $ as a valuation of $\mathbb C(Y)=\mathbb C(X)$. By the definition of $\mathcal I_\infty $, the section spaces defining $\Gamma ^\infty (\theta ,\varphi )$ are naturally identified with

\[ \mathrm{H}^0\left(Y,\pi ^*L^ k\otimes \mathcal O_ Y(-\lceil kD\rceil )\right). \]

Thus we may compute $\Gamma ^\infty (\theta ,\varphi )$ on $Y$.

In this case,

\[ \Gamma ^{\infty }(\theta ,\varphi )= \Biggl \{ (\nu (s),k):k\in \mathbb {N}, s\in \mathrm{H}^0\left(Y,\pi ^*L^ k\otimes \mathcal{O}_ Y(-\lceil k D \rceil )\right)^{\times } \Biggr \} \]

Since $\pi ^*L-D$ is nef and big by 5.18.5, our assertion follows from the same argument as proposition 19.4.7. $\square$

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