Theorem 19.4.517CC4ADAC32DC

Proof. Replacing $\varphi $ by $P_{\theta }[\varphi ]_{\mathcal{I}}$, we may assume that $\varphi $ is $\mathcal{I}$-good.

Step 1. We first reduce to the case where $\varphi $ has analytic singularities.

By theorem 18.1.2, we can find a sequence $(\varphi _ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ with analytic singularities such that $\varphi _ j\xrightarrow {d_ S}\varphi $. It follows from theorem 19.4.30 that

Therefore, by 19.7.10,

In view of theorem 16.9.55, it suffices to prove (19.4.52) with $\varphi _ j$ in place of $\varphi $.

Step 2. Assume that $\varphi $ has analytic singularities. In view of proposition 19.4.32 and theorem 5.18.8, after replacing $X$ by a modification, we may assume that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $F$.

Perturbing $L$ by an ample $\mathbb {Q}$-line bundle by proposition 19.4.39, we may assume that $\theta _{\varphi }$ is a Kähler current. Therefore, $L-F$ is ample by 5.18.5. Finally, by rescaling, we may assume that $F$ is a divisor and $L$ is a line bundle.

By theorem 19.4.41, we know that

By definition,

In view of proposition 5.16.15, it remains to show that

The $\geq $ direction is trivial, we prove the converse. Observe that

As $L-F$ is ample, for large enough $k$, the line bundle $k(L-F)$ is globally generated. Hence we can choose a section of $k(L-F)$ which does not vanish at the generic point of $E$. Viewed as a section of $L^ k\otimes \mathcal O_ X(-kF)$, it has vanishing order exactly $\operatorname{ord}_ E(kF)$ along $E$. Therefore,

Thus, (19.4.54) follows. $\square$