16.9.54 Continuity of invariantsF6973807BD19

Proof. First observe that since $d_ S$ is a pseudometric, it suffices to prove (16.9.56) when $I=\mathbb {Z}_{>0}$ as partially ordered sets.

By corollary 16.9.36, after adding a Kähler form to $\theta $, we may assume that the masses of $\varphi _ j$ and of $\varphi $ are bounded from below by a positive constant.

By theorem 16.9.49, we may assume that $\varphi _ j$ and $\varphi $ are both $\mathcal{I}$-model and hence model. When proving (16.9.56), we are free to pass to subsequences.

By proposition 16.9.17 and a sandwich argument, we may assume that the sequence $(\varphi _ j)_ j$ is either increasing or decreasing. In the increasing case, the Lelong numbers $\nu (\varphi _ j,E)$ decrease to a limit at least $\nu (\varphi ,E)$, while $L^1$-convergence gives the opposite inequality by upper semicontinuity of Lelong numbers. In the decreasing case, (16.9.56) follows from proposition 16.4.4. $\square$

Proof. The latter part of (16.9.58) is just a special case of theorem 16.9.30. It remains to prove the former part.

We may therefore assume that $\int _ X \theta _{\varphi _ j}^ n>0$ for all $j\in I$ after passing to a tail. Then by theorem 16.9.49, we have

Therefore, the first part of (16.9.58) follows again from theorem 16.9.30. $\square$

Proof. Fix $\lambda '>\lambda >0$. We want to find $j_0>0$ so that for $j\geq j_0$, (16.9.60) holds.

Step 1. We first assume that $\varphi $ has analytic singularities.

Let $\pi \colon Y\rightarrow X$ be a log resolution of $\varphi $ and let $E_1,\ldots ,E_ N$ be all prime divisors in the polar locus of $\varphi $ on $Y$. Recall that by theorem 5.16.18, a non-zero local holomorphic function $f$ lies in the right-hand side of (16.9.60) if and only if for all $i=1,\ldots ,N$,

where $A_ X$ denotes the log discrepancy. Similarly, if $f$ lies in the left-hand side of (16.9.60), then for all $i=1,\ldots ,N$,

As Lelong numbers are continuous with respect to $d_ S$ by theorem 16.9.55, we can find $j_0>0$ so that when $j\geq j_0$, $\lambda '\nu (\varphi _ j,E_ i)\geq \lambda \nu (\varphi ,E_ i)$ for all $i$. In particular, (16.9.61) follows.

Step 2. We handle the general case.

By corollary 16.9.36, we are free to increase $\theta $ and assume that $\theta _{\varphi }$ is a Kähler current.

Take a quasi-equisingular approximation $(\psi _ k)_ k$ of $\varphi $ in $\mathrm{PSH}(X,\theta )$. The existence is guaranteed by theorem 5.18.11. Take $\lambda ''\in (\lambda ,\lambda ')$. Then by definition of quasi-equisingular approximations (cf. (5.18.9)), we can find $k>0$ so that

Observe that $\psi _ k\geq \varphi $ for all $k>0$. By proposition 16.9.42, we then find that for any $k>0$, we have $\varphi _ j\lor \psi _ k\xrightarrow {d_ S}\varphi \lor \psi _ k=\psi _ k$ as $j\to \infty $. By Step 1, we can find $j_0>0$ so that for $j\geq j_0$,

It follows that for $j\geq j_0$,

This completes the proof. $\square$