18.9 Analytic closureE3C1856717ED

Proof. We use the notation of corollary 18.7.9: $m_ j=2^ j$, $a_ j$ has type $(1/m_ j,\mathcal I(m_ ju))$, $c_ j=(1+\epsilon _ j)^{-1}\uparrow 1$ is rational, and $b_ j=c_ ja_ j$. The ordinary and ideal envelopes of $b_ j$ agree and decrease to $P_\omega [u]_{\mathcal I}$. Thus the two envelope coordinates of $[b_ j]$ converge to those of $[u]$ exactly when $[u]\in \mathcal G_\omega $. It remains to show that the other coordinates always converge.

Fix $t>0$ and $a>1$, and choose $1<s<a$. For large $j$, one has $ac_ j>s$ and $m_ j\geq st/(s-1)$, so lemma 5.35.4 gives

Since $u\preceq a_ j\preceq b_ j$, the reverse relation $\mathcal I(atu)\subseteq \mathcal I(tb_ j)$ holds for every $j$. The factor $c_ j$ enters the weight in the first inclusion, which is why we chose $s<a$.

For a divisor $E$ on a fixed modification $\pi $, the Bergman divisorial estimate gives

where $A(E)$ is the log discrepancy, computed over a coordinate neighborhood of the center of $E$. Each coefficient therefore converges to $\nu _ E(u)$, and the summability in lemma 18.8.1 gives $d_\pi ([b_ j],[u])\to 0$ by dominated convergence. This proves convergence of all the remaining coordinates.

Finally, analytic classes have equal envelopes by proposition 18.4.8. The map $[u]\mapsto (P_\omega [u],P_\omega [u]_{\mathcal I})$ is continuous into $L^1\times L^1$, whose diagonal is closed. Thus every limit of a net of analytic classes lies in $\mathcal G_\omega $. The sequence $b_ j$ gives the converse and the sequential assertion. $\square$