18.4 Ideal projection3458DC26AA54

Proof. Write $q=P_\theta [u]_{\mathcal I}$. By Choquet's lemma, some sequence from the defining family has regularized supremum $q$. Replacing its terms by successive finite maxima, we obtain by lemma 5.37.4 an increasing sequence $v_ j$ that remains in the family and converges to $q$ almost everywhere.

Fix $x\in X$ and $\lambda >0$. On a coordinate ball about $x$, choose one smooth function $r$ such that every $v_ j+r$ and $q+r$ is plurisubharmonic; this is possible because all these functions have complex Hessian bounded below by the same smooth form $-\theta $. The increasing-limit assertion of theorem 5.37.3 gives

for all large $j$; the second equality holds because smooth additions do not change multiplier ideals. Hence $q\sim _{\mathcal I}u$.

Since $q\leq 0$, it belongs to its own defining family. The pointwise supremum is therefore at least $q$, and it is at most its regularization $q$; this proves the assertion about regularization. The defining families for $P_\theta [q]_{\mathcal I}$ and $P_\theta [u]_{\mathcal I}$ coincide, which proves idempotence. Finally, if $\sup _ Xq<0$, then $q-\sup _ Xq$ would be a larger competitor, which is impossible. $\square$

Proof. By proposition 5.43.18, pullback identifies the potentials in the two backgrounds. Surjectivity of $\mu $ identifies their nonpositivity constraints, and lemma 5.43.24 identifies their all-weight multiplier-ideal classes. Hence pullback gives a bijection of the complete competitor families. The ideal envelope is itself a competitor and is their greatest member by theorem 18.4.2. Its pullback is therefore the greatest competitor upstairs, which proves the identity. $\square$

Proof. Let $h\leq 0$ be a competitor for the ordinary envelope, so that $h\preceq u$. The function $\max (h,u-\sup _ Xu)$ is nonpositive, ordinarily equivalent to $u$, and dominates $h$. Since ordinarily equivalent potentials have the same multiplier ideals, it is a competitor for the ideal envelope. Taking regularized suprema gives $P_\theta [u]\leq P_\theta [u]_{\mathcal I}$; the two outer inequalities follow directly from the defining families.

Projection and monotonicity of local integrability now give, for every $\lambda >0$,

Thus the ordinary envelope preserves all multiplier ideals, and it has the same ideal envelope as $u$. For $q=P_\theta [u]_{\mathcal I}$, the inequalities already proved and idempotence give $q\leq P_\theta [q]\leq P_\theta [q]_{\mathcal I}=q$. $\square$

Proof. By the last identity in corollary 18.4.4, $P_\theta [u]_{\mathcal I}$ is an ordinary model, that is, an envelope fixed point. The mass estimate for envelope fixed points in proposition 16.28.7 applies. $\square$

Proof. Let $h$ be a competitor for $P_\theta [u]_{\mathcal I}$. Since $h\sim _{\mathcal I}u\preceq _{\mathcal I}v$, the Lelong-number criterion in theorem 5.37.3 shows that $\max (h,v-\sup _ Xv)$ is multiplier-ideal equivalent to $v$. This function is nonpositive and dominates $h$, so it is a competitor for $P_\theta [v]_{\mathcal I}$; taking the supremum over $h$ proves that the envelope preserves the order. Multiplication by $a$ is a bijection between the defining families in the scaling identity. For the sum, $P_{\theta _1}[u_1]_{\mathcal I}+P_{\theta _2}[u_2]_{\mathcal I}$ is nonpositive and $(\theta _1+\theta _2)$-plurisubharmonic, and it is multiplier-ideal equivalent to $u_1+u_2$ by projection and lemma 5.37.4. It is therefore a competitor for the envelope on the right. $\square$

Proof. Set $q=P_\theta [a]_{\mathcal I}$. Projection gives $q\sim _{\mathcal I}a$, so lemma 5.38.11 gives $q\preceq a$. Since $q\leq 0$, it is a competitor for $P_\theta [a]$. Together with proposition 18.4.4, this yields

Hence the two envelopes are equal and ordinarily equivalent to $a$. Since they differ from $a$ by a bounded function, they inherit its analytic presentation. $\square$

Proof. Set $q=P_\Omega [v]_{\mathcal I}$. Projection gives $q\sim _{\mathcal I}v$, and lemma 5.38.5 gives $q\preceq v$. Thus $q$ is a competitor for $P_\Omega [v]$. Together with proposition 18.4.4, this yields $v-\sup _ Zv\leq P_\Omega [v]=q\preceq v$. $\square$