Pome848269EDF931848269EDF931
Two constraints act on limits of partial Fubini–Study potentials. The continuous weight is an obstacle on the set where sections are measured, and the selecting potential imposes a multiplier-ideal constraint on the whole manifold. The partial envelope is the largest potential satisfying both. Since an obstacle on a non-pluripolar compact set bounds every competitor globally, the partial envelope lies below a translate of the multiplier-ideal envelope of the selecting potential; conversely, a translate of that envelope lying below the obstacle is a competitor. The partial envelope therefore differs from the multiplier-ideal envelope by a bounded function.
Let $\theta $ be a smooth real $(1,1)$-form on $X$, let $\varphi \in \mathrm{PSH}(X,\theta )$, let $K\subseteq X$ be compact and non-pluripolar, and let $w\in C(K,\mathbb R)$. Recall that theorem 5.37.3 characterizes the order $\preceq _{\mathcal I}$ by Lelong numbers on local modifications. As in section 5.40, an inequality holds quasi-everywhere if it holds outside a pluripolar set. The partial envelope of $\varphi $ with obstacle $w$ on $K$ is
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18.32.1
Theorem 18.32.2.04BABC8924B6 The function $E=E_{\theta ,K}[\varphi ]_{\mathcal I}(w)$ is $\theta $-plurisubharmonic, and it is the largest member of its defining family. Let $q=P_\theta [\varphi ]_{\mathcal I}$, and let $C$ be a constant as in lemma 5.40.1, so that $\sup _ Xa\leq C$ for every $a\in \mathrm{PSH}(X,\theta )$ with $a\leq w$ quasi-everywhere on $K$. Then
In particular, $E\sim q$ and
Proof. By theorem 18.4.2, $q\leq 0$ and $q\sim _{\mathcal I}\varphi $. Hence $q+\min _ Kw$ belongs to the defining family. Lemma 5.40.1 bounds the entire family above by $C$, so its regularized supremum $E$ is $\theta $-plurisubharmonic, and $E\geq q+\min _ Kw$.
For the upper bound, let $a$ be a member of the family. Since $\sup _ Xa\leq C$, proposition 18.4.4, proposition 18.4.6 give
Taking the regularized supremum over $a$ proves $E\leq q+C$.
The two bounds show that $E-q$ is bounded. Hence $E$ and $q$ have the same multiplier ideals, and these are the multiplier ideals of $\varphi $.
It remains to show that $E$ itself satisfies the obstacle condition, which is not automatic after regularization. By Choquet's lemma, $E$ is the regularization of the supremum of a countable subfamily. Replace its members by their successive finite maxima. Finite maxima are again members of the family: the multiplier-ideal order is preserved by theorem 5.37.3, and a finite union of pluripolar exceptional sets is pluripolar. We obtain an increasing sequence $a_ j$ in the family whose limit has regularization $E$. Outside a countable union of pluripolar subsets of $K$, every $a_ j$ is at most $w$, and hence so is $\lim _ ja_ j$. By the local upper-envelope theorem recalled in section 5.40, $E$ differs from $\lim _ ja_ j$ only on a pluripolar set. Thus $E\leq w$ quasi-everywhere on $K$.
We have shown that $E$ belongs to its defining family. It dominates every member by construction, so it is the largest member. In particular, the supremum in (18.32.1) equals $E$ at every point even before regularization. $\square$
Proposition 18.32.3.ABD0D0DB843C The partial envelope depends only on the multiplier-ideal class of $\varphi $, and it is increasing in the obstacle and in $\varphi $ with respect to $\preceq _{\mathcal I}$. For $w,w'\in C(K)$ and $c\in \mathbb R$,
The second line stands for the two corresponding pointwise inequalities, which make sense also at the poles. Let $\theta _0,\theta _1$ be smooth real $(1,1)$-forms on $X$, and let $\varphi _ i\in \mathrm{PSH}(X,\theta _ i)$ and $w_ i\in C(K)$ for $i=0,1$. For $0<t<1$ put $\theta _ t=(1-t)\theta _0+t\theta _1$, and define $\varphi _ t$ and $w_ t$ in the same way. Then
Finally, for a smooth real function $f$, changing the background and the obstacle gives the identity
Proof. The first two assertions hold because the defining families are equal or nested. Adding a constant $c$ is a bijection between the families for $w$ and $w+c$, which proves the translation rule. Applying monotonicity and translation to $w'-\| w-w'\| _{C(K)}\leq w\leq w'+\| w-w'\| _{C(K)}$ gives the two inequalities in the second line.
For concavity, the left-hand side is $\theta _ t$-plurisubharmonic, and it lies below $w_ t$ outside the union of two pluripolar exceptional sets. By theorem 18.32.2, its two summands have the multiplier-ideal classes of $\varphi _0$ and $\varphi _1$, so the addition and scaling rules of lemma 5.37.4 show that the left-hand side has the multiplier-ideal class of $\varphi _ t$. It is therefore a member of the family defining the right-hand side. Finally, $a\mapsto a-f$ is a bijection between the two families in the last identity, and regularization commutes with subtraction of a continuous function. $\square$
In the next section the partial envelope bounds every limit of partial Fubini–Study potentials. The point is to show that both constraints, the multiplier-ideal order and the obstacle on $K$, survive $L^1$ limits.
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