Pome

Proposition 18.32.3ABD0D0DB843C

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 $c\in \mathbb R$,

\[ \begin{split} E_{\theta ,K}[\varphi ]_{\mathcal I}(w+c) & =E_{\theta ,K}[\varphi ]_{\mathcal I}(w)+c,\\ \bigl |E_{\theta ,K}[\varphi ]_{\mathcal I}(w) -E_{\theta ,K}[\varphi ]_{\mathcal I}(w')\bigr | & \leq \| w-w'\| _{C(K)}. \end{split} \]

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

\[ (1-t)E_{\theta _0,K}[\varphi _0]_{\mathcal I}(w_0) +tE_{\theta _1,K}[\varphi _1]_{\mathcal I}(w_1) \leq E_{\theta _ t,K}[\varphi _ t]_{\mathcal I}(w_ t). \]

Finally, for a smooth real function $f$, changing the background and the obstacle gives the identity

\[ E_{\theta +\mathrm{dd}^{\mathrm{c}}f,K}[\varphi -f]_{\mathcal I}(w-f|_ K) =E_{\theta ,K}[\varphi ]_{\mathcal I}(w)-f. \]

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. $\square$

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