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16.6.26 Properties of the $\mathcal{I}$-envelope4FA5F6B29982

Let $\theta ,\theta _1,\theta _2$ be smooth closed real $(1,1)$-forms on $X$.

We have the following properties of the $\mathcal{I}$-envelope.

Proposition 16.6.27.F7C47263CB3C

  1. 6B3F6E8F1EC1

    Suppose that $\varphi \in \mathrm{PSH}(X,\theta )$ and $\lambda \in \mathbb {R}_{>0}$. Then

    \[ P_{\lambda \theta }[\lambda \varphi ]_{\mathcal{I}}=\lambda P_{\theta }[\varphi ]_{\mathcal{I}}. \]
  2. 0F43574FD82C

    Suppose that $\varphi _1\in \mathrm{PSH}(X,\theta _1)$ and $\varphi _2\in \mathrm{PSH}(X,\theta _2)$. Then

    \[ P_{\theta _1+\theta _2}[\varphi _1+\varphi _2]_{\mathcal{I}}\geq P_{\theta _1}[\varphi _1]_{\mathcal{I}}+P_{\theta _2}[\varphi _2]_{\mathcal{I}}. \]
  3. 56C6F951EFD4

    Suppose that $\varphi _1\in \mathrm{PSH}(X,\theta _1)$ and $\varphi _2\in \mathrm{PSH}(X,\theta _2)$. Then

    \[ P_{\theta _1+\theta _2}[\varphi _1+\varphi _2]_{\mathcal{I}}\sim _{\mathcal{I}} P_{\theta _1}[\varphi _1]_{\mathcal{I}}+P_{\theta _2}[\varphi _2]_{\mathcal{I}}. \]
  4. 14196330587C

    Suppose that $\varphi _1,\varphi _2\in \mathrm{PSH}(X,\theta )$. Then

    \[ P_{\theta }[\varphi _1\lor \varphi _2]_{\mathcal{I}}\sim _{\mathcal{I}} P_{\theta }[\varphi _1]_{\mathcal{I}}\lor P_{\theta }[\varphi _2]_{\mathcal{I}}. \]

Proof. 1 This follows by rescaling the candidates in the definition of the $\mathcal{I}$-envelope.

2 Suppose that $\psi _1\in \mathrm{PSH}(X,\theta _1)$ and $\psi _2\in \mathrm{PSH}(X,\theta _2)$ satisfy

\[ \psi _ i\leq 0,\quad \psi _ i \sim _{\mathcal{I}} \varphi _ i \]

for $i=1,2$. Then thanks to proposition 16.6.5,

\[ \psi _1+\psi _2\leq 0,\quad \psi _1+\psi _2 \sim _{\mathcal{I}} \varphi _1+\varphi _2. \]

It follows that

\[ \psi _1+\psi _2\leq P_{\theta _1+\theta _2}[\varphi _1+\varphi _2]_{\mathcal{I}}. \]

Since $\psi _1$ and $\psi _2$ are arbitrary, we conclude.

3 and 4: These follow easily from proposition 16.6.11 and proposition 16.6.5. $\square$

Lemma 16.6.28.2A3BCF353C03 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi \preceq \psi $. Then

\[ P_{\theta }[\varphi ]_{\mathcal{I}}\leq P_{\theta }[\psi ]_{\mathcal{I}}. \]

Proof. We may assume that $\psi \leq 0$. Observe that $P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi \sim _{\mathcal{I}} \psi $ as a consequence of proposition 5.16.8 and proposition 16.6.11. So

\[ P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi \leq P_{\theta }[\psi ]_{\mathcal{I}}. \]

The desired inequality follows. $\square$

Proposition 16.6.29.2D7AF550FFF7 Consider a decreasing net $(\varphi _ i)_{i\in I}$ of model potentials, with $\varphi _ i\in \mathrm{PSH}(X,\theta )_{>0}$ for all $i\in I$. Set

\[ \varphi \coloneqq \inf _{i\in I}\varphi _ i. \]

Suppose that $\varphi \not\equiv -\infty $ and $\int _ X \theta _{\varphi }^ n>0$. Then

\[ \inf _{i\in I} P_{\theta }[\varphi _ i]_{\mathcal{I}}=P_{\theta }[\varphi ]_{\mathcal{I}}. \]

In particular, if the $\varphi _ i$'s are all $\mathcal{I}$-model, then so is $\varphi $.

Proof. Let $\eta =\inf _{i\in I} P_{\theta }[\varphi _ i]_{\mathcal{I}}$. We have $\eta \geq P_{\theta }[\varphi ]_{\mathcal{I}}$ as a consequence of 16.6.28. In particular, $\eta \in \mathrm{PSH}(X,\theta )$.

By proposition 16.4.4, we have

\[ \lim _{i\in I} \int _ X\theta _{\varphi _ i}^ n=\int _ X\theta _{\varphi }^ n>0. \]

We may assume that for each $i\in I$, $\int _ X\theta _{\varphi _ i}^ n>\int _ X\theta _{\varphi }^ n$, since otherwise $\varphi _ i=\varphi $ by theorem 16.3.15, and hence $P_{\theta }[\varphi _ i]_{\mathcal{I}}=P_{\theta }[\varphi ]_{\mathcal{I}}$ due to (16.6.19); the same holds for all $j\geq i$, and there is nothing to prove.

So by 13.13.30, we can find a decreasing net $\epsilon _ i\searrow 0$ ($i\in I$) with $\epsilon _ i\in (0,1)$ and $\psi _ i\in \mathrm{PSH}(X,\theta )$ such that for all $i\in I$,

\[ (1-\epsilon _ i) \varphi _ i+\epsilon _ i\psi _ i\leq \varphi . \]

We can, for example, take

\[ \epsilon _ i^{-1}=\frac{1}{2}\left( 1+\left(\frac{\int _ X \theta _{\varphi _ i}^ n}{\int _ X \theta _{\varphi _ i}^ n-\int _ X \theta _{\varphi }^ n}\right)^{1/n} \right). \]

By proposition 16.6.27 and 16.6.28, we have

\[ \eta +\epsilon _ i P_{\theta }[\psi _ i]_{\mathcal{I}}\leq (1-\epsilon _ i) \eta +\epsilon _ i P_{\theta }[\psi _ i]_{\mathcal{I}}\leq (1-\epsilon _ i) P_{\theta }[\varphi _ i]_{\mathcal{I}}+\epsilon _ i P_{\theta }[\psi _ i]_{\mathcal{I}}\leq P_{\theta }[\varphi ]_{\mathcal{I}}. \]

Each $P_{\theta }[\psi _ i]_{\mathcal{I}}$ has supremum $0$, so proposition 5.17.3 shows that the family is bounded in $L^1$. Hence $\epsilon _ iP_{\theta }[\psi _ i]_{\mathcal{I}}\to 0$ in $L^1$, and the preceding inequality gives $\eta \leq P_{\theta }[\varphi ]_{\mathcal{I}}$ almost everywhere. By proposition 5.14.6, the inequality holds everywhere. $\square$

Proposition 16.6.30.0C703BE43035 Let $(\varphi _ i)_{i\in I}$ be an increasing net in $\mathrm{PSH}(X,\theta )_{>0}$ uniformly bounded from above. Let $\varphi \coloneqq \operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ i$. Then

\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta }[\varphi _ i]_{\mathcal{I}}=P_{\theta }[\varphi ]_{\mathcal{I}}. \]

In particular, if the $\varphi _ i$'s are all $\mathcal{I}$-model, then so is $\varphi $.

Proof. Let $\eta =\operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta }[\varphi _ i]_{\mathcal{I}}$. Then $\eta \leq P_{\theta }[\varphi ]_{\mathcal{I}}$ as a consequence of 16.6.28.

By corollary 13.13.12, we have

\[ \lim _{i\in I} \int _ X\theta _{\varphi _ i}^ n=\int _ X\theta _{\varphi }^ n>0. \]

We may assume that for each $i\in I$, $\int _ X\theta _{\varphi _ i}^ n<\int _ X\theta _{\varphi }^ n$, since otherwise $P_{\theta }[\varphi _ i]=P_{\theta }[\varphi ]$ by theorem 16.3.15, and hence $P_{\theta }[\varphi _ i]_{\mathcal{I}}=P_{\theta }[\varphi ]_{\mathcal{I}}$ due to (16.6.19); the same holds for all $j\geq i$, and there is nothing to prove.

So by 13.13.30, we can find a decreasing net $\epsilon _ i\searrow 0$ ($i\in I$) as in (16.4.10) with $\epsilon _ i\in (0,1)$ and $\psi _ i\in \mathrm{PSH}(X,\theta )$ such that for all $i\in I$,

\[ (1-\epsilon _ i) \varphi +\epsilon _ i\psi _ i\leq \varphi _ i. \]

By proposition 16.6.27 and 16.6.28, we have

\[ P_{\theta }[\varphi ]_{\mathcal{I}}+\epsilon _ i P_{\theta }[\psi _ i]_{\mathcal{I}}\leq (1-\epsilon _ i) P_{\theta }[\varphi ]_{\mathcal{I}}+\epsilon _ i P_{\theta }[\psi _ i]_{\mathcal{I}}\leq P_{\theta }[\varphi _ i]_{\mathcal{I}}\leq \eta . \]

As above, $\epsilon _ iP_{\theta }[\psi _ i]_{\mathcal{I}}\to 0$ in $L^1$. It follows that $P_{\theta }[\varphi ]_{\mathcal{I}}\leq \eta $ almost everywhere, hence everywhere by proposition 5.14.6. $\square$

Remark 16.6.31.0F2E59A9BE30 One could also define the following interpolation between the $\mathcal{I}$-envelope and the $P$-envelope: Suppose $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, $j\in \{ 0,\ldots ,n\} $. Then we let

\[ \begin{split} P_{\theta ,j}[\varphi ]\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\left\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0,\varphi \preceq \psi , \int _ X \theta _{\varphi }^ j\wedge \theta _{P_{\theta }[\varphi ]_{\mathcal{I}}}^{n-j}\right.\\ \left.=\int _ X \theta _{\psi }^ j\wedge \theta _{P_{\theta }[\psi ]_{\mathcal{I}}}^{n-j} \right\} . \end{split} \]

Based on the techniques developed in section 16.7, one could show that $P_{\theta ,j}[\bullet ]$ is a projection operator. The cases $j=0$ and $j=n$ recover the $\mathcal{I}$- and $P$-envelopes respectively; the intermediate operators $P_{\theta ,j}$ have not, to our knowledge, been systematically studied in the literature.

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