Let $X$ be a connected compact Kähler manifold of dimension $n$ and $K \subseteq X$ be a closed non-pluripolar set. Let $\theta $ be a smooth closed real $(1,1)$-form on $X$ representing a pseudo-effective cohomology class.
Definition 18.28.2.28F090ED5EFA Given a function $f\colon K\rightarrow [-\infty ,\infty ]$ and $\varphi \in \mathrm{PSH}(X,\theta )$, we introduce the relative $P$-envelope of $\varphi $ (with respect to $K,f,\theta $) as
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18.28.3
\begin{equation} P_{\theta ,K}[\varphi ](f)\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq \varphi \Bigr \} . \end{equation}
Similarly, we define the relative $\mathcal{I}$-envelope of $\varphi $ (with respect to $K,f,\theta $) as
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18.28.4
\begin{equation} P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _{\mathcal{I}} \varphi \Bigr \} . \end{equation}
Observe that when $f$ is bounded from below, neither envelope is identically $-\infty $.
The envelopes contain essentially all envelope operators that we have studied here:
Example 18.28.5.D0D0EA7C293D When $K=X$ and $f=0$, we have
\[ P_{\theta ,X}[\varphi ](0)=P_{\theta }[\varphi ],\quad P_{\theta ,X}[\varphi ]_{\mathcal{I}}(0)=P_{\theta }[\varphi ]_{\mathcal{I}}. \]
When $K=X$, we have
\[ P_{\theta ,X}[\varphi ](f)=P_{\theta }[\varphi ](f). \]
These definitions reduce to the envelopes studied in section 16.1. On the other hand, when $\varphi =V_{\theta }$, we have
\[ \begin{aligned} P_{\theta ,K}[V_{\theta }](f) & =P_{\theta ,K}[V_{\theta }]_{\mathcal{I}}(f)\\ & =\operatorname *{sup}_{}{}\! {}^{*}\left\{ \eta \in \mathrm{PSH}(X,\theta ): \begin{array}{l} \eta |_ K \leq f \textup{ quasi-everywhere} \end{array} \right\} . \end{aligned} \]
In an even more special case where $\varphi =V_{\theta }$ and $K=X$, we find that
\[ P_{\theta ,X}[V_{\theta }](f)=\operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta \leq f \textup{ quasi-everywhere}\Bigr \} . \]
This is nothing but $P_{\theta }(f)$ as we studied in definition 13.13.18.
Proposition 18.28.6.C9561913A3E8 Let $f,g\colon K\rightarrow [-\infty ,\infty ]$ be functions and $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then we have the following:
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We have
\[ P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f); \]
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assume that $f$ is bounded from above, then
\[ \begin{aligned} \left.P_{\theta ,K}[\varphi ](f)\right|_ K\leq & f\textup{ quasi-everywhere},\\ \left. P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\right|_ K\leq & f\textup{ quasi-everywhere}; \end{aligned} \]
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suppose that $\varphi \preceq _{\mathcal{I}}\psi $, then
\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq P_{\theta ,K}[\psi ]_{\mathcal{I}}(f); \]
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if $f\leq g$, then
\[ P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[\varphi ](g),\quad P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(g); \]
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for any $t \in (0,1)$, we have
\[ \begin{aligned} P_{\theta ,K}\left[t\varphi +(1-t)\psi \right](f)\geq & tP_{\theta ,K}[\varphi ](f)+(1-t)P_{\theta ,K}[\psi ](f),\\ P_{\theta ,K}\left[t\varphi +(1-t)\psi \right]_{\mathcal{I}}(f)\geq & tP_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)+(1-t)P_{\theta ,K}[\psi ]_{\mathcal{I}}(f). \end{aligned} \]
Proof.
1 This follows from $\eta \preceq \varphi \Rightarrow \eta \preceq _{\mathcal I}\varphi $.
2 This follows from proposition 5.14.5 and 5.17.4.
3 This follows by inclusion of the defining candidate sets.
4 This follows by inclusion of the defining candidate sets.
5 If $\eta ,\eta '\in \mathrm{PSH}(X,\theta )$ are such that
\[ \begin{split} & \eta |_ K\leq f\textup{ quasi-everywhere},\quad \eta '|_ K\leq f\textup{ quasi-everywhere},\\ & \eta \preceq \varphi ,\quad \eta '\preceq \psi , \end{split} \]
then for $t\in (0,1)$,
\[ \left.\left(t\eta +(1-t)\eta '\right)\right|_ K\leq f\textup{ quasi-everywhere},\quad t\eta +(1-t)\eta ' \preceq t\varphi +(1-t)\psi . \]
Therefore,
\[ t\eta +(1-t)\eta '\leq P_{\theta ,K}\left[t\varphi +(1-t)\psi \right](f). \]
Since $\eta $ and $\eta '$ are arbitrary, the first inequality follows. The second follows from the same argument.
$\square$
Similar to the usual envelopes studied in section 16.1, the $P$-partial envelopes are well-behaved only when $\varphi $ has positive mass.
Lemma 18.28.7.0F9B18404274 Suppose that $f\colon K\rightarrow (-\infty ,\infty ]$ is bounded from below, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then
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18.28.8
\begin{equation} P_{\theta ,K}[\varphi ](f)= \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq \varphi \Bigr \} . \end{equation}
Proof.
By our assumption, we can find $C>0$ so that $\varphi |_ K-C\leq f$. Then if $\eta $ is a candidate in (18.28.3), $\eta \lor (\varphi -C)$ is also a candidate, and $\eta \lor (\varphi -C)\in \mathrm{PSH}(X,\theta )_{>0}$. Therefore, $\eta \lor (\varphi -C)$ is also a candidate for the right-hand side of (18.28.8). The equality (18.28.8) follows.
$\square$
Note that in the most important cases, we can actually replace the condition $\eta \leq f$ quasi-everywhere in (18.28.3) by everywhere.
Lemma 18.28.9.A86A369D4CBF Let $f\colon K\rightarrow (-\infty ,\infty ]$ be bounded from below, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then
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18.28.10
\begin{equation} \begin{split} P_{\theta ,K}[\varphi ](f)=& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ and } \eta \preceq \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ and } \eta \preceq \varphi \Bigr \} . \end{split} \end{equation}
Proof.
Both $\geq $ directions in (18.28.10) are trivial.
Conversely, let $\eta $ be a candidate in (18.28.8). Let $S\subseteq X$ be a pluripolar set so that $\eta |_{K\setminus S}\leq f|_{K\setminus S}$.
We now claim that we can choose $\chi \in \mathrm{PSH}(X,\theta )_{>0}$ with $\chi |_ S\equiv -\infty $ and $\chi \leq \eta $.
Assume this for the moment. For any $\delta \in (0,1)$, we have
\[ (1-\delta )\eta |_ K+\delta \chi |_ K\leq f,\quad (1-\delta )\eta +\delta \chi \preceq \varphi . \]
Hence, $(1-\delta )\eta +\delta \chi $ is dominated by the right-hand side of (18.28.10). Letting $\delta \to 0+$, we conclude that $\eta $ is dominated by the right-hand side of (18.28.10) and hence, (18.28.10) follows.
It remains to construct $\chi $. Since $\varphi $ has positive mass, the class $\{ \theta \} $ is big. Fix a Kähler form $\omega _0$ on $X$ and choose $\rho \in \mathrm{PSH}(X,\theta )$ with $\theta _{\rho }\geq 2\epsilon _0\omega _0$ for some $\epsilon _0>0$. By theorem 5.13.20, there is a quasi-psh function $q$ such that $q|_ S\equiv -\infty $. Choose $A>0$ so that $A\omega _0+\mathrm{dd}^{\mathrm{c}}q\geq 0$. For $a>0$ small enough, $\rho +aq\in \mathrm{PSH}(X,\theta )$ and still has pole $-\infty $ on $S$. Replacing $\rho $ by $\rho +aq$, we may therefore assume that $\rho |_ S\equiv -\infty $. Since $\int _ X \theta _{\eta }^ n>0$, fixing such a $\rho $ and taking $\epsilon \in (0,1)$ small enough, we have
\[ \int _ X \theta ^ n_{\epsilon \rho +(1-\epsilon )V_{\theta }}+\int _ X \theta _{\eta }^ n>\int _ X \theta _{V_{\theta }}^ n. \]
Thus, by proposition 16.3.25 and theorem 16.3.35, we have
\[ \Bigl (\epsilon \rho +(1-\epsilon )V_{\theta }\Bigr ) \land \eta \in \mathrm{PSH}(X,\theta )_{>0}. \]
It suffices to define this function as $\chi $.
$\square$
Corollary 18.28.11.712499DFC37F Suppose that $f\colon K\rightarrow \mathbb {R}$ is bounded, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then there is a constant $C=C(\sup _ K f,K,X)>0$ such that
\[ P_{\theta ,K}[\varphi ](f)\leq C,\quad P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq C. \]
Proof.
Since $K$ is non-pluripolar, 5.17.4 then gives a constant $C>0$, depending only on $\sup _ K f$, $K$ and $X$, such that every $\eta \in \mathrm{PSH}(X,\theta )$ satisfying $\eta |_ K\leq f$ quasi-everywhere satisfies $\eta \leq C$ on $X$.
This applies to the candidates defining both envelopes. Hence both candidate families are locally uniformly bounded above, so their upper semicontinuous suprema are genuine $\theta $-psh functions. By proposition 18.28.6 2, both envelopes are dominated by $f$ quasi-everywhere on $K$. Applying 5.17.4 once more gives
\[ P_{\theta ,K}[\varphi ](f)\leq C,\quad P_{\theta ,K}[\varphi ]_{\mathcal I}(f)\leq C. \]
$\square$
Proposition 18.28.12.A1C36A6249C7 Let $f\colon K\rightarrow (-\infty ,\infty ]$ be a function bounded from below and $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Suppose that $\varphi \preceq _{P} \psi $. Then
\[ P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[\psi ](f). \]
Proof.
Let $\eta \in \mathrm{PSH}(X,\theta )_{>0}$ be such that $\eta |_ K\leq f$ quasi-everywhere and $\eta \preceq \varphi $. Then $\eta \preceq _ P \psi $, and thanks to corollary 16.8.30 for any $C>0$, we have $\eta \land (\psi +C)\in \mathrm{PSH}(X,\theta )_{>0}$. But then $\eta \land (\psi +C)$ is a candidate in the definition of $P_{\theta ,K}[\psi ](f)$, and we find
\[ \eta \land (\psi +C)\leq P_{\theta ,K}[\psi ](f). \]
By corollary 16.8.30 again, we get
\[ \eta =P_{\theta }[\psi ](\eta )\leq P_{\theta ,K}[\psi ](f). \]
Our assertion follows now from 18.28.7.
$\square$
Corollary 18.28.13.72097BEFEC59 Let $f\colon K\rightarrow (-\infty ,\infty ]$ be a function bounded from below and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then
\[ \begin{aligned} P_{\theta ,K}[\varphi ](f)= & \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ and } \eta \preceq _ P \varphi \Bigr \} . \end{aligned} \]
Proof.
We first apply proposition 18.28.12,
\[ \begin{split} P_{\theta ,K}[\varphi ](f)= & P_{\theta ,K}\left[P_{\theta }[\varphi ]\right](f)\\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} . \end{split} \]
Next observe that by 18.28.7, the latter can also be written as
\[ \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} . \]
Next we apply 18.28.9; the remaining statements follow.
$\square$
Corollary 18.28.14.96B842B9BF11 Let $f\colon K\rightarrow (-\infty ,\infty ]$ be a function bounded from below and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ be $\mathcal{I}$-good. Then
\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)=P_{\theta ,K}[\varphi ](f). \]
Proof.
In fact, since $\varphi $ is $\mathcal{I}$-good, a potential $\eta \in \mathrm{PSH}(X,\theta )$ satisfies $\eta \preceq _ P \varphi $ if and only if $\eta \preceq _{\mathcal{I}} \varphi $. Therefore, our assertion follows from corollary 18.28.13 and (18.28.4).
$\square$
Next, we make the following observations about the singularity types of our envelopes:
Lemma 18.28.15.18E5130ACB80 Let $f \colon K\rightarrow \mathbb {R}$ be a bounded function, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. We have
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18.28.16
\begin{equation} P_{\theta ,K}[\varphi ](f)\sim P_{\theta }[\varphi ],\quad P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\sim P_{\theta }[\varphi ]_{\mathcal{I}}. \end{equation}
Proof.
We first prove the $P$-part of (18.28.16).
If $\eta \in \mathrm{PSH}(X,\theta )$ with $\eta \preceq \varphi $ and $\eta \leq 0$, then $\eta +\inf _ K f$ is a candidate in the definition (18.28.3), and hence,
\[ \eta +\inf _ K f \leq P_{\theta ,K}[\varphi ](f). \]
It follows that
\[ P_{\theta }[\varphi ]+\inf _ K f \leq P_{\theta ,K}[\varphi ](f). \]
Conversely, let $\eta \in \mathrm{PSH}(X,\theta )$ be such that
\[ \eta |_ K\leq f,\quad \eta \preceq \varphi . \]
Since $K$ is non-pluripolar, by 5.17.4 we can find $C=C(\sup _ K f,K, X)>0$ so that $\eta \leq C$. Hence $\eta -C$ is a candidate in the definition of $P_{\theta }[\varphi ]$, and hence
\[ \eta -C\leq P_{\theta }[\varphi ]. \]
By 18.28.9, this implies that
\[ P_{\theta ,K}[\varphi ](f) \leq P_{\theta }[\varphi ] + C. \]
The proof of the $\mathcal{I}$-part is similar, but we spell it out since this comparison will be used below. If $\eta \in \mathrm{PSH}(X,\theta )$ satisfies $\eta \leq 0$ and $\eta \preceq _{\mathcal{I}}\varphi $, then $\eta +\inf _ K f$ is a candidate in (18.28.4). Hence
\[ P_{\theta }[\varphi ]_{\mathcal{I}}+\inf _ K f\leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f). \]
Conversely, let $\eta \in \mathrm{PSH}(X,\theta )$ be a candidate in (18.28.4). Since $K$ is non-pluripolar and $f$ is bounded, the same compactness estimate gives a constant $C>0$, independent of $\eta $, such that $\eta \leq C$. Then $\eta -C\leq 0$ and $\eta -C\preceq _{\mathcal{I}}\varphi $. By corollary 16.8.16,
\[ \eta -C\leq P_{\theta }[\varphi ]_{\mathcal{I}}. \]
Taking the upper envelope over all such $\eta $ gives
\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq P_{\theta }[\varphi ]_{\mathcal{I}}+C. \]
This completes the proof.
$\square$
Corollary 18.28.17.CE1417F951E3 Let $f\colon K\rightarrow \mathbb {R}$ be a bounded function, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then
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18.28.18
\begin{equation} P_{\theta ,K}[\varphi ](f)=P_{\theta ,X}[\varphi ]\left( P_{\theta ,K}[V_{\theta }](f) \right). \end{equation}
Proof.
Take any $\eta \in \mathrm{PSH}(X,\theta )$ with $\eta \preceq \varphi $ and $\eta |_ K\leq f$ quasi-everywhere, then
\[ \eta \leq P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[V_{\theta }](f). \]
Therefore,
\[ \eta \leq P_{\theta ,X}[\varphi ]\left( P_{\theta ,K}[V_{\theta }](f) \right). \]
So we conclude the $\leq $ direction in (18.28.18).
For the reverse direction, take $\eta \in \mathrm{PSH}(X,\theta )_{>0}$ such that
\[ \eta \preceq \varphi ,\quad \eta \leq P_{\theta ,K}[V_{\theta }](f) \quad \textup{quasi-everywhere}. \]
Then $\eta |_ K\leq f$ quasi-everywhere by proposition 18.28.6. Hence
\[ \eta \leq P_{\theta ,K}[\varphi ](f). \]
By corollary 18.28.13, the $\geq $ direction in (18.28.18) follows.
$\square$
Pome
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