Recall that the $P$-envelope is defined in definition
16.3.5.
Definition 16.8.2.4E959BD934C4 For $\varphi ,\psi \in \mathrm{QPSH}(X)$, we say $\varphi $ is $P$-more singular than $\psi $ and write $\varphi \preceq _ P \psi $ if for some (equivalently, every, due to 16.8.4 below) closed smooth real $(1,1)$-form $\theta $ on $X$ such that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$, we have
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16.8.3
\begin{equation} P_{\theta }[\varphi ]\leq P_{\theta }[\psi ]. \end{equation}
If $\varphi \preceq _ P \psi $ and $\psi \preceq _ P \varphi $, we write $\varphi \sim _ P \psi $ and say that $\varphi $ and $\psi $ have the same $P$-singularity type.
Recall that $\mathrm{PSH}(X,\theta )_{>0}$ was introduced in (
13.13.1). Note that if $\varphi \preceq \psi $, then $\varphi \preceq _ P \psi $. So the $P$-preorder is
coarser than $\preceq $.
The $P$-equivalence relation can also be formulated using the language of relative full mass classes developed in 16.5. In fact, $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$ satisfies $\varphi \sim _ P \psi $ if and only if $\varphi ,\psi \in \mathcal{E}(X,\theta ;\phi )$ for a common model potential $\phi \in \mathrm{PSH}(X,\theta )_{>0}$.
The condition (
16.8.3) is independent of the choice of $\theta $:
Lemma 16.8.4.A3C93A5CDBF4 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. For any Kähler form $\omega $ on $X$, the following are equivalent:
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$P_{\theta }[\varphi ]\leq P_{\theta }[\psi ]$;
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$P_{\theta +\omega }[\varphi ]\leq P_{\theta +\omega }[\psi ]$.
In particular, $\preceq _ P$ defines a preorder on $\mathrm{QPSH}(X)$.
Proof.
1 $\implies $ 2. Assume 1. Observe that
\[ \varphi \preceq P_{\theta }[\varphi ]\leq P_{\theta +\omega }[\varphi ]. \]
Taking the $P_{\theta +\omega }$-envelope, we find
\[ P_{\theta +\omega }[\varphi ] \leq P_{\theta +\omega }\Bigl [P_{\theta }[\varphi ]\Bigr ] \leq P_{\theta +\omega }\Bigl [P_{\theta +\omega }[\varphi ]\Bigr ]=P_{\theta +\omega }[\varphi ], \]
where the equality follows from theorem 16.3.15. In particular,
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16.8.5
\begin{equation} P_{\theta +\omega }[\varphi ]=P_{\theta +\omega }\Bigl [P_{\theta }[\varphi ]\Bigr ]. \end{equation}
A similar formula holds for $\psi $. So we see that 2 holds.
2 $\implies $ 1. Assume 2. By (16.8.5), we may assume that $\varphi $ and $\psi $ are both model potentials in $\mathrm{PSH}(X,\theta )_{>0}$.
Observe that $\varphi \lor \psi \preceq P_{\theta +\omega }[\psi ]$. It follows that $P_{\theta +\omega }[\varphi \lor \psi ]\leq P_{\theta +\omega }[\psi ]$. The reverse inequality is trivial, so
\[ P_{\theta +\omega }[\varphi \lor \psi ]= P_{\theta +\omega }[\psi ]. \]
From the direction we have proved, for any $C\geq 1$,
\[ P_{\theta +C\omega }[\varphi \lor \psi ]= P_{\theta +C\omega }[\psi ]. \]
So by proposition 16.3.10,
\[ \int _ X \Bigl (\theta +C\omega +\mathrm{dd}^{\mathrm{c}}(\varphi \lor \psi )\Bigr )^ n=\int _ X\left(\theta +C\omega +\mathrm{dd}^{\mathrm{c}}\psi \right)^ n. \]
Since both sides are polynomials in $C$, the equality extends to $C=0$, namely,
\[ \int _ X \theta _{\varphi \lor \psi }^ n=\int _ X \theta _{\psi }^ n. \]
In particular, $\varphi \lor \psi \leq P_{\theta }[\psi ]=\psi $ by (16.3.16). So 1 follows.
$\square$
As a consequence of 16.8.4, we can define the $P$-preorder at the level of currents. Given closed positive $(1,1)$-currents $T=\theta _{\varphi }$, $S=\theta '_{\psi }$, we write $T\preceq _ P S$ (resp. $T\sim _ P S$) if $\varphi \preceq _ P \psi $ (resp. $\varphi \sim _ P \psi $). This definition is independent of the decompositions of $T$ and $S$.
As a first example of $P$-equivalence, we have:
Example 16.8.6.14285897F9E7 Let $\theta $ be a closed smooth real $(1,1)$-form on $X$ and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then
\[ \varphi \sim _ P P_{\theta }[\varphi ]{}^{\href{#sgpt-a0000048053}{1}}. \]
This follows immediately from theorem 16.3.15.
We give a very useful criterion of the $P$-equivalence in terms of the non-pluripolar masses.
Proposition 16.8.7.C7C64ABB4647 Let $\theta $ be a closed smooth real $(1,1)$-form on $X$ representing a big cohomology class. Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$ and $\varphi \preceq _ P \psi $. Then the following are equivalent:
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$\varphi \sim _ P \psi $;
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for each $j=0,\ldots ,n$, we have
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16.8.8
\begin{equation} \int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\psi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{equation}
Assume furthermore that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$, then these conditions are equivalent to the following:
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We have
\[ \int _ X \theta _{\varphi }^ n=\int _ X \theta _{\psi }^ n. \]
Recall that $V_{\theta }$ is introduced in (
13.13.7).
Proof.
We first prove the equivalence between 1 and 1 when $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$.
1 $\implies $ 1. Assume that $\varphi \sim _ P \psi $. By 16.8.4, we have
\[ P_{\theta }[\varphi ]=P_{\theta }[\psi ]. \]
So 1 follows from proposition 16.3.10.
1 $\implies $ 1. Since $\varphi \preceq _ P\psi $, we have $P_{\theta }[\varphi ]\leq P_{\theta }[\psi ]$. Moreover, proposition 16.3.10 shows that $P_{\theta }[\varphi ]$ and $P_{\theta }[\psi ]$ have the same positive mass. By theorem 16.3.15, $P_{\theta }[\psi ]$ is a candidate in the characterization of $P_{\theta }\bigl [P_{\theta }[\varphi ]\bigr ]=P_{\theta }[\varphi ]$. Hence $P_{\theta }[\psi ]\leq P_{\theta }[\varphi ]$, and therefore $P_{\theta }[\varphi ]=P_{\theta }[\psi ]$. So 1 follows.
Let us come back to the general case with $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$.
1 $\implies $ 2. Fix $j\in \{ 0,\ldots ,n\} $. We argue (16.8.8).
Take a Kähler form $\omega $ on $X$. By 16.8.4, for each $\epsilon >0$, we have
\[ P_{\theta +\epsilon \omega }[\varphi ]= P_{\theta +\epsilon \omega }[\psi ]. \]
It follows from proposition 16.3.10 that
\[ \begin{aligned} \int _ X \left(\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\psi \right)^ j\wedge \theta _{V_{\theta }}^{n-j}=& \int _ X \Bigl (\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta +\epsilon \omega }[\psi ] \Bigr )^ j\wedge \theta _{V_{\theta }}^{n-j}\\ =& \int _ X \Bigl (\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta +\epsilon \omega }[\varphi ] \Bigr )^ j\wedge \theta _{V_{\theta }}^{n-j} \\ =& \int _ X \left(\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi \right)^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{aligned} \]
Since the two extremes are both polynomials in $\epsilon $ by the multi-linearity of the non-pluripolar products proposition 13.11.4 6, we conclude that the same holds when $\epsilon =0$, that is, (16.8.8) holds.
2 $\implies $ 1. Assume (16.8.8) holds for all $j=0,\ldots ,n$. Then the same holds with $P_{\theta }[\psi ]$ in place of $\psi $, thanks to proposition 16.3.10. Therefore, for each $t\in (0,1)$, we have
\[ \int _ X \theta _{t\varphi +(1-t)V_{\theta }}^ n= \int _ X \theta _{tP_{\theta }[\psi ]+(1-t)V_{\theta }}^ n \]
by the binomial expansion. Furthermore, $t\varphi +(1-t)V_{\theta }\preceq tP_{\theta }[\psi ]+(1-t)V_{\theta }$. By the implication 1 $\implies $ 1, we have
\[ t\varphi +(1-t)V_{\theta }\sim _ P t P_{\theta }[\psi ]+(1-t)V_{\theta } \]
for each $t\in (0,1)$.
Fix a Kähler form $\omega $ on $X$. From the implication 1 $\implies $ 1, we have
\[ \int _ X (\theta +\omega )_{t\varphi +(1-t)V_{\theta }}^ n= \int _ X (\theta +\omega )_{tP_{\theta }[\psi ]+(1-t)V_{\theta }}^ n. \]
Since both sides are polynomials in $t$ by the multi-linearity of the non-pluripolar products proposition 13.11.4 6, the same holds when $t=1$. From the implication 1 $\implies $ 1 again, we have $\varphi \sim _ P P_{\theta }[\psi ]\sim _ P \psi $.
$\square$
Next we introduce a different preorder.
Proposition 16.8.9.EE590FE41862 Given $\varphi ,\psi \in \mathrm{QPSH}(X)$, the following are equivalent:
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For any $k\in \mathbb {Z}_{>0}$, we have
\[ \mathcal{I}(k\varphi )\subseteq \mathcal{I}(k\psi ); \]
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for any $\lambda \in \mathbb {R}_{>0}$, we have
\[ \mathcal{I}(\lambda \varphi )\subseteq \mathcal{I}(\lambda \psi ); \]
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for any modification $\pi \colon Y\rightarrow X$ and any $y\in Y$, we have
\[ \nu (\pi ^*\varphi ,y)\geq \nu (\pi ^*\psi ,y); \]
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for any proper bimeromorphic morphism $\pi \colon Y\rightarrow X$ from a Kähler manifold and any $y\in Y$, we have
\[ \nu (\pi ^*\varphi ,y)\geq \nu (\pi ^*\psi ,y); \]
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for any prime divisor $E$ over $X$, we have
\[ \nu (\varphi ,E)\geq \nu (\psi ,E). \]
Proof.
The proof is almost identical to that of proposition 16.6.2.
$\square$
Definition 16.8.10.5C5512F45CC2 Let $\varphi ,\psi \in \mathrm{QPSH}(X)$, we say $\varphi $ is $\mathcal{I}$-more singular than $\psi $ and write $\varphi \preceq _{\mathcal{I}} \psi $ if the equivalent conditions in proposition 16.8.9 are satisfied.
It is clear that $\preceq _{\mathcal{I}}$ is a preorder on $\mathrm{QPSH}(X)$.
Note that $\varphi \preceq _{\mathcal{I}}\psi $ and $\psi \preceq _{\mathcal{I}}\varphi $ both hold if and only if $\varphi \sim _{\mathcal{I}}\psi $ in the sense of definition 16.6.3.
Given closed positive $(1,1)$-currents $T=\theta _{\varphi }$, $S=\theta '_{\psi }$, we write $T\preceq _{\mathcal{I}} S$ (resp. $T\sim _{\mathcal{I}} S$) if $\varphi \preceq _{\mathcal{I}} \psi $ (resp. $\varphi \sim _{\mathcal{I}} \psi $). This definition is independent of the decompositions of $T$ and $S$.
Lemma 16.8.11.4A5CF38738F2 Let $\theta $ be a closed smooth real $(1,1)$-form on $X$ representing a big cohomology class. Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Then
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16.8.12
\begin{equation} P_{\theta }[\varphi \lor \psi ]=P_{\theta }\Bigl [ P_{\theta }[\varphi ]\lor P_{\theta }[\psi ] \Bigr ]. \end{equation}
Proof.
Since $\varphi \lor \psi \preceq P_{\theta }[\varphi ]\lor P_{\theta }[\psi ]$, the $\leq $ direction of (16.8.12) follows. Conversely, it suffices to show that
\[ P_{\theta }[\varphi \lor \psi ]\geq P_{\theta }[\varphi ]\lor P_{\theta }[\psi ], \]
which follows from $\varphi \preceq \varphi \lor \psi $, $\psi \preceq \varphi \lor \psi $, and the monotonicity of the $P$-envelope with respect to $\preceq $.
$\square$
Lemma 16.8.13.005A077A0167 Let $\varphi ,\psi \in \mathrm{QPSH}(X)$. Then the following are equivalent:
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$\varphi \preceq _ P \psi $ (resp. $\varphi \preceq _{\mathcal{I}} \psi $);
1D8449F6F20B
$\varphi \lor \psi \sim _ P\psi $ (resp. $\varphi \lor \psi \sim _{\mathcal{I}}\psi $).
Proof.
Take a closed real smooth $(1,1)$-form $\theta $ on $X$ such that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. We only prove the $P$ case, the $\mathcal{I}$ case is similar.
2 $\implies $ 1. Assume 2. By 2 and example 16.8.6, $P_{\theta }[\varphi \lor \psi ]=P_{\theta }[\psi ]\sim _ P \psi $. But $\varphi \preceq P_{\theta }[\varphi \lor \psi ]$, so 1 follows.
1 $\implies $ 2. Assume 1. We may assume that $\varphi ,\psi $ are both model by 16.8.11. Then $\varphi \leq \psi $ and 2 follows.
$\square$
Corollary 16.8.14.FE600747C81E Let $\varphi ,\psi \in \mathrm{QPSH}(X)$. Assume that $\varphi \preceq _ P \psi $. Then $\varphi \preceq _{\mathcal{I}}\psi $.
Therefore, the preorder $\preceq _{\mathcal{I}}$ is coarser than $\preceq _ P$. Given $\varphi ,\psi \in \mathrm{QPSH}(X)$, we therefore have the implications:
\[ \varphi \preceq \psi \implies \varphi \preceq _ P \psi \implies \varphi \preceq _{\mathcal{I}} \psi . \]
Proof.
Thanks to 16.8.13, we have $\varphi \lor \psi \sim _ P \psi $. Take a closed smooth real $(1,1)$-form $\theta $ on $X$ such that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Then by proposition 16.6.18, we have
\[ \psi \sim _{\mathcal{I}} P_{\theta }[\psi ]= P_{\theta }\left[\varphi \lor \psi \right]\sim _{\mathcal{I}} \varphi \lor \psi . \]
Our assertion follows from the $\mathcal{I}$-version of 16.8.13.
$\square$
Next we give a few extra characterizations of the $P$-envelope.
Corollary 16.8.15.564E8A0E68F9 Let $\theta $ be a closed smooth real $(1,1)$-form on $X$ representing a big cohomology class. Assume that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then
\[ \begin{aligned} P_{\theta }[\varphi ]=& \sup \left\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0, \psi \sim _ P \varphi \right\} \\ =& \sup \left\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0, \psi \preceq _ P \varphi \right\} . \end{aligned} \]
Just for comparison, let us recall a few other characterizations of the $P$-envelope for $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$:
\[ \begin{aligned} P_{\theta }[\varphi ]=& \operatorname *{sup}_{}{}\! {}^{*}\left\{ \psi \in \mathrm{PSH}(X,\theta ):\psi \leq 0,\psi \preceq \varphi \right\} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\left\{ \psi \in \mathrm{PSH}(X,\theta ):\psi \leq 0,\psi \sim \varphi \right\} \\ =& \operatorname *{sup}_{C\in \mathbb {Z}_{>0}}{}\! {}^{*} (\varphi +C)\land V_{\theta }\\ =& \sup \left\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0,\varphi \preceq \psi , \int _ X \theta _{\varphi }^ n=\int _ X \theta _{\psi }^ n\right\} . \end{aligned} \]
Proof.
Note that $\psi \sim _ P \varphi $ implies that $\psi \in \mathrm{PSH}(X,\theta )_{>0}$ by proposition 16.3.10. We observe that
\[ \begin{split} & \sup \left\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0, \psi \sim _ P \varphi \right\} \\ =& \sup \left\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0, \varphi \preceq \psi , \psi \sim _ P \varphi \right\} \end{split} \]
since for any candidate $\psi $ for the first supremum, $\psi \lor (\varphi -\sup _ X \varphi )$ is larger than $\psi $, and is a candidate for the second supremum thanks to 16.8.13. So the first equality is a direct consequence of proposition 16.8.7 and theorem 16.3.15.
Next we prove the second equality. We only need to show that for any $\psi \in \mathrm{PSH}(X,\theta )$ with $\psi \leq 0$ and $\psi \preceq _ P \varphi $, we have $\psi \leq P_{\theta }[\varphi ]$.
By 16.8.13 and example 16.8.6, we know that $P_{\theta }[\varphi ]\lor \psi \sim _ P \varphi $ and $P_{\theta }[\varphi ]\lor \psi \leq 0$. It follows from the first equality that $\psi \leq P_{\theta }[\varphi ]$.
$\square$
Similarly, we have a new characterization of the $\mathcal{I}$-envelope.
Corollary 16.8.16.69E7C4B20FAE Let $\theta $ be a closed smooth real $(1,1)$-form on $X$ representing a pseudo-effective cohomology class. Assume that $\varphi \in \mathrm{PSH}(X,\theta )$. Then
\[ P_{\theta }[\varphi ]_{\mathcal{I}}=\sup \left\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0, \psi \preceq _{\mathcal{I}} \varphi \right\} . \]
Proof.
It suffices to show that for any $\psi \in \mathrm{PSH}(X,\theta )$ with $\psi \leq 0$ and $\psi \preceq _{\mathcal{I}}\varphi $, we have $\psi \leq P_{\theta }[\varphi ]_{\mathcal{I}}$. By 16.8.13 and proposition 16.6.11, we know that $P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi \sim _{\mathcal{I}}\varphi $. Therefore, $P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi $ is a candidate in the definition of $P_{\theta }[\varphi ]_{\mathcal{I}}$ (cf. (16.6.8)), so
\[ \psi \leq P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi \leq P_{\theta }[\varphi ]_{\mathcal{I}}. \]
We conclude the proof.
$\square$
Proposition 16.8.17.2E8F613E7CBE Let $\theta $ be a closed real smooth $(1,1)$-form on $X$ and $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then the following are equivalent:
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$\varphi \preceq _{\mathcal{I}}\psi $;
6B5F8E56CB73
$P_{\theta }[\varphi ]_{\mathcal{I}}\leq P_{\theta }[\psi ]_{\mathcal{I}}$.
Proof.
1 $\implies $ 2. This follows immediately from corollary 16.8.16.
2 $\implies $ 1. It follows from proposition 16.6.11 that
\[ \varphi \sim _{\mathcal{I}} P_{\theta }[\varphi ]_{\mathcal{I}}\leq P_{\theta }[\psi ]_{\mathcal{I}} \sim _{\mathcal{I}} \psi . \]
We conclude 1.
$\square$
Example 16.8.18.C7A2AD00BFF3 Let us continue our example example 16.3.22, where $X=\mathbb {P}^1$, $\omega $ is the Fubini–Study metric and $\varphi \in \mathrm{PSH}(X,\omega )$ has log-log singularity at $0$. We have shown that $P_{\omega }[\varphi ]=0$ in (16.3.23), so $\varphi \sim _{P} 0$ and hence $\varphi \sim _{\mathcal{I}} 0$. In particular, $P$-equivalence is not equivalent to the equivalence of singularity types.
On the other hand, consider a potential $\psi \in \mathrm{PSH}(X,\omega )$ with log singularity at $0$, as in example 5.20.12. We know that $\nu (\psi ,0)=1$ from the explicit expression (5.20.13). So $\psi \not\sim _{\mathcal{I}} 0$ and hence $\psi \not\sim _{P} 0$.
Moreover, $\psi \preceq _{P} \varphi $ and hence $\psi \preceq _{\mathcal{I}}\varphi $.
We give an example showing that $P$-equivalence is not equivalent to $\mathcal{I}$-equivalence.
Example 16.8.19.325A49CBCF74 Let $X=\mathbb {P}^1$ and $\omega $ be the Fubini–Study metric. Let $K\subseteq \mathbb {P}^1$ be a polar Cantor set carrying an atom free probability measure $\mu $ supported on $K$ (see [Page 31, Car83]). Write $\mu =\omega +\mathrm{dd}^{\mathrm{c}}\varphi $ for some $\omega $-subharmonic function $\varphi $. Since $\mu $ is atom free, we know that all Lelong numbers of $\varphi $ are $0$. On the other hand, $\varphi $ has $0$ non-pluripolar mass since $K$ is pluripolar.
Then observe that $\varphi \sim _{\mathcal{I}} 0$ while $\varphi \not\sim _ P 0$.
For later use, we give the following definition.
Definition 16.8.20.33F50367BB68 Let $L$ be a pseudo-effective line bundle on $X$. An elementary metric on $L$ is a psh metric $h$ on $L$ such that there is a generalized Fubini–Study metric $h'$ on $L$ such that
\[ \mathrm{dd}^{\mathrm{c}}h\sim _{P} \mathrm{dd}^{\mathrm{c}}h'. \]
The set of elementary metrics on $L$ is denoted by $\mathrm{Ele}(L)$.
We also say $\mathrm{dd}^{\mathrm{c}}h$ is elementary. If we have fixed a Hermitian metric $h_0$ on $L$, and if we represent $h$ as $h_0\exp (-\varphi )$, we also say the quasi-psh function $\varphi $ is elementary.
Recall that the generalized Fubini–Study metrics are defined in definition
5.20.15.
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