Now we state a more natural version of the monotonicity theorem theorem 13.13.11.
Proposition 16.8.22 (Monotonicity theorem).82DCD8E3B1BA Let $\theta _1,\ldots ,\theta _ n$ be closed real smooth $(1,1)$-forms on $X$. Let $\varphi _ i,\psi _ i\in \mathrm{PSH}(X,\theta _ i)$ for $i=1,\ldots ,n$. Assume that $\varphi _ i\preceq _ P \psi _ i$ for each $i$. Then
\[ \int _ X \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}\leq \int _ X \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{n,\psi _ n}. \]
In other words, the non-pluripolar mass is a decreasing quantity with respect to the $P$-singularity types of the potentials.
Proof.
Fix a Kähler form $\omega $ on $X$. For each $i=1,\ldots ,n$, since $\varphi _ i\preceq _ P\psi _ i$, we have
\[ P_{\theta _ i+\epsilon \omega }[\varphi _ i]\leq P_{\theta _ i+\epsilon \omega }[\psi _ i] \]
for all $\epsilon >0$. Therefore, by proposition 16.3.10 and our previous monotonicity theorem theorem 13.13.11, we have
\[ \int _ X (\theta _1+\epsilon \omega )_{\varphi _1}\wedge \cdots \wedge (\theta _ n+\epsilon \omega )_{\varphi _ n}\leq \int _ X (\theta _1+\epsilon \omega )_{\psi _1}\wedge \cdots \wedge (\theta _ n+\epsilon \omega )_{\psi _ n}. \]
Letting $\epsilon \to 0+$ and applying the multi-linearity of the non-pluripolar products proposition 13.11.4 6, we find the desired inequality.
$\square$
The $P$-preorder allows us to derive a natural generalization of 13.13.30.
Theorem 16.8.23.68E00C8C52A1 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$, $\varphi \preceq _ P \psi $. Then for any
3BF94E656512
16.8.24
\begin{equation} a\in \left( 1, \left(\frac{\int _ X \theta _{\psi }^ n}{\int _ X \theta _{\psi }^ n-\int _ X \theta _{\varphi }^ n}\right)^{1/n} \right), \end{equation}
the function
\[ \eta \coloneqq P_{\theta }\Bigl (a\varphi +(1-a)\psi \Bigr ) \]
lies in $\mathrm{PSH}(X,\theta )_{>0}$, satisfies $\eta \preceq _ P \varphi $ and
9690BB77C920
16.8.25
\begin{equation} a^{-1}\eta +\left(1-a^{-1}\right)\psi \leq \varphi . \end{equation}
If in addition, $\varphi \sim _ P \psi $, then $\eta \sim _ P \varphi $ as well.
In (16.8.24), the fraction is understood as $\infty $ if either $\int _ X \theta _{\psi }^ n=\int _ X \theta _{\varphi }^ n$ or $n=0$, exactly as in 13.13.30.
Proof.
By our assumption and (16.3.8), we have $\varphi \preceq P_{\theta }[\varphi ]\leq P_{\theta }[\psi ]$. Due to proposition 16.3.10, the interval (16.8.24) remains unchanged after replacing $\psi $ by $P_{\theta }[\psi ]$. Fix $a$ in this interval. So by 13.13.30 and the discussions afterwards,
\[ \eta '\coloneqq P_{\theta }\Bigl (a\varphi +(1-a)P_{\theta }[\psi ]\Bigr )\in \mathrm{PSH}(X,\theta )_{>0}. \]
We first check that $\eta $ is a genuine $\theta $-psh function. Choose a closed non-pluripolar set $K\subseteq X$ on which $\varphi $ and $\psi $ are bounded. Then the obstacle $a\varphi +(1-a)\psi $ is bounded above on $K$. Every candidate in the definition of $\eta $ is therefore uniformly bounded above on $X$ by 5.17.4, so $\eta \not\equiv \infty $ and hence $\eta \in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $. On the other hand, since $\psi \preceq P_{\theta }[\psi ]$, we can find $C>0$ such that $\psi \leq P_{\theta }[\psi ]+C$. Then
\[ \eta '-(a-1)C\leq a\varphi +(1-a)\psi \]
quasi-everywhere, hence $\eta \geq \eta '-(a-1)C$. Thus $\eta '\preceq \eta $, and the monotonicity theorem theorem 13.13.11 gives $\eta \in \mathrm{PSH}(X,\theta )_{>0}$.
By (13.13.20),
\[ \eta \leq a\varphi +(1-a)\psi \]
quasi-everywhere. Equivalently, (16.8.25) holds quasi-everywhere. Since both sides of (16.8.25) are $\theta $-psh, proposition 5.14.6 upgrades the inequality to an everywhere inequality.
It remains to prove the assertion about the $P$-order. Using proposition 16.4.2, we get
\[ a^{-1} P_{\theta }[\eta ]+\left(1-a^{-1}\right)P_{\theta }[\psi ] \leq P_{\theta }[\varphi ]. \]
By the assumption $\varphi \preceq _ P\psi $, we also have $P_{\theta }[\varphi ]\leq P_{\theta }[\psi ]$. Therefore,
\[ a^{-1} P_{\theta }[\eta ]\leq P_{\theta }[\varphi ]-\left(1-a^{-1}\right)P_{\theta }[\psi ]\leq a^{-1} P_{\theta }[\varphi ] \]
quasi-everywhere and hence everywhere by proposition 5.14.6. It follows that $P_{\theta }[\eta ]\leq P_{\theta }[\varphi ]$, namely $\eta \preceq _ P\varphi $.
Finally assume that $\varphi \sim _ P\psi $. Then $\int _ X\theta _{P_{\theta }[\psi ]}^ n=\int _ X\theta _{\varphi }^ n$ by proposition 16.3.10. Applying corollary 13.13.42 gives
\[ \int _ X\theta _{\eta '}^ n=\int _ X\theta _{\varphi }^ n. \]
Since $\eta '\preceq \eta $, the monotonicity theorem theorem 13.13.11 gives
\[ \int _ X\theta _{\eta }^ n\geq \int _ X\theta _{\varphi }^ n. \]
The reverse inequality follows from $\eta \preceq _ P\varphi $ and proposition 16.8.22. Hence $\eta \sim _ P\varphi $ by proposition 16.8.7.
$\square$
Next we show that the $P$ and $\mathcal{I}$-preorders are preserved by some natural operations.
Lemma 16.8.26.FB1663F2B6E1 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$. Given two quasi-plurisubharmonic functions $\varphi ,\psi $ on $X$, then the following are equivalent:
8ED2C9D6D32D
$\varphi \preceq _{P} \psi $;
754FFF454EEB
$\pi ^*\varphi \preceq _{P} \pi ^*\psi $.
The same holds with $\mathcal{I}$ in place of $P$.
Proof.
In the $P$-case, this follows from proposition 16.4.1, while in the $\mathcal{I}$-case, this follows from proposition 16.6.10.
$\square$
Proposition 16.8.27.C42706880DED Let $\varphi ,\psi ,\varphi ',\psi '\in \mathrm{QPSH}(X)$. Assume that
\[ \varphi \preceq _ P \psi ,\quad \varphi '\preceq _ P \psi '. \]
Then
\[ \varphi +\varphi '\preceq _ P \psi +\psi '. \]
The same holds with $\preceq _{\mathcal{I}}$ in place of $\preceq _ P$.
Proof.
Take a Kähler form $\omega $ on $X$ such that $\varphi ,\psi ,\varphi ',\psi '\in \mathrm{PSH}(X,\omega )_{>0}$. The statement for $\preceq _{\mathcal{I}}$ is a simple consequence of proposition 5.16.8. We only need to handle the case of $\preceq _{P}$.
Step 1. We first show that
\[ P_{\omega }[\varphi ]+P_{\omega }[\varphi ']\sim _ P \varphi +\varphi '. \]
In fact, we clearly have
\[ P_{\omega }[\varphi ]+P_{\omega }[\varphi ']\succeq \varphi +\varphi '. \]
So by proposition 16.8.7, it suffices to show that they have the same mass. We compute
\[ \begin{aligned} & \int _ X \Bigl (2\omega +\mathrm{dd}^{\mathrm{c}}P_{\omega }[\varphi ]+\mathrm{dd}^{\mathrm{c}}P_{\omega }[\varphi ']\Bigr )^ n\\ =& \sum _{j=0}^ n \binom {n}{j} \int _ X \Bigl (\omega +\mathrm{dd}^{\mathrm{c}}P_{\omega }[\varphi ]\Bigr )^ j\wedge \Bigl (\omega +\mathrm{dd}^{\mathrm{c}}P_{\omega }[\varphi ']\Bigr )^{n-j}\\ =& \sum _{j=0}^ n \binom {n}{j} \int _ X \omega _{\varphi }^ j\wedge \omega _{\varphi '}^{n-j}\\ =& \int _ X \left(2\omega +\mathrm{dd}^{\mathrm{c}}\varphi +\mathrm{dd}^{\mathrm{c}}\varphi '\right)^ n, \end{aligned} \]
where we applied proposition 16.3.10 to the mixed products on the third line.
Step 2. By Step 1, we may assume that $\varphi ,\psi ,\varphi ',\psi '$ are all model potentials, after replacing them by their $P_{\omega }$-envelopes. So $\varphi \leq \psi $ and $\varphi '\leq \psi '$. Our assertion follows.
$\square$
Proposition 16.8.28.1C6DED516857 Let $(\varphi _ i)_{i\in I}$, $(\psi _ i)_{i\in I}$ be uniformly bounded from above non-empty families in $\mathrm{QPSH}(X)$. Assume that there exists a closed smooth real $(1,1)$-form $\theta $ such that $\varphi _ i,\psi _ i\in \mathrm{PSH}(X,\theta )$ and $\varphi _ i\preceq _ P \psi _ i$ for all $i\in I$. Then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i\preceq _ P \operatorname *{sup}_{i\in I}{}\! {}^{*}\psi _ i. \]
The same holds with $\preceq _{\mathcal{I}}$ in place of $\preceq _ P$.
Proof.
By adding a Kähler form to $\theta $, we may assume that $\varphi _ i,\psi _ i\in \mathrm{PSH}(X,\theta )_{>0}$ for all $i\in I$. The statement for $\preceq _{\mathcal{I}}$ is a simple consequence of corollary 5.16.9, we only have to consider the statement for $\preceq _ P$.
Step 1. We first handle the case where $I$ is a directed set and $(\varphi _ i)_{i\in I}$ and $(\psi _ i)_{i\in I}$ are increasing nets.
In this case, our assertion follows simply from proposition 16.4.9.
Step 2. We handle the case where $I$ is finite. We may assume that $I=\{ 0,1\} $. It suffices to show that
\[ P_{\theta }[\varphi _0]\lor P_{\theta }[\varphi _1]\sim _ P \varphi _0\lor \varphi _1, \]
which follows from 16.8.11.
Step 3. The general case can be reduced to the two cases handled in Step 1 and Step 2. More precisely, by applying proposition 5.14.3 to the two families and taking the union of the resulting countable subsets, we can find a countable subset $J\subseteq I$ such that
\[ \operatorname *{sup}_{j\in J}{}\! {}^{*} \varphi _ j=\operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i,\quad \operatorname *{sup}_{j\in J}{}\! {}^{*} \psi _ j=\operatorname *{sup}_{i\in I}{}\! {}^{*} \psi _ i. \]
We may replace $I$ by $J$ and assume that $I$ is countable. If $I$ is finite, Step 2 applies directly. Otherwise, write $I=\mathbb {Z}_{>0}$ and set
\[ \tilde{\varphi }_ m=\varphi _1\lor \cdots \lor \varphi _ m,\quad \tilde{\psi }_ m=\psi _1\lor \cdots \lor \psi _ m. \]
By Step 2, $\tilde{\varphi }_ m\preceq _ P\tilde{\psi }_ m$ for every $m$. The two new sequences are increasing and have the same upper envelopes as the original families, so Step 1 applies.
$\square$
We now refine 16.3.32 using the $P$-equivalence relation.
Proposition 16.8.29.8C181599D520 Let $\varphi ,\psi ,\varphi ',\psi '\in \mathrm{PSH}(X,\theta )$ for some closed smooth real $(1,1)$-form $\theta $ on $X$. Assume that
\[ \varphi \sim _ P \varphi ',\quad \psi \sim _ P\psi ',\quad \varphi '\land \psi '\in \mathrm{PSH}(X,\theta )_{>0}. \]
Then
\[ \varphi \land \psi \in \mathrm{PSH}(X,\theta )_{>0},\quad \varphi \land \psi \sim _ P \varphi '\land \psi '. \]
Proof.
We first observe that $\varphi ,\varphi ',\psi ,\psi '\in \mathrm{PSH}(X,\theta )_{>0}$ by assumption. Let
\[ \phi \coloneqq P_{\theta }[\varphi ]=P_{\theta }[\varphi '],\quad \gamma =P_{\theta }[\psi ]=P_{\theta }[\psi ']. \]
Since $\varphi '\preceq \phi $ and $\psi '\preceq \gamma $, after adding a common constant to $\varphi '$ and $\psi '$, we may assume that $\varphi '\leq \phi $ and $\psi '\leq \gamma $. Then $\varphi '\land \psi '\leq \phi \land \gamma $, so the assumption $\varphi '\land \psi '\in \mathrm{PSH}(X,\theta )_{>0}$ implies $\phi \land \gamma \in \mathrm{PSH}(X,\theta )_{>0}$ by monotonicity. It follows from 16.3.32 that
\[ \int _ X \theta _{\varphi '\land \psi '}^ n=\int _ X \theta _{\phi \land \gamma }^ n. \]
Next, we apply 16.3.32 again to conclude that $\varphi \land \psi \in \mathrm{PSH}(X,\theta )$ and
\[ \int _ X \theta _{\varphi \land \psi }^ n=\int _ X \theta _{\phi \land \gamma }^ n=\int _ X \theta _{\varphi '\land \psi '}^ n>0. \]
Since $\varphi \land \psi \preceq \phi \land \gamma $ and the two potentials have the same positive mass, theorem 16.3.15 gives $P_{\theta }[\varphi \land \psi ]=\phi \land \gamma $. The same argument gives $P_{\theta }[\varphi '\land \psi ']=\phi \land \gamma $, hence $\varphi \land \psi \sim _ P\varphi '\land \psi '$.
$\square$
We can formulate a generalization of proposition 16.5.54 using the $P$-partial order.
Corollary 16.8.30.21CC5153F56F Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Assume that $\psi \preceq _ P \varphi $. Then for any $C>0$, $\psi \land (\varphi +C)\in \mathrm{PSH}(X,\theta )_{>0}$ and
42D232E5CD85
16.8.31
\begin{equation} P_{\theta }[\varphi ](\psi )=\operatorname *{sup}_{C>0}{}\! {}^{*} \psi \land (\varphi +C)=\psi . \end{equation}
Proof.
Since $\psi \preceq _ P\varphi $, by proposition 16.8.29, we get
\[ \psi \land (\varphi +C)\in \mathrm{PSH}(X,\theta )_{>0} \]
and $\psi \land (\varphi +C)\sim _ P\psi $ for every $C>0$.
Let
\[ \eta \coloneqq \operatorname *{sup}_{C>0}{}\! {}^{*}\psi \land (\varphi +C). \]
Then $\eta \sim _ P \psi $. By corollary 16.3.14, we have
\[ \theta _\eta ^ n\leq \mathds {1}_{\{ \eta =\psi \} }\, \theta _\psi ^ n. \]
The domination principle theorem 13.13.43 gives $\eta \geq \psi $. The reverse inequality is trivial, so $\eta =\psi $.
$\square$
The $P$-equivalence relation characterizes when a subgeodesic exists. See definition 17.10.2 for the notion of subgeodesics.
Theorem 16.8.32.1CA975F04423 Let $\varphi _0,\varphi _1\in \mathrm{PSH}(X,\theta )_{>0}$. Then the following are equivalent:
8E345DA2D98A
There is a subgeodesic from $\varphi _0$ to $\varphi _1$;
5498C9F0BF2D
$\varphi _0\sim _ P\varphi _1$.
Proof.
2 $\implies $ 1. This follows from proposition 17.10.24.
1 $\implies $ 2. Let $(\varphi _ t)_{t\in (0,1)}$ be a subgeodesic from $\varphi _0$ to $\varphi _1$.
Step 1. We assume that $\varphi _0\geq \varphi _1$.
We first replace the given subgeodesic by the geodesic from $\varphi _0$ to $\varphi _1$. Note that the geodesic is necessarily a subgeodesic as well by 17.10.22.
Observe that $t\mapsto \varphi _ t$ is decreasing due to 17.10.28.
Let $\varphi _ t=\varphi _1$ for all $t>1$. Then by the gluing lemma 17.10.16, we find that $(\varphi _ t)_{t\geq 0}$ is a subgeodesic ray.
Next, we consider the Legendre transform
\[ \Gamma _{\tau }\coloneqq \inf _{t\geq 0}(\varphi _ t-t\tau ),\quad \tau \in \mathbb {R}. \]
It follows from Kiselman's principle proposition 5.14.15 that $\Gamma _{\tau }\in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $. Note that for $\tau >0$, we clearly have $\Gamma _{\tau }\equiv -\infty $. On the other hand, for $\tau \leq 0$,
\[ \Gamma _{\tau }=\inf _{t\in [0,1]}(\varphi _ t-t\tau )\in \mathrm{PSH}(X,\theta ). \]
By Legendre inversion (cf. corollary 21.11.26), for $t>0$,
\[ \varphi _ t=\sup _{\tau <0} \left(\Gamma _{\tau }+t\tau \right). \]
Fix a Kähler form $\omega $ on $X$. It follows from proposition 16.8.28 that for each $t>0$,
\[ \varphi _ t\sim _ P \operatorname *{sup}_{\tau <0}{}\! {}^{*} P_{\theta +\omega }[\Gamma _{\tau }]. \]
The right-hand side is independent of $t$. Here by adding $\omega $, we no longer have to worry about the possibility where $\Gamma _{\tau }$ has vanishing mass.
Write
\[ \varphi _0=\operatorname *{sup}_{t\in (0,1)}{}\! {}^{*}\varphi _ t. \]
By proposition 16.8.28 again, we find that
\[ \varphi _0\sim _ P \operatorname *{sup}_{\tau <0}{}\! {}^{*} P_{\theta +\omega }[\Gamma _{\tau }] \]
as well. So $\varphi _0\sim _ P \varphi _1$.
Step 2. We prove the general case.
Observe that $(\varphi _ t\lor \varphi _1)_{t\in (0,1)}$ is a subgeodesic from $\varphi _0\lor \varphi _1$ to $\varphi _1$. By Step 1,
\[ \varphi _0\lor \varphi _1\sim _ P \varphi _1. \]
Hence $\varphi _0\preceq _ P \varphi _1$ by 16.8.13. The converse is proved similarly. Hence 2 follows.
$\square$
Restating the 1 $\implies $ 2 direction in the theorem in terms of the complexification, we find the following interesting result.
Let $S=\{ z\in \mathbb {C}:0< \operatorname{Re}z <1\} $. We write $p_1\colon X\times S\rightarrow X$ for the natural projection.
Corollary 16.8.33.4EE47D80CC25 Let $\Phi \in \mathrm{PSH}(X\times S,p_1^*\theta )$. Assume that for any $c\in \mathbb {R}$, $x\in X$ and $z\in S$, we have
\[ \Phi (x,z)=\Phi (x,z+\mathrm{i}c). \]
Then $\int _ X (\theta +\mathrm{dd}^{\mathrm{c}}\Phi _ z)^ n$ is independent of $z\in S$, where $\Phi _ z \in \mathrm{PSH}(X,\theta )$ is given by $\Phi _ z(x)=\Phi (x,z)$.
Proof.
Fix a Kähler form $\omega $ on $X$. We may regard $\Phi $ as an element of $\mathrm{PSH}(X\times S,p_1^*(\theta +\omega ))$. For any $0<a<b<1$, the restriction $(\Phi _ t)_{t\in [a,b]}$ is then a subgeodesic from $\Phi _ a$ to $\Phi _ b$ in the class $\theta +\omega $. The slices have positive mass in this class, so theorem 16.8.32 gives $\Phi _ a\sim _ P\Phi _ b$. By proposition 16.8.22, this directly implies
\[ \int _ X \theta _{\Phi _ a}^ n=\int _ X \theta _{\Phi _ b}^ n. \]
Since $\Phi $ is invariant under imaginary translations, $\Phi _ z$ depends only on $\operatorname{Re}z$, so the mass is independent of $z\in S$.
$\square$
Pome
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