PomeRemark 13.13.2.218A02F6B9E0 Suppose that $X$ is a connected complex manifold of dimension $0$, namely, $X$ is a single point. In this case, by definition, the non-pluripolar product $\theta _{\varphi }^ n$ is given by the current of integration at the unique point. So $\mathrm{PSH}(X,\theta )_{>0}=\mathrm{PSH}(X,\theta )\cong \mathbb {R}$ in this case and $\int _ X \theta _{\varphi }^ n=1$ for all $\varphi \in \mathrm{PSH}(X,\theta )$.
13.13 Properties of non-pluripolar products01C01D21DDD2
01C01D21DDD2
Let $X$ be a connected compact Kähler manifold of dimension $n$ and $\theta ,\theta _1,\ldots ,\theta _ n$ be closed real smooth $(1,1)$-forms on $X$.
We write
44C4E200865B
13.13.1
\begin{equation} \mathrm{PSH}(X,\theta )_{>0}=\left\{ \varphi \in \mathrm{PSH}(X,\theta ): \int _ X\theta _{\varphi }^ n>0 \right\} . \end{equation}
The non-pluripolar product $\theta _{\varphi }^ n$ is well-defined thanks to proposition 13.11.14.
Proposition 13.13.3.C0B519026258 Assume that $\mathrm{PSH}(X,\theta )_{>0}$ is non-empty. Then the cohomology class $\{ \theta \} $ is big.
Proposition 13.13.4.365687A6A087 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$ and $T_1,\ldots ,T_ n$ be closed positive $(1,1)$-currents on $X$. Then
\[ \int _ Y (\pi ^*T_1)\wedge \cdots \wedge (\pi ^*T_ n)=\int _ X T_1\wedge \cdots \wedge T_ n. \]
Similarly, if $S_1,\ldots ,S_ n$ are closed positive $(1,1)$-currents on $Y$, then
\[ \int _ Y S_1\wedge \cdots \wedge S_ n=\int _ X \left(\pi _*S_1\right)\wedge \cdots \wedge \left(\pi _*S_ n\right). \]
Proof. The assertions follow from proposition 13.11.4 1 on the Zariski open set where $\pi $ is an isomorphism. The exceptional locus is pluripolar, and the relevant non-pluripolar products put no mass on pluripolar sets by proposition 13.11.4 4. $\square$
Theorem 13.13.5.7F7BE8292E76 Let $\varphi _0,\varphi _1\in \mathrm{PSH}(X,\theta )$. Then the map
\[ [0,1] \ni t\mapsto \log \int _ X \theta _{t\varphi _1+(1-t)\varphi _0}^ n \]
is concave.
More generally, given closed positive $(1,1)$-currents $T_1,\ldots ,T_ n$ on $X$, we have
\[ \int _ X T_1\wedge \cdots \wedge T_ n\geq \prod _{i=1}^ n \left( \int _ X T_ i^ n \right)^{1/n}. \]
Remark 13.13.6.22841C35D44B Here and in the sequel, when we write expressions like $t\varphi +(1-t)\psi $ for $\varphi ,\psi \in \mathrm{QPSH}(X)$, we will follow the convention that when $t=0$, the value is $\psi $ and when $t=1$, the value is $\varphi $.
We shall write
6F52E700CD98
13.13.7
\begin{equation} V_{\theta }=\sup \left\{ \varphi \in \mathrm{PSH}(X,\theta ):\varphi \leq 0 \right\} . \end{equation}
Note that the upper semicontinuous regularization $V_{\theta }^*$ of $V_{\theta }$ is itself a candidate of (13.13.7) if $\mathrm{PSH}(X,\theta )\neq \varnothing $ as follows from proposition 5.14.2, therefore, $V_{\theta }=V_{\theta }^*\in \mathrm{PSH}(X,\theta )$ if $\mathrm{PSH}(X,\theta )\neq \varnothing $.
The function $V_{\theta }$ should be regarded as a canonical representative of the least singular potentials in $\mathrm{PSH}(X,\theta )$. We recall the following result:
Lemma 13.13.8.F77E450F00FD Suppose that $\{ \theta \} $ is big. The potential $V_{\theta }$ is locally bounded on $X\setminus \operatorname{nK}(\{ \theta \} )$.
Proof. By definition of $\operatorname{nK}(\{ \theta \} )$, it suffices to show that if $\varphi \in \mathrm{PSH}(X,\theta )$ has analytic singularities and $\theta _{\varphi }$ is a Kähler current, then $V_{\theta }$ is locally bounded outside $\{ \varphi =-\infty \} $. After replacing $\varphi $ by $\varphi -\sup _ X\varphi $, the function $\varphi $ is a candidate in (13.13.7); hence $V_{\theta }\geq \varphi $ and $V_{\theta }\leq 0$. Since $\varphi $ is locally bounded outside its pole set, so is $V_{\theta }$. $\square$
Theorem 13.13.9 (Semicontinuity theorem).B3DC3180FF35 Let $\varphi _ j, \varphi _ j^ k\in \mathrm{PSH}(X,\theta _ j)$ ($k\in \mathbb {N}$, $j=1,\ldots ,n$). Let $\chi _ k,\chi $ ($k\in \mathbb {N}$) be non-negative uniformly bounded quasi-continuous functions on $X$ such that $(\chi _ k)_ k$ converges to $\chi $ in capacity. Assume that for any $j=1,\ldots ,n$, as $k\to \infty $, either $\varphi _ j^ k$ decreases to $\varphi _ j\in \mathrm{PSH}(X,\theta _ j)$ or increases to $\varphi _ j\in \mathrm{PSH}(X,\theta _ j)$ almost everywhere. Then
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13.13.10
\begin{equation} \varliminf _{k\to \infty }\int _ X\chi _ k \, \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k} \geq \int _ X \chi \, \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \end{equation}
If in addition,
\[ \lim _{k\to \infty }\int _ X\theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k} =\int _ X \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}, \]
then
\[ \chi _ k \, \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k}\rightharpoonup \chi \, \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n} \]
weakly as measures3. In particular, the limit in (13.13.10) exists and equality holds in (13.13.10).
Theorem 13.13.11 (Monotonicity theorem).0E5F6133B048 Let $\varphi _ j,\psi _ j\in \mathrm{PSH}(X,\theta _ j)$ for $j=1,\ldots ,n$. Assume that $\varphi _ j \succeq \psi _ j$4 for every $j$, then
\[ \int _{X} \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n} \geq \int _{X} \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{n,\psi _ n}. \]
In particular, if $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$ with $\varphi \succeq \psi $, then
\[ \int _ X \theta _{\varphi }^ n\geq \int _ X \theta _{\psi }^ n. \]
In other words, the non-pluripolar mass increases as the potentials become less singular.
By this theorem, the non-pluripolar mass $\int _ X \theta _{\varphi }^ n$ can be used as a rough measure of the singularities of $\varphi \in \mathrm{PSH}(X,\theta )$. In section 16.2, we shall refine this measure by defining the notion of $P$-envelope.
As a corollary, we obtain thatCorollary 13.13.12.37623491DEDC Fix a directed set $I$. For each $j=1,\ldots ,n$, take an increasing net $(\varphi _ j^ i)_{i\in I}$ in $\mathrm{PSH}(X,\theta _ j)$, uniformly bounded from above. Set
\[ \varphi _ j\coloneqq \operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ j^ i,\quad j=1,\ldots ,n. \]
Then
18A95DBDA47A
13.13.13
\begin{equation} \lim _{i\in I}\int _ X \theta _{1,\varphi _1^ i}\wedge \cdots \wedge \theta _{n,\varphi _ n^ i} = \int _ X \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \end{equation}
In particular, if the nets are indeed sequences, we have
\[ \theta _{1,\varphi _1^ i}\wedge \cdots \wedge \theta _{n,\varphi _ n^ i} \rightharpoonup \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \]
Proof. The final assertion follows from (13.13.13) together with theorem 13.13.9. It remains to establish (13.13.13).
We may assume that $I$ is infinite as there is nothing to prove otherwise. Thanks to theorem 13.13.11, we already know the $\leq $ inequality in (13.13.13). We prove the reverse inequality. When $I\cong \mathbb {Z}_{>0}$ as directed sets, the reverse inequality follows from theorem 13.13.9. In general, by Choquet's lemma proposition 5.14.3, we can find a countable infinite subset $R\subseteq I$ such that
\[ \operatorname *{sup}_{r\in R}{}\! {}^{*}\varphi ^ r_ j=\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi ^ i_ j \]
for all $j=1,\ldots ,n$. We fix a bijection $R\cong \mathbb {Z}_{>0}$. For any $j=1,\ldots ,n$, we will then denote elements $\varphi _ j^ r$ ($r\in R$) by $\varphi _ j^1,\varphi _ j^2,\dots $. We shall write
\[ \psi _ j^ a=\varphi _ j^1\lor \cdots \lor \varphi _ j^ a \]
for each $a\in \mathbb {Z}_{>0}$.
For every $a$ we may choose $i_ a\in I$ dominating the finitely many indices corresponding to $\varphi _ j^1,\ldots ,\varphi _ j^ a$ for all $j$. It then follows from the monotonicity theorem that
\[ \lim _{i\in I}\int _ X \theta _{1,\varphi _1^ i}\wedge \cdots \wedge \theta _{n,\varphi _ n^ i}\geq \lim _{a\to \infty }\int _ X \theta _{1,\psi _1^ a}\wedge \cdots \wedge \theta _{n,\psi _ n^ a}. \]
From the special case mentioned above, we know that the right-hand side is exactly the right-hand side of (13.13.13), so we conclude. $\square$
Lemma 13.13.14.F56651914DE1 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then
37A13FA36078
13.13.15
\begin{equation} \theta _{\varphi \lor \psi }^ n\geq \mathds {1}_{\{ \varphi \geq \psi \} }\, \theta _{\varphi }^ n+\mathds {1}_{\{ \varphi <\psi \} }\, \theta _{\psi }^ n. \end{equation}
In particular, if $\varphi \leq \psi $, then
\[ \mathds {1}_{\{ \varphi =\psi \} }\, \theta _{\varphi }^ n\leq \mathds {1}_{\{ \varphi =\psi \} }\, \theta _{\psi }^ n. \]
Proof. For each $k\in \mathbb {N}$, we set
\[ \psi _ k=\psi \lor (V_{\theta }-k),\quad \varphi _ k=\varphi \lor (V_{\theta }-k). \]
For each $t>0$ and each $k\in \mathbb {N}$, we have
\[ \begin{split} \theta _{\psi _ k\lor (\varphi _ k+t)}^ n\geq & \mathds {1}_{\{ \psi _ k>\varphi _ k+t\} }\, \theta _{\psi _ k\lor (\varphi _ k+t)}^ n+\mathds {1}_{\{ \psi _ k<\varphi _ k+t\} }\, \theta _{\psi _ k\lor (\varphi _ k+t)}^ n\\ =& \mathds {1}_{\{ \psi _ k>\varphi _ k+t\} }\, \theta _{\psi _ k}^ n+\mathds {1}_{\{ \psi _ k<\varphi _ k+t\} }\, \theta _{\varphi _ k}^ n, \end{split} \]
where the equality follows from the plurifine locality of non-pluripolar products proposition 13.11.4 1. We observe that for any $t>0$, $\psi _ k\lor (\varphi _ k+t)\sim \psi _ k\lor \varphi _ k$. Therefore, the monotonicity theorem theorem 13.13.11 guarantees that
\[ \int _ X \theta _{\psi _ k\lor (\varphi _ k+t)}^ n=\int _ X\theta _{\psi _ k\lor \varphi _ k}^ n. \]
As $t\to 0+$,
\[ \theta _{\psi _ k\lor (\varphi _ k+t)}^ n\rightharpoonup \theta _{\psi _ k\lor \varphi _ k}^ n \]
by theorem 13.13.9.
Letting $t\to 0+$, we conclude that
\[ \theta _{\psi _ k\lor \varphi _ k}^ n\geq \mathds {1}_{\{ \psi _ k>\varphi _ k\} }\, \theta _{\psi _ k}^ n+\mathds {1}_{\{ \psi _ k\leq \varphi _ k\} }\, \theta _{\varphi _ k}^ n. \]
In particular, multiplying both sides by $\mathds {1}_{\{ \min \{ \varphi ,\psi \} >V_{\theta }-k\} }$ and applying proposition 13.11.4 1 again, we find that
\[ \begin{split} \mathds {1}_{\{ \min \{ \varphi ,\psi \} >V_{\theta }-k\} }\, \theta _{\psi \lor \varphi }^ n\geq & \mathds {1}_{\{ \min \{ \varphi ,\psi \} >V_{\theta }-k\} \cap \{ \psi \leq \varphi \} }\, \theta _{\varphi }^ n \\ & +\mathds {1}_{\{ \min \{ \varphi ,\psi \} >V_{\theta }-k\} \cap \{ \psi >\varphi \} }\, \theta _{\psi }^ n. \end{split} \]
Letting $k\to \infty $, we conclude (13.13.15). $\square$
Corollary 13.13.16.7CB7A3C51640 Let $(\varphi _ i)_{i\in I}$ be a net in $\mathrm{PSH}(X,\theta )$, and $\varphi \in \mathrm{PSH}(X,\theta )$ so that $\varphi _ i\to \varphi $ in $L^1$. Assume that $\varphi _ i\leq 0$ for all $i\in I$. Then
D30B25375476
13.13.17
\begin{equation} \varlimsup _{i\in I} \int _{\{ \varphi _ i=0\} }\theta _{\varphi _ i}^ n\leq \int _{\{ \varphi =0\} } \theta _{\varphi }^ n. \end{equation}
Proof. Observe that $\varphi \leq 0$ by proposition 5.17.3.
Step 1. We first assume that $I\cong \mathbb {Z}_{>0}$ as directed sets. Namely, $(\varphi _ i)_{i>0}$ is a sequence.
For each $k>0$, let
\[ \psi _ k=\operatorname *{sup}_{j\geq k}{}\! {}^{*} \varphi _ j. \]
Then $(\psi _ k)_{k}$ is a decreasing sequence in $\mathrm{PSH}(X,\theta )$ with limit $\varphi $, see corollary 5.14.8 for example.
Thanks to 13.13.14, for each $k>0$, we have
\[ \mathds {1}_{\{ \varphi _ k=\psi _ k\} }\, \theta _{\varphi _ k}^ n\leq \mathds {1}_{\{ \varphi _ k=\psi _ k\} }\, \theta _{\psi _ k}^ n. \]
Multiplying both sides with $\mathds {1}_{\{ \varphi _ k=0\} }$, we find that
\[ \mathds {1}_{\{ \varphi _ k=0\} }\, \theta _{\varphi _ k}^ n\leq \mathds {1}_{\{ \varphi _ k=0\} }\, \theta _{\psi _ k}^ n. \]
Take $b,C>0$. For each $k>0$, we have
\[ \begin{split} \int _{\{ \varphi _ k=0\} } \theta _{\varphi _ k}^ n\leq \int _{\{ \varphi _ k=0\} } \theta _{\psi _ k}^ n\leq \int _{\{ \psi _ k=0\} } \theta _{\psi _ k}^ n\\ =\int _{\{ \psi _ k=0\} } \theta _{\psi _ k\lor (V_{\theta }-C)}^ n\leq \int _ X \mathrm{e}^{b\psi _ k}\, \theta _{\psi _ k\lor (V_{\theta }-C)}^ n, \end{split} \]
where the equality part follows from proposition 13.11.4 1.
Letting $k\to \infty $ with the help of theorem 13.13.9, we conclude that
\[ \varlimsup _{k\to \infty }\int _{\{ \varphi _ k=0\} } \theta _{\varphi _ k}^ n\leq \int _ X \mathrm{e}^{b\varphi }\, \theta _{\varphi \lor (V_{\theta }-C)}^ n. \]
Letting $b\to \infty $ and then $C\to \infty $, we conclude (13.13.17).
Step 2. We now handle the general net case.
Fix a volume form on $X$, so that we can talk about the $L^1$-norm on $X$ in the sequel.
Suppose that (13.13.17) fails. Then there is $\delta >0$ such that, for every $i_0\in I$, one can find $i\geq i_0$ satisfying
\[ \int _{\{ \varphi _ i=0\} }\theta _{\varphi _ i}^ n>\int _{\{ \varphi =0\} }\theta _{\varphi }^ n+\delta . \]
Since $\varphi _ i\to \varphi $ in $L^1$, for every $k>0$ there is $a_ k\in I$ such that
\[ \| \varphi _ i-\varphi \| _{L^1}<k^{-1},\quad i\geq a_ k. \]
We can then recursively choose an increasing sequence $(i_ k)_{k>0}$ in $I$ with $i_ k\geq a_ k$ and
\[ \int _{\{ \varphi _{i_ k}=0\} }\theta _{\varphi _{i_ k}}^ n>\int _{\{ \varphi =0\} }\theta _{\varphi }^ n+\delta . \]
This sequence converges to $\varphi $ in $L^1$, contradicting Step 1. $\square$
Definition 13.13.18.434FC63292E2 Given a function $f\colon X\rightarrow [-\infty ,\infty ]$, we define $P_{\theta }(f)$ as follows:
B0CB5D62EE36
13.13.19
\begin{equation} P_{\theta }(f)\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\left\{ \varphi \in \mathrm{PSH}(X,\theta ): \varphi \leq f\textup{ quasi-everywhere} \right\} {}^{\href{#sgpt-a0000013831}{5}}. \end{equation}
We also observe that
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13.13.20
\begin{equation} P_{\theta }(f)\leq f\quad \textup{quasi-everywhere}. \end{equation}
This is a consequence of proposition 5.14.3 and proposition 5.14.5.
Lemma 13.13.21.2904B26253E2 Fix a closed smooth real $(1,1)$-form $\theta $ on $X$ representing a big cohomology class. Let $f\colon X\rightarrow \mathbb {R}$ be a lower semicontinuous function. Assume that $P_{\theta }(f)\not\equiv \infty $. Then $P_{\theta }(f)$ is locally bounded outside $\operatorname{nK}(\{ \theta \} )$. Set $U\coloneqq \{ P_{\theta }(f)<f\} \setminus \operatorname{nK}(\{ \theta \} )$, then
0D2A1EE39CF8
13.13.22
\begin{equation} \Bigl (\theta |_ U+\mathrm{dd}^{\mathrm{c}}P_{\theta }(f)|_ U\Bigr )^ n=0. \end{equation}
Proof. First observe that $V_{\theta }+\inf _ X f$ is a candidate in the definition (13.13.19). So
\[ P_{\theta }(f)\geq V_{\theta }+\inf _ X f, \]
and in particular $P_{\theta }(f)\not\equiv -\infty $.
In particular, $P_{\theta }(f)\in \mathrm{PSH}(X,\theta )$. It is therefore bounded from above. By 13.13.8, it is locally bounded away from $\operatorname{nK}(\{ \theta \} )$.
Now fix $x\in U$, and $\epsilon >0$ so that
\[ P_{\theta }(f)(x)<f(x)-\epsilon . \]
Take holomorphic balls $B\Subset B'\Subset U$ containing $x$. After shrinking $B$, we can guarantee that
BB745DB26624
13.13.23
\begin{equation} \left.P_{\theta }(f)\right|_{B}<f(x)-\epsilon ,\quad f|_ B> f(x)-\epsilon . \end{equation}
Let $\varphi \in \mathrm{PSH}(X,\theta )$ be the balayage of $P_{\theta }(f)$ on $B$, as constructed in proposition 5.17.9. By the domination principle proposition 5.14.22, we find that
\[ \varphi |_{B}\leq f(x)-\epsilon <f|_{B}. \]
Since $\varphi =P_{\theta }(f)$ outside $B$, we conclude that
\[ \varphi \leq f\quad \textup{quasi-everywhere}. \]
From this it follows that $\varphi \leq P_{\theta }(f)$, and hence equality holds. But then (13.13.22) holds automatically on $B$ since $\varphi $ is constructed by balayage. Since $x$ and $B$ are arbitrary, we conclude (13.13.22). $\square$
Theorem 13.13.24.A60C61FA5696 Let $f\colon X\rightarrow [-\infty ,\infty ]$ be a quasi-continuous function.
Assume that $P_{\theta }(f)\not\equiv \pm \infty $. Then $P_{\theta }(f)\in \mathrm{PSH}(X,\theta )$ and
426FF7E9832C
13.13.25
\begin{equation} \int _{\{ P_{\theta }(f)<f\} } \theta _{P_{\theta }(f)}^ n=0. \end{equation}
Thanks to (13.13.20), we can rewrite (13.13.25) as
\[ \int _{\{ P_{\theta }(f)\neq f\} } \theta _{P_{\theta }(f)}^ n=0. \]
We shall say that $\theta _{P_{\theta }(f)}^ n$ is carried by the contact set $\{ P_{\theta }(f)=f\} $. 1
Proof. Step 1. We first reduce to the case where $f$ is bounded.
Step 1.1. Reduce to the case where $f\leq 0$.
Take $C\in \mathbb {R}$ so that $P_{\theta }(f)\leq C$. Then
\[ P_{\theta }(f)=P_{\theta }\Bigl ( \min \{ f,C\} \Bigr ). \]
Thus we can replace $f$ by $\min \{ f,C\} $. Replacing $f$ by $f-\sup _ X f$, we reduce easily to the case where $f\leq 0$.
Step 1.2. Reduce to the case where $f$ is bounded. Assume that in this case, the theorem is known. We prove (13.13.25) for $f\leq 0$.
For each $j>0$, we have
1F8BEEBD94D3
13.13.26
\begin{equation} \int _{\{ P_{\theta }(f_ j)<f_ j\} } \theta _{P_{\theta }(f_ j)}^ n=0, \end{equation}
where $f_ j=f\lor (-j)$.
Now fix $C>0$ and two open sets $G'\Subset G\Subset X\setminus \operatorname{nK}(\{ \theta \} )$. Fix a smooth function $\chi \colon X\rightarrow [0,1]$ so that $\chi |_{G'}\equiv 1$ and $\chi $ is supported in $G$.
Define
\[ U_ C\coloneqq G \cap \left\{ P_{\theta }(f)>V_{\theta }-C \right\} ,\quad U'_ C\coloneqq G' \cap \left\{ P_{\theta }(f)>V_{\theta }-C \right\} . \]
Then $U_ C$ is $\mathcal{F}$-open. It follows from proposition 13.11.4 1 that for any $j>0$,
\[ \mathds {1}_{U_ C}\, \theta _{P_{\theta }(f_ j)\lor (V_{\theta }-C)}^ n=\mathds {1}_{U_ C}\, \theta _{P_{\theta }(f_ j)}^ n. \]
In particular, thanks to (13.13.26),
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13.13.27
\begin{equation} \int _{U_ C} \left( 1-\mathrm{e}^{P_{\theta }(f_ j)-f_ j} \right)\, \theta _{P_{\theta }(f_ j)\lor (V_{\theta }-C)}^ n=0. \end{equation}
Note that the $P_{\theta }(f_ j)\lor (V_{\theta }-C)$'s for various $j$ are uniformly bounded from below on $U_ C$, thanks to 13.13.8.
Next we claim that $(P_{\theta }(f_ j))_ j$ is decreasing and converges to $P_{\theta }(f)$.
The sequence is decreasing; let us compute its limit. In fact, $P_{\theta }(f_ j)\leq f_ j$ quasi-everywhere. It follows that $\inf _ j P_{\theta }(f_ j)\leq f$ quasi-everywhere and hence
\[ \inf _{j>0} P_{\theta }(f_ j)\leq P_{\theta }(f). \]
The reverse inequality is trivial, and our assertion follows.
Since $P_{\theta }(f)\not\equiv -\infty $, we know that the set $\{ f=-\infty \} $ is pluripolar. It follows that $P_{\theta }(f_ j)-f_ j$ converges to the following function $g\colon X\rightarrow [-\infty ,\infty )$ in capacity:
\[ g(x)=\begin{cases} P_{\theta }(f)(x)-f(x),& \textup{if } f(x)\neq -\infty ;\\ -\infty ,& \textup{otherwise}. \end{cases} \]
Therefore,
\[ 1-\mathrm{e}^{P_{\theta }(f_ j)-f_ j}\xlongrightarrow {\mathrm{C}} 1-\mathrm{e}^{g}. \]
Next we claim that
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13.13.28
\begin{equation} \int _{U'_ C} \left( 1-\mathrm{e}^{g} \right)\, \theta _{P_{\theta }(f)}^ n=0. \end{equation}
Since $g\leq 0$ quasi-everywhere, the left-hand side of (13.13.28) is non-negative. It suffices to prove the $\leq $ direction.
We wish to let $j\to \infty $ directly in (13.13.27), but since $U_ C$ is not open in general, this cannot be done directly. In the sequel, we shall slightly enlarge $U_ C$ to get an open set and then take the limit.
Fix $\epsilon >0$. We can find an open subset $W\Subset X\setminus \operatorname{nK}(\{ \theta \} )$ so that
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13.13.29
\begin{equation} \operatorname{Cap}_{C^{-1}\theta }(W\setminus U_ C)<\epsilon . \end{equation}
For example, we can construct $W$ as follows. By example 13.12.5, we can find an open set $A\subseteq G$ so that $\operatorname{Cap}_{C^{-1}\theta }(A)<\epsilon $ and $P_{\theta }(f)-V_{\theta }$ is continuous on $G\setminus A$. Then $U_ C\cap (G\setminus A)$ is relatively open in $G\setminus A$, so we may choose an open set $W_0\subseteq G$ such that
\[ W_0\cap (G\setminus A)=U_ C\cap (G\setminus A). \]
Taking $W=W_0\cup A$, we get $U_ C\subseteq W$ and $W\setminus U_ C\subseteq A$, hence (13.13.29).
Thanks to (13.13.29) and (13.13.27), we have
\[ \int _ W \chi \left( 1-\mathrm{e}^{P_{\theta }(f_ j)-f_ j} \right)\, \theta _{P_{\theta }(f_ j)\lor (V_{\theta }-C)}^ n\leq C^ n\epsilon . \]
Letting $j\to \infty $ and applying theorem 13.12.8, we find that
\[ \int _{U'_ C} \left( 1-\mathrm{e}^{g} \right)\, \theta _{P_{\theta }(f)\lor (V_{\theta }-C)}^ n\leq \int _ W \chi \left( 1-\mathrm{e}^{g} \right)\, \theta _{P_{\theta }(f)\lor (V_{\theta }-C)}^ n\leq C^ n\epsilon . \]
Since $\epsilon >0$ is arbitrary, (13.13.28) follows.
Now letting $C\to \infty $ in (13.13.28), we find
\[ \int _{G'}\left( 1-\mathrm{e}^{g} \right)\, \theta _{P_{\theta }(f)}^ n=0. \]
Since $G'\Subset X\setminus \operatorname{nK}(\{ \theta \} )$ is arbitrary, and $\operatorname{nK}(\{ \theta \} )\subseteq X$ is a proper analytic subset, we finally conclude
\[ \int _ X \left( 1-\mathrm{e}^{g} \right)\, \theta _{P_{\theta }(f)}^ n=0. \]
In other words, $P_{\theta }(f)=f$ almost everywhere with respect to $\theta _{P_{\theta }(f)}^ n$. This proves (13.13.25).
Step 2. We now assume that $-C'\leq f\leq 0$ for some $C'>0$.
For each $j>0$, take an open subset $V_ j\subseteq X$ so that
F0D9C03CCEDA$\operatorname{Cap}_{C'^{-1}\theta }(V_ j)\leq 2^{-j-1}$, and
AF1E921609BC$f|_{X\setminus V_ j}$ is continuous.
Without loss of generality, we may assume that the $V_ j$'s are decreasing. Take a continuous function $f_ j\colon X\rightarrow [-C',0]$ extending $f|_{X\setminus V_ j}$. This is always possible thanks to Tietze's extension theorem. For each $j>0$, we let
\[ h_ j\coloneqq \sup _{k\geq j} f_ k. \]
Then $h_ j$ is lower semicontinuous and $h_ j$ agrees with $f$ outside $V_ j$. Set
\[ h=\inf _{j>0} h_ j. \]
Then the set $\{ h\neq f\} $ is contained in the intersection of the $V_ j$'s and hence is a pluripolar set, thanks to theorem 13.12.4. In particular, $P_{\theta }(f)=P_{\theta }(h)$.
Now we can apply 13.13.21 to conclude that
\[ \int _{X\setminus \operatorname{nK}(\{ \theta \} )} \left(1-\mathrm{e}^{P_{\theta }(h_ j)-h_ j} \right) \theta _{P_{\theta }(h_ j)}^ n=0 \]
for each $j>0$.
Fix two open subsets $G'\Subset G\Subset X\setminus \operatorname{nK}(\{ \theta \} )$. Note that $-C'\leq h_ j\leq 0$. In particular,
\[ V_{\theta }-C'\leq P_{\theta }(h_ j)\leq V_{\theta }. \]
Hence,
\[ \begin{aligned} & \int _ G \left| 1-\mathrm{e}^{P_{\theta }(h_ j)-f}\right|\, \theta _{P_{\theta }(h_ j)}^ n\\ =& \int _{G\cap V_ j} \left|1-\mathrm{e}^{P_{\theta }(h_ j)-f}\right|\, \theta _{P_{\theta }(h_ j)}^ n+\int _{G\setminus V_ j} \left|1-\mathrm{e}^{P_{\theta }(h_ j)-h_ j}\right|\, \theta _{P_{\theta }(h_ j)}^ n\\ =& \int _{G\cap V_ j} \left|1-\mathrm{e}^{P_{\theta }(h_ j)-f}\right|\, \theta _{P_{\theta }(h_ j)}^ n\\ \leq & \int _{G\cap V_ j} \left( 1+\mathrm{e}^{C'}\right)\, \theta _{P_{\theta }(h_ j)}^ n\\ \leq & \left( 1+\mathrm{e}^{C'}\right) \cdot 2^{-j-1} C'^ n. \end{aligned} \]
Now we can apply the same arguments as in Step 1.2 to conclude that
\[ \int _{G'}\left(1-\mathrm{e}^{P_{\theta }(f)-f}\right)\, \theta _{P_{\theta }(f)}^ n=0. \]
Since $G'\Subset X\setminus \operatorname{nK}(\{ \theta \} )$, (13.13.25) follows. $\square$
Lemma 13.13.30.1CC7CDBE65A9 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$, $\varphi \preceq \psi $ and $\int _ X \theta _{\varphi }^ n>0$. Then for any
D1F48E296303
13.13.31
\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){}^{\href{#sgpt-a0000014934}{2}}, \end{equation}
there is $\eta \in \mathrm{PSH}(X,\theta )_{>0}$ such that
CD0083D5C7BC
13.13.32
\begin{equation} a^{-1}\eta +\left(1-a^{-1}\right)\psi \leq \varphi . \end{equation}
We write
23948CFE8AEE
13.13.33
\begin{equation} \begin{aligned} P_{\theta }\Bigl (a\varphi +(1-a)\psi \Bigr )\coloneqq & \operatorname *{sup}_{}{}\! {}^{*}\left\{ \eta \in \mathrm{PSH}(X,\theta ): a^{-1}\eta +\left(1-a^{-1}\right)\psi \leq \varphi \right\} \\ \in & \mathrm{PSH}(X,\theta ). \end{aligned} \end{equation}
Note that if we regard $a\varphi +(1-a)\psi $ as a function defined outside the pluripolar set $\{ \varphi =\psi =-\infty \} $ on $X$, then (13.13.33) coincides with the envelope in the sense of (13.13.19).
Observe that
8921383DCB12
13.13.34
\begin{equation} a^{-1}P_{\theta }\Bigl (a\varphi +(1-a)\psi \Bigr )+(1-a^{-1})\psi \leq \varphi . \end{equation}
In fact, this equation holds outside a pluripolar set by proposition 5.14.5, hence it holds everywhere by proposition 5.14.6.
Finally, observe that
\[ P_{\theta }\Bigl (a\varphi +(1-a)\psi \Bigr )\preceq P_{\theta }\Bigl (a\varphi +(1-a)\varphi \Bigr )=\varphi . \]
If $\varphi \leq \psi $, the same argument gives
\[ P_{\theta }\Bigl (a\varphi +(1-a)\psi \Bigr )\leq \varphi . \]
Proof. Without loss of generality, we may assume that $\varphi \leq \psi \leq 0$.
Step 1. We show the existence of $\eta \in \mathrm{PSH}(X,\theta )$ satisfying (13.13.32).
Step 1.1. We first consider the case where $\psi \sim \varphi $. In this case, we can take $\eta =\varphi -C$ enough $C>0$. But since by (13.13.34), we also have $P_{\theta }(a\varphi +(1-a)\psi )\leq \psi $, we find that
\[ P_{\theta }\Bigl (a\varphi +(1-a)\psi \Bigr )\sim \varphi \sim \psi \]
in this case.
Step 1.2. We make a first reduction of the problem.
Now for each $\gamma \in \mathrm{PSH}(X,\theta )$, we let
\[ P_{\theta }[\gamma ]\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta \leq \min \{ \gamma +C,0\} \textup{ for some }C>0 \Bigr \} {}^{\href{#sgpt-a0000015147}{6}}. \]
Observe that due to corollary 13.13.12 and theorem 13.13.11, we have
00E5A7157825
13.13.35
\begin{equation} \int _ X \theta _{P_{\theta }[\gamma ]}^ n=\int _ X \theta _{\gamma }^ n. \end{equation}
Further, $0\geq P_{\theta }[\gamma ]\geq \gamma $ if $\gamma \leq 0$.
We now define an increasing transfinite sequence $(\psi _{\alpha })_{\alpha }$ consisting of non-positive functions in $\mathrm{PSH}(X,\theta )$ as follows:
1A558B3DDD3E$\psi _0=\psi $;
F7EF40CB8904$\psi _{\alpha +1}=P_{\theta }[\psi _{\alpha }]$;
04DDC5115F70if $\alpha $ is a limit ordinal, we define
\[ \psi _{\alpha }=\operatorname *{sup}_{\beta <\alpha }{}\! {}^{*} \psi _{\beta }. \]
Observe that
\[ \int _ X \theta _{\psi _{\alpha }}^ n=\int _ X \theta _{\psi }^ n \]
by corollary 13.13.12 and (13.13.35). Since $\mathrm{PSH}(X,\theta )$ is indeed a set, the transfinite sequence must stabilize, say at some $\phi \in \mathrm{PSH}(X,\theta )$. Note that $0 \geq \phi \geq \psi $, and $P_{\theta }[\phi ]=\phi $.
Note that the interval (13.13.31) remains the same if we replace $\psi $ by $\phi $. Take $a$ in this interval, it suffices to prove the existence of $\eta \in \mathrm{PSH}(X,\theta )$ so that
8E31B7D70F87
13.13.36
\begin{equation} a^{-1}\eta +\left(1-a^{-1}\right)\phi \leq \varphi . \end{equation}
Let us record the following observation for later use. Suppose that $\tau \in \mathrm{PSH}(X,\theta )$ and $\tau \preceq \phi $. Then
CEDD9EE9573E
13.13.37
\begin{equation} \sup _{\{ \phi \neq -\infty \} } (\tau -\phi )=\sup _ X \tau . \end{equation}
Observe that $\phi \leq 0$, so on the set $\{ \phi \neq -\infty \} $, we have
\[ \tau -\phi \geq \tau . \]
So the $\geq $ direction in (13.13.37) follows from corollary 5.15.21. Conversely, by assumption, we can find a constant $C>0$ so that
\[ \tau -\sup _ X \tau \leq \min \{ \phi +C,0 \} . \]
It follows that $\tau -\sup _ X \tau \leq P_{\theta }[\phi ]=\phi $. Therefore, the reverse inequality follows, and (13.13.37) follows.
For each $k>0$, we introduce
\[ \varphi _ k=\varphi \lor \left(\phi -k\right),\quad \eta _ k\coloneqq P_{\theta }\Bigl ( a \varphi _ k+(1-a)\phi \Bigr ). \]
Since $\varphi _ k\sim \phi $, using Step 1.1, we have $\eta _ k\in \mathrm{PSH}(X,\theta )$ and $\eta _ k\sim \phi $ as well.
Note that $(\eta _ k)_ k$ is decreasing in $k$. Let
\[ \eta \coloneqq \inf _{k\in \mathbb {N}} \eta _ k. \]
Note that $\eta $ automatically satisfies (13.13.36). It remains to show that $\eta \not\equiv -\infty $.
Step 1.3. We prove that $\eta \not\equiv -\infty $.
Suppose, for contradiction, that $\sup _ X \eta _ k\to -\infty $. For each $k>0$, let
\[ \gamma _ k\coloneqq a^{-1}\eta _ k+\left(1-a^{-1}\right)\phi ,\quad D_ k\coloneqq \left\{ \gamma _ k=\varphi _ k\right\} . \]
We claim that
87D0AE27AE70
13.13.38
\begin{equation} \theta _{\eta _ k}^ n\leq a^ n\mathds {1}_{D_ k}\, \theta _{\varphi _ k}^ n. \end{equation}
Since $\gamma _ k\leq \varphi _ k$ with equality on $D_ k$, 13.13.14 gives
\[ \mathds {1}_{D_ k}\, \theta _{\gamma _ k}^ n\leq \mathds {1}_{D_ k}\, \theta _{\varphi _ k}^ n. \]
On the other hand,
\[ \theta _{\gamma _ k}=a^{-1}\theta _{\eta _ k}+\left(1-a^{-1}\right)\theta _{\phi }, \]
so positivity and multilinearity of the non-pluripolar product imply
\[ a^{-n}\theta _{\eta _ k}^ n\leq \theta _{\gamma _ k}^ n. \]
Finally, it follows from theorem 13.13.24 that
\[ \mathds {1}_{D_ k}\, \theta _{\eta _ k}^ n=\theta _{\eta _ k}^ n. \]
Putting these results together, we conclude (13.13.38).
Fix $j>k>0$. Note that
2263D1F9CA2F
13.13.39
\begin{equation} \int _{\{ \varphi \leq \phi -k \} } \theta _{\varphi _ j}^ n=\int _ X \theta _{\varphi _ j}^ n-\int _{\{ \varphi >\phi -k \} } \theta _{\varphi _ j}^ n=\int _ X \theta _{\phi }^ n-\int _{\{ \varphi >\phi -k \} } \theta _{\varphi }^ n, \end{equation}
where we applied theorem 13.13.11 and proposition 13.11.4 1 in the second equality.
Next, we compute
\[ \begin{aligned} \int _{\left\{ \eta _ j\leq \phi -ak \right\} } \theta _{\eta _ j}^ n\leq & a^ n \int _{\left\{ \eta _ j\leq \phi -ak \right\} } \mathds {1}_{D_ j}\, \theta _{\varphi _ j}^ n & \textup{by (\href{../../tag/87D0AE27AE70/}{13.13.38})}\\ \leq & a^ n \int _{\left\{ a\varphi _ j+(1-a)\phi \leq \phi -ak \right\} \cap \left\{ \phi \neq -\infty \right\} } \theta _{\varphi _ j}^ n\\ = & a^ n \int _{\left\{ \varphi _ j\leq \phi -k \right\} } \theta _{\varphi _ j}^ n\\ = & a^ n \int _{\left\{ \varphi \leq \phi -k \right\} } \theta _{\varphi _ j}^ n\\ \leq & a^ n\left( \int _ X \theta _{\phi }^ n- \int _{\left\{ \varphi >\phi -k \right\} } \theta _{\varphi }^ n \right)& \textup{ by (\href{../../tag/2263D1F9CA2F/}{13.13.39})}. \end{aligned} \]
Next thanks to (13.13.37),
\[ \sup _{ \left\{ \phi \neq -\infty \right\} } \left(\eta _ j-\phi \right)=\sup _ X \eta _ j\to -\infty . \]
In particular, for a fixed $k$, if $j$ is large enough, we have
\[ \left\{ \eta _ j\leq \phi -ak \right\} =X. \]
Therefore, for a fixed $k$, for any large enough $j$,
\[ \int _ X \theta _{\phi }^ n=\int _ X \theta _{\eta _ j}^ n\leq a^ n\left( \int _ X \theta _{\phi }^ n- \int _{\{ \varphi >\phi -k\} } \theta _{\varphi }^ n\right). \]
Letting $k\to \infty $, we find
\[ \int _ X \theta _{\phi }^ n\leq a^ n \left(\int _ X \theta _{\phi }^ n-\int _ X \theta _{\varphi }^ n \right). \]
This is a contradiction with our choice of $a$.
Step 2. Next we argue that $P_{\theta }(a\varphi +(1-a)\psi )\in \mathrm{PSH}(X,\theta )_{>0}$. Choose
\[ a'\in \left( a, \left(\frac{\int _ X \theta _{\psi }^ n}{\int _ X \theta _{\psi }^ n-\int _ X \theta _{\varphi }^ n}\right)^{1/n} \right). \]
It follows from (13.13.33) that
1C8D5126D276
13.13.40
\begin{equation} P_{\theta }\Bigl (a\varphi +(1-a)\psi \Bigr )\geq \frac{a}{a'} P_{\theta }\Bigl (a'\varphi +(1-a')\psi \Bigr )+\frac{a'-a}{a'}\varphi . \end{equation}
Therefore, by theorem 13.13.11, we have
675086676ABE
13.13.41
\begin{equation} \int _ X \theta _{P_{\theta }\left(a\varphi +(1-a)\psi \right)}^ n\geq \frac{(a'-a)^ n}{a'^ n}\int _ X \theta _{\varphi }^ n>0. \end{equation}
We conclude the proof. $\square$
Corollary 13.13.42.992A84CBAC5D Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$, $\varphi \preceq \psi $. Assume that $\int _ X \theta _{\varphi }^ n=\int _ X \theta _{\psi }^ n$. Then for any $\epsilon \in (0,1)$, there is $\eta \in \mathrm{PSH}(X,\theta )$ such that
F009B4D18B32$\int _ X \theta _{\eta }^ n=\int _ X \theta _{\varphi }^ n$;
6009D3BD92A6$\epsilon \eta +(1-\epsilon )\psi \leq \varphi $.
In fact, we can take
\[ \eta =P_{\theta }\left(\epsilon ^{-1}\varphi +(1-\epsilon ^{-1})\psi \right). \]
Proof. Fix $\epsilon \in (0,1)$. We define $\eta $ as in the statement of the corollary. By 13.13.30, $\eta \in \mathrm{PSH}(X,\theta )_{>0}$. Observe that 2 is satisfied by (13.13.34).
Thanks to (13.13.41), for each $a'>\epsilon ^{-1}$, we have
\[ \int _ X \theta _{\eta }^ n>\left(\frac{a'-\epsilon ^{-1}}{a'}\right)^ n\int _ X \theta _{\varphi }^ n. \]
Letting $a'\to \infty $, we conclude that
\[ \int _ X \theta _{\eta }^ n\geq \int _ X \theta _{\varphi }^ n. \]
On the other hand, since $\eta \preceq \psi $, using the monotonicity theorem theorem 13.13.11 we find that
\[ \int _ X \theta _{\eta }^ n\leq \int _ X \theta _{\psi }^ n= \int _ X \theta _{\varphi }^ n. \]
Hence,
\[ \int _ X \theta _{\eta }^ n= \int _ X \theta _{\varphi }^ n. \]
Therefore, 1 holds as well. $\square$
Theorem 13.13.43 (Domination principle).B798B54E6F2E Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Assume that
92DC05470A2E
13.13.44
\begin{equation} \exists \phi \in \mathrm{PSH}(X,\theta )\quad \textup{s.t.} \quad \varphi \preceq \phi , \quad \psi \preceq \phi ,\quad \int _ X \theta _{\varphi }^ n=\int _ X \theta _{\psi }^ n=\int _ X \theta _{\phi }^ n, \end{equation}
and
99773FAEB14E
13.13.45
\begin{equation} \int _{\{ \varphi <\psi \} } \theta _{\varphi }^ n=0. \end{equation}
Then $\varphi \geq \psi $.
Proof. Take the common less singular potential $\phi$ in the hypotheses as the reference potential. The inequalities $\varphi\preceq\phi$, $\psi\preceq\phi$ and the two mass equalities say exactly that $\varphi,\psi\in\mathcal E(X,\theta,\phi)$. The reference mass is positive, because it equals the positive mass of $\varphi$. The second hypothesis says that the Monge--Ampère measure of $\varphi$ vanishes on $\{\varphi<\psi\}$. The relative full-mass domination principle therefore gives $\varphi\geq\psi$. This identifies every hypothesis of the earlier proof; in particular the common full-mass reference cannot be dropped.
Remark 13.13.46.B80857C56A45 Using the terminologies to be introduced in section 16.1, we can reformulate a special case of theorem 13.13.43 as follows: Suppose that $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ is a model potential and $\varphi ,\psi \in \mathcal{E}(X,\theta ;\phi )$. Assume that (13.13.45) holds then $\varphi \geq \psi $.
Equivalently, using the terminologies of section 16.7, the assumption (13.13.44) can be reformulated as $\varphi \sim _ P \psi $.
Proof. Thanks to the monotonicity theorem theorem 13.13.11,
\[ \int _ X \theta _{\varphi \lor \psi }^ n=\int _ X \theta _{\phi }^ n. \]
We may replace $\psi $ by $\varphi \lor \psi $ and assume that $\varphi \leq \psi $. Without loss of generality, we may also assume that $\psi \leq 0$.
Now fix $a>1$. We define
\[ \eta _ a\coloneqq P_{\theta }\Bigl ( a\varphi +(1-a)\psi \Bigr ). \]
Note that $\eta _ a\in \mathrm{PSH}(X,\theta )$ by corollary 13.13.42 and
\[ \int _ X \theta _{\eta _ a}^ n=\int _ X \theta _{\phi }^ n. \]
Furthermore, $\eta _ a\leq P_{\theta }(a\varphi +(1-a)\varphi )=\varphi $.
Define
\[ \gamma _ a\coloneqq a^{-1}\eta _ a+(1-a^{-1})\psi ,\quad D_ a\coloneqq \{ \gamma _ a=\varphi \} . \]
Then $\gamma _ a\leq \varphi $ with equality on $D_ a$. Therefore,
\[ \begin{aligned} a^{-n}\, \theta _{\eta _ a}^ n\leq & a^{-n}\, \theta _{\eta _ a}^ n+\mathds {1}_{D_ a} (1-a^{-1})^ n \, \theta _{\psi }^ n& \\ =& \mathds {1}_{D_ a} a^{-n}\, \theta _{\eta _ a}^ n+\mathds {1}_{D_ a} (1-a^{-1})^ n \, \theta _{\psi }^ n & \textup{ by theorem \href{../../tag/A60C61FA5696/}{13.13.24}}\\ \leq & \mathds {1}_{D_ a} \theta _{\gamma _ a}^ n & \\ \leq & \mathds {1}_{D_ a} \theta _{\varphi }^ n & \textup{ by \href{../../tag/F56651914DE1/}{13.13.14}}\\ =& \mathds {1}_{D_ a\cap \{ \varphi =\psi \} }\, \theta _{\varphi }^ n & \textup{ by (\href{../../tag/99773FAEB14E/}{13.13.45})}. \end{aligned} \]
Note that on $D_ a\cap \{ \varphi =\psi \} $, we have $\eta _ a=\varphi $. We deduce that
\[ \theta _{\eta _ a}^ n=\mathds {1}_{\{ \eta _ a=\varphi \} }\, \theta _{\eta _ a}^ n \leq \theta _{\varphi }^ n, \]
where the inequality follows from 13.13.14.
But the two ends have the same mass, and hence
\[ \theta _{\eta _ a}^ n=\theta _{\varphi }^ n=\mathds {1}_{\{ \eta _ a=\varphi \} }\, \theta _{\varphi }^ n. \]
Therefore,
\[ \int _ X \mathrm{e}^{\eta _ a}\, \theta _{\varphi }^ n=\int _ X \mathrm{e}^{\varphi }\, \theta _{\varphi }^ n>0. \]
Note that $\eta _ a$ is decreasing in $a$. The above equation shows that
\[ \eta \coloneqq \inf _{a>1} \eta _ a\not\equiv -\infty . \]
On the other hand, if $x\in X$ is such that $\varphi (x)<\psi (x)$, we then have
\[ \eta _ a(x)\leq a\varphi (x)-(a-1)\psi (x)= \psi (x)+a\Bigl ( \varphi (x)-\psi (x) \Bigr ). \]
Letting $a\to \infty $, we find that $\eta (x)=-\infty $. Therefore, $\{ \varphi \neq \psi \} $ is pluripolar, and hence empty by proposition 5.14.6. Our assertion follows. $\square$
Lemma 13.13.47.3AA4A74713F0 For any $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, there is $\psi \in \mathrm{PSH}(X,\theta )$ such that
B1CBA0B9F2D3$\theta _{\psi }$ is a Kähler current, and
9583FAA82351$\psi \leq \varphi $.
In particular, there is an increasing sequence $(\varphi _ i)_{i>0}$ in $\mathrm{PSH}(X,\theta )$ converging almost everywhere to $\varphi $ such that $\theta _{\varphi _ i}$ is a Kähler current for all $i\geq 1$.
Proof. Using 13.13.30, we can find $\epsilon \in (0,1)$ and $\gamma \in \mathrm{PSH}(X,\theta )$ such that
\[ \epsilon V_{\theta }+(1-\epsilon )\gamma \leq \varphi . \]
We observe that the cohomology class $\{ \theta \} $ is big as a consequence of proposition 13.13.3. Therefore, we can take $\eta \in \mathrm{PSH}(X,\theta )$ such that $\theta _{\eta }$ is a Kähler current and $\eta \leq 0$. Since $\eta \leq V_{\theta }$, we may take
\[ \psi \coloneqq \epsilon \eta +(1-\epsilon )\gamma . \]
Then $\psi $ clearly satisfies 1 and 2.
For the latter claim, it suffices to take
\[ \varphi _ i=\left(1-(i+1)^{-1}\right)\varphi +(i+1)^{-1}\psi \]
for each $i>0$. $\square$
Lemma 13.13.48.72A62384B9EC Let $L$ be a holomorphic line bundle on $X$ with $\theta \in c_1(L)$. Assume that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then there exists $k_0>0$ such that for each $k\geq k_0$, we have
\[ \mathrm{H}^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)\neq 0. \]
Proof. By 13.13.47, there is $\psi \in \mathrm{PSH}(X,\theta )$ with $\theta _{\psi }$ a Kähler current and $\psi \leq \varphi $. Since $\mathcal{I}(k\psi )\subseteq \mathcal{I}(k\varphi )$, it suffices to prove the assertion with $\psi $ in place of $\varphi $. In this case, the result follows from Hörmander's $L^2$-estimate ([Hor66]), see [Theorem 13.21, Dem12]. $\square$
In the remainder of this section, we sharpen theorem 13.13.24 for $C^{1,\overline{1}}$ barrier functions.
Theorem 13.13.49 (Tosatti, Chu–Zhou).FE0721C3DE01 Let $X$ be a compact Kähler manifold with a Kähler form $\omega $ and let $f\in C^{1,1}(X)$. Then
\[ P_\omega (f)\in C^{1,1}(X), \]
and there is a constant $C=C(X,\omega ,\| f\| _{C^{1,1}})$ such that
\[ \| P_\omega (f)\| _{C^{1,1}}\leq C. \]
Proposition 13.13.50 (Darvas–Rubinstein).4BEA034735FD Let $\omega $ be a Kähler form on $X$, and let $f_1,f_2\in C^{1,\overline{1}}(X)$. Then $P_\omega (\min \{ f_1,f_2\} )\in C^{1,\overline{1}}(X)$.
Lemma 13.13.51.9453F71CEE8F Assume that $\omega $ is a Kähler form on $X$ and let $f_1,f_2\in C^{1,\overline{1}}(X)$. Then the functions $f_1$, $f_2$ and $P_\omega (\min \{ f_1,f_2\} )$ coincide up to second order at almost every point of the contact set $\{ P_\omega (\min \{ f_1,f_2\} )=f_1=f_2\} $. In particular,
\[ \mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_1=f_2\right\} }\, \omega _{f_1}^ n=\mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_1=f_2\right\} }\, \omega _{f_2}^ n. \]
We also have for $j\in \{ 1,2\} $,
93FB8527BF9B
13.13.52
\begin{equation} \begin{aligned} \omega ^ n_{P_\omega \left(\min \{ f_1,f_2\} \right)} = & \mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_1\right\} }\, \omega ^ n_{f_1} + \mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_2\right\} }\, \omega ^ n_{f_2} \\ & - \mathds {1}_{\left\{ P_\omega (\min \{ f_1,f_2\} )=f_1=f_2\right\} }\, \omega ^ n_{f_ j}\\ \geq & \mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_ j\right\} }\, \omega ^ n_{f_ j}. \end{aligned} \end{equation}
Here $\omega _{f_ i}\coloneqq \omega +\mathrm{dd}^{\mathrm{c}}f_ i\in L^ n(X)$, since $f_ i\in C^{1,\overline{1}}(X)$. Therefore, $\omega _{f_ i}^ n$ makes sense as an absolutely continuous Radon measure.
Proof. Set $\varphi \coloneqq P_\omega (\min \{ f_1,f_2\} )$. Since $\min \{ f_1,f_2\} \in C^0(X)$, we have $\varphi \leq \min \{ f_1,f_2\} $. By proposition 13.13.50, $\varphi \in C^{1,\overline{1}}(X)$. As $f_1,f_2,\varphi \in C^{1,\overline{1}}(X)$, the measures $\omega ^ n_{f_1}$, $\omega ^ n_{f_2}$ and $\omega ^ n_\varphi $ are all absolutely continuous.
Step 1: We show that at almost every contact point, $\varphi $ and $f_ i$ agree to order two.
For $i\in \{ 1,2,3\} $ set
\[ \Psi _1\coloneqq \varphi -f_1,\quad \Psi _2\coloneqq \varphi -f_2,\quad \Psi _3\coloneqq f_1-f_2. \]
Each $\Psi _ i$ lies in $C^{1,\overline{1}}(X)$.
We claim that for $i=1,2,3$, we have
12862DAB63AB
13.13.53
\begin{equation} \Psi _ i\textup{ vanishes to second order at almost every point of }\{ \Psi _ i=0\} . \end{equation}
To prove this, write $\{ \Psi _ i=0\} $ as the disjoint union $A_ i\sqcup B_ i\sqcup C_ i\sqcup D_ i$ of the following four sets:
$A_ i\coloneqq \{ \Psi _ i=0\} \cap \{ \mathrm{d}\Psi _ i\neq 0\} $;
$B_ i\coloneqq \{ \Psi _ i=0\} \cap \{ \textup{the classical real Hessian of }\Psi _ i\textup{ does not exist}\} $;
$C_ i\coloneqq \{ \Psi _ i=0\} \cap \{ \mathrm{d}\Psi _ i=0\} \cap \{ \textup{the classical real Hessian exists and is non-zero}\} $;
$D_ i\coloneqq \{ \Psi _ i=0\} \cap \{ \mathrm{d}\Psi _ i=0\} \cap \{ \textup{the classical real Hessian exists and equals zero}\} $.
It suffices to prove $A_ i$, $B_ i$, $C_ i$ are Lebesgue null sets.
The set $A_ i$ is a $C^1$ real hypersurface in $X$, hence has Lebesgue measure zero.
Since $\Psi _ i\in W^{2,1}_{\mathrm{loc}}$, the classical pointwise real Hessian of $\Psi _ i$ exists almost everywhere, so $B_ i$ is Lebesgue null.
Finally $C_ i$ is Lebesgue null by Stampacchia's lemma. This proves (13.13.53).
Specialising (13.13.53) to $\Psi _1,\Psi _2$, the functions $\varphi $ and $f_ i$ agree to order two at almost every point of the contact set $\{ \varphi =f_ i\} $. Consequently $\omega +\mathrm{dd}^{\mathrm{c}}\varphi $ and $\omega +\mathrm{dd}^{\mathrm{c}}f_ i$ have the same matrix of coefficients almost everywhere on $\{ \varphi =f_ i\} $, and therefore so do their $n$-fold wedge powers:
A9866D01984D
13.13.54
\begin{equation} \mathds {1}_{\{ \varphi =f_ i\} }\, \omega ^ n_\varphi = \mathds {1}_{\{ \varphi =f_ i\} }\, \omega ^ n_{f_ i},\qquad i\in \{ 1,2\} . \end{equation}
The analogous statement applied to $\Psi _3$ gives
6FB38BF4B16C
13.13.55
\begin{equation} \mathds {1}_{\{ f_1=f_2\} }\, \omega ^ n_{f_1} = \mathds {1}_{\{ f_1=f_2\} }\, \omega ^ n_{f_2}. \end{equation}
Step 2. The barrier $\min \{ f_1,f_2\} $ is continuous, hence quasi-continuous, so theorem 13.13.24 yields
77FFDFF752EB
13.13.56
\begin{equation} \int _{X\setminus \left(\{ \varphi =f_1\} \cup \{ \varphi =f_2\} \right)}\omega ^ n_\varphi = \int _{\left\{ \varphi <\min \{ f_1,f_2\} \right\} }\omega ^ n_\varphi =0. \end{equation}
Set $V_1\coloneqq \{ f_1<f_2\} $ and $V_2\coloneqq \{ f_1>f_2\} $, both open. We may decompose
\[ \{ \varphi =f_1\} \cup \{ \varphi =f_2\} = \bigl (V_1\cap \{ \varphi =f_1\} \bigr ) \sqcup \bigl (V_2\cap \{ \varphi =f_2\} \bigr ) \sqcup \{ \varphi =f_1=f_2\} . \]
This combined with (13.13.56) and (13.13.54) yields
\[ \begin{aligned} \omega ^ n_\varphi ={}& \mathds {1}_{V_1\cap \{ \varphi =f_1\} }\, \omega ^ n_\varphi + \mathds {1}_{V_2\cap \{ \varphi =f_2\} }\, \omega ^ n_\varphi + \mathds {1}_{\{ \varphi =f_1=f_2\} }\, \omega ^ n_\varphi \\ ={}& \mathds {1}_{V_1\cap \{ \varphi =f_1\} }\, \omega ^ n_{f_1} + \mathds {1}_{V_2\cap \{ \varphi =f_2\} }\, \omega ^ n_{f_2} + \mathds {1}_{\{ \varphi =f_1=f_2\} }\, \omega ^ n_{f_ j}. \end{aligned} \]
On the other hand,
\[ \mathds {1}_{V_ j\cap \{ \varphi =f_ j\} }\, \omega ^ n_{f_ j} = \mathds {1}_{\{ \varphi =f_ j\} }\, \omega ^ n_{f_ j} - \mathds {1}_{\{ \varphi =f_1=f_2\} }\, \omega ^ n_{f_ j}. \]
Therefore,
\[ \omega ^ n_\varphi = \mathds {1}_{\{ \varphi =f_1\} }\, \omega ^ n_{f_1} + \mathds {1}_{\{ \varphi =f_2\} }\, \omega ^ n_{f_2} + \mathds {1}_{\{ \varphi =f_1=f_2\} } \, \bigl (\omega ^ n_{f_ j}-\omega ^ n_{f_1}-\omega ^ n_{f_2}\bigr ). \]
Theorem 13.13.57 (Di Nezza–Trapani).6245F87DD88E Let $\theta _1,\dots ,\theta _ n$ be closed real smooth $(1,1)$-forms on $X$ representing pseudo-effective cohomology classes. For each $i=1,\dots ,n$, let $\varphi _ i\in \mathrm{PSH}(X,\theta _ i)$ and $f_ i\in C^{1,\overline{1}}(X)$ be functions with $\varphi _ i\leq f_ i$. Then
F5A7BEAB8003
13.13.58
\begin{equation} \mathds {1}_{\bigcap _ i\{ \varphi _ i=f_ i\} }\, \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n} = \mathds {1}_{\bigcap _ i\{ \varphi _ i=f_ i\} }\, \theta _{1,f_1}\wedge \cdots \wedge \theta _{n,f_ n}. \end{equation}
Proof. Step 1. We first reduce to the case where $\theta _1=\cdots =\theta _ n$, $\varphi _1=\cdots =\varphi _ n$ and $f_1=\cdots =f_ n$.
Assume for the moment that this special case is known. Let us handle the general case.
For $\lambda =(\lambda _1,\dots ,\lambda _ n)\in (\mathbb {R}_{>0})^ n$, we set
\[ \theta _\lambda \coloneqq \sum _{j=1}^ n\lambda _ j\theta _ j,\quad \varphi _\lambda \coloneqq \sum _{j=1}^ n\lambda _ j\varphi _ j,\quad f_\lambda \coloneqq \sum _{j=1}^ n\lambda _ j f_ j. \]
Then $\varphi _\lambda \in \mathrm{PSH}(X,\theta _\lambda )$, $f_\lambda \in C^{1,\overline{1}}(X)$, $\varphi _\lambda \leq f_\lambda $, and
\[ \bigcap _{j=1}^ n\{ \varphi _ j=f_ j\} \subseteq \{ \varphi _\lambda =f_\lambda \} . \]
So
\[ \mathds {1}_{\bigcap _ j\{ \varphi _ j=f_ j\} }(\theta _\lambda )^ n_{\varphi _\lambda } = \mathds {1}_{\bigcap _ j\{ \varphi _ j=f_ j\} }(\theta _\lambda )^ n_{f_\lambda }. \]
Both sides are polynomials in $\lambda _1,\dots ,\lambda _ n$; comparing the coefficient of $\lambda _1\cdots \lambda _ n$ gives (13.13.58).
From now on, we assume
\[ \theta =\theta _1=\dots =\theta _ n,\quad \varphi =\varphi _1=\cdots =\varphi _ n\in \mathrm{PSH}(X,\theta ), \]
and
\[ f=f_1=\cdots =f_ n\in C^{1,\overline{1}}(X) \]
with
\[ \varphi \leq f. \]
We wish to prove that
1FE4032CE3EE
13.13.59
\begin{equation} \mathds {1}_{\left\{ \varphi =f\right\} }\, \theta _{\varphi }^ n=\mathds {1}_{\left\{ \varphi =f\right\} }\, \theta _{f}^ n. \end{equation}
Step 2. We first handle the special case where $f\in \mathrm{PSH}(X,\theta )$.
Step 2.1. We assume that $\theta $ is a Kähler form, denoted by $\omega $ for clarity.
Pick a decreasing sequence $(\phi _ j)_ j$ in $C^{\infty }(X)$7 with pointwise limit $\varphi $ and set
\[ \psi _ j\coloneqq P_{\omega }\left(\min \{ \phi _ j,f\} \right). \]
Then $\psi _ j\in \mathrm{PSH}(X,\omega )$, and
\[ \varphi \leq \psi _ j\leq \min \{ \phi _ j,f\} \leq f, \]
and hence $\psi _ j\searrow \varphi $. Applying 13.13.51 with $f_1=\phi _ j$ and $f_2=f$ gives
88AC8C43CD82
13.13.60
\begin{equation} \omega ^ n_{\psi _ j}\geq \mathds {1}_{\{ \psi _ j=f\} }\, \omega ^ n_ f \geq \mathds {1}_{\{ \varphi =f\} }\, \omega ^ n_ f. \end{equation}
Fix $C>0$ with $\min _ X f>-C$ and $\epsilon >0$. Let
\[ h_ j^{C,\epsilon }\coloneqq \frac{(\psi _ j+C)\lor 0}{\left((\psi _ j+C)\lor 0 \right)+\epsilon },\qquad h^{C,\epsilon }\coloneqq \frac{(\varphi +C)\lor 0}{\left((\varphi +C)\lor 0\right)+\epsilon }. \]
These functions are both quasi-continuous with values in $[0,1]$. Observe that $h_ j^{C,\epsilon }$ converges in capacity to $h^{C,\epsilon }$ as $j\to \infty $.
Now fix $g\in C^0(X)$, $g\geq 0$. Note that $h_ j^{C,\epsilon }=0$ on the set $\{ \psi _ j\leq -C\} $ and $h^{C,\epsilon }=0$ on the set $\{ \varphi \leq -C\} $. Using locality of the non-pluripolar product proposition 13.11.4 1, we have
\[ h_ j^{C,\epsilon }g \, \omega _{\psi _ j}^ n=h_ j^{C,\epsilon }g \, \omega _{\psi _ j\lor (-C)}^ n,\quad h^{C,\epsilon }g \, \omega _{\varphi }^ n=h^{C,\epsilon }g \, \omega _{\varphi \lor (-C)}^ n. \]
Therefore,
\[ \begin{aligned} \int _ X g\, \omega _{\varphi }^ n\geq & \int _ X h^{C,\epsilon }g \, \omega _{\varphi }^ n & \\ =& \int _ X h^{C,\epsilon }g \, \omega _{\varphi \lor (-C)}^ n & \\ =& \lim _{j\to \infty } \int _ X h_ j^{C,\epsilon }g \, \omega _{\psi _ j\lor (-C)}^ n & \textup{by theorem \href{../../tag/272876B00528/}{13.12.8}}\\ =& \lim _{j\to \infty } \int _ X h_ j^{C,\epsilon }g \, \omega _{\psi _ j}^ n & \\ \geq & \lim _{j\to \infty } \int _{\{ \varphi =f\} } h_ j^{C,\epsilon }g \, \omega _{f}^ n & \textup{by }(\href{../../tag/88AC8C43CD82/}{13.13.60})\\ =& \int _{\{ \varphi =f\} }h^{C,\epsilon }g \, \omega _{f}^ n, \end{aligned} \]
where the last equality follows from the dominated convergence theorem. By our choice of $C$, we have $h^{C,\epsilon }>0$ on $\{ \varphi =f\} $, and hence when $\epsilon \to 0+$, $h^{C,\epsilon }\to 1$ everywhere on this set. Letting $\epsilon \to 0+$, we find
\[ \int _ X g\, \omega _{\varphi }^ n\geq \int _{\{ \varphi =f\} }g \, \omega _{f}^ n \]
by the dominated convergence theorem.
By the Riesz representation theorem, we conclude that
\[ \omega _{\varphi }^ n\geq \mathds {1}_{\{ \varphi =f\} }\, \omega _{f}^ n. \]
Then by 13.13.14, the reverse inequality also holds, and (13.13.59) follows in this case.
Step 2.2. We no longer assume that $\theta $ is a Kähler form.
Choose a Kähler form $\omega $ on $X$ so that $\theta +\omega $ is a Kähler form and $f\in \mathrm{PSH}(X,\theta +\omega )$. Fix $g\in C^0(X)$ and consider, for $t\geq 0$,
\[ Q(t)\coloneqq \int _{\{ \varphi =f\} }g\, (\theta +t\omega +\mathrm{dd}^{\mathrm{c}}\varphi )^ n - \int _{\{ \varphi =f\} }g\, (\theta +t\omega +\mathrm{dd}^{\mathrm{c}}f)^ n. \]
By multilinearity of the non-pluripolar product, $Q$ is a polynomial in $t$. Step 2.1 applied to $\theta +t\omega $ for $t>1$ gives $Q(t)\equiv 0$ for $t>1$, hence $Q\equiv 0$, and in particular $Q(0)=0$. Since $g$ is arbitrary, (13.13.59) holds.
Step 3. We handle the general case. We no longer assume that $f\in \mathrm{PSH}(X,\theta )$.
Choose a Kähler form $\omega $ on $X$ so that $\theta +\omega $ is a Kähler form. Then
\[ \varphi \leq P_{\theta +\omega }(f)\leq f. \]
By 13.13.51 applied to $f_1=f_2=f$, we find that $f$ and $P_{\theta +\omega }(f)$ agree to the second order on the contact set $\{ P_{\theta +\omega }(f)=f \} $, so
\[ \mathds {1}_{\left\{ P_{\theta +\omega }(f)=f \right\} }\, \theta _{P_{\theta +\omega }(f)}^ n=\mathds {1}_{\left\{ P_{\theta +\omega }(f)=f \right\} }\, \theta _{f}^ n.{}^{\href{#sgpt-a0000018446}{8}} \]
However, Step 2 implies that
\[ \mathds {1}_{\left\{ P_{\theta +\omega }(f)=\varphi \right\} }\, \theta _{\varphi }^ n=\mathds {1}_{\left\{ P_{\theta +\omega }(f)=\varphi \right\} }\, \theta _{P_{\theta +\omega }(f)}^ n. \]
Combining these results, we conclude (13.13.59). $\square$
Corollary 13.13.61.77CAD5891555 Assume that $\mathrm{PSH}(X,\theta )\neq \varnothing $. For any $f\in C^{1,\overline{1}}(X)$, we have
\[ \theta ^ n_{P_\theta (f)} = \mathds {1}_{\{ P_\theta (f)=f\} }\, \theta ^ n_ f. \]
In particular,
\[ \theta _{V_{\theta }}^ n=\mathds {1}_{\{ V_{\theta }=0\} }\, \theta ^ n. \]
Proof. From theorem 13.13.57, we find
\[ \mathds {1}_{\{ P_\theta (f)=f\} }\, \theta ^ n_{P_\theta (f)}=\mathds {1}_{\{ P_\theta (f)=f\} }\, \theta ^ n_ f. \]
By the concentration property theorem 13.13.24, our assertion follows. $\square$
Theorem 13.13.62 (Di Nezza–Trapani).700E44C7083B Let $\theta $ be a smooth closed real $(1,1)$-form on $X$ representing a big cohomology class. Let $f\in C^{1,\overline{1}}(X)$. Then
\[ P_\theta (f)\in C^{1,\overline{1}}\bigl (X\setminus \operatorname{nK}(\{ \theta \} )\bigr ). \]
In particular, $V_{\theta }\in C^{1,\overline{1}}\bigl (X\setminus \operatorname{nK}(\{ \theta \} )\bigr )$.
We refer to [DNT23] for the proof.
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