Let $X$ be a connected compact Kähler manifold of dimension $n$. Let $\textup{Käh}(X)$ be the set of Kähler forms on $X$ with the partial order given as follows: We say $\omega \preceq \omega '$ if $\omega \geq \omega '$. Note that the partially ordered set $\textup{Käh}(X)$ is a directed set.
Let $\theta $ be a closed smooth real $(1,1)$-form representing a pseudo-effective cohomology class.
Definition 22.1.2.657951EB240B We define
\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )=\varprojlim _{\omega \in \textup{Käh}(X)} \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0}{}^{\href{#sgpt-a0000106603}{1}} \]
in the category of sets, where the transition maps are given as follows: Suppose that $\omega ,\omega '\in \textup{Käh}$ and $\omega \geq \omega '$. Then the transition map is defined in proposition 17.14.6:
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22.1.3
\begin{equation} P_{\theta +\omega }[\bullet ]_{\mathcal{I}}\colon \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega ')_{>0}\rightarrow \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0}. \end{equation}
Note that the transition maps are compatible:
\[ P_{\theta +\omega ''}[P_{\theta +\omega '}[\varphi ]_\mathcal {I}]_\mathcal {I}=P_{\theta +\omega ''}[\varphi ]_\mathcal {I},\quad \forall \, \varphi \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0} \]
for $\omega \leq \omega '\leq \omega ''$, since both sides are $\mathcal{I}$-model and are $\mathcal{I}$-equivalent to $\varphi $.
Recall that $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ is defined in definition 17.14.1.
In general, when we denote an element in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ by $\Gamma $, its component in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0}$ ($\omega \in \textup{Käh}(X)$) will be written either as $\Gamma ^{\theta +\omega }$ or as $P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}$.
Note that $\Gamma ^{\theta +\omega }_{\max }$ is independent of the choice of $\omega \in \textup{Käh}(X)$. We denote this common value by $\Gamma _{\max }$.
Proposition 22.1.5.B80F3474DB2F Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. Then given $\omega ,\omega '\in \textup{Käh}(X)$ with $\omega \geq \omega '$, we have
\[ P_{\theta +\omega }\left[ \Gamma ^{\theta +\omega '}_{-\infty } \right]=P_{\theta +\omega }\left[ \Gamma ^{\theta +\omega '}_{-\infty } \right]_{\mathcal{I}} = \Gamma ^{\theta +\omega }_{-\infty }. \]
Proof.
Since for any $\tau <\Gamma _{\max }$, the potential $\Gamma ^{\theta +\omega '}_{\tau }$ is $\mathcal{I}$-good by example 18.1.11, it follows that
\[ P_{\theta +\omega }\left[ \Gamma ^{\theta +\omega '}_{\tau } \right]=P_{\theta +\omega }\left[ \Gamma ^{\theta +\omega '}_{\tau } \right]_{\mathcal{I}}=\Gamma ^{\theta +\omega }_{\tau } \]
for all $\tau <\Gamma _{\max }$. Our assertion follows from proposition 16.4.9 and proposition 16.6.30.
$\square$
Proposition 22.1.6.2CB4FA8703B1 There is a natural injective map
\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}\hookrightarrow \mathrm{PSH}^{\mathrm{NA}}(X,\theta ),\quad \Gamma \mapsto \left(P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}\right)_{\omega \in \textup{Käh}(X)}. \]
In the sequel, we do not distinguish an element of $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ from its image in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. Note that given $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$, the value of $\Gamma _{\max }$ does not depend on whether we view it as an element of $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ or of $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$.
Proof.
The map is well-defined by the compatibility of the transition maps. It suffices to argue its injectivity. Suppose that $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ and
\[ P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}=P_{\theta +\omega }[\Gamma ']_{\mathcal{I}} \]
for some Kähler form $\omega $ on $X$. Then $\Gamma _{\max }=\Gamma '_{\max }$ and for any $\tau <\Gamma _{\max }$, we have
\[ \Gamma _{\tau }\sim _{\mathcal{I}}\Gamma '_{\tau } \]
by proposition 16.8.17. It follows again from proposition 16.8.17 that
\[ \Gamma _{\tau }=\Gamma '_{\tau }. \]
This completes the proof.
$\square$
Definition 22.1.7.A92D6DFAC0CF Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. We define its volume as follows:
\[ \operatorname{vol}\Gamma \coloneqq \lim _{\omega \in \textup{Käh}(X)} \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{-\infty }\right)^ n\in [0,\infty ). \]
Observe that the net is decreasing, so the limit exists.
Proposition 22.1.8.47A4DA9EBA71 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$. Then
\[ \operatorname{vol}\Gamma =\int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{-\infty }\right)^ n. \]
Proof.
This follows from proposition 16.4.4, corollary 16.4.8 and proposition 17.12.19.
$\square$
Lemma 22.1.9.587EEDE88F4F Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, let $\omega $ be a Kähler form on $X$, and let $0<t\leq 1$. Then for every $\tau <\Gamma _{\max }$, we have
\[ \begin{aligned} & \int _ X \left(\theta +t\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +t\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +t\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +t\omega }_{\tau }\right)^ n\\ \leq {}& \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{\tau }\right)^ n. \end{aligned} \]
Proof.
For $\sigma \in \{ -\infty ,\tau \} $, put
\[ S_ t^\sigma =\theta +t\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +t\omega }_{\sigma }. \]
By the transition compatibility and proposition 22.1.5, example 18.1.11, proposition 16.3.10, projecting $\Gamma ^{\theta +t\omega }_{\sigma }$ to the class $\theta +\omega $ does not change the total mass computed in the larger class. Thus
\[ \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{\sigma }\right)^ n = \int _ X \left(S_ t^\sigma +(1-t)\omega \right)^ n. \]
Expanding the right-hand side and subtracting the two identities for $\sigma =-\infty $ and $\sigma =\tau $, we get
\[ \begin{aligned} & \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{\tau }\right)^ n\\ ={}& \sum _{a=0}^ n \binom {n}{a}(1-t)^{n-a} \left(\int _ X (S_ t^{-\infty })^ a\wedge \omega ^{n-a}-\int _ X (S_ t^\tau )^ a\wedge \omega ^{n-a}\right). \end{aligned} \]
Since $\Gamma ^{\theta +t\omega }_{-\infty }\geq \Gamma ^{\theta +t\omega }_{\tau }$, each summand is non-negative by the monotonicity theorem for mixed non-pluripolar products. Keeping only the term $a=n$ gives the desired inequality.
$\square$
Lemma 22.1.10.9A2678C2C223 The image of the canonical injection
\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}\hookrightarrow \mathrm{PSH}^{\mathrm{NA}}(X,\theta ) \]
is given by the set of $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ with positive volume.
Proof.
By proposition 22.1.8, every element in the image has positive volume.
Conversely, take $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ with positive volume. We want to construct $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ representing $\Gamma $.
Fix a Kähler form $\omega $ on $X$. Define
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22.1.11
\begin{equation} \Gamma '_{\tau }\coloneqq \lim _{k\to \infty }\Gamma ^{\theta +k^{-1}\omega }_{\tau },\quad \tau <\Gamma _{\max }. \end{equation}
We claim that it suffices to show
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22.1.12
\begin{equation} \varliminf _{k\to \infty }\int _ X \left(\theta +k^{-1}\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +k^{-1}\omega }_{\tau }\right)^ n>0 \end{equation}
for some $\tau <\Gamma _{\max }$. If this holds, then the argument of 17.12.6 implies that the same holds for all $\tau <\Gamma _{\max }$. Then proposition 16.4.4, proposition 16.6.29 guarantee that $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ and represents $\Gamma $.
It remains to argue (22.1.12). Let $\epsilon =\operatorname{vol}\Gamma >0$. Take $\tau <\Gamma _{\max }$ so that
\[ \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{\tau }\right)^ n<\epsilon /2. \]
Therefore, by 22.1.9, for any $k\geq 1$,
\[ \int _ X \left(\theta +k^{-1}\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +k^{-1}\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +k^{-1}\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +k^{-1}\omega }_{\tau }\right)^ n<\epsilon /2. \]
By the definition of $\operatorname{vol}\Gamma $, the first integral in the preceding line tends to $\epsilon $ as $k\to \infty $. Hence
\[ \varliminf _{k\to \infty }\int _ X \left(\theta +k^{-1}\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +k^{-1}\omega }_{\tau }\right)^ n\geq \epsilon /2>0, \]
which proves (22.1.12).
$\square$
Example 22.1.13.796F447D96B6 Fix $\varphi \in \mathrm{PSH}(X,\theta )$. We can define an associated non-Archimedean metric
\[ \Gamma ^{\varphi }\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ) \]
as follows:
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$\Gamma ^{\varphi }_{\max }=0$;
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for any $\omega \in \textup{Käh}(X)$ and any $\tau <0$, we set
\[ \Gamma ^{\varphi ,\theta +\omega }_{\tau }=P_{\theta +\omega }[\varphi ]_{\mathcal{I}}. \]
Such non-Archimedean metrics are called homogeneous non-Archimedean metrics.
Observe that
\[ \operatorname{vol}\Gamma ^{\varphi }=\operatorname{vol}\theta _{\varphi }. \]
See proposition 18.1.27 and the footnote there.
Definition 22.1.14.CCAAC9DA4E2F Let $\omega $ be a closed real smooth positive $(1,1)$-form on $X$. We define the map
\[ P_{\theta +\omega }[\bullet ]_{\mathcal{I}}\colon \mathrm{PSH}^{\mathrm{NA}}(X,\theta )\rightarrow \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega ) \]
as follows: Given $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, we define $P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}$ as the element such that for any $\omega '\in \textup{Käh}(X)$, we have
\[ P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}^{\theta +\omega +\omega '}=\Gamma ^{\theta +\omega +\omega '}. \]
It is straightforward to check that under the identification of proposition 22.1.6, the map $P_{\theta +\omega }[\bullet ]_{\mathcal{I}}$ extends the map (22.1.3).
Proposition 22.1.15.3943AA7119C5 The maps $P_{\theta +\omega }[\bullet ]_{\mathcal{I}}$ in definition 22.1.14 together induce a bijection
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22.1.16
\begin{equation} \mathrm{PSH}^{\mathrm{NA}}(X,\theta )\xrightarrow {\sim } \varprojlim _{\omega \in \textup{Käh}(X)}\mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega ). \end{equation}
Proof.
It is a tautology that the maps $P_{\theta +\omega }[\bullet ]_{\mathcal{I}}$ in definition 22.1.14 are compatible with the transition maps. So the map (22.1.16) is well-defined. It is injective by the same argument as proposition 22.1.6. We argue the surjectivity.
By unfolding the definitions, an object in the target of (22.1.16) is an assignment: With each $\omega \in \textup{Käh}(X)$, we associate a family $(\Gamma ^{\omega ,\omega '})_{\omega '\in \textup{Käh}(X)}$ satisfying:
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$\Gamma ^{\omega ,\omega '}\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega +\omega ')_{>0}$ for each $\omega ,\omega '\in \textup{Käh}(X)$;
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for each $\omega ,\omega ',\omega ''\in \textup{Käh}(X)$ satisfying $\omega ''\geq \omega '$, we have
\[ P_{\theta +\omega +\omega ''}\left[\Gamma ^{\omega ,\omega '}\right]_{\mathcal{I}}=\Gamma ^{\omega ,\omega ''}; \]
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for each $\omega ,\omega ',\omega ''\in \textup{Käh}(X)$ satisfying $\omega \leq \omega '$, we have
\[ P_{\theta +\omega '+\omega ''}\left[\Gamma ^{\omega ,\omega ''}\right]_{\mathcal{I}}=\Gamma ^{\omega ',\omega ''}. \]
The preimage of such an object is given by the family $(\Gamma ^{\theta +\omega })_{\omega \in \textup{Käh}(X)}$ given by
\[ \Gamma ^{\theta +\omega }=\Gamma ^{\omega /2,\omega /2}. \]
The fact that the image of $\Gamma $ is as expected follows directly from the definitions.
$\square$
With an almost identical argument involving proposition 16.4.4, we get
Proposition 22.1.17.817C1BB3E570 The maps $P_{\theta +\omega }[\bullet ]_{\mathcal{I}}$ in definition 22.1.14 and the injective maps proposition 22.1.6 together induce bijections
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22.1.18
\begin{equation} \mathrm{PSH}^{\mathrm{NA}}(X,\theta )\xrightarrow {\sim } \varprojlim _{\omega }\mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0}\xrightarrow {\sim } \varprojlim _{\omega }\mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega ), \end{equation}
where $\omega $ runs over either the partially ordered set of all smooth closed real positive $(1,1)$-forms with positive volume on $X$ or $\textup{Käh}(X)$.
Corollary 22.1.19.347399E47277 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a compact Kähler manifold $Y$. Then $\pi ^*$ induces a bijection
\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )\xrightarrow {\sim } \mathrm{PSH}^{\mathrm{NA}}(Y,\pi ^*\theta ). \]
Proof.
This follows immediately from proposition 22.1.17.
$\square$
It is immediate to verify that $\pi ^*$ in corollary 22.1.19 extends the map proposition 17.14.4.
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