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22.1.20 Operations on non-Archimedean metricsC20FEECB51B7

Let $X$ be a connected compact Kähler manifold of dimension $n$ and $\theta ,\theta ',\theta ''$ be closed real smooth $(1,1)$-forms on $X$ representing big cohomology classes.

This section relies heavily on section 17.15. We shall use the notions introduced in that section without further explanations.

Definition 22.1.21.DA2917901B54 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$. We say $\Gamma \leq \Gamma '$ if for some $\omega \in \textup{Käh}(X)$, we have

\[ \Gamma ^{\theta +\omega } \leq \Gamma '{}^{\theta '+\omega }. \]
This notion is independent of the choice of $\omega $ thanks to 17.15.3. Moreover, we have the following:

Proposition 22.1.22.8C5A487B9CCC Let $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $\omega $ be a closed smooth positive $(1,1)$-form on $X$. Then the following are equivalent:

  1. A1B3616B9907

    $\Gamma \leq \Gamma '$;

  2. 3EAAE9BE6F17

    $P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}\leq P_{\theta +\omega }[\Gamma ']_{\mathcal{I}}$.

Proof. This follows immediately from 17.15.3. $\square$

Observe that this definition coincides with the corresponding definition in definition 17.15.1 when $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$.

Proposition 22.1.23.F7D709A21AE1 Let $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. Assume that $\Gamma \leq \Gamma '$. Then

\[ \operatorname{vol}\Gamma \leq \operatorname{vol}\Gamma '. \]

Proof. It suffices to show that for any Kähler form $\omega $ on $X$, we have

\[ \operatorname{vol}\Gamma ^{\theta +\omega }_{-\infty }\leq \operatorname{vol}\Gamma '{}^{\theta +\omega }_{-\infty }, \]

which is an immediate consequence of theorem 13.13.11. $\square$

Definition 22.1.24.93939D1E527C Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$. Then we define $\Gamma +\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\theta ')$ as the unique element such that for any $\omega \in \textup{Käh}(X)$, we have

\[ \left( \Gamma +\Gamma ' \right)^{\theta +\theta '+2\omega }=\Gamma ^{\theta +\omega }+\Gamma '^{\theta '+\omega }. \]
This definition yields an element in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta +\theta ')$ by 17.15.9 and it extends the definition in definition 17.15.4 by 17.15.9 as well.

Proposition 22.1.25.06DFF1DE1519 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$. Suppose that $\omega ,\omega '$ are two smooth closed positive $(1,1)$-forms on $X$. Then

\[ P_{\theta +\omega +\theta '+\omega '}[\Gamma +\Gamma ']_{\mathcal{I}}=P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}+P_{\theta '+\omega '}[\Gamma ']_{\mathcal{I}}. \]

Proof. This is a direct consequence of 17.15.9. $\square$

Proposition 22.1.26.E9E94041F68D The operation $+$ is commutative and associative: For any $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$ and $\Gamma ''\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta '')$, we have

\[ \Gamma +\Gamma '=\Gamma '+\Gamma ,\quad (\Gamma +\Gamma ')+\Gamma ''=\Gamma +(\Gamma '+\Gamma ''). \]

Proof. This is a direct consequence of proposition 17.15.7. $\square$

Definition 22.1.27.D1E32864F8F0 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $C\in \mathbb {R}$. We define $\Gamma +C\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ as the unique element such that for any $\omega \in \textup{Käh}(X)$, we have

\[ \left( \Gamma +C \right)^{\theta +\omega }=\Gamma ^{\theta +\omega }+C. \]
It is obvious from definition 17.15.10 that $\Gamma +C\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. It is also obvious that this definition extends definition 17.15.10.

Proposition 22.1.28.ECFDC5EDD4F2 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $C\in \mathbb {R}$. Suppose that $\omega $ is a smooth closed positive $(1,1)$-form on $X$. Then

\[ P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}+C=P_{\theta +\omega }[\Gamma +C]_{\mathcal{I}}. \]

Proof. This is clear by definition. $\square$

Proposition 22.1.29.87CB041BFCA3 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$ and $C,C'\in \mathbb {R}$. Then

  1. 80F4957A059F

    $(\Gamma +\Gamma ')+C=\Gamma +(\Gamma '+C)=(\Gamma +C)+\Gamma '$;

  2. 036FE960D3D0

    $\Gamma +(C+C')=(\Gamma +C)+C'$.

Proof. This is a direct consequence of proposition 17.15.11. $\square$

Proposition 22.1.30.11099644479B Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $C\in \mathbb {R}$. Then

\[ \operatorname{vol}\Gamma =\operatorname{vol}(\Gamma +C). \]

Proof. It suffices to show that for each Kähler form $\omega $ on $X$,

\[ \operatorname{vol}\Gamma ^{\theta +\omega }_{-\infty }=\operatorname{vol}(\Gamma +C)^{\theta +\omega }_{-\infty }, \]

which is obvious. $\square$

Definition 22.1.31.B819F1A85E09 Let $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, we define $\Gamma \lor \Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ as the unique element such that for any $\omega \in \textup{Käh}(X)$, we have

\[ \left( \Gamma \lor \Gamma ' \right)^{\theta +\omega }=\Gamma ^{\theta +\omega }\lor \Gamma '^{\theta +\omega }. \]
It follows from 17.15.19 that $\Gamma \lor \Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and this definition extends the corresponding definition in definition 17.15.12.

Proposition 22.1.32.9CACCE9D7690 Let $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $\omega $ be a closed smooth positive $(1,1)$-form on $X$. Then

\[ P_{\theta +\omega }[\Gamma \lor \Gamma ']_{\mathcal{I}}=P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}\lor P_{\theta +\omega }[\Gamma ']_{\mathcal{I}}. \]

Proof. This is a direct consequence of 17.15.19. $\square$

In particular, given a finite non-empty family $(\Gamma ^ i)_{i\in I}$ in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, we then define $\bigvee _{i\in I}\Gamma ^ i$ in the obvious way.

Proof. This is a direct consequence of corollary 17.15.17. $\square$

Definition 22.1.34.C172DAF1B3E6 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. Assume that

AB5F1151338C 22.1.35
\begin{equation} \sup _{i\in I}\Gamma ^ i_{\max }<\infty . \end{equation}

Then we define $\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ as the unique element such that for any $\omega \in \textup{Käh}(X)$, we have

\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)^{\theta +\omega }=\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^{i,\theta +\omega }. \]
It follows immediately from 17.15.30 that $\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and this definition extends definition 17.15.25. Moreover, this definition clearly extends definition 22.1.31 as well.

Proposition 22.1.36.74CC0BF97B41 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Assume that $\omega $ is a closed smooth positive $(1,1)$-form on $X$. Then

\[ P_{\theta +\omega }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right]_{\mathcal{I}}= \operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[ \Gamma ^{i} \right]_{\mathcal{I}}. \]

Proof. This is a direct consequence of 17.15.30. $\square$

We also have a non-Archimedean version of Choquet's lemma.

Proposition 22.1.37.21737B0303A5 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Then there exists a countable subfamily $I'\subseteq I$ such that

\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i=\operatorname *{sup}_{i\in I'}{}\! {}^{*}\Gamma ^ i. \]

Proof. Fix a Kähler form $\omega _0$ on $X$. For each $m\geq 1$, proposition 17.15.31 gives a countable subfamily $I_ m\subseteq I$ such that

\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}P_{\theta +m^{-1}\omega _0}[\Gamma ^ i]_{\mathcal{I}}=\operatorname *{sup}_{i\in I_ m}{}\! {}^{*}P_{\theta +m^{-1}\omega _0}[\Gamma ^ i]_{\mathcal{I}}. \]

Set $I'=\bigcup _{m\geq 1}I_ m$. This is countable. We claim that it works.

Let $\omega '\in \textup{Käh}(X)$. Choose $m$ so large that $\omega '\geq m^{-1}\omega _0$. By proposition 22.1.36 and the compatibility of the transition maps,

\[ \begin{aligned} \operatorname *{sup}_{i\in I}{}\! {}^{*}P_{\theta +\omega '}[\Gamma ^ i]_{\mathcal{I}} =& P_{\theta +\omega '}\left[\operatorname *{sup}_{i\in I}{}\! {}^{*}P_{\theta +m^{-1}\omega _0}[\Gamma ^ i]_{\mathcal{I}}\right]_{\mathcal{I}}\\ =& P_{\theta +\omega '}\left[\operatorname *{sup}_{i\in I'}{}\! {}^{*}P_{\theta +m^{-1}\omega _0}[\Gamma ^ i]_{\mathcal{I}}\right]_{\mathcal{I}}\\ =& \operatorname *{sup}_{i\in I'}{}\! {}^{*}P_{\theta +\omega '}[\Gamma ^ i]_{\mathcal{I}}. \end{aligned} \]

Thus the two non-Archimedean metrics have the same component in every class $\theta +\omega '$, and the assertion follows. $\square$

Proposition 22.1.38.938848E420F9 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Let $C\in \mathbb {R}$. Then

\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} (\Gamma ^ i+C)=\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^ i+C. \]

Suppose that $(\Gamma '^ i)_{i\in I}$ is another family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$ satisfying (22.1.35). Suppose that $\Gamma ^ i\leq \Gamma '^ i$ for all $i\in I$. Then

\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\leq \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma '^ i. \]

Proof. This is an immediate consequence of proposition 17.15.32. $\square$

Proposition 22.1.39.15E20104530B Let $(\Gamma ^ i)_{i\in I}$ be an increasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Then

AA5696D80091 22.1.40
\begin{equation} \operatorname{vol}\left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^ i \right)=\lim _{i\in I} \operatorname{vol}\Gamma ^ i. \end{equation}

Proof. The $\geq $ direction in (22.1.40) is a direct consequence of proposition 22.1.23. It remains to prove the reverse inequality.

Note that (22.1.40) holds when $\operatorname{vol}\Gamma ^ i>0$ for each $i\in I$, as a consequence of proposition 17.15.22, corollary 16.9.12 and theorem 16.9.57.

In particular, for each Kähler form $\omega $ on $X$, we have

\[ \operatorname{vol}\left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i,\theta +\omega } \right)=\lim _{i\in I} \operatorname{vol}\Gamma ^{i,\theta +\omega }. \]

For our purpose, we need to show that for any $\epsilon >0$, we can find $\omega $ so that

\[ \sup _{i\in I} \operatorname{vol}\Gamma ^{i,\theta +\omega }<\sup _{i\in I} \operatorname{vol}\Gamma ^{i}+\epsilon . \]

We shall show that it is possible to choose $\omega $ so that the stronger statement holds:

\[ \operatorname{vol}\Gamma ^{i,\theta +\omega }<\operatorname{vol}\Gamma ^{i}+\epsilon ,\quad \forall i\in I. \]

Equivalently, we need to choose $\omega $ so that for any Kähler form $\omega '$ on $X$ dominated by $\omega $, we have

\[ \operatorname{vol}\Gamma ^{i,\theta +\omega }_{-\infty }<\operatorname{vol}\Gamma ^{i,\theta +\omega '}_{-\infty }+\epsilon /2,\quad \forall i\in I. \]

Fix a Kähler form $\omega _0$ on $X$ and a Kähler form $\Omega $ such that $\Omega \geq \theta +\omega _0$. It is enough to take $\omega =\delta \omega _0$ for some $\delta \in (0,1)$. If $\omega '$ is dominated by $\omega $, then $\theta +\omega '\leq \Omega $. Set

\[ T_ i\coloneqq \theta +\omega '+\mathrm{dd}^{\mathrm{c}}\Gamma ^{i,\theta +\omega '}_{-\infty }. \]

By the transition compatibility and proposition 17.12.19, the potential $\Gamma ^{i,\theta +\omega }_{-\infty }$ has the same $P$-singularity type in the class $\theta +\omega $ as $\Gamma ^{i,\theta +\omega '}_{-\infty }$. Hence its mass can be computed as

\[ \int _ X\left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{i,\theta +\omega '}_{-\infty }\right)^ n=\int _ X \left(T_ i+\omega -\omega '\right)^ n. \]

Therefore,

\[ \begin{aligned} \operatorname{vol}\Gamma ^{i,\theta +\omega }_{-\infty }-\operatorname{vol}\Gamma ^{i,\theta +\omega '}_{-\infty } =& \int _ X \left(T_ i+\omega -\omega '\right)^ n-\int _ X T_ i^ n\\ =& \sum _{a=0}^{n-1} \binom {n}{a} \int _ X T_ i^ a\wedge \left( \omega -\omega '\right)^{n-a}\\ \leq & \sum _{a=0}^{n-1} \binom {n}{a} \int _ X \Omega ^ a\wedge \omega ^{n-a}. \end{aligned} \]

In the last inequality we used the facts that $0\leq \omega -\omega '\leq \omega $ and $\theta +\omega '\leq \Omega $, and the monotonicity theorem theorem 13.13.11. We can choose $\omega $ so that the right-hand side is less than $\epsilon /2$. Our assertion then follows. $\square$

Definition 22.1.41.65E2E07550F1 Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. Assume that

827FE14EA1A0 22.1.42
\begin{equation} \inf _{i\in I}\Gamma ^ i_{\max }>-\infty , \end{equation}

then we define $\inf _{i\in I}\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ as the unique element such that for each $\omega \in \textup{Käh}(X)$, we have

341067287939 22.1.43
\begin{equation} \left( \inf _{i\in I}\Gamma ^ i \right)^{\theta +\omega }=\inf _{i\in I} \Gamma ^{i,\theta +\omega }. \end{equation}
We observe that
\[ \left( \inf _{i\in I}\Gamma ^ i \right)^{\theta +\omega }\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0}. \]

This follows from proposition 17.15.40. Moreover, by 17.15.41, we have $\inf _{i\in I}\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, and this definition extends definition 17.15.37.

In general,

\[ \operatorname{vol}\left( \inf _{i\in I}\Gamma ^ i \right)\leq \lim _{i\in I} \operatorname{vol}\Gamma ^ i \]

as a consequence of proposition 22.1.23. But the reverse inequality fails in general.

Proposition 22.1.44.07B251D899EF Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.42). Assume that $\omega $ is a closed smooth positive $(1,1)$-form on $X$. Then

\[ P_{\theta +\omega }\left[ \inf _{i\in I} \Gamma ^{i} \right]_{\mathcal{I}}= \inf _{i\in I} P_{\theta +\omega }\left[ \Gamma ^{i} \right]_{\mathcal{I}}. \]

Proof. This follows from 17.15.41. $\square$

Proposition 22.1.45.C8734FD15163 Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.42). Let $C\in \mathbb {R}$. Then

\[ \inf _{i\in I} (\Gamma ^ i+C)=\inf _{i\in I} \Gamma ^ i+C. \]

Suppose that $(\Gamma '^ i)_{i\in I}$ is another decreasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$ satisfying (22.1.42). Suppose that $\Gamma ^ i\leq \Gamma '^ i$ for all $i\in I$. Then

\[ \inf _{i\in I}\Gamma ^ i\leq \inf _{i\in I}\Gamma '^ i. \]

Proof. This is clear by definition. $\square$

Definition 22.1.46.EB1311598A32 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $\lambda \in \mathbb {R}_{>0}$. Then we define $\lambda \Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\lambda \theta )$ as the unique element such that for any $\omega \in \textup{Käh}(X)$, we have

\[ \left( \lambda \Gamma \right)^{\lambda \theta +\omega }=\lambda \Gamma ^{\theta +\lambda ^{-1}\omega }. \]
It follows immediately from 17.15.36 that $\lambda \Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\lambda \theta )$ and this definition extends definition 17.15.33.

Proposition 22.1.47.68FB6124EEB2 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $\lambda \in \mathbb {R}_{>0}$. Then for any closed smooth positive $(1,1)$-form $\omega $ on $X$, we have

\[ P_{\lambda \theta +\omega }[\lambda \Gamma ]_{\mathcal{I}}=\lambda P_{\theta +\lambda ^{-1}\omega }[\Gamma ]_{\mathcal{I}}. \]

Proof. This follows immediately from 17.15.36. $\square$

Proposition 22.1.48.EDFB2682E78E Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$, $C\in \mathbb {R}$ and $\lambda ,\lambda '>0$, we have

\[ \begin{aligned} \lambda (\Gamma +\Gamma ')=& \lambda \Gamma +\lambda \Gamma ',\\ (\lambda \lambda ')\Gamma =& \lambda (\lambda '\Gamma ),\\ \lambda (\Gamma +C)=& \lambda \Gamma +\lambda C. \end{aligned} \]

Suppose that $(\Gamma ^ i)_{i\in I}$ is a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Then

\[ \lambda \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)=\operatorname *{sup}_{i\in I}{}\! {}^{*}(\lambda \Gamma ^ i). \]

If $(\Gamma ^ i)_{i\in I}$ is a decreasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.42), then

\[ \lambda \left( \inf _{i\in I}\Gamma ^ i \right)=\inf _{i\in I}(\lambda \Gamma ^ i). \]

Proof. Everything except the last assertion follows from proposition 17.15.35. The last assertion follows directly from the definition. $\square$

Proposition 22.1.49.1743F9AEA475 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $\lambda \in \mathbb {R}_{>0}$. Then

\[ \operatorname{vol}\left( \lambda \Gamma \right)=\lambda ^ n \operatorname{vol}\Gamma . \]

Proof. This is clear by definition. $\square$

Definition 22.1.50.C0E37055D126 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. Let $Y\subseteq X$ be an irreducible analytic subset. We say that the trace operator of $\Gamma $ along $Y$ is well-defined if

\[ \nu \left(\Gamma _{\tau }^{\theta +\omega },Y\right)=0 \]

for all sufficiently negative $\tau $ and every $\omega \in \textup{Käh}(X)$. We define

\[ \left(\operatorname{Tr}_ Y(\Gamma )\right)_{\max }\coloneqq \sup \left\{ \tau <\Gamma _{\max }: \nu \left(\Gamma _{\tau }^{\theta +\omega },Y\right)=0\right\} . \]

Here $\omega $ is any Kähler form on $X$; this number is independent of the choice of $\omega $ by the compatibility of the transition maps.

In this case, we define $\operatorname{Tr}_ Y(\Gamma )\in \mathrm{PSH}^{\mathrm{NA}}(\widetilde{Y},\theta |_{\widetilde{Y}})$3 as the unique element such that for any $\omega \in \textup{Käh}(\widetilde{Y})$, the component

6697E3349082 22.1.51
\begin{equation} \operatorname{Tr}_ Y(\Gamma )^{\theta |_{\widetilde{Y}}+\omega }\in \mathrm{PSH}^{\mathrm{NA}}(\widetilde{Y},\theta |_{\widetilde{Y}}+\omega )_{>0} \end{equation}

is defined as follows:

  1. 5517A8C43D97

    We let

    284D786E2B4E 22.1.52
    \begin{equation} \left( \operatorname{Tr}_ Y(\Gamma )^{\theta |_{\widetilde{Y}}+\omega } \right)_{\max }=\left(\operatorname{Tr}_ Y(\Gamma )\right)_{\max }; \end{equation}
  2. E0B260D169B4

    for each $\tau \in \mathbb {R}$ less than $\left(\operatorname{Tr}_ Y(\Gamma )\right)_{\max }$, we define

    \[ \operatorname{Tr}_ Y(\Gamma )^{\theta |_{\widetilde{Y}}+\omega }_{\tau }\coloneqq P_{\theta |_{\widetilde{Y}}+\omega }\left[ \operatorname{Tr}_ Y^{\theta +\tilde{\omega }}\left(\Gamma ^{\theta +\tilde{\omega }}_{\tau }\right) \right]_{\mathcal{I}}, \]

    where $\tilde{\omega }$ is an arbitrary Kähler form on $X$ such that $\omega \geq \tilde{\omega }|_{\widetilde{Y}}$.

It follows from [Proposition 3.6, GK20] that $\widetilde{Y}$ is a normal Kähler space and hence $\tilde{\omega }$ exists. We observe that the choice of the trace operator $\operatorname{Tr}_ Y^{\theta +\tilde{\omega }}\left(\Gamma ^{\theta +\tilde{\omega }}_{\tau }\right)$ is irrelevant since two different choices are $\mathcal{I}$-equivalent. Moreover, (22.1.51) holds as a consequence of proposition 20.2.8 and proposition 20.2.21. It is therefore clear that $\operatorname{Tr}_ Y(\Gamma )\in \mathrm{PSH}^{\mathrm{NA}}(\widetilde{Y},\theta |_{\widetilde{Y}})$. Here when $\widetilde{Y}$ is not smooth, we understand $\mathrm{PSH}^{\mathrm{NA}}(\widetilde{Y},\theta |_{\widetilde{Y}})$ as the set of non-Archimedean metrics on a resolution of $\widetilde{Y}$, which is independent of the choice of the resolution by proposition 20.2.21. See section 9.10 for more details.

Proposition 22.1.53.E7009D90BCC6 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a compact Kähler manifold $Y$. Then all definitions in this section are invariant under pullback to $Y$.

The meaning is clear in most cases. In the case of the trace operator, this means the following: Suppose that $Z\subseteq X$ is an analytic subset and $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ has non-trivial restriction to $Z$. Suppose that $Z$ is not contained in the non-isomorphism locus of $\pi $ so that the strict transform $W$ of $Z$ is defined. If we write $\Pi \colon W\rightarrow Z$ for the restriction of $\pi $ and $\widetilde{\Pi }\colon \widetilde{W}\rightarrow \widetilde{Z}$ the strict transform of $\Pi $, then we have
\[ \widetilde{\Pi }^* \operatorname{Tr}_ Z(\Gamma )=\operatorname{Tr}_ W(\pi ^*\Gamma ). \]

The relevant notations are summarized in the following diagram:

Mathematical diagram from the source

Proof. We only prove the assertion for the trace operator, as the other proofs are similar.

We shall use the notations above. Observe that for any closed positive smooth $(1,1)$-form $\omega $ on $X$ with positive mass, we have

\[ \left(\widetilde{\Pi }^* \operatorname{Tr}_ Z(\Gamma )\right)_{\max }=\left( \operatorname{Tr}_ Z(\Gamma )\right)_{\max }=\sup \left\{ \tau <\Gamma _{\max }: \nu \left(\Gamma ^{\theta +\omega }_{\tau },Z\right)=0 \right\} , \]

and

\[ \begin{aligned} \left(\operatorname{Tr}_ W(\pi ^*\Gamma )\right)_{\max }=& \sup \left\{ \tau <\Gamma _{\max }: \nu \left((\pi ^*\Gamma )^{\pi ^*\theta +\pi ^*\omega }_{\tau },W\right)=0 \right\} \\ =& \sup \left\{ \tau <\Gamma _{\max }: \nu \left(\pi ^*\Gamma ^{\theta +\omega }_{\tau },W\right)=0 \right\} \\ =& \sup \left\{ \tau <\Gamma _{\max }: \nu \left(\Gamma ^{\theta +\omega }_{\tau },Z\right)=0 \right\} . \end{aligned} \]

Here we implicitly used proposition 22.1.17. Therefore,

\[ \left(\widetilde{\Pi }^* \operatorname{Tr}_ Z(\Gamma )\right)_{\max }=\left(\operatorname{Tr}_ W(\pi ^*\Gamma )\right)_{\max }. \]

Let $\tau \in \mathbb {R}$ be less than this common value. Take a Kähler form $\omega _{\widetilde{Z}}$ (resp. $\omega _{\widetilde{W}}$) on $\widetilde{Z}$ (resp. $\widetilde{W}$). Take a Kähler form $\omega _ Y$ on $Y$ (resp. $\omega _ X$ on $X$) such that

\[ \omega _{\widetilde{Z}}\geq \omega _ X|_{\widetilde{Z}},\quad \omega _{\widetilde{W}} \geq \omega _ Y|_{\widetilde{W}},\quad \omega _ Y \geq \pi ^*\omega _ X. \]

We want to show that, after projection to the common background $(\pi ^*\theta )|_{\widetilde{W}}+\omega _{\widetilde{W}}$,

\[ P_{(\pi ^*\theta )|_{\widetilde{W}}+\omega _{\widetilde{W}}}\left[\widetilde{\Pi }^*\left(\operatorname{Tr}_ Z(\Gamma )^{\theta |_{\widetilde{Z}}+\omega _{\widetilde{Z}}}_\tau \right)\right]_{\mathcal I} \sim _ P \left(\operatorname{Tr}_ W(\pi ^*\Gamma )\right)^{(\pi ^*\theta )|_{\widetilde{W}}+\omega _{\widetilde{W}}}_\tau . \]

Unfolding the definitions, we reduce to

\[ \widetilde{\Pi }^* \operatorname{Tr}_ Z^{\theta +\omega _ X}\left( \Gamma ^{\theta +\omega _ X}_{\tau } \right)\sim _ P \operatorname{Tr}_{W}^{\pi ^*\theta +\omega _ Y}\left( \left( \pi ^*\Gamma \right)^{\pi ^*\theta +\omega _ Y}_{\tau }\right). \]

Using proposition 20.2.21, this is equivalent to

\[ \widetilde{\Pi }^* \operatorname{Tr}_ Z\left( \Gamma ^{\theta +\omega _ X}_{\tau } \right)\sim _ P \operatorname{Tr}_{W}\left( \left( \pi ^*\Gamma \right)^{\pi ^*\theta +\pi ^*\omega _ X}_{\tau }\right). \]

This is a consequence of 20.2.24. $\square$

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