22.1 Non-Archimedean pluripotential theory1E9C5D0261FD

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

for all $\tau <\Gamma _{\max }$. Our assertion follows from proposition 16.4.9 and proposition 16.6.30. $\square$

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

for some Kähler form $\omega $ on $X$. Then $\Gamma _{\max }=\Gamma '_{\max }$ and for any $\tau <\Gamma _{\max }$, we have

by proposition 16.8.17. It follows again from proposition 16.8.17 that

This completes the proof. $\square$

Proof. This follows from proposition 16.4.4, corollary 16.4.8 and proposition 17.12.19. $\square$

Proof. For $\sigma \in \{ -\infty ,\tau \} $, put

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

Expanding the right-hand side and subtracting the two identities for $\sigma =-\infty $ and $\sigma =\tau $, we get

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. $\square$

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

We claim that it suffices to show

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

Therefore, by 22.1.9, for any $k\geq 1$,

By the definition of $\operatorname{vol}\Gamma $, the first integral in the preceding line tends to $\epsilon $ as $k\to \infty $. Hence

which proves (22.1.12). $\square$

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:

1. 5067A37E976E

   $\Gamma ^{\omega ,\omega '}\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega +\omega ')_{>0}$ for each $\omega ,\omega '\in \textup{Käh}(X)$;

2. CDB9ED278B54

   for each $\omega ,\omega ',\omega ''\in \textup{Käh}(X)$ satisfying $\omega ''\geq \omega '$, we have

   \[ P_{\theta +\omega +\omega ''}\left[\Gamma ^{\omega ,\omega '}\right]_{\mathcal{I}}=\Gamma ^{\omega ,\omega ''}; \]
3. 3EBB3E3A9CF6

   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

The fact that the image of $\Gamma $ is as expected follows directly from the definitions. $\square$

Proof. This follows immediately from proposition 22.1.17. $\square$

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

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

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

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

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

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

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

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

which is obvious. $\square$

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

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

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

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

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,

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

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

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

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

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

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

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

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

Therefore,

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. $\square$

Proof. This follows from 17.15.41. $\square$

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

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

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

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

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

and

Here we implicitly used proposition 22.1.17. Therefore,

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

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

Unfolding the definitions, we reduce to

Using proposition 20.2.21, this is equivalent to

This is a consequence of 20.2.24. $\square$

Proof. We observe that (22.1.57) holds by the computations in proposition 19.6.19 and theorem 19.5.26 1. It is independent of the choice of the flag. Furthermore, (22.1.58) follows immediately from theorem 19.5.26 1.

Assume first that $\Gamma $ is bounded. Since the Duistermaat–Heckman measure has bounded support in this case (cf. theorem 19.6.13), we conclude that $\operatorname{DH}(\Gamma )$ is uniquely determined.

By translating the parameter, it suffices to treat the case $\Gamma _{\max }=0$. Indeed, if $a=\Gamma _{\max }$ and

then $\widetilde\Gamma _{\max }=0$, and the associated test function is translated by $a$. Consequently,

Thus independence of the flag and the moment identity for $\widetilde\Gamma $ imply the corresponding assertions for $\Gamma $.

For each $\epsilon >0$, we define $\Gamma ^{\epsilon }\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ as follows:

1. 36B347B9186A

   Let $\Gamma ^{\epsilon }_{\max }=0$, and

2. 12A1009F3470

   we set

   \[ \Gamma ^{\epsilon }_{\tau }= \begin{cases} \Gamma _{-\infty },& \text{ if }\tau \leq -\epsilon ^{-1},\\ P_{\theta }\left[ (1+\epsilon \tau )\Gamma _{\tau }-\epsilon \tau \Gamma _{-\infty } \right],& \text{ if }\tau \in \left(-\epsilon ^{-1},0\right). \end{cases} \]

Let us check that $\Gamma ^\epsilon $ is indeed an element of $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$. The concavity and monotonicity in $\tau $ follow from proposition 16.4.2. Since the $\Gamma _\tau $'s are $\mathcal I$-model, they are $\mathcal I$-good by example 18.1.11; hence

is $\mathcal I$-good by proposition 18.1.17. Therefore its $P$-envelope is also its $\mathcal I$-envelope, and the corresponding level is $\mathcal I$-model. The value at $-\infty $ is $\Gamma _{-\infty }$. Thus $\Gamma ^\epsilon \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$.

For each fixed $\tau <0$, the argument of theorem 17.13.21 Step 3.3 gives

By proposition 16.4.4 and corollary 16.9.14, this convergence is in $d_ S$. Hence theorem 19.5.26 4 gives

for every $\tau <0$. Since

for all $\epsilon $, the underlying convex body is fixed. Therefore 19.6.22 applies and gives

It follows that $\operatorname{DH}(\Gamma )$ is independent of the choice of the flag. $\square$

Proof. We may assume that $X$ admits a smooth flag $Y_{\bullet }$.

Assume 1. Let

Since $\Gamma _\tau \leq \Gamma ^ i_{-\infty }$ for every $i$ and every $\tau <\Gamma _{\max }$, we have

By proposition 16.4.4, the decreasing limit $\eta $ is model and

Using (22.1.61), this equals

Since $\Gamma _{-\infty }\leq \eta $ and both potentials are model, proposition 16.8.7 gives

Thus

We want to derive (22.1.62) from 19.6.22. It boils down to prove the following: For any $\tau <\Gamma _{\max }$, we have

For fixed $\tau <\Gamma _{\max }$, the potentials $\Gamma ^ i_\tau $ decrease to $\Gamma _\tau $. By proposition 16.4.4, their masses converge to the mass of $\Gamma _\tau $; hence $\Gamma ^ i_\tau \xrightarrow {d_ S}\Gamma _\tau $ by corollary 16.9.14. Applying the continuity of the partial Okounkov body map, theorem 19.5.26 4, gives

The proof under the assumption 2 is similar. We only need to apply 19.6.23 instead of 19.6.22. $\square$

Proof. Recall that

By (22.1.69), we have $\sigma (v)(\log |\delta |^2)=1$ and $\sigma (v)(\Gamma _{\tau })=v(\Gamma _{\tau })$ for all $\tau \leq \Gamma _{\max }$. So we have

Lastly, since $\sigma (v)$ is a divisorial valuation on $X \times \Delta $, by corollary 5.16.9, we conclude (22.1.85). $\square$

Proof. We continue to use the notations in definition 22.1.86. The fact that $\tilde{A}\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ follows from 22.1.88 and the cocycle property of the $\mathcal{I}$-projection. In order to prove (22.1.90), we fix $v\in X^{\operatorname{div}}$ and $\omega \in \textup{Käh}(X)$. The inductive construction in 22.1.88 uses only finite maxima and positive affine combinations of the potentials $A_{\tau _ i}$. On divisorial valuations, analytification commutes with both operations. Moreover, $P_{\theta +\omega }[\bullet ]_{\mathcal I}$ does not change divisorial valuations. Hence

is the ordinary upper concave envelope of $\tau \mapsto A_\tau ^{\mathrm{An}}(v)$. The Legendre-type supremum over $\tau $ is therefore unchanged by passing from $A$ to $A'$, which proves (22.1.90). $\square$

Proof. After replacing $L$ with a sufficiently high power, we may assume that $L$ is a line bundle. Take a very ample line bundle $H$ on $X$. By Siu's uniform global generation theorem [Siu98], [Theorem 6.27, Dem12] there exists $b>0$ large enough so that $H^{b}\otimes L^ k\otimes \mathcal{I}(k\varphi )$ is globally generated for all $k>0$. Let $\{ s_ i\} _ i$ be a finite set of global sections that generate the sheaf $H^{b}\otimes L^ k\otimes \mathcal{I}(k\varphi )$. Then

It follows that $v\mapsto -k^{-1}v\left(\mathcal{I}(k\varphi ) \right)$ lies in $\mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}((L+\frac{b}{k}H)^{\mathrm{An}})$. Using (22.1.81), we conclude that $\varphi ^{\mathrm{An}}\in \mathrm{PSH}(L^{\mathrm{An}})$. $\square$

Proof. The result follows from (22.1.87), proposition 22.1.73 and 22.1.92. $\square$

Proof. It suffices to argue that $I$ can be expanded as in (22.1.95). To see this, assume that $a\in I$. We can write $a=a_0+a_1t+\cdots +a_ mt^ m$ with $a_ i\in R$. Then our assumption implies that $\sum _ i a_ i \rho ^ i t^ i\in I$ as well for all $\rho \in S^1$. So by the Lagrange interpolation formula, $a_ it^ i\in I$ for all $i$. Therefore, we can write $I$ as $I_0+I_1t+I_2t^2+\cdots $ for some ideals $I_0\subseteq I_1\subseteq \cdots $ in $R$. But as $R$ is noetherian, there is $m\geq 0$ so that $I_{m'}=I_ m$ for $m'>m$. (22.1.95) follows. $\square$

Proof. Let $q:X\times (\mathbb {P}^1\setminus \{ 0\} )\rightarrow X$ be the natural projection. Since $\mathbb C^* \subset \mathbb {P}^1\setminus \{ 0\} $, we can glue $q^*\mathcal{J}$ with $\mathcal{I}$ to get an analytic coherent ideal sheaf on $X\times \mathbb {P}^1$. By the GAGA principle, this ideal sheaf is necessarily algebraic, hence so is its restriction to $X\times \mathbb {C}$. $\square$

Proof. The argument for this is exactly the same as the one in [Lemma 5.6, BBJ21] with Nadel's vanishing theorem replaced by the family version proved by Matsumura in [Theorem 1.7, Mat16]. $\square$

Proof. We may assume that $L$ is a line bundle. Observe that the extension is unique if it exists by theorem 22.1.72.

Let $p_1\colon X\times \mathbb {C}\rightarrow X$ be the projection. For each $t>0$, we have $\ell _ t\in \mathcal{E}(X,\theta ;\phi )$, hence $\ell _ t\sim _ P\phi $ by theorem 16.3.15 and $\ell _ t\sim _{\mathcal{I}}\phi $ by corollary 16.8.14. Thanks to proposition 5.16.22 and 20.2.56, for each $m\in \mathbb {Z}_{>0}$, we have

In particular, $\mathcal{I}(m\Phi )$ admits an $S^1$-invariant coherent extension to $X\times \mathbb {C}$, obtained by gluing it with $p_1^*\mathcal{I}(m\phi )$ over $X\times \mathbb {C}^*$.

From 22.1.94 and 22.1.96, we get that

where the $\mathfrak {a}_ i$'s are coherent ideal sheaves on $X$.

Using 22.1.97, there exists an ample line bundle $T$ over $X$ such that $p_1^* (T\otimes L^ m)\otimes \mathcal{I}(m\Phi )$ is globally generated, which is equivalent to $T\otimes L^ m\otimes \mathfrak {a}_ i$ being globally generated for all $i$.⁴

We define

From the right-hand side of the formula, $\varphi _ m$ can be extended to an element in $\mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}((L+m^{-1}T)^{\mathrm{An}})$, which we denote by the same symbol.

For $v\in X^{\operatorname{div}}$,

and the right-hand side defines an element in $\mathrm{PSH}(L^{\mathrm{An}})$ by definition, since $\{ \varphi _{2^ m}\} _ m$ is decreasing by the subadditivity of multiplier ideals theorem 5.16.14. $\square$

Proof. Observe that the extension is unique if it exists by theorem 22.1.72. We may assume that $\Gamma _{\max }=0$ without loss of generality.

When $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$, our assertion follows from proposition 22.1.98 and proposition 22.1.84.

In general, fix an ample line bundle $H$ on $X$ and a Kähler form $\omega \in c_1(H)$. Then we know that

for any $m\in \mathbb {Z}_{>0}$. Therefore, $\Gamma ^{\mathrm{An}}\in \mathrm{PSH}(L^{\mathrm{An}})$ by proposition 22.1.74. $\square$

Proof. The map (22.1.102) is order preserving by definition. Let us take $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ with $\Gamma ^{\mathrm{An}}\leq \Gamma '^{\mathrm{An}}$. Fix a Kähler form $\omega $ on $X$.

Let $v\in X^{\operatorname{div}}$ and $t \in \mathbb {Q}_{>0}$. Then, using (22.1.83) and the homogeneity of divisorial valuations, we notice that

A similar equality holds for $\Gamma '$. Therefore, by corollary 21.11.26, we have

for all $\tau \in \mathbb {R}$. It follows that

by proposition 16.8.9. Since both sides are $\mathcal{I}$-model, proposition 16.8.17 gives

for all $\tau \in \mathbb {R}$. Hence $\Gamma \leq \Gamma '$ by the definition of the order. $\square$

Proof. Let us write

where $m\in \mathbb {Z}_{>0}$, $s_{1},\ldots , s_ M$ are a finite number of sections of $L^ m$ and $\lambda _1,\ldots ,\lambda _ M\in \mathbb {Q}$.

Write $I_{\tau }$ for the set of $i$ such that $m^{-1}\lambda _ i=\tau $. We denote the finitely many $\tau $ so that $I_{\tau }$ is non-empty as $\tau _0> \cdots > \tau _ N$. For each $i=0,\ldots ,N$, we write

We define $A$ as the piecewise linear curve associated with the $A_{\tau _ i}$'s and the $\tau _ i$'s. For every $v\in X^{\mathrm{An}}$,

where we used the convention $\log |s_ j|(v)=-v(s_ j)$.

The final assertion follows from 22.1.89 and theorem 22.1.72. $\square$

Proof. Take a Kähler form $\omega $ on $X$. We need to show that

Therefore, after replacing $\theta $ by $\theta +\omega $, we may assume that $\Gamma _ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ for all $i\in I$ and $\inf _{i\in I}\Gamma _ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$. Fix $v\in X^{\operatorname{div}}$. By theorem 22.1.72, it suffices to prove that

Write $\Gamma =\inf _{i\in I}\Gamma _ i$. We first compare the two sides inside the Legendre-type supremum. If $\tau <\Gamma _{\max }$, then by the definition of the infimum test curve,

The net $(\Gamma _{i,\tau })_ i$ decreases to $\Gamma _{\tau }$. By 17.15.41, $\Gamma _{\tau }$ is stable under $P_{\theta }[\bullet ]$. Hence proposition 16.4.4 gives convergence of the masses, and corollary 16.9.14 yields

Hence theorem 16.9.55 gives

If $\tau >\Gamma _{\max }$, then $\Gamma _{\tau }\equiv -\infty $, and also $\inf _ i\Gamma _{i,\tau }\equiv -\infty $, since $\Gamma _{\max }=\inf _ i(\Gamma _ i)_{\max }$. Thus the same equality holds for $\tau >\Gamma _{\max }$ as well, with value $-\infty $. The possible endpoint $\tau =\Gamma _{\max }$ does not affect the supremum: for any test curve $\Lambda $ and any $v\in X^{\operatorname{div}}$,

because $\Lambda _{\Lambda _{\max }}\leq \Lambda _{\tau }$ for $\tau <\Lambda _{\max }$, hence

and we let $\tau \nearrow \Lambda _{\max }$.

Consequently, in (22.1.110) we may replace $\left(\inf _ i\Gamma _{i,\tau }\right)^{\mathrm{An}}(v)$ by $\inf _ i\Gamma _{i,\tau }^{\mathrm{An}}(v)$ inside the supremum. The equality then follows from proposition 21.11.36 applied to the closed convex functions $-\Gamma _{i,\bullet }^{\mathrm{An}}(v)$. $\square$

Proof. The map (22.1.102) is an order preserving injection by 22.1.105. It remains to prove that it is surjective. Let $\phi \in \mathrm{PSH}(L^{\mathrm{An}})$. We want to construct $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ with $\Gamma ^{\mathrm{An}}=\phi $.

Let $H$ be an ample line bundle and $(\epsilon _ i)_ i$ be a decreasing sequence of rational numbers with limit $0$, $\phi _ i\in \mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}((L+\epsilon _ i H)^{\mathrm{An}})$ such that

The existence of these data is guaranteed by theorem 22.1.75. Fix a Kähler form $\omega \in c_1(H)$.

Thanks to 22.1.107, we can find $\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\epsilon _ i \omega )$ with $(\Gamma ^ i)^{\mathrm{An}}=\phi _ i$. It follows from 22.1.105 that

Therefore, for any $\omega '\in \textup{Käh}(X)$, after discarding finitely many $i$ so that $\omega '\geq \epsilon _ i\omega $, the sequence $(P_{\theta +\omega '}\left[ \Gamma ^{i} \right]_{\mathcal{I}})_ i$ is decreasing. This does not change its infimum. We let

Note that the infimum is defined thanks to (22.1.104). It follows from proposition 22.1.109 that

From this, we conclude that for $\omega ',\omega ''\in \textup{Käh}(X)$ with $\omega '\leq \omega ''$, we have

Indeed, both sides have analytification $\phi $, by (22.1.103) and (22.1.102). The injectivity part of 22.1.105 gives our assertion.

It follows that $(\Gamma ^{\theta +\omega '})_{\omega '\in \textup{Käh}(X)}$ defines an element $\Gamma $ in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ and $\Gamma ^{\mathrm{An}}=\phi $. $\square$

Proof. We first reduce to the positive mass case. Choose a Kähler form $\omega _0$ on $X$ and set $\widehat{\Gamma }=P_{\theta +\omega _0}[\Gamma ]_{\mathcal{I}}$. Replacing $\theta $ by $\theta +\omega _0$ and $\Gamma $ by $\widehat{\Gamma }$ does not change $\Gamma ^{\mathrm{An}}$, by (22.1.103); it also only replaces $\operatorname{Tr}_ Y(\Gamma )$ by the corresponding $\mathcal{I}$-projection on $\widetilde{Y}$, whose analytification is unchanged. Hence we may assume that $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$. Let $\phi =\Gamma ^{\mathrm{An}}\in \mathrm{PSH}(L^{\mathrm{An}})$. By 22.1.116, $\phi (v_{Y,\mathrm{triv}})\neq -\infty $.

Take an ample line bundle $S$ on $X$ and a Kähler form $\omega $ in $c_1(S)$. Write $\phi $ as the decreasing limit of a sequence $(\phi ^{i})_ i$ of elements in $\mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}((L+i^{-1}S)^{\mathrm{An}})$ as in theorem 22.1.75.

Take $\Gamma ^{i}\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +i^{-1}\omega )$ such that $\Gamma ^{i,\mathrm{An}}=\phi ^{i}$. Note that by 22.1.10, $\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +i^{-1}\omega )_{>0}$.

Write $\widehat{\Gamma }^ i\coloneqq P_{\theta +\omega }[\Gamma ^ i]_{\mathcal{I}}$ and $\widehat{\Gamma }\coloneqq P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}$. Applying proposition 22.1.109 to the decreasing sequence $(\widehat{\Gamma }^ i)_ i$ and then using the injectivity in 22.1.105, we get $\inf _ i\widehat{\Gamma }^ i=\widehat{\Gamma }$. Hence, by the componentwise definition of this infimum, for any $\tau <\Gamma _{\max }$, we have

Now proposition 16.4.4 and corollary 16.9.14 give

for all $\tau <\Gamma _{\max }$.

By 22.1.116 again, each $\widehat{\Gamma }^{i}$ has non-trivial restriction to $Y$. By proposition 20.2.22, for any Kähler form $\omega '$ on $\widetilde{Y}$ satisfying $\omega '\geq \omega |_{\widetilde{Y}}$ we have

for any $\tau <(\operatorname{Tr}_ Y(\Gamma ))_{\max }$. Thanks to theorem 16.9.55, for every $c\operatorname{ord}_ F\in Y^{\operatorname{div}}$ and every such $\tau $,

Applying proposition 21.11.36 to the Legendre transforms in $\tau $, and using again that $\mathcal{I}$-projection does not change analytification, we get

Since also $\Gamma ^{\mathrm{An}}=\inf _ i \Gamma ^{i,\mathrm{An}}$, it suffices to prove (22.1.114) with $\Gamma ^ i$ in place of $\Gamma $.

In other words, we have reduced to the case where $\phi \in \mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}(L^{\mathrm{An}})$ and $L$ is big.

Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ with $\Gamma ^{\mathrm{An}}=\phi $. By 22.1.107, we can find a concave curve $(\Gamma '_{\tau })_{\tau \leq \Gamma _{\max }}$ such that $\Gamma '_{\tau }$ is a generalized Fubini–Study metric for each $\tau \leq \Gamma _{\max }$ and such that

Here the second equality follows from example 18.1.10, since generalized Fubini–Study metrics have analytic singularities. With the convention that $\operatorname{Tr}_ Y(-\infty )=-\infty $ and $(-\infty )^{\mathrm{An}}\equiv -\infty $, it follows that for any $c\operatorname{ord}_ F\in \widetilde{Y}^{\operatorname{div}}$,

The third equality follows from example 20.2.15, since $\Gamma '_{\tau }$ has analytic singularities. The fourth equality follows from $\Gamma _{\tau }=P_{\theta }[\Gamma '_{\tau }]$, proposition 20.2.21 and corollary 16.8.14. It remains to justify the second line. Namely, we want to show that for any generalized Fubini–Study metric $\varphi $ whose restriction to $\widetilde{Y}$ is not identically $-\infty $, we have

We can immediately reduce to the case where $\varphi $ is a Fubini–Study metric, and then to the case

where $s$ is a holomorphic section of $L$, not vanishing identically on $Y$, in which case (22.1.115) is obvious. $\square$

Proof. The equivalence between 1 and 2 is a simple consequence of the maximum principle [Lemma 1.4(i), BJ22].

To see the equivalence between 1 and 3, it suffices to observe that for any $\varphi \in \mathrm{PSH}(X,\theta )$,

This completes the proof. $\square$