22.1.101 The comparison theoremF1438A60D3D4

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. $\square$