Theorem 22.1.113D613676A36CA

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. $\square$