PomeProposition 10.31.25.4F5454D09F78 For every $z\in \mathcal E_\eta $, the rooftop $P_\theta (\varphi ,z)$ is not identically $-\infty $ and belongs to $\mathcal E_\varphi $.
10.31.24 Relative projection and supersolutions4FA2BA45E658
To pass from the minimal class to the reference $\varphi $, we project a solution $z\in \mathcal E_ V$ onto $\mathcal E_\varphi $. The projection is the increasing limit $P_\theta [\varphi ](z)$ of the rooftops $P_\theta (\varphi +C,z)$ as $C\to \infty $, in the notation of lemma 13.19.4. The small stretches of lemma 13.20.1 first show that the rooftop $P_\theta (\varphi ,z)$ is not identically $-\infty $ and has the full upper volume of $\varphi $. Contact support then bounds the measure of the projection by the measure of $z$ on the set where the two coincide. The argument uses only that the reference of $z$ lies above $\varphi $, so we formulate it for an arbitrary ordinary model above $\varphi $; this will also allow the reference to vary.
Let $\eta \in \mathrm{PSH}(X,\theta )$ satisfy $P_\theta [\eta ]=\eta \geq \varphi $, and write $\mathcal E_\eta =\mathcal E_{\mathrm{env}}(X,\theta ,\eta )$.
Proof. First suppose $z\leq \eta $; recall also that $\varphi \leq \eta \leq 0$. For $b>1$ sufficiently close to one, lemma 13.20.1 gives
\[ h_ b=Q_ b(\varphi ,\eta )\leq \varphi , \]
which is not identically $-\infty $. Set $a=1-b^{-1}$. The stretching lemma lemma 16.28.63, now applied in $\mathcal E_\eta $, gives $z_{1/a}=Q_{1/a}(z,\eta )\in \mathcal E_\eta $. The potential
\[ k_ a=a z_{1/a}+(1-a)h_ b \]
lies below both $\varphi $ and $z$. Indeed, $z_{1/a}\leq \eta $ and $(1-a)h_ b+a\eta \leq \varphi $ give the first bound, while $h_ b\leq \eta $ and $a z_{1/a}+(1-a)\eta \leq z$ give the second. Thus $p=P_\theta (\varphi ,z)$ is not identically $-\infty $. Volume addition and monotonicity give
\[ (1-a)^ n\overline{\operatorname {vol}}_\theta (h_ b) \leq \overline{\operatorname {vol}}_\theta (k_ a) \leq \overline{\operatorname {vol}}_\theta (p) \leq \overline{\operatorname {vol}}_\theta (\varphi ). \]
As $b\downarrow 1$, we have $a\downarrow 0$, and the first term tends to the last by lemma 13.20.1. The upper volumes of $p$ and $\varphi $ are therefore equal. Since $p\leq \varphi $, theorem 13.18.5 gives $p\in \mathcal E_\varphi $.
For general $z$, the envelope identity in $\mathcal E_\eta $ gives $z-\sup _ Xz\leq \eta $. Set $S=\max (\sup _ Xz,0)$. The preceding case gives $P_\theta (\varphi ,z-S)\in \mathcal E_\varphi $, and
\[ P_\theta (\varphi ,z-S)\leq P_\theta (\varphi ,z)\leq \varphi . \]
Upward stability finishes the proof. $\square$
Proposition 10.31.26.AABBF080DAD2 Let $z\in \mathcal E_\eta $ and let $Q=P_\theta [\varphi ](z)=\bigl (\lim _{C\to \infty }P_\theta (\varphi +C,z)\bigr )^*$ be its projection onto the relative class of $\varphi $. Then $Q\in \mathcal E_\varphi $, $Q\leq z$, and
\[ \mu _ Q\leq \mathbf1_{\{ Q=z\} }\mu _ z. \]
Proof. For $C\geq 0$ put $p_ C=P_\theta (\varphi +C,z)$. The preceding proposition gives $p_0\in \mathcal E_\varphi $. Since $p_0\leq p_ C\preceq \varphi $, upward stability gives $p_ C\in \mathcal E_\varphi $. The envelope identity, together with $p_ C\leq z$, yields the uniform bound
\[ p_ C\leq \varphi +\sup _ Xp_ C\leq \varphi +\sup _ Xz. \]
Consequently the increasing regularized limit $Q$ is not identically $-\infty $, belongs to $\mathcal E_\varphi $ and is at most $z$. The monotone convergence holds in capacity by lemma 5.29.19.
Finally, lemma 13.19.4, applied with $u=\varphi $ and $w=z$, gives
$\square$Corollary 10.31.28.1B6B55C4EF1E For every nonzero finite non-pluripolar measure $\nu $, there is $Q\in \mathcal E_\varphi $ such that
\[ 0\ne \mathrm{e}^{-Q}\mu _ Q\leq \nu . \]
Proof. By theorem 10.31.22, there is $z\in \mathcal E_ V$ with $\mu _ z=\mathrm{e}^ z\nu $. Since $V$ is an ordinary model and $V\geq \varphi $, the preceding proposition applies with $\eta =V$ and gives $Q=P_\theta [\varphi ](z)\in \mathcal E_\varphi $. On the contact set $\{ Q=z\} $,
\[ \mu _ Q\leq \mathbf1_{\{ Q=z\} }\mathrm{e}^ z\nu \leq \mathrm{e}^ Q\nu . \]
The weights are finite off the pole sets, which carry no mass. Thus $\mathrm{e}^{-Q}\mu _ Q\leq \nu $, and this measure is nonzero because $\mu _ Q(X)\geq \ell >0$. $\square$
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