Pome29C5C6037D7DFix a resolution $\pi :Y\to X$. The envelopes of section 5.40 and definition 5.40.4 are defined on $X$ by the same formulas: for an obstacle $f:X\to [-\infty ,+\infty ]$, $P_\theta (f)$ is the upper semicontinuous regularization of the supremum of all $u\in \mathrm{PSH}(X,\theta )$ with $u\leq f$ pointwise, and $P_\theta [u]$ is defined as in definition 5.40.4. In the next proposition we assume that the family of these competitors is nonempty and locally uniformly bounded above. The descent bijection identifies the competitors on $X$ and on $Y$; the remaining issue is whether upper semicontinuous regularization commutes with pullback.
Proposition 9.11.7.622B386B559D For such an obstacle $f:X\to [-\infty ,+\infty ]$, the envelope $P_\theta (f)$ belongs to $\mathrm{PSH}(X,\theta )$ and
The same identity holds for the ordinary envelope $P_\theta [u]$, for every $u\in \mathrm{PSH}(X,\theta )$.
Proof. The preceding theorem identifies the competitor families. Let $s$ be their pointwise supremum on $X$; its pullback is the pointwise supremum upstairs. The regularization of $s\circ \pi $ on $Y$ is $\pi ^*\theta $-plurisubharmonic and not identically $-\infty $, since there is a competitor below it. By the theorem, it descends to a $\theta $-plurisubharmonic function $q$. Since $q\geq s$ and $q$ is upper semicontinuous, $q\geq s^*$.
For the reverse inequality, fix $x\in X$ and $y\in \pi ^{-1}(x)$. For each neighborhood $U$ of $x$, the unregularized supremum upstairs is bounded by $\sup _ U s$ on the open neighborhood $\pi ^{-1}(U)$ of $y$. Therefore $q(x)\leq \sup _ U s$, and taking the infimum over $U$ gives $q(x)\leq s^*(x)$. This proves the identity and shows that $P_\theta (f)=s^*$ is $\theta $-plurisubharmonic. The proof for $P_\theta [u]$ is identical, because pullback preserves the ordinary singularity order together with its constants. $\square$
We give the normalized space $\{ u\in \mathrm{PSH}(X,\theta ):\sup _ Xu=0\} $ the $L^1$ topology of the pullbacks to $Y$. By theorem 5.25.2 and descent, it is compact. On a fixed resolution, different smooth positive volume forms give equivalent norms. Under a further modification, pullback is continuous by lemma 5.43.19 and bijective by descent, hence a homeomorphism between the normalized compact spaces. Since any two resolutions are dominated by a common one, the topology does not depend on the resolution; the norms on different models are not compared.
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