Proposition 16.8.28.1C6DED516857 Let $(\varphi _ i)_{i\in I}$, $(\psi _ i)_{i\in I}$ be uniformly bounded from above non-empty families in $\mathrm{QPSH}(X)$. Assume that there exists a closed smooth real $(1,1)$-form $\theta $ such that $\varphi _ i,\psi _ i\in \mathrm{PSH}(X,\theta )$ and $\varphi _ i\preceq _ P \psi _ i$ for all $i\in I$. Then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i\preceq _ P \operatorname *{sup}_{i\in I}{}\! {}^{*}\psi _ i. \]
The same holds with $\preceq _{\mathcal{I}}$ in place of $\preceq _ P$.
Proof.
By adding a Kähler form to $\theta $, we may assume that $\varphi _ i,\psi _ i\in \mathrm{PSH}(X,\theta )_{>0}$ for all $i\in I$. The statement for $\preceq _{\mathcal{I}}$ is a simple consequence of corollary 5.16.9, we only have to consider the statement for $\preceq _ P$.
Step 1. We first handle the case where $I$ is a directed set and $(\varphi _ i)_{i\in I}$ and $(\psi _ i)_{i\in I}$ are increasing nets.
In this case, our assertion follows simply from proposition 16.4.9.
Step 2. We handle the case where $I$ is finite. We may assume that $I=\{ 0,1\} $. It suffices to show that
\[ P_{\theta }[\varphi _0]\lor P_{\theta }[\varphi _1]\sim _ P \varphi _0\lor \varphi _1, \]
which follows from 16.8.11.
Step 3. The general case can be reduced to the two cases handled in Step 1 and Step 2. More precisely, by applying proposition 5.14.3 to the two families and taking the union of the resulting countable subsets, we can find a countable subset $J\subseteq I$ such that
\[ \operatorname *{sup}_{j\in J}{}\! {}^{*} \varphi _ j=\operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i,\quad \operatorname *{sup}_{j\in J}{}\! {}^{*} \psi _ j=\operatorname *{sup}_{i\in I}{}\! {}^{*} \psi _ i. \]
We may replace $I$ by $J$ and assume that $I$ is countable. If $I$ is finite, Step 2 applies directly. Otherwise, write $I=\mathbb {Z}_{>0}$ and set
\[ \tilde{\varphi }_ m=\varphi _1\lor \cdots \lor \varphi _ m,\quad \tilde{\psi }_ m=\psi _1\lor \cdots \lor \psi _ m. \]
By Step 2, $\tilde{\varphi }_ m\preceq _ P\tilde{\psi }_ m$ for every $m$. The two new sequences are increasing and have the same upper envelopes as the original families, so Step 1 applies.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.