Curves over different backgrounds are related by projecting each slice to the ideal envelope in the larger background. For these projections to form a consistent system, projecting in two steps must agree with projecting directly. This follows because each projection preserves multiplier ideals, while in each background a multiplier-ideal class has a unique representative fixed by the ideal envelope. The same fact gives injectivity: equal projected slices have equal multiplier ideals, hence come from equal fixed representatives.
Proposition 16.29.5.7DD8E9433762 Let $\beta \geq \theta $ be a smooth real $(1,1)$-form on $X$. Slicewise projection, $\Gamma \mapsto (P_\beta [\Gamma _\tau ]_{\mathcal I})_{\tau <c}$, defines an injective and order-preserving map
\[ P_\beta [\cdot ]_{\mathcal I}:\mathscr T_{a,c}(\theta )\longrightarrow \mathscr T_{a,c}(\beta ). \]
For every smooth real $(1,1)$-form $\beta '$ with $\beta \geq \beta '\geq \theta $, these maps satisfy
\[ P_\beta \bigl [P_{\beta '}[\Gamma ]_{\mathcal I}\bigr ]_{\mathcal I}=P_\beta [\Gamma ]_{\mathcal I}. \]
Proof.
Since $\beta \geq \theta $, every slice is $\beta $-plurisubharmonic, and by theorem 18.4.2 its projection is a $\beta $-plurisubharmonic function fixed by $P_\beta [\cdot ]_{\mathcal I}$. Decreasingness, constancy for $\tau \leq a$, and order preservation follow from proposition 18.4.6.
For concavity, let $r,t<c$ and $0<s<1$, and put $\tau =(1-s)r+s t$. By the same proposition and the concavity of the original curve,
\[ \begin{split} (1-s)P_\beta [\Gamma _ r]_{\mathcal I}+sP_\beta [\Gamma _ t]_{\mathcal I} & \leq P_\beta [(1-s)\Gamma _ r+s\Gamma _ t]_{\mathcal I}\\ & \leq P_\beta [\Gamma _\tau ]_{\mathcal I}. \end{split} \]
Here the first inequality follows by combining the scaling and addition formulas of that proposition, with backgrounds $(1-s)\beta $ and $s\beta $.
Each projection preserves the multiplier ideals at every weight. Hence the two sides of the transition identity are $P_\beta [\cdot ]_{\mathcal I}$-fixed representatives of the same multiplier-ideal class, and they are equal. If two projected curves agree, their original slices have the same multiplier ideals; since those slices are $P_\theta [\cdot ]_{\mathcal I}$-fixed, they agree as well. This proves injectivity.
$\square$
Let $\mathscr H(X)$ denote the set of Hermitian forms on $X$; for $\eta \in \mathscr H(X)$ we also write $\eta >0$.
Definition 16.29.6.949E1D79FF19 An envelope system over $\theta $, with parameters $a<c$, is a family
\[ \boldsymbol \Gamma =(\Gamma ^\eta )_{\eta \in \mathscr H(X)}, \qquad \Gamma ^\eta \in \mathscr T_{a,c}(\theta +\eta ), \]
such that
101FB6DC67C6
16.29.7
\begin{equation} \Gamma ^\eta =P_{\theta +\eta }[\Gamma ^{\eta '}]_{\mathcal I} \quad \hbox{whenever }\eta \geq \eta '. \end{equation}
Write $\mathscr S_{a,c}(\theta )$ for the set of these systems, ordered componentwise. For $\Gamma \in \mathscr T_{a,c}(\theta )$, set
\[ \mathsf E(\Gamma )^\eta =P_{\theta +\eta }[\Gamma ]_{\mathcal I} \qquad (\eta \in \mathscr H(X)). \]
A system is realized in the background $\theta $ if it equals $\mathsf E(\Gamma )$ for some $\Gamma \in \mathscr T_{a,c}(\theta )$.
By the transition proposition, $\mathsf E(\Gamma )$ is an envelope system, and the map $\mathsf E$ is injective. All components of an envelope system have the same multiplier ideals at each $\tau <c$: compare two components $\Gamma ^\eta $ and $\Gamma ^{\eta '}$ with the component indexed by $\eta +\eta '$.
If $u\in \mathrm{PSH}(X,\theta )$, the constant family $\Gamma _\tau =P_\theta [u]_{\mathcal I}$, $\tau <c$, with cutoff $c$, lies in $\mathscr T_{a,c}(\theta )$, and its projections form the envelope system $\mathsf E(\Gamma )$.
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