Pome

Lemma 16.29.212478F9CE0209

Lemma 16.29.21.2478F9CE0209 Suppose $\Gamma _\tau ^{\epsilon h}$ is $E$-invariant for every $\epsilon >0$ and $\tau <c$. There is a unique envelope system $\boldsymbol \Psi \in \mathscr S_{a,c}(\kappa )$ such that

\[ \Gamma _\tau ^{\epsilon h}=\pi ^*\Psi _\tau ^{\epsilon \kappa } \quad (\epsilon >0,\ \tau <c). \]

Writing $\Gamma =\mathsf R\boldsymbol \Gamma $ and $\psi =\mathsf R\boldsymbol \Psi $, we have $\Gamma _\tau =\pi ^*\psi _\tau $ for every $\tau <c$.

Proof. The descent in proposition 16.29.20 gives $\Gamma _\tau ^{\epsilon h}=\pi ^*\phi _\tau ^\epsilon $ with $\phi _\tau ^\epsilon \in \mathrm{PSH}(B,(1+\epsilon )\kappa )$. Normalization, decrease, concavity and constancy for $\tau \leq a$ descend pointwise. The fiber-envelope identity descends both ideal-model fixedness and scalar compatibility:

\[ \phi _\tau ^{\epsilon _2} =P_{(1+\epsilon _2)\kappa } [\phi _\tau ^{\epsilon _1}]_{\mathcal I} \quad (\epsilon _2\geq \epsilon _1>0). \]

In particular, the positive scalar slices have the same ideals at every weight.

For a Hermitian form $\xi >0$ on $B$, choose $\delta >0$ with $\delta \kappa \leq \xi $, fixed for the whole parameter interval, and define

\[ \Psi _\tau ^\xi =P_{\kappa +\xi } [\phi _\tau ^\delta ]_{\mathcal I}. \]

The definition is independent of $\delta $ because the scalar inputs have the same ideals. It reproduces the scalar components, and proposition 16.29.5 proves that each component is a curve in $\mathscr T_{a,c}(\kappa +\xi )$. Projection preserves the ideals, so comparing the complete competitor families proves compatibility for $\xi \geq \xi '$. Thus $\boldsymbol \Psi $ is a full envelope system. Every possible extension is forced by this formula, which proves uniqueness.

The retraction theorem applies to both systems. Taking their decreasing limits along $h/j$ and $\kappa /j$ in the scalar identity gives $\Gamma _\tau =\pi ^*\psi _\tau $. In particular the limits, including their values at poles, agree. For any other Hermitian metric $k$ on $B$, constants $0<l\leq r$ with $l\kappa \leq k\leq r\kappa $ give

\[ \phi _\tau ^{l/j}\leq \Psi _\tau ^{k/j}\leq \phi _\tau ^{r/j}. \]

Both outer sequences decrease to $\psi _\tau $. This also identifies the transverse limit directly, without assuming that it keeps the ideals of the positive scalar slices. $\square$

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