Theorem 16.29.22.2DB4FF5FB5C1 Let $\boldsymbol \Gamma \in \mathscr S_{a,c}(h)$ have $E$-invariant scalar components as in lemma 16.29.21. Let $\psi =\mathsf R\boldsymbol \Psi $ be the transverse limiting curve. If
\[ \underline{\operatorname {vol}}_\kappa (\psi _ a)>0, \]
then for every $\tau <c$ and every Hermitian form $\eta >0$ on $X$,
\[ \Gamma _\tau ^\eta =P_{h+\eta }[\pi ^*\psi _\tau ]_{\mathcal I}. \]
Thus $\boldsymbol \Gamma =\mathsf E(\mathsf R\boldsymbol \Gamma )$.
Proof.
The closed Kähler background $\kappa $ is big and $B$ has the bounded mass property. Apply theorem 13.21.7 to the full transverse system. Its initial-volume hypothesis is exactly the displayed one, and it gives
\[ \Psi _\tau ^{\epsilon \kappa } =P_{(1+\epsilon )\kappa }[\psi _\tau ]_{\mathcal I} \quad (\epsilon >0,\ \tau <c). \]
The fiber-envelope identity yields $\Gamma _\tau ^{\epsilon h}=P_{(1+\epsilon )h}[\pi ^*\psi _\tau ]_{\mathcal I}$. For a general $\eta >0$, choose $\epsilon h\leq \eta $. Compatibility and the nested-background identity of proposition 16.29.5 then give
\[ \Gamma _\tau ^\eta =P_{h+\eta }[\Gamma _\tau ^{\epsilon h}]_{\mathcal I} =P_{h+\eta }[\pi ^*\psi _\tau ]_{\mathcal I}. \]
$\square$
Pome
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