Example 16.29.24.0631D73C4C88 Fix $0<\lambda _0<\lambda _*<1$ and $r>0$ with $r(c-a)<\lambda _*-\lambda _0$. For $\tau <c$, set
\[ \lambda (\tau )=\lambda _0+r\max (\tau -a,0),\qquad u_\tau =\lambda (\tau )u_0, \]
and, for every Hermitian form $\eta >0$ on $X$, define
\[ \Gamma _\tau ^\eta =P_{h+\eta }[\pi ^*u_\tau ]_{\mathcal I}. \]
These components form an envelope system with $E$-invariant scalar slices. Its transverse limiting curve is $\psi _\tau =P_\kappa [u_\tau ]_{\mathcal I}$, and
\[ \underline{\operatorname {vol}}_\kappa (\psi _\tau ) =\overline{\operatorname {vol}}_\kappa (\psi _\tau ) =(1-\lambda (\tau ))^2>0. \]
The system is recovered from its limit. Its ideal data cannot be represented by analytic singularities.
Proof.
The function $\lambda $ is positive, increasing and convex, and $u_0\leq 0$, so $u_\tau $ decreases, is pointwise concave and is constant for $\tau \leq a$. Moreover,
\[ \kappa +\mathrm{dd}^{\mathrm{c}}u_\tau =(1-\lambda (\tau ))\kappa +\lambda (\tau )\sum _ jq_ j[H_ j]>0, \qquad h+\mathrm{dd}^{\mathrm{c}}\pi ^*u_\tau \geq (1-\lambda _*)h. \]
Set $q_\tau =P_\kappa [u_\tau ]_{\mathcal I}$. The order, scaling and addition inequalities of proposition 18.4.6 show that $q_\tau $ remains decreasing and concave; it is constant on the same left half-line. Thus $q\in \mathscr T_{a,c}(\kappa )$. Apply the same argument upstairs to $Q_\tau =P_ h[\pi ^*u_\tau ]_{\mathcal I}$. The fiber-envelope identity gives $Q_\tau =\pi ^*q_\tau $, and the nested-background identity identifies the displayed family with $\mathsf E(Q)$. It is therefore an envelope system, its limit is $Q$ by $\mathsf R\mathsf E=\mathrm{id}$, and its scalar slices are invariant.
Apply 13.24.8 with background $\kappa $, coefficients $a_ j=\lambda (\tau )q_ j$ and residual $(1-\lambda (\tau ))\kappa $. It gives the asserted exact lower and upper volumes of $q_\tau $. The transverse recovery theorem applies, also confirming recovery directly.
For every $j$, the divisorial coefficient of $u_\tau $ along $H_ j$ is positive. Projection preserves all ideals and hence these coefficients. Analytic singularities on a compact manifold have only finitely many divisorial components, since their defining coherent ideals have this property. Thus no analytic singularity type has the same all-weight ideals. The same conclusion holds upstairs along the distinct divisors $\pi ^{-1}(H_ j)$, by submersion pullback.
$\square$
Pome
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