Theorem 16.29.17.761E4875D2B4 For the Kähler background $\kappa $ of lemma 16.29.16 and every $a<c$, there is a system $\boldsymbol \Gamma \in \mathscr S_{a,c}(\kappa )$ for which
\[ \mathsf E\mathsf R(\boldsymbol \Gamma ) \neq \boldsymbol \Gamma . \]
Every slice $\Gamma ^\eta _\tau $, $\tau <c$, of every component has multiplier ideals
\[ \mathcal I(t\Gamma ^\eta _\tau ) =\mathcal O_ X(-\lfloor t\rfloor F)\qquad (t>0), \]
whereas its limiting slices have multiplier ideals
\[ \mathcal I\bigl (t(\mathsf R\boldsymbol \Gamma )_\tau \bigr ) =\mathcal O_ X(-\lfloor t\rfloor (F+D)). \]
In particular, on the system space $\mathscr S_{a,c}(\kappa )$, which carries no mass condition, $\mathsf E$ is not surjective and $\mathsf R$ is not injective.
Proof.
For each Hermitian form $\eta $, the nef approximation of lemma 16.29.16 gives a smooth function $f_\eta $ such that $\theta +\eta +\mathrm{dd}^{\mathrm{c}}f_\eta \geq 0$. Then
\[ u_\eta =\psi _ F+f_\eta \in \mathrm{PSH}(X,\kappa +\eta ). \]
Let every slice $\Gamma ^\eta _\tau $, $\tau <c$, of the component $\Gamma ^\eta $ be $q_\eta =P_{\kappa +\eta }[u_\eta ]_{\mathcal I}$, with cutoff $c$. The multiplier ideals of $q_\eta $ are those of $\psi _ F$ and do not depend on the smooth function $f_\eta $. Hence the compatibility condition follows by comparing these ideals in the larger background, and the constant curves form a system with the required parameters.
Let $q$ be its limiting slice under $\mathsf R$, and let $\nu (q,F)$ be its generic Lelong number along $F$. Each $q_\eta $ has generic Lelong number one along $F$, so lemma 5.38.7, applied to a decreasing cofinal sequence, gives $\nu (q,F)\geq 1$. By lemma 5.38.1,
\[ \kappa +\mathrm{dd}^{\mathrm{c}}q-[F]\geq 0. \]
By the Poincaré–Lelong identity for $\psi _ F$, this current equals $\theta +\mathrm{dd}^{\mathrm{c}}(q-\psi _ F)$, where the difference $q-\psi _ F$ is first taken as a locally integrable function. Since the current is positive, this difference has a canonical $\theta $-plurisubharmonic representative, and by lemma 16.29.16 it differs from $\psi _ D$ by a constant. Hence
\[ q=\psi _ F+\psi _ D+C. \]
The divisors $F$ and $D$ are smooth and meet transversely, since one is a fiber and the other a section. A direct integrability calculation in local product coordinates therefore gives the stated ideals for every real $t>0$. In particular, the Lelong number at a generic point of $D$ has increased from zero to one, and proposition 16.29.11 shows that recovery fails.
Finally, since $\mathsf R\mathsf E=\mathrm{id}$, the system $\boldsymbol \Gamma $ and the realized system $\mathsf E\mathsf R\boldsymbol \Gamma \neq \boldsymbol \Gamma $ have the same image under $\mathsf R$. Thus $\mathsf R$ is not injective, and $\boldsymbol \Gamma $ is not in the image of $\mathsf E$.
$\square$
Pome
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