Lemma 18.20.1.F61F12847624 For $u_ i\in \mathrm{PSH}(C_ i,\sigma _ i)$ put
\[ u=\sum _ i u_ i\circ \pi _ i,\qquad U=\Psi +u\circ p, \qquad q_ i=P_{\sigma _ i}[u_ i],\quad r_ i=P_{\sigma _ i}[u_ i]_{\mathcal I}. \]
Then
\[ \begin{aligned} P_{\Theta _\alpha }[U]& =\Psi +\sum _ iq_ i\circ \pi _ i\circ p,\\ P_{\Theta _\alpha }[U]_{\mathcal I} & =\Psi +\sum _ ir_ i\circ \pi _ i\circ p. \end{aligned} \]
Every ordinary representative of either of these envelopes has zero extended mass, so their upper and lower singularity volumes are zero.
Proof.
On $B$, sums of ordinary competitors on the curves give the lower bound $P_{\Gamma _\alpha }[u\circ \pi ]\geq \sum _ iq_ i\circ \pi _ i\circ \pi $. For the upper bound, replace a competitor by its maximum with a sufficiently low translate of $u\circ \pi $. Its restriction to every slice $Y\times \{ t\} $ is then not identically $-\infty $, and since $\alpha $ restricts to zero there, it is a $\gamma _\sigma $-plurisubharmonic ordinary competitor for $u$. The case of the zero obstacle in proposition 18.16.2 bounds it by $\sum _ iq_ i\circ \pi _ i$. This proves the identity for the ordinary envelope on $B$.
Take the atomic Green representatives $g_ i$ of the proof of proposition 18.16.2. The pullback $g$ of their sum to $B$ belongs to $\mathrm{PSH}(B,\Gamma _\alpha )$, is multiplier-ideal equivalent to $u\circ \pi $, and is ordinarily equivalent to $\sum _ ir_ i\circ \pi _ i\circ \pi $. Its complex Hessian is a countable current of divisors along coordinate fibers, minus a smooth form; this divisor current has finite mass, as one sees with a fixed product Hermitian metric and the finite atomic masses on the curves. By lemma 5.38.5, every ideal competitor is therefore at least as singular as $g$; if the divisor current is zero, $g$ is smooth and this is immediate by compactness. Thus every such competitor is bounded by $P_{\Gamma _\alpha }[g]=P_{\Gamma _\alpha }[\sum _ ir_ i\circ \pi _ i\circ \pi ]$, which equals $\sum _ ir_ i\circ \pi _ i\circ \pi $ by the ordinary identity just proved and $P_{\sigma _ i}[r_ i]=r_ i$. This sum is itself an ideal competitor, which proves the identity for the ideal envelope on $B$.
Apply proposition 18.14.1 to $M\to B$ and the divisor $B\times D$ with weight one. Its induced zero obstacle is zero because $\sup _ Z\psi =0$, which gives both identities upstairs. The mass assertion of the same proposition, applied to each of the two envelopes, shows that every representative of its ordinary type has zero mass, so both singularity volumes are zero.
$\square$
Pome
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