Corollary 20.2.50B590E31078F3

Proof. First observe that the set $\{ x\in X:\nu (\varphi ,x)>0\} $ is a countable union of proper analytic subsets by Siu's semicontinuity theorem theorem 5.16.7. It follows that a very general element in $\Lambda $ is not contained in this set.

Fix an ample line bundle $L$ so that there is a smooth Hermitian metric $h_ L$ such that $c_1(L,h_ L)+\mathrm{dd}^{\mathrm{c}}\varphi $ is a Kähler current. Thanks to theorem 20.2.45 and the connectedness assumption on $\Lambda $, we can find a co-pluripolar set $\Lambda '\subseteq \Lambda $ such that each $H\in \Lambda '$ satisfies the following:

1. 51A1BB87BE61

   $H$ is smooth and connected;

2. FB6AE3B7BE82

   $\nu (\varphi ,H)=0$;

3. 0B98F656C5AC

   $\mathcal{I}(k\varphi |_ H)=\operatorname{Res}_{H}(\mathcal{I}(k\varphi ))$ for all $k\in \mathbb {Z}_{>0}$.

Put $\alpha _ H\coloneqq c_1(L,h_ L)|_ H$. It follows from theorem 20.2.39 and theorem 18.1.46 that

Since $\varphi |_ H\preceq _ P \operatorname{Tr}_ H(\varphi )$ by proposition 20.2.12, we also have $\varphi |_ H\preceq _{\mathcal I}\operatorname{Tr}_ H(\varphi )$. Hence

The trace is $\mathcal I$-good, so the right-hand side agrees with its $P$-envelope. The equality of the masses above and theorem 16.3.15 then imply that the two sides have the same $P$-singularity type, hence