Proposition 10.19.42555699F5657

Proof. The form $\theta _ B$ is closed, and $\pi $ is a proper holomorphic submersion with connected fibers, so proposition 5.43.5 gives the asserted bijection. Since $\pi $ is surjective, an ordinary singularity inequality holds upstairs exactly when it holds downstairs, with the same constant.

By lemma 5.37.7, multiplier ideals of potentials on $B$ pull back to those of the composed potentials on $X$, at every positive weight. Restriction to local holomorphic sections shows conversely that an inclusion of multiplier ideals on $X$ implies the corresponding inclusion on $B$. Hence the ideal orders and equivalences correspond as well.

The competitors defining both envelopes correspond under the bijection, including the bound by zero and the required ordinary or ideal condition, and their pointwise suprema commute with pullback. Upper semicontinuous regularization also commutes with pullback: in a product chart, the supremum of a pulled-back function over a sufficiently small product neighborhood equals its supremum over the corresponding base neighborhood. This proves the two envelope identities.

Finally, write $u=v\circ \pi $ and, for positive integers $j$, set $u_ j=\max \{ u,-j\} $, which descends to $v_ j=\max \{ v,-j\} $. Since $\theta _ B$ is semipositive, these are bounded admissible potentials. By lemma 5.28.4, $(\theta +\mathrm{dd}^{\mathrm{c}}u_ j)^3=0$ because the base has dimension two. The definition of the extended non-pluripolar measure by increasing truncations now proves the last assertion. $\square$