Descent also applies to potentials and preserves both ordinary singularity types and multiplier ideals, so the singularity envelopes in these backgrounds are computed on the base. The three-dimensional Monge–Ampère measures of these potentials nevertheless vanish, because their currents have only two horizontal directions.
We use the ordinary singularity order $\preceq $ and equivalence $\sim $ of definition 16.7.1, the ideal order $\preceq _{\mathcal I}$ and equivalence $\sim _{\mathcal I}$, which compare the multiplier ideals $\mathcal I(ku)$ at all weights $k>0$, and the ordinary and multiplier-ideal envelopes $P_\theta [u]$ and $P_\theta [u]_{\mathcal I}$ of definition 5.40.4, definition 18.4.1.
Proposition 10.19.4.2555699F5657 Let $a,b\geq 0$, put $\theta _ B=a\sigma _1+b\sigma _2$ and $\theta =\pi ^*\theta _ B$. Pullback is a bijection
\[ \mathrm{PSH}(B,\theta _ B)\longrightarrow \mathrm{PSH}(X,\theta ),\qquad v\longmapsto v\circ \pi . \]
It preserves the ordinary singularity order and equivalence, as well as the ideal order and equivalence. For every $v\in \mathrm{PSH}(B,\theta _ B)$,
\[ P_\theta [v\circ \pi ]=P_{\theta _ B}[v]\circ \pi , \qquad P_\theta [v\circ \pi ]_{\mathcal I}=P_{\theta _ B}[v]_{\mathcal I}\circ \pi . \]
Nevertheless, every $u\in \mathrm{PSH}(X,\theta )$ satisfies
\[ \mathrm{MA}^{\mathrm{ext}}_\theta (u)=0. \]
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.
$\square$
Thus the ordinary and ideal envelopes coincide in these backgrounds exactly when they coincide on the base, while every three-dimensional Monge–Ampère measure vanishes. Criteria from the closed theory that require positive volume do not apply to these classes, which are not big. The Hermitian background $\omega $ behaves differently: its Monge–Ampère equations with smooth positive densities are solvable, even though the infimum of its masses is zero.
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.