Proposition 18.14.11195E1B9B18D

Proof. We first study the induced obstacle and bound the base competitors, then describe all competitors upstairs by divisorial removal and descent, and finally prove the mass assertion.

Step 1: the induced obstacle. Since $p$ is open, $Y\setminus Z=p(M\setminus H)$ is open and nonempty. The function $F-\Psi $ is bounded below by $\min _ MF-\sup _ M\Psi $ and tends to $+\infty $ near $H$, because $\Psi \to -\infty $ there. On every fiber that meets $M\setminus H$, its infimum is therefore attained outside $H$. A local holomorphic section through a minimizer gives upper semicontinuity of $\widehat F$. Conversely, let $y_ j\to y$ with bounded fiber minima. By compactness, a subsequence of the minimizers converges, and its limit lies outside $H$ because $F-\Psi $ remains bounded along it; this gives lower semicontinuity. If $y\in Z$, the same limit would lie in $H$, a contradiction. Thus $\widehat F\to +\infty $ at $Z$, and $\widehat F$ is continuous in the extended sense.

Step 2: a uniform bound for the base competitors. Choose a relatively compact nonempty ball $U$ in the base and a holomorphic section $s$ of $p$ on a neighborhood of $\overline U$ with image outside $H$. There is a finite $C_ U$ with $\widehat F\leq F\circ s-\Psi \circ s\leq C_ U$ on $U$. Fix a smooth probability volume form $\mathrm dV$ on $Y$. For every $b\in \mathrm{PSH}(Y,\beta )$ below $\widehat F$, normalized compactness in theorem 5.25.2 gives

Hence every such $b$ satisfies $\sup _ Yb\leq K_ F=C_ U+C_\beta /\int _ U\mathrm dV$. A sufficiently low translate of $w$ belongs to both base competitor families. Their regularized suprema are therefore $\beta $-plurisubharmonic functions, not identically $-\infty $, and by the extended continuity of $\widehat F$ they still lie below $\widehat F$.

Step 3: the competitors upstairs. The pullback of $w$ is quasi-plurisubharmonic, so its generic Lelong numbers are nonnegative, and $\nu (u,H_ j)=\lambda _ j+\nu (w\circ p,H_ j)\geq \lambda _ j$. Let $v$ be an ordinary competitor for $u$, or an ideal competitor with $v\sim _{\mathcal I}u$. By the ordinary order or by theorem 5.37.3, its generic Lelong number along $H_ j$ is at least $\lambda _ j$. Divisorial removal in lemma 5.38.1 gives

Hence $v-\Psi $ has a quasi-plurisubharmonic representative $a$ satisfying $p^*\beta +\mathrm{dd}^{\mathrm{c}}a\geq 0$. Descent along the compact connected fibers, proposition 5.43.5, gives $a=b\circ p$ for a unique $b\in \mathrm{PSH}(Y,\beta )$, and $v=\Psi +b\circ p$ for the canonical representatives. Conversely, every potential of this form is $\Theta $-plurisubharmonic.

For such $v$, one has $v\preceq u$ if and only if $b\preceq w$. Indeed, canceling $\Psi $ outside $H$ gives the base inequality almost everywhere, by Fubini's theorem in product charts, hence everywhere by lemma 5.24.3; the reverse implication follows by pullback and addition. Similarly, $v\sim _{\mathcal I}u$ if and only if $b\sim _{\mathcal I}w$: lemma 5.37.4 permits both addition and cancellation of $\Psi $, the pullback formula in lemma 5.37.7 then gives one direction, and restriction of the extended ideals along any local section gives the other. This comparison of ideals does not need a section outside $H$.

If the divisors have relative simple normal crossings, the ideals have an explicit form. Here this means that near every point there are product coordinates $(y,z)$ with $p(y,z)=y$ in which the components through the point are distinct coordinate hyperplanes in the fiber variables. Then

To see this, work in such coordinates. Up to a smooth correction, $\Psi $ is the sum of the weights $\lambda _ j\log |z_ j|^2$ over the components through the point. Since $b$ is locally bounded above, integrability forces vanishing to order at least $\lfloor t\lambda _ j\rfloor $ along each hyperplane. After this product of coordinate powers is factored out, the remaining fiber weight is

It is integrable and bounded below by a positive constant on a small polydisk. Integrability of the quotient against this weight therefore implies integrability against the pullback of $\mathrm e^{-tb}$ alone, and lemma 5.37.7 puts the quotient in the extension of $\mathcal I(tb)$. Conversely, local generators of that ideal are weighted-integrable against the integrable product weight, by Fubini's theorem. Outside $H$ the formula is the same pullback lemma. For one smooth divisor of weight one transverse to the fibers, the first factor is the ideal sheaf of the divisor raised to the power $\lfloor t\rfloor $.

Step 4: the envelope identities. The bound $v\leq F$ is equivalent to $b\leq \widehat F$, with no constraint over $Z$. The competitor families upstairs and on the base therefore correspond to each other, which gives (18.14.2) outside $H$, since regularization commutes with pullback in product coordinates there. On $H$ both sides equal $-\infty $: every competitor is at most $\Psi +K_ F$, by the bound $b\leq K_ F$, and this upper semicontinuous function equals $-\infty $ on $H$, so regularization does not change the values there.

Step 5: equality below all obstacles. Suppose that the two base envelopes agree below every smooth obstacle. Uniform obstacle comparison extends this equality to continuous real obstacles. The functions $G_ j=\min (\widehat F,j)$ are continuous and real-valued. Every base competitor below $\widehat F$ has finite supremum and hence lies below $G_ j$ for every sufficiently large integer $j$. Each full competitor family is thus the union of its subfamilies below the $G_ j$, and equality for all $G_ j$ gives equality below $\widehat F$ after taking regularized suprema. By the reduction, the envelopes upstairs agree below every smooth obstacle.

Step 6: zero mass. Take any $b\in \mathrm{PSH}(Y,\beta )$ and write $a=\Psi +b\circ p$. On a relatively compact chart outside $H$, the functions $b_ j=\max (b,-j)$ are locally bounded and quasi-plurisubharmonic, and there

Since $n>d$, the $n$th bounded product of this current is zero by lemma 5.28.4. This identity does not require $b_ j$ to be $\beta $-plurisubharmonic: proposition 5.28.2 defines the products for locally bounded quasi-plurisubharmonic potentials. By plurifine locality, this zero product agrees with $\mathrm{MA}^{\mathrm{ext}}_\Theta (a)$ on $\{ b\circ p>-j\} $ in the chart. These sets cover the locus where $b\circ p$ is finite, while the remaining pole set and $H$ receive no extended mass. Thus $\mathrm{MA}^{\mathrm{ext}}_\Theta (a)=0$.

By divisorial removal and descent, every representative ordinarily equivalent to $u$ has this form. By the identity already proved, the ideal model is also of this form, with base potential $P_{\beta ,\widehat0}[w]_{\mathcal I}$, where $\widehat0$ is the induced obstacle for $F=0$. The same removal and descent apply to every representative of its ordinary type. All these masses are zero, which proves both volume assertions. $\square$