16.24 Divisorial constraints under envelopesA7364A53065E

Proof. Condition (2) is a special case of (1). Assume (2) and let $E$ be a prime divisor over an open subset $U\subset X$ (possibly $U=X$). The inclusions (2) restrict to $U$, and a smaller ideal has a larger order along $E$, so $\operatorname {ord}_ E\mathcal I(tv)\geq \operatorname {ord}_ E\mathcal I(tu)$ for every positive integer $t$. Dividing by $t$ and letting $t\to \infty $, corollary 6.12.14, applied on $U$, gives $\nu (v,E)\geq \nu (u,E)$; this is (3). Conversely, assume (3), let $t>0$, and let $f$ be a germ of $\mathcal I(tv)$ at a point $x$. The valuative criterion, theorem 6.12.16, gives $\varepsilon >0$ and a neighborhood $V$ of $x$ such that

for every prime divisor $E$ over an open neighborhood of $x$ contained in $V$ whose center contains $x$; the second inequality is (3) for these divisors. Since $t\nu (u,E)=\nu (tu,E)$ by proposition 6.7.9, the same criterion, applied to $tu$, shows that $f\in \mathcal I(tu)$; this is (1). Applying the equivalence in both directions proves the assertions about equalities. $\square$

Proof. Let $\pi \colon Y\to U$ be a smooth model of an open subset $U\subset X$ (possibly $U=X$) on which $E$ is a divisor, and let $Z\subset Y$ be a nowhere dense analytic subset outside which $\pi $ is a biholomorphism onto an open subset of $U$; such a set exists by the definition of a smooth model, and it has Lebesgue measure zero. We first show that $v\circ \pi $ is the regularized supremum of the family $(v_ j\circ \pi )$. By theorem 5.10.2, a countable subfamily has the same regularized supremum $v$, and its increasing finite maxima $w_ k$ satisfy $\sup _ kw_ k=v$ almost everywhere. The functions $w_ k\circ \pi $ are $\pi ^*\theta $-plurisubharmonic, increase with $k$, and are locally uniformly bounded above; let $W$ be their regularized supremum, so that $W=\sup _ kw_ k\circ \pi $ almost everywhere by theorem 5.10.2. If $N\subset U$ is a Lebesgue null set outside which $\sup _ kw_ k=v$, then $\pi ^{-1}(N)\setminus Z$ is a null set, because $\pi $ restricts to a biholomorphism of $Y\setminus Z$ onto an open set; hence $\sup _ kw_ k\circ \pi =v\circ \pi $ outside the null set $(\pi ^{-1}(N)\setminus Z)\cup Z$. Thus the quasi-plurisubharmonic functions $W$ and $v\circ \pi $ agree almost everywhere, and therefore everywhere. Since every $v_ j$ is at most $v$, we get $W\leq (\sup _ jv_ j\circ \pi )^*\leq v\circ \pi =W$, so $v\circ \pi $ is the regularized supremum of the whole family $(v_ j\circ \pi )$.

It remains to compare Lelong numbers. Near a point $y\in E$, adding to all these functions the pullback of a smooth local potential of $\theta $ near $\pi (y)$ makes them plurisubharmonic, so proposition 6.7.9 gives $\nu (v\circ \pi ,y)=\inf _ j\nu (v_ j\circ \pi ,y)$ at every point $y\in E$. Taking the infimum over $y\in E$, which defines the generic Lelong number (proposition 6.9.7) and commutes with the infimum over $j$, we obtain the formula.

For the remaining assertions, let $\varphi $ and $\psi $ be quasi-plurisubharmonic functions on $X$, and recall from proposition 6.7.9 that Lelong numbers at a point are additive and positively homogeneous, and that the Lelong number of a maximum is the minimum of the Lelong numbers. Taking infima over $y\in E$, we obtain $\nu (c\varphi ,E)=c\nu (\varphi ,E)$ for $c>0$ and $\nu (\max \{ \varphi ,\psi \} ,E)=\min \{ \nu (\varphi ,E),\nu (\psi ,E)\} $. For sums we use that, by proposition 6.9.7, the infima defining $\nu (\varphi ,E)$, $\nu (\psi ,E)$ and $\nu (\varphi +\psi ,E)$ are attained at every point of $E$ outside a countable union of proper analytic subsets of $E$. Such a point exists, since by the same proposition the complement in $E$ of such a union is dense in $E$, and in particular nonempty. At a point of $E$ where the three infima are attained, the additivity at a point gives $\nu (\varphi +\psi ,E)=\nu (\varphi ,E)+\nu (\psi ,E)$. By these three rules and proposition 16.24.1, maxima, sums and positive multiples preserve $\sim _{\mathcal I}$, and by the formula just proved, so do upper envelopes of termwise equivalent families. $\square$