Proposition 18.26.175F93C15013C0

Proof. Place the polydisk of lemma 18.26.15 in a holomorphic coordinate chart of $X$. Choose a smooth cutoff function $0\leq \chi \leq 1$, compactly supported in that polydisk and equal to one on a neighborhood of $S$, and extend $H=\chi h$ by zero. Then $H$ is quasi-plurisubharmonic and not identically $-\infty $. Indeed, outside $S$ the gradients of $h$ are locally bounded, so the terms containing derivatives of $\chi $ in

have bounded coefficients on their compact support, while the first term is positive. Hence $\mathrm{dd}^{\mathrm{c}}H\geq -C\omega $ for some $C>0$. Take $0<t<1/C$ and put $u=tH$. Then $u\leq 0$, $\sup _ Xu=0$, the pole set of $u$ is $S$, and

Set $u_ k=\max (u,-k)$. By Stokes' theorem in the closed background, the bounded currents $\omega +\mathrm{dd}^{\mathrm{c}}u_ k$ have total Monge–Ampère mass $A=\int _ X\omega ^ n$. Near $S$, where $\chi =1$,

For large $k$, the whole support of the measure on the right lies in this neighborhood; these measures have mass $t^ nM$ and their supports converge to $S$. Take a subsequence along which both these local measures and the global measures $(\omega +\mathrm{dd}^{\mathrm{c}}u_ k)^ n$ converge weakly. The global limit has mass at least $t^ nM$ on $S$. Off $S$, the function $u$ is locally bounded, and $u_ k=u$ eventually on every compact set, so there the global limit is the non-pluripolar measure of $u$. It follows that

The lower bound follows from the current inequality above and bounded local products off the Lebesgue-null set $S$.

In a closed Kähler background, bounded changes of a potential do not change its non-pluripolar mass. Both singularity volumes of $u$ therefore equal this positive number. By theorem 13.19.5, $p=P_\omega [u]$ is a normalized ordinary model with the same volumes. In particular $p\ne 0$, and $d_\omega (p,0)=A-\overline{\operatorname {vol}}_\omega (p)>0$. Taking for $\delta $ half the lower volume of $p$ puts both models in $\mathcal M_\delta (\omega )$. Also $u\leq p\leq 0$, so the pole set of $p$ is contained in $S$.

Fix a smooth modification $\pi $, and let $E_ i$ be the irreducible divisorial components of its exceptional locus, away from which $\pi $ is an isomorphism. The polar current $R$ of $p\circ \pi $ is closed and positive by proposition 5.46.7. On the set where $\pi $ is an isomorphism, $R$ is carried by the inverse image of $S$, which has Hausdorff dimension zero in each relatively compact chart; by corollary 5.34.3, $R$ vanishes there. The support theorem for closed positive currents therefore gives

see [Chapter III, Corollaries 2.11 and 2.14, pp. 141–143, DemCADG]. Components of codimension at least two carry no closed $(1,1)$-current of order zero, and the sum is finite by compactness. Since $R\leq \pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(p\circ \pi )$, its coefficients satisfy $b_ i\leq \nu (p\circ \pi ,E_ i)$.

If $\pi (E_ i)$ is not contained in $S$, some regular point of $E_ i$ maps outside $S$. Near such a point $u\circ \pi $, and hence $p\circ \pi $, is bounded below, so the generic Lelong number of $p\circ \pi $ along $E_ i$ vanishes and $b_ i=0$. Every remaining image $\pi (E_ i)$ is an irreducible analytic set contained in the set $S$ of Hausdorff dimension zero, hence a point $a=(a_1,\ldots ,a_ n)$. Near that point, $u=t\max _ jg(z_ j)$. At a general point of $E_ i$, with a local equation $\zeta $ of $E_ i$, the first coordinate satisfies

A local holomorphic $m$th root of the unit $f$ turns this expression into $\zeta '^ m$ for another local equation $\zeta '$ of $E_ i$. Taking circular means gives

Since $p\circ \pi \geq u\circ \pi \geq t\, g(z_1\circ \pi )$ there, again $b_ i=0$. Thus $R=0$ for every modification.

The same local comparison gives zero Lelong numbers for $u$, and hence for $p\geq u$, at every point. Skoda's local integrability theorem, recalled in [SGPT], then makes $\mathrm e^{-sp}$ locally integrable for every $s>0$, so all multiplier ideals of $p$ are trivial. The ideal envelope of $p$ is therefore the zero model, and since $p\ne 0$, $p$ is not an ideal model. The zero model has analytic singularities, and $\delta _\omega ^{\mathrm{pol}}(p,0)=0$ puts $p$ in the closure of the analytic models for the polar comparison. This proves the final assertions. $\square$