Lemma 5.38.81792691862B8

Proof. Since adding a constant does not change Lelong numbers, we may assume that $\sup _ Zv=0$. By theorem 5.25.2, these normalized potentials have a common $L^1$ bound. Cover $Z$ by finitely many triples of concentric coordinate balls, a smaller, an intermediate and a larger one, such that the smaller balls cover $Z$. On each larger ball choose a fixed smooth quadratic function $r$ with $\mathrm{dd}^{\mathrm{c}}r\geq \beta $, so that $p=v+r$ is plurisubharmonic there. The functions $p$ obtained in this way have uniformly bounded local $L^1$ norms.

Write $d=\dim Z$ and let $\kappa $ be the Euclidean form in the chart. Choose a nonnegative smooth cutoff $\chi $ supported in the larger ball and equal to one on an intermediate ball. Positivity and distributional integration by parts give

Every point of the smaller ball is the center of a Euclidean ball of a fixed radius contained in the intermediate ball. The local Lelong monotonicity formula, recalled in Proposition 6.7.1, bounds the Lelong number at this center by a fixed multiple of the trace mass of the intermediate ball. The finite cover gives one bound on $Z$. Since smooth additions and additive constants do not change Lelong numbers, this proves the assertion for the original potentials. $\square$