Theorem 5.37.33F1FC2F620B4

Theorem 5.37.3.3F1FC2F620B4 Let $u,v$ be plurisubharmonic functions near a point of $\mathbb C^ n$. By a local modification we mean a modification $\pi $, with smooth source, of a neighborhood of this point.

1. The inclusions $\mathcal I(ku)\subseteq \mathcal I(kv)$ for every real $k>0$ are equivalent to the inequalities $\nu (u\circ \pi ,y)\geq \nu (v\circ \pi ,y)$ at every point $y$ of the source of every local modification $\pi $. Likewise, equality of the multiplier ideals at every positive weight is equivalent to equality of all these Lelong numbers.

2. If $p_ j$ is an increasing sequence of plurisubharmonic functions converging almost everywhere to a plurisubharmonic function $p$, then, for each point $x$ and each $\lambda >0$,

   \[ \mathcal I(\lambda p_ j)_ x=\mathcal I(\lambda p)_ x \quad \text{for all sufficiently large }j. \]