PomeLemma 6.14.7.7110958B611B Let $x$ be a point of a complex manifold $X$ at which $X$ has positive dimension, let $\mathfrak a\subset \mathcal O_ X$ be a coherent ideal sheaf with $\mathfrak a_ x\neq 0$, generated near $x$ by $g_1,\ldots ,g_ s$, and let $c>0$. If $w$ is quasi-plurisubharmonic near $x$ and $w=c\log \sum _ i|g_ i|^2+O(1)$ near $x$, then $\nu (w,x)=c\operatorname {ord}_ x\mathfrak a$.
Lemma 6.14.77110958B611B
Proof. Choose a connected coordinate neighborhood $V$ of $x$ on which the $g_ i$ are defined and generate $\mathfrak a$, on which $w=v+f$ with $v$ plurisubharmonic and $f$ smooth and bounded, and on which $|w-c\log \sum _ i|g_ i|^2|\leq M$ for a constant $M$. Let $I$ be the set of indices $i$ for which the germ of $g_ i$ at $x$ is not zero. It is not empty, since these germs generate $\mathfrak a_ x\neq 0$, and for $i\notin I$ the function $g_ i$ vanishes identically on $V$, by the identity theorem. Since $0\leq \log \sum _{i\in I}|g_ i|^2-\max _{i\in I}\log |g_ i|^2\leq \log s$, the function $v$ differs from $c\max _{i\in I}\log |g_ i|^2$ by a bounded function on $V$. The latter function is plurisubharmonic by proposition 5.10.3, proposition 5.9.5, so proposition 6.7.1 and proposition 6.7.9 give
\[ \nu (w,x)=\nu (v,x)=c\, \nu \Bigl (\max _{i\in I}\log |g_ i|^2,x\Bigr )=c\min _{i\in I}\operatorname {ord}_ xg_ i=c\operatorname {ord}_ x\mathfrak a, \]
where the first equality is the definition of Lelong numbers of quasi-plurisubharmonic functions (corollary 6.9.3) and the last one is the description of $\operatorname {ord}_ x\mathfrak a$ by generators in section 6.9. $\square$
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