PomeDefinition 5.16.1.A8F9CC5C117D Let $\varphi \in \mathrm{PSH}(X)$ and $x\in X$. The Lelong number $\nu (\varphi ,x)$ of $\varphi $ at $x$ is defined as follows: If $x$ is a point on a $0$-dimensional connected component of $X$, we set
\[ \nu (\varphi ,x)=0. \]
Otherwise, take an open neighborhood $U$ of $x$ in $X$ and a biholomorphic map $F\colon U\rightarrow \Omega $, where $\Omega $ is a domain in $\mathbb {C}^ n$. Then we define
D4BCC2E76B62
5.16.2
\begin{equation} \nu (\varphi ,x)\coloneqq \sup \left\{ \gamma \in \mathbb {R}_{\geq 0}: \varphi (F^{-1}(y))\leq \gamma \log |y-F(x)|^2+\mathcal{O}(1)\textup{ as }y\to F(x) \right\} . \end{equation}
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