Definition 9.9.17A62ADAFC2A7

Definition 9.9.1.7A62ADAFC2A7 Let $X$ be a complex space. A function $u\colon X\to [-\infty ,+\infty )$ is locally plurisubharmonic if every point of $X$ has a local model, which identifies a neighborhood of the point with an analytic subset of a domain $\Omega \subset \mathbb C^ N$, and a ball $V\subset \Omega $ around the point such that $u$ is the restriction to $X\cap V$ of a function on $V$ that is plurisubharmonic or identically $-\infty $. We write $u\in \mathrm{PSH}(X)$ if, in addition, $u$ is not identically $-\infty $ on any irreducible component of $X$, the components being defined as in corollary 9.3.16. Locally plurisubharmonic functions are allowed to be identically $-\infty $ on some components; we use them in this generality when only local properties are needed.