Plurisubharmonic functions

Let $\mathrm{\Omega }\subseteq {\mathbb{C}}^m$ be a domain. A function $u:\mathrm{\Omega }\to [-\mathrm{\infty },\mathrm{\infty })$ is plurisubharmonic when:

1. $u$ is upper semicontinuous;
2. on no connected component of $\mathrm{\Omega }$ is $u$ identically $-\mathrm{\infty }$;
3. for every $a\in \mathrm{\Omega }$ and every nonzero $v\in {\mathbb{C}}^m$, the function $u_{a,v}(\lambda ):=u(a+\lambda v)$ is subharmonic or identically $-\mathrm{\infty }$ on each connected component of $\{\lambda \in \mathbb{C}:a+\lambda v\in \mathrm{\Omega }\}\mathrm{.}$

This is the standard upper-semicontinuous convention from This page uses the standard upper-semicontinuous subharmonic convention, transported from the plane definition Subharmonic functions on plane domains to affine complex lines.

The case $v=0$ is excluded because then the pullback is constant and carries no information. A line restriction is allowed to be identically $-\mathrm{\infty }$ even though $u$ itself is excluded from being identically $-\mathrm{\infty }$ on a component of $\mathrm{\Omega }$.