Plurisubharmonic function

Summary

A plurisubharmonic function on an open subset of $\C^n$ is an upper semicontinuous function whose restriction to every complex line is subharmonic. For $C^2$ functions this means that the complex Hessian $\left(\partial^2 u / \partial z_j \partial \bar z_k\right)$ is positive semidefinite, that is, the Levi form is nonnegative. If $f$ is holomorphic, then $\log\abs{f}$ is plurisubharmonic, and this is the main source of examples. Plurisubharmonic functions are the basic objects of pluripotential theory, and they characterise pseudoconvexity and domains of holomorphy.