Plurisubharmonic Function - Formal Definition

Formal Definition

A function

with domain is called plurisubharmonic if it is upper semi-continuous, and for every complex line

with

the function is a subharmonic function on the set

In full generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space as follows. An upper semi-continuous function

is said to be plurisubharmonic if and only if for any holomorphic map the function

is subharmonic, where denotes the unit disk.

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