Plurisubharmonic Function - Properties

Properties

• The set of plurisubharmonic functions form a convex cone in the vector space of semicontinuous functions, i.e.

• if is a plurisubharmonic function and a positive real number, then the function is plurisubharmonic,
• if and are plurisubharmonic functions, then the sum is a plurisubharmonic function.

• Plurisubharmonicity is a local property, i.e. a function is plurisubharmonic if and only if it is plurisubharmonic in a neighborhood of each point.
• If is plurisubharmonic and a monotonically increasing, convex function then is plurisubharmonic.
• If and are plurisubharmonic functions, then the function is plurisubharmonic.
• If is a monotonically decreasing sequence of plurisubharmonic functions

then so is .

• Every continuous plurisubharmonic function can be obtained as a limit of monotonically decreasing sequence of smooth plurisubharmonic functions. Moreover, this sequence can be chosen uniformly convergent.
• The inequality in the usual semi-continuity condition holds as equality, i.e. if is plurisubharmonic then

(see limit superior and limit inferior for the definition of lim sup).

• Plurisubharmonic functions are subharmonic, for any Kähler metric.

• Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if is plurisubharmonic on the connected open domain and

for some point then is constant.