# Plurisubharmonic function

In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Riemannian manifold) plurisubharmonic functions can be defined in full generality on complex analytic spaces.

## Formal definition

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.

### Differentiable plurisubharmonic functions

If is of (differentiability) class , then is plurisubharmonic if and only if the hermitian matrix , called Levi matrix, with entries

is positive semidefinite.

Equivalently, a -function f is plurisubharmonic if and only if is a positive (1,1)-form.

## Examples

Relation to Kähler manifold: On n-dimensional complex Euclidean space , is plurisubharmonic. In fact, is equal to the standard Kähler form on up to constant multiplies. More generally, if satisfies

for some Kähler form , then is plurisubharmonic, which is called Kähler potential.

Relation to Dirac Delta: On 1-dimensional complex Euclidean space , is plurisubharmonic. If is a C-class function with compact support, then Cauchy integral formula says

which can be modified to

.

It is nothing but Dirac measure at the origin 0 .

## History

Plurisubharmonic functions were defined in 1942 by Kiyoshi Oka [1] and Pierre Lelong.[2]

## Properties

• 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 is plurisubharmonic.

• Every continuous plurisubharmonic function can be obtained as the limit of a monotonically decreasing sequence of smooth plurisubharmonic functions. Moreover, this sequence can be chosen uniformly convergent.[3]
• 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).

for some point then is constant.

## Applications

In complex analysis, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy and Stein manifolds.

## Oka theorem

The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by Kiyoshi Oka in 1942.[1]

A continuous function is called exhaustive if the preimage is compact for all . A plurisubharmonic function f is called strongly plurisubharmonic if the form is positive, for some Kähler form on M.

Theorem of Oka: Let M be a complex manifold, admitting a smooth, exhaustive, strongly plurisubharmonic function. Then M is Stein. Conversely, any Stein manifold admits such a function.

## References

• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
• Robert C. Gunning. Introduction to Holomorphic Functions in Several Variables, Wadsworth & Brooks/Cole.