Plurisubharmonic function


In mathematics, plurisubharmonic 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 plurisubharmonic functions can be defined in full generality on complex analytic spaces.

Formal definition

A function
with domain
is called plurisubharmonic if it is upper semi-continuous, and for every complex line
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 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 -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 multiples. 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.
More Examples
Plurisubharmonic functions were defined in 1942 by
Kiyoshi Oka and Pierre Lelong.

Properties

then is plurisubharmonic.
.
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.
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.