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 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 complex matrices|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. These can be readily generated by applying the ddbar lemma to Kähler forms on a Kähler manifold.
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

  • The set of plurisubharmonic functions has the following properties like a convex cone:
  • 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 an increasing convex function then is plurisubharmonic.
  • If and are plurisubharmonic functions, then the function is plurisubharmonic.
  • The pointwise limit of a decreasing sequence of plurisubharmonic functions is plurisubharmonic.
  • Every continuous plurisubharmonic function can be obtained as the limit of a 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.
  • Plurisubharmonic functions are subharmonic, for any Kähler metric.
  • Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if is plurisubharmonic on the domain and for some point then is constant.

    Applications

In several complex variables, 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 a Stein manifold. Conversely, any Stein manifold admits such a function.