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

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

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 is subharmonic denotes the unit disk.

Differentiable plurisubharmonic functions

, called Levi matrix, with entries

is positive semidefinite.

-function f is a positive (1,1)-form.

Examples

Relation to Kähler manifold: is equal to the standard Kähler form satisfies

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: is a C∞-class function with compact support, then Cauchy integral formula says

which can be modified to

1. .

It is nothing but Dirac measure at the origin 0 .

More Examples

• is an analytic function on an open set is plurisubharmonic on that open set.

• Convex functions are plurisubharmonic.

• is a domain of holomorphy is plurisubharmonic.

History

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:

  is plurisubharmonic,

• 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.

• .)

• 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.

• Plurisubharmonic functions are subharmonic, for any Kähler metric.

• Therefore, plurisubharmonic functions satisfy the maximum principle is plurisubharmonic on the domain 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.

is called exhaustive. A plurisubharmonic function f is called strongly plurisubharmonic 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.

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.

• Klimek, Pluripotential Theory, Clarendon Press 1992.

External links

Notes