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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
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 X 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 f is of (differentiability) class C^2, then f is plurisubharmonic if and only if the hermitian matrix, called Levi matrix, with entries is positive semidefinite. Equivalently, a C^2-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 multiples. More generally, if g satisfies for some Kähler form \omega, then g 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 f 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
History
Plurisubharmonic functions were defined in 1942 by Kiyoshi Oka and Pierre Lelong.
Properties
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 \omega 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.
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