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.
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
An upper semi-continuous function
is said to be plurisubharmonic if for any holomorphic map
denotes the unit disk.
is plurisubharmonic if and only if the hermitian matrix
, called Levi matrix, with entries is positive semidefinite.
Relation to Kähler manifold: On n-dimensional complex Euclidean space
is equal to the standard Kähler form on
up to constant multiples.
satisfies for some Kähler form
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 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[1] and Pierre Lelong.
[2] In several complex variables, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy and Stein manifolds.
The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by Kiyoshi Oka in 1942.
is called exhaustive if the preimage
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.
Conversely, any Stein manifold admits such a function.