Plurisubharmonic function: Difference between revisions

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==Formal definition==

A [[function (mathematics)|function]]

:<math>f \colon G \to {\mathbb{R}}\cup\{-\infty\},</math>

with ''___domain'' <math>G \subset {\mathbb{C}}^n</math>

is called '''plurisubharmonic''' if it is [[semi-continuous function|upper semi-continuous]], and for every [[complex number|complex]] line

:<math>\{ a + b z \mid z \in {\mathbb{C}} \}\subset {\mathbb{C}}^n</math> with <math>a, b \in {\mathbb{C}}^n</math>

the function <math>z \mapsto f(a + bz)</math> is a [[subharmonic function]] on the set

:<math>\{ z \in {\mathbb{C}} \mid a + b z \in G \}.</math>

In ''full generality'', the notion can be defined on an arbitrary [[complex manifold]] or even a [[Complex analytic space]] <math>X</math> as follows. An [[semi-continuity|upper semi-continuous function]]

:<math>f \colon X \to {\mathbb{R}} \cup \{ - \infty \}</math>

is said to be plurisubharmonic if and only if for any [[holomorphic map]]

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Equivalently, a <math>C^2</math>-function ''f'' is plurisubharmonic if and only if <math>\sqrt{-1}\partial\bar\partial f</math> is a [[positive form|positive (1,1)-form]].

== Examples ==

'''Relation to Kähler manifold:''' On n-dimensional complex Euclidean space <math>\mathbb{C}^n</math> , <math>f(z) = |z|^2</math> is plurisubharmonic. In fact, <math>\sqrt{-1}\partial\overline{\partial}f</math> is equal to the standard [[Kähler form]] on <math>\mathbb{C}^n</math>　 up to constant multiplies. More generally, if <math>g</math> satisfies

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It is nothing but [[Dirac measure]] at the origin 0 .

== History ==

Plurisubharmonic functions were defined in 1942 by

[[Kiyoshi Oka]] <ref name=oka> K. Oka, ''Domaines pseudoconvexes,'' Tohoku Math. J. '''49''' (1942), 15&ndash;52.</ref> and [[Pierre Lelong]]. <ref> P. Lelong, ''Definition des fonctions plurisousharmoniques,'' C. R. Acd. Sci. Paris '''215''' (1942), 398&ndash;400.</ref>

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*The inequality in the usual [[semi-continuity]] condition holds as equality, i.e. if <math>f</math> is plurisubharmonic then

: <math>\limsup_{x\to x_0}f(x) =f(x_0)</math>

(see [[limit superior and limit inferior]] for the definition of ''lim sup'').

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*Therefore, plurisubharmonic functions satisfy the [[maximum principle]], i.e. if <math>f</math> is plurisubharmonic on the [[connected space|connected]] open ___domain <math>D</math> and

: <math>\sup_{x\in D}f(x) =f(x_0)</math>

for some point <math>x_0\in D</math> then <math>f</math> is constant.

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In [[complex analysis]], plurisubharmonic functions are used to describe [[pseudoconvexity|pseudoconvex domains]], [[___domain of holomorphy|domains of holomorphy]] and [[Stein manifold]]s.

== Oka theorem ==

The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by [[Kiyoshi Oka]] in 1942. <ref name=oka> K. Oka, ''Domaines pseudoconvexes,'' Tohoku Math. J. 49 (1942), 15-52.</ref>

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* 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.

==External links==

* {{springer|title=Plurisubharmonic function|id=p/p072930}}

== Notes ==

<references />