Plurisubharmonic function: Difference between revisions

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{{Short description|Type of function in complex analysis}}

In [[mathematics]], '''plurisubharmonic''' functions (sometimes abbreviated as '''psh''', '''plsh''', or '''plush''' functions) form an important class of [[function (mathematics)|functions]] used in [[complex analysis]]. On a [[Kähler manifold]], plurisubharmonic functions form a subset of the [[subharmonic function]]s. However, unlike subharmonic functions (which are defined on a [[Riemannian manifold]]) plurisubharmonic functions can be defined in full generality on [[complex analytic space]]s.

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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 for any [[holomorphic map]]

<math>\varphi\colon\Delta\to X</math> the function <math>f\circ\varphi \colon \Delta \to {\mathbb{R}} \cup \{ - \infty \}</math> is [[subharmonic function|subharmonic]], where <math>\Delta\subset{\mathbb{C}}</math> denotes the [[unit disk]].

===Differentiable plurisubharmonic functions===

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'''More Examples'''

* If <math>f</math> is an analytic function on an [[open set]], then <math>\log|f|</math> is plurisubharmonic on that open set.

* [[Convex function|Convex functions]] are plurisubharmonic.

* If <math>\Omega</math> is a [[___domain of holomorphy]] then <math>-\log (dist(z,\Omega^c))</math> is plurisubharmonic.

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:* if <math>f_1</math> and <math>f_2</math> are plurisubharmonic functions, then the sum <math>f_1+f_2</math> 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.

*If <math>f</math> is plurisubharmonic and <math>\phivarphi:\mathbb{R}\to\mathbb{R}</math> aan increasing, convex function then <math>\phivarphi \circ f</math> is plurisubharmonic. (<math>\varphi(-\infty)</math> is interpreted as <math>\lim_{x \rightarrow -\infty} \varphi(x)</math>.)

*If <math>f_1</math> and <math>f_2</math> are plurisubharmonic functions, then the function <math>\max(f_1,f_2)</math> is plurisubharmonic.

*IfThe <math>f_1,f_2,\dots</math>pointwise islimit of a decreasing sequence of plurisubharmonic functions then its pointwise limit 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.<ref>R. E. Greene and H. Wu, ''<math>C^\infty</math>-approximations of convex, subharmonic, and plurisubharmonic functions'', Ann. Scient. Ec. Norm. Sup. 12 (1979), 47&ndash;84.</ref>

*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>.

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==Applications==

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

==Oka theorem==