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| 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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| ==Formal definition==
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| A [[function (mathematics)|function]]
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| :<math>f \colon G \to {\mathbb{R}}\cup\{-\infty\},</math>
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| with ''domain'' <math>G \subset {\mathbb{C}}^n</math>
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| is called '''plurisubharmonic''' if it is [[semi-continuous function|upper semi-continuous]], and for every [[complex number|complex]] line | |
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| :<math>\{ a + b z \mid z \in {\mathbb{C}} \}\subset {\mathbb{C}}^n</math> with <math>a, b \in {\mathbb{C}}^n</math>
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| the function <math>z \mapsto f(a + bz)</math> is a [[subharmonic function]] on the set
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| :<math>\{ z \in {\mathbb{C}} \mid a + b z \in G \}.</math>
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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]]
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| :<math>f \colon X \to {\mathbb{R}} \cup \{ - \infty \}</math>
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| is said to be plurisubharmonic if and only if for any [[holomorphic]] map
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| <math>\varphi\colon\Delta\to X</math> the function
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| :<math>f\circ\varphi \colon \Delta \to {\mathbb{R}} \cup \{ - \infty \}</math>
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| is [[subharmonic function|subharmonic]], where <math>\Delta\subset{\mathbb{C}}</math> denotes the unit disk.
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| ===Differentiable plurisubharmonic functions===
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| If <math>f</math> is of (differentiability) class <math>C^2</math>, then <math>f</math> is plurisubharmonic, if and only if the hermitian matrix <math>L_f=(\lambda_{ij})</math>, called Levi matrix, with
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| entries
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| : <math>\lambda_{ij}=\frac{\partial^2f}{\partial z_i\partial\bar z_j}</math>
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| is positive semidefinite.
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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]].
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| == History ==
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| Plurisubharmonic functions were defined in 1942 by
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| [[Kiyoshi Oka]] <ref name=oka> K. Oka, ''Domaines pseudoconvexes,'' Tohoku Math. J. '''49''' (1942), 15–52.</ref> and [[Pierre Lelong]]. <ref> P. Lelong, ''Definition des fonctions plurisousharmoniques,'' C. R. Acd. Sci. Paris '''215''' (1942), 398–400.</ref>
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| ==Properties==
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| *The set of plurisubharmonic functions form a [[convex cone]] in the [[vector space]] of semicontinuous functions, i.e.
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| :* if <math>f</math> is a plurisubharmonic function and <math>c>0</math> a positive real number, then the function <math>c\cdot f</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.
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| *Plurisubharmonicity is a ''local property'', i.e. a function is plurisubharmonic if and only if it is plurisubharmonic in a neighborhood of each point.
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| *If <math>f</math> is plurisubharmonic and <math>\phi:\mathbb{R}\to\mathbb{R}</math> a monotonically increasing, convex function then <math>\phi\circ f</math> is plurisubharmonic.
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| *If <math>f_1</math> and <math>f_2</math> are plurisubharmonic functions, then the function <math>f(x):=\max(f_1(x),f_2(x))</math> is plurisubharmonic.
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| *If <math>f_1,f_2,\dots</math> is a monotonically decreasing sequence of plurisubharmonic functions
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| then so is <math>f(x):=\lim_{n\to\infty}f_n(x)</math>.
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| *Every continuous plurisubharmonic function can be obtained as a limit of monotonically 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), 47–84.</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
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| : <math>\limsup_{x\to x_0}f(x) =f(x_0)</math>
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| (see [[limit superior and limit inferior]] for the definition of ''lim sup''). | |
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| * Plurisubharmonic functions are [[Subharmonic function|subharmonic]], for any [[Kähler manifold|Kähler metric]].
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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
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| : <math>\sup_{x\in D}f(x) =f(x_0)</math> | |
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| for some point <math>x_0\in D</math> then <math>f</math> is constant.
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| ==Applications==
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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.
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| == Oka theorem ==
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| 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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| A continuous function <math>f:\; M \mapsto {\Bbb R}</math>
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| is called ''exhaustive'' if the preimage <math>f^{-1}(]-\infty, c])</math>
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| is compact for all <math>c\in {\Bbb R}</math>. A plurisubharmonic
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| function ''f'' is called ''strongly plurisubharmonic''
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| if the form <math>\sqrt{-1}(\partial\bar\partial f-\omega)</math>
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| is [[positive form|positive]], for some [[Kähler manifold|Kähler form]]
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| <math>\omega</math> on ''M''.
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| '''Theorem of Oka:''' Let ''M'' be a complex manifold,
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| admitting a smooth, exhaustive, strongly plurisubharmonic function.
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| Then ''M'' is [[Stein manifold|Stein]]. Conversely, any
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| [[Stein manifold]] admits such a function.
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| ==References==
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| * Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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| * Robert C. Gunning. Introduction to Holomorphic Functions in Several Variables, Wadsworth & Brooks/Cole.
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| ==External links==
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| * {{springer|title=Plurisubharmonic function|id=p/p072930}}
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| == Notes ==
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| <references />
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| [[Category:Subharmonic functions]]
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| [[Category:Several complex variables]]
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