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.

==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>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

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:<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 for any [[holomorphic map]]

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

is said to be plurisubharmonic if and only 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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is [[Definiteness of a matrix#Definitions for complex matrices|positive semidefinite]].

Equivalently, a <math>C^2</math>-function ''f'' is plurisubharmonic if and only if <math>\sqrt{-1}i\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}i\partial\overline{\partial}f</math> is equal to the standard [[Kähler form]] on <math>\mathbb{C}^n</math> up to constant multiples. More generally, if <math>g</math> satisfies

::<math>\sqrt{-1}i\partial\overline{\partial}g=\omega</math>

for some Kähler form <math>\omega</math>, then <math>g</math> 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 <math>\mathbb{C}^1</math> , <math>u(z) = \log(|z)|</math> is plurisubharmonic. If <math>f</math> is a C^∞-class function with [[compact support]], then [[Cauchy integral formula]] says

::<math>f(0)=-\frac{\sqrt{-1}}{2\pi i}\int_D\frac{\partial f}{\partial\bar{z}}\frac{dzd\bar{z}}{z},</math>

which can be modified to

::<math>\frac{\sqrt{-1}i}{\pi}\partial\overline{\partial}\log|z|=dd^c\log|z|</math>.

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

'''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[[___domain of Holomorphyholomorphy]] then <math>-\log (dist(z,\Omega^c))</math> is plurisubharmonic.

* Harmonic functions are not necessarily plurisubharmonic

==History==

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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,

:* 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> a monotonicallyan 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>f(x):=\max(f_1(x),f_2(x))</math> is plurisubharmonic.

*IfThe <math>f_1,f_2,\dots</math>pointwise islimit of a monotonically decreasing sequence of plurisubharmonic functions is plurisubharmonic.

*Every continuous plurisubharmonic function can be obtained as the limit of a 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&ndash;84.</ref>

then <math>f(x):=\lim_{n\to\infty}f_n(x)</math> is plurisubharmonic.

*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>.

*Every continuous plurisubharmonic function can be obtained as the limit of a 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&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>

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

* Plurisubharmonic functions are [[Subharmonic function|subharmonic]], for any [[Kähler manifold|Kähler metric]].

*Therefore, plurisubharmonic functions satisfy the [[maximum principle]], i.e. if <math>f</math> is plurisubharmonic on the [[connected___domain space(mathematical analysis)|connected___domain]] open<math>D</math> ___domainand<math>\sup_{x\in D}f(x) =f(x_0)</math> for some point <math>x_0\in D</math> andthen <math>f</math> is constant.

: <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.

==Applications==

In [[Severalseveral 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==

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is compact for all <math>c\in {\mathbb R}</math>. A plurisubharmonic

function ''f'' is called ''strongly plurisubharmonic''

if the form <math>\sqrt{-1}i(\partial\bar\partial f-\omega)</math>

is [[positive form|positive]], for some [[Kähler manifold|Kähler form]]

<math>\omega</math> on ''M''.