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<title>Plurisubharmonic function</title>

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Plurisubharmonic function

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Mathematics" title="Mathematics">mathematics</a>, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> used in <a href="Complex_analysis" title="Complex analysis">complex analysis</a>. On a <a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a>, plurisubharmonic functions form a subset of the <a href="Subharmonic_function" title="Subharmonic function">subharmonic functions</a>. However, unlike subharmonic functions (which are defined on a <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>) plurisubharmonic functions can be defined in full generality on <a href="Complex_analytic_space" class="mw-redirect" title="Complex analytic space">complex analytic spaces</a>.

<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

A <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon G\to {\mathbb {R} }\cup \{-\infty \},}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon G\to {\mathbb {R} }\cup \{-\infty \},}</annotation>

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</math><img src="./835c829aa7b00e09dcc0c1fd603df4c604098acb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.118ex; height:2.843ex;" alt="{\displaystyle f\colon G\to {\mathbb {R} }\cup \{-\infty \},}" loading="lazy">

with domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\subset {\mathbb {C} }^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle G\subset {\mathbb {C} }^{n}}</annotation>

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</math><img src="./a669d69f08cb6302f1ba79303e9c98c14cb13083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.822ex; height:2.343ex;" alt="{\displaystyle G\subset {\mathbb {C} }^{n}}" loading="lazy"> is called plurisubharmonic if it is <a href="Semi-continuous_function" class="mw-redirect" title="Semi-continuous function">upper semi-continuous</a>, and for every <a href="Complex_number" title="Complex number">complex</a> line

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<annotation encoding="application/x-tex">{\displaystyle \{a+bz\mid z\in {\mathbb {C} }\}\subset {\mathbb {C} }^{n},}</annotation>

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</math><img src="./b154c54f4c0b8f126e222d3418d148d37987ed7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.667ex; height:2.843ex;" alt="{\displaystyle \{a+bz\mid z\in {\mathbb {C} }\}\subset {\mathbb {C} }^{n},}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in {\mathbb {C} }^{n},}">

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<annotation encoding="application/x-tex">{\displaystyle a,b\in {\mathbb {C} }^{n},}</annotation>

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</math><img src="./d1e985ff01d8824bba652cccd7501c0ba8309fd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.645ex; height:2.676ex;" alt="{\displaystyle a,b\in {\mathbb {C} }^{n},}" loading="lazy"></dd></dl>

the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto f(a+bz)}">

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<annotation encoding="application/x-tex">{\displaystyle z\mapsto f(a+bz)}</annotation>

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</math><img src="./e1b212da0d22241e01645c9cd54929644c0bd773.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.946ex; height:2.843ex;" alt="{\displaystyle z\mapsto f(a+bz)}" loading="lazy"> is a <a href="Subharmonic_function" title="Subharmonic function">subharmonic function</a> on the set

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<annotation encoding="application/x-tex">{\displaystyle \{z\in {\mathbb {C} }\mid a+bz\in G\}.}</annotation>

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</math><img src="./7cde1b32eddaa710a084203de6837c91b270cb5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.339ex; height:2.843ex;" alt="{\displaystyle \{z\in {\mathbb {C} }\mid a+bz\in G\}.}" loading="lazy"></dd></dl>

In full generality, the notion can be defined on an arbitrary <a href="Complex_manifold" title="Complex manifold">complex manifold</a> or even a <a href="Complex_analytic_space" class="mw-redirect" title="Complex analytic space">complex analytic space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> as follows. An <a href="Semi-continuity" title="Semi-continuity">upper semi-continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to {\mathbb {R} }\cup \{-\infty \}}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to {\mathbb {R} }\cup \{-\infty \}}</annotation>

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</math><img src="./a20f5e5b94325f0f895ad75faf195b90ed5ed8cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.624ex; height:2.843ex;" alt="{\displaystyle f\colon X\to {\mathbb {R} }\cup \{-\infty \}}" loading="lazy"> is said to be plurisubharmonic if for any <a href="Holomorphic_map" class="mw-redirect" title="Holomorphic map">holomorphic map</a>

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<annotation encoding="application/x-tex">{\displaystyle \varphi \colon \Delta \to X}</annotation>

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</math><img src="./234ddbb00a040136d86f6017eafb85174ff80838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.084ex; height:2.676ex;" alt="{\displaystyle \varphi \colon \Delta \to X}" loading="lazy"> the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \varphi \colon \Delta \to {\mathbb {R} }\cup \{-\infty \}}">

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<annotation encoding="application/x-tex">{\displaystyle f\circ \varphi \colon \Delta \to {\mathbb {R} }\cup \{-\infty \}}</annotation>

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</math><img src="./dfb32871655578eb365aac2ba2b03976bdf42131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.295ex; height:2.843ex;" alt="{\displaystyle f\circ \varphi \colon \Delta \to {\mathbb {R} }\cup \{-\infty \}}" loading="lazy"> is <a href="Subharmonic_function" title="Subharmonic function">subharmonic</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subset {\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta \subset {\mathbb {C} }}</annotation>

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</math><img src="./ddffd5b44c54475e28aa630f8634cf1523460e3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.712ex; height:2.176ex;" alt="{\displaystyle \Delta \subset {\mathbb {C} }}" loading="lazy"> denotes the <a href="Unit_disk" title="Unit disk">unit disk</a>.

<div class="mw-heading mw-heading3"><h3 id="Differentiable_plurisubharmonic_functions">Differentiable plurisubharmonic functions</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is of (differentiability) class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle C^{2}}</annotation>

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</math><img src="./4fd6a5946b7e916352b0afc557f992328bac85e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{2}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is plurisubharmonic if and only if the hermitian matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{f}=(\lambda _{ij})}">

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<annotation encoding="application/x-tex">{\displaystyle L_{f}=(\lambda _{ij})}</annotation>

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</math><img src="./292f41919957d289e8d6a3fb861fdaac91bb01fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.459ex; height:3.009ex;" alt="{\displaystyle L_{f}=(\lambda _{ij})}" loading="lazy">, called Levi matrix, with

entries

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{ij}={\frac {\partial ^{2}f}{\partial z_{i}\partial {\bar {z}}_{j}}}}</annotation>

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</math><img src="./9d2867a7882d2e895821179567779bc1c2a0d7b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.49ex; height:6.509ex;" alt="{\displaystyle \lambda _{ij}={\frac {\partial ^{2}f}{\partial z_{i}\partial {\bar {z}}_{j}}}}" loading="lazy"></dd></dl>

is <a href="Definiteness_of_a_matrix" class="mw-redirect" title="Definiteness of a matrix">positive semidefinite</a>.

Equivalently, a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle C^{2}}</annotation>

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</math><img src="./4fd6a5946b7e916352b0afc557f992328bac85e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{2}}" loading="lazy">-function f is plurisubharmonic if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\partial {\bar {\partial }}f}">

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<annotation encoding="application/x-tex">{\displaystyle i\partial {\bar {\partial }}f}</annotation>

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</math><img src="./d3cb1339cfa73c9cb93e48e5081521ada1a32f8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.887ex; height:3.009ex;" alt="{\displaystyle i\partial {\bar {\partial }}f}" loading="lazy"> is a <a href="Positive_form" title="Positive form">positive (1,1)-form</a>.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

Relation to Kähler manifold: On n-dimensional complex Euclidean space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation>

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</math><img src="./a53b4e76242764d1bca004168353c380fef25258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" loading="lazy"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=|z|^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)=|z|^{2}}</annotation>

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</math><img src="./f8b5e74e3179788a2dd4c66b39d42b9e380c012f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.711ex; height:3.343ex;" alt="{\displaystyle f(z)=|z|^{2}}" loading="lazy"> is plurisubharmonic. In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\partial {\overline {\partial }}f}">

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<annotation encoding="application/x-tex">{\displaystyle i\partial {\overline {\partial }}f}</annotation>

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</math><img src="./ea4a75ed8fa0d7e9f3ccf414a8e935432c6e5206.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.887ex; height:3.343ex;" alt="{\displaystyle i\partial {\overline {\partial }}f}" loading="lazy"> is equal to the standard <a href="K%C3%A4hler_form" class="mw-redirect" title="Kähler form">Kähler form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation>

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</math><img src="./a53b4e76242764d1bca004168353c380fef25258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" loading="lazy"> up to constant multiples. More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> satisfies

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<annotation encoding="application/x-tex">{\displaystyle i\partial {\overline {\partial }}g=\omega }</annotation>

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</math><img src="./415b7007fdb840e45bb508ab7322aeb16ffe003e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.268ex; height:3.343ex;" alt="{\displaystyle i\partial {\overline {\partial }}g=\omega }" loading="lazy"></dd></dl></dd></dl>

for some Kähler form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> is plurisubharmonic, which is called Kähler potential. These can be readily generated by applying the <a href="Ddbar_lemma" title="Ddbar lemma">ddbar lemma</a> to Kähler forms on a Kähler manifold.

Relation to Dirac Delta: On 1-dimensional complex Euclidean space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{1}}</annotation>

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</math><img src="./1c544f118d69a9759f9a7c13ab24aff652187876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{1}}" loading="lazy"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(z)=\log |z|}">

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<annotation encoding="application/x-tex">{\displaystyle u(z)=\log |z|}</annotation>

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</math><img src="./923773e11819edaced752321fc3396a8a15a2e02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.066ex; height:2.843ex;" alt="{\displaystyle u(z)=\log |z|}" loading="lazy"> is plurisubharmonic. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a C∞-class function with <a href="Compact_support" class="mw-redirect" title="Compact support">compact support</a>, then <a href="Cauchy_integral_formula" class="mw-redirect" title="Cauchy integral formula">Cauchy integral formula</a> says

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<annotation encoding="application/x-tex">{\displaystyle f(0)={\frac {1}{2\pi i}}\int _{D}{\frac {\partial f}{\partial {\bar {z}}}}{\frac {dzd{\bar {z}}}{z}},}</annotation>

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</math><img src="./dd63538c9f454ee9211b9a235fbab9342d334e85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.891ex; height:6.009ex;" alt="{\displaystyle f(0)={\frac {1}{2\pi i}}\int _{D}{\frac {\partial f}{\partial {\bar {z}}}}{\frac {dzd{\bar {z}}}{z}},}" loading="lazy"></dd></dl></dd></dl>

which can be modified to

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<annotation encoding="application/x-tex">{\displaystyle {\frac {i}{\pi }}\partial {\overline {\partial }}\log |z|=dd^{c}\log |z|}</annotation>

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</math><img src="./91483fdac15b1adf09a874bb30e52a0bd477fc65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.706ex; height:5.176ex;" alt="{\displaystyle {\frac {i}{\pi }}\partial {\overline {\partial }}\log |z|=dd^{c}\log |z|}" loading="lazy">.</dd></dl></dd></dl>

It is nothing but <a href="Dirac_measure" title="Dirac measure">Dirac measure</a> at the origin 0 .

More Examples

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is an analytic function on an <a href="Open_set" title="Open set">open set</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log |f|}">

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<annotation encoding="application/x-tex">{\displaystyle \log |f|}</annotation>

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</math><img src="./38c6997d5e72c1c55ddf72365d5edd7bf4bc2042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.931ex; height:2.843ex;" alt="{\displaystyle \log |f|}" loading="lazy"> is plurisubharmonic on that open set.</li>

<li><a href="Convex_function" title="Convex function">Convex functions</a> are plurisubharmonic.</li>

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> is a <a href="Domain_of_holomorphy" title="Domain of holomorphy">domain of holomorphy</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\log(dist(z,\Omega ^{c}))}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\log(dist(z,\Omega ^{c}))}</annotation>

</semantics>

</math><img src="./df4bdf31cdc71dae7b611d0000f8615ca0176399.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.478ex; height:2.843ex;" alt="{\displaystyle -\log(dist(z,\Omega ^{c}))}" loading="lazy"> is plurisubharmonic.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

Plurisubharmonic functions were defined in 1942 by

<a href="Kiyoshi_Oka" title="Kiyoshi Oka">Kiyoshi Oka</a><a href="#cite_note-oka-1">[1]</a> and <a href="Pierre_Lelong" title="Pierre Lelong">Pierre Lelong</a>.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<ul><li>The set of plurisubharmonic functions has the following properties like a <a href="Convex_cone" title="Convex cone">convex cone</a>:</li></ul>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a plurisubharmonic function and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c>0}</annotation>

</semantics>

</math><img src="./2ba126f626d61752f62eaacaf11761a54de4dc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}" loading="lazy"> a positive real number, then the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\cdot f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\cdot f}</annotation>

</semantics>

</math><img src="./884124fc59ee5e45dff4e39cd99dea42087fb0e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.965ex; height:2.509ex;" alt="{\displaystyle c\cdot f}" loading="lazy"> is plurisubharmonic,</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}}</annotation>

</semantics>

</math><img src="./50dfd257a51e037112c917f8a9e47c9c053466df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}}</annotation>

</semantics>

</math><img src="./cc886fdaa7adc9be11ff4a5076da5e0943bcff58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{2}}" loading="lazy"> are plurisubharmonic functions, then the sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}+f_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}+f_{2}}</annotation>

</semantics>

</math><img src="./7875d43d94bfdb30c7a2a5d72fea4a6960bc8787.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.227ex; height:2.509ex;" alt="{\displaystyle f_{1}+f_{2}}" loading="lazy"> is a plurisubharmonic function.</li></ul></dd></dl>

<ul><li>Plurisubharmonicity is a local property, i.e. a function is plurisubharmonic if and only if it is plurisubharmonic in a neighborhood of each point.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is plurisubharmonic and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi :\mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./1da2ccce20f693de147b1da37f3edb402a4496a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.428ex; height:2.676ex;" alt="{\displaystyle \varphi :\mathbb {R} \to \mathbb {R} }" loading="lazy"> an increasing convex function then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \circ f}</annotation>

</semantics>

</math><img src="./48bc0467914ba2ece3280bc232bea7570f4043d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.993ex; height:2.676ex;" alt="{\displaystyle \varphi \circ f}" loading="lazy"> is plurisubharmonic. (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (-\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (-\infty )}</annotation>

</semantics>

</math><img src="./2fae9e0fbb0cfd79745e80a52b524b884f3927d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.461ex; height:2.843ex;" alt="{\displaystyle \varphi (-\infty )}" loading="lazy"> is interpreted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\rightarrow -\infty }\varphi (x)}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow -\infty }\varphi (x)}</annotation>

</semantics>

</math><img src="./41b046a41049453aca28a842b2990ea7826993f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.551ex; height:4.009ex;" alt="{\displaystyle \lim _{x\rightarrow -\infty }\varphi (x)}" loading="lazy">.)</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}}</annotation>

</semantics>

</math><img src="./50dfd257a51e037112c917f8a9e47c9c053466df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}}</annotation>

</semantics>

</math><img src="./cc886fdaa7adc9be11ff4a5076da5e0943bcff58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{2}}" loading="lazy"> are plurisubharmonic functions, then the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(f_{1},f_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \max(f_{1},f_{2})}</annotation>

</semantics>

</math><img src="./1761de4558044d8515c24b20fb84df9fa8142344.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.556ex; height:2.843ex;" alt="{\displaystyle \max(f_{1},f_{2})}" loading="lazy"> is plurisubharmonic.</li>

<li>The pointwise limit of a decreasing sequence of plurisubharmonic functions is plurisubharmonic.</li>

<li>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.<a href="#cite_note-3">[3]</a></li>

<li>The inequality in the usual <a href="Semi-continuity" title="Semi-continuity">semi-continuity</a> condition holds as equality, i.e. if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is plurisubharmonic then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \limsup _{x\to x_{0}}f(x)=f(x_{0})}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \limsup _{x\to x_{0}}f(x)=f(x_{0})}</annotation>

</semantics>

</math><img src="./0c5b6612b8dc05137fbd8ed2ea50c9aa6f2595f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.494ex; height:4.676ex;" alt="{\displaystyle \limsup _{x\to x_{0}}f(x)=f(x_{0})}" loading="lazy">.</li>

<li>Plurisubharmonic functions are <a href="Subharmonic_function" title="Subharmonic function">subharmonic</a>, for any <a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler metric</a>.</li>

<li>Therefore, plurisubharmonic functions satisfy the <a href="Maximum_principle" title="Maximum principle">maximum principle</a>, i.e. if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is plurisubharmonic on the <a href="Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> and<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{x\in D}f(x)=f(x_{0})}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sup _{x\in D}f(x)=f(x_{0})}</annotation>

</semantics>

</math><img src="./ca4d11a90e0e3ef00c92b3777bbcef45a5d756c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.876ex; height:4.509ex;" alt="{\displaystyle \sup _{x\in D}f(x)=f(x_{0})}" loading="lazy"> for some point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in D}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in D}</annotation>

</semantics>

</math><img src="./b107b130543b591ac3037f4bc152da314ec51586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.149ex; height:2.509ex;" alt="{\displaystyle x_{0}\in D}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is constant.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

In <a href="Several_complex_variables" class="mw-redirect" title="Several complex variables">several complex variables</a>, plurisubharmonic functions are used to describe <a href="Pseudoconvexity" title="Pseudoconvexity">pseudoconvex domains</a>, <a href="Domain_of_holomorphy" title="Domain of holomorphy">domains of holomorphy</a> and <a href="Stein_manifold" title="Stein manifold">Stein manifolds</a>.

<div class="mw-heading mw-heading2"><h2 id="Oka_theorem">Oka theorem</h2></div>

The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by <a href="Kiyoshi_Oka" title="Kiyoshi Oka">Kiyoshi Oka</a> in 1942.<a href="#cite_note-oka-1">[1]</a>

A continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\;M\mapsto {\mathbb {R} }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\;M\mapsto {\mathbb {R} }}</annotation>

</semantics>

</math><img src="./5a3e58a69cd852d5ab90b2f55cb9ee735d0b64ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.595ex; height:2.509ex;" alt="{\displaystyle f:\;M\mapsto {\mathbb {R} }}" loading="lazy">

is called exhaustive if the preimage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}((-\infty ,c])}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}((-\infty ,c])}</annotation>

</semantics>

</math><img src="./92c8aebf8bf641686544c3cbc9ffa7d51749f3f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.187ex; height:3.176ex;" alt="{\displaystyle f^{-1}((-\infty ,c])}" loading="lazy">

is compact for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in {\mathbb {R} }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\in {\mathbb {R} }}</annotation>

</semantics>

</math><img src="./af8e76c1cb20897acedfa02805b3248ea9e9ab40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in {\mathbb {R} }}" loading="lazy">. A plurisubharmonic

function f is called strongly plurisubharmonic

if the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(\partial {\bar {\partial }}f-\omega )}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i(\partial {\bar {\partial }}f-\omega )}</annotation>

</semantics>

</math><img src="./d199be2f0e0388612c225fe51e2b42f28d609934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.983ex; height:3.176ex;" alt="{\displaystyle i(\partial {\bar {\partial }}f-\omega )}" loading="lazy">

is <a href="Positive_form" title="Positive form">positive</a>, for some <a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler form</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> on M.

Theorem of Oka: Let M be a complex manifold,

admitting a smooth, exhaustive, strongly plurisubharmonic function.

Then M is <a href="Stein_manifold" title="Stein manifold">Stein</a>. Conversely, any

<a href="Stein_manifold" title="Stein manifold">Stein manifold</a> admits such a function.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBremermann1956" class="citation journal cs1">Bremermann, H. J. (1956). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1956-0079100-2">"Complex Convexity"</a>. Transactions of the American Mathematical Society. 82 (1): 17–51. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1956-0079100-2">10.1090/S0002-9947-1956-0079100-2</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1992976">1992976</a>.</cite></li>

<li>Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.</li>

<li><a href="Robert_C._Gunning" title="Robert C. Gunning">Robert C. Gunning</a>. Introduction to Holomorphic Functions in Several Variables, Wadsworth & Brooks/Cole.</li>

<li>Klimek, Pluripotential Theory, Clarendon Press 1992.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Plurisubharmonic_function">"Plurisubharmonic function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

<li id="cite_note-oka-1">^ <a href="#cite_ref-oka_1-0">a</a> <a href="#cite_ref-oka_1-1">b</a> <cite id="CITEREFOka1942" class="citation cs2">Oka, Kiyoshi (1942), <a rel="nofollow" class="external text" href="https://www.nara-wu.ac.jp/aic/gdb/nwugdb/oka/ko_ron/f/n06/p001.html">"Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes"</a>, Tohoku Mathematical Journal, First Series, 49: 15–52, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0040-8735">0040-8735</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0060.24006">0060.24006</a></cite> note:In the treatise, it is referred to as the pseudoconvex function, but this means the plurisubharmonic function, which is the subject of this page, not the pseudoconvex function of convex analysis.<a href="#CITEREFBremermann1956">Bremermann (1956)</a>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFLelong1942" class="citation journal cs1">Lelong, P. (1942). <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k3167n/f398.item">"Definition des fonctions plurisousharmoniques"</a>. C. R. Acad. Sci. Paris. 215: 398–400.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> R. E. Greene and H. Wu, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>

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</math><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">-approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Scient. Ec. Norm. Sup. 12 (1979), 47–84.

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