Plurisubharmonic function: Difference between revisions

Revision as of 20:42, 21 May 2024

Revision as of 21:07, 21 May 2024

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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 .

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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''.