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Concept
Plurisubharmonic function
Résumé
In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Riemannian manifold) plurisubharmonic functions can be defined in full generality on complex analytic spaces.
A function
with domain
is called plurisubharmonic if it is upper semi-continuous, and for every complex line
with
the function is a subharmonic function on the set
In full generality, the notion can be defined on an arbitrary complex manifold or even a complex analytic space as follows. An upper semi-continuous function
is said to be plurisubharmonic if and only if for any holomorphic map
the function
is subharmonic, where denotes the unit disk.
If is of (differentiability) class , then is plurisubharmonic if and only if the hermitian matrix , called Levi matrix, with
entries
is positive semidefinite.
Equivalently, a -function f is plurisubharmonic if and only if is a positive (1,1)-form.
Relation to Kähler manifold: On n-dimensional complex Euclidean space , is plurisubharmonic. In fact, is equal to the standard Kähler form on up to constant multiples. More generally, if satisfies
for some Kähler form , then 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 , is plurisubharmonic. If is a C∞-class function with compact support, then Cauchy integral formula says
which can be modified to
It is nothing but Dirac measure at the origin 0 .
More Examples
If is an analytic function on an open set, then is plurisubharmonic on that open set.
Convex functions are plurisubharmonic
If is a Domain of Holomorphy then is plurisubharmonic
Harmonic functions are not necessarily plurisubharmonic
Plurisubharmonic functions were defined in 1942 by
Kiyoshi Oka and Pierre Lelong.
Source officielle
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