Plurisubharmonic function

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.

Formal definition

A function

with domain is called plurisubharmonic if it is upper semi-continuous, and for every complex line


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.

Differentiable plurisubharmonic functions

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 multiplies. More generally, if satisfies

for some Kähler form , then is plurisubharmonic, which is called Kähler potential.

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 .


Plurisubharmonic functions were defined in 1942 by Kiyoshi Oka [1] and Pierre Lelong.[2]


  • if is a plurisubharmonic function and a positive real number, then the function is plurisubharmonic,
  • if and are plurisubharmonic functions, then the sum is a plurisubharmonic function.

then is plurisubharmonic.

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

for some point then is constant.


In complex analysis, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy and Stein manifolds.

Oka theorem

The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by Kiyoshi Oka in 1942.[1]

A continuous function is called exhaustive if the preimage is compact for all . A plurisubharmonic function f is called strongly plurisubharmonic if the form is positive, for some Kähler form on M.

Theorem of Oka: Let M be a complex manifold, admitting a smooth, exhaustive, strongly plurisubharmonic function. Then M is Stein. Conversely, any Stein manifold admits such a function.


External links


  1. 1 2 K. Oka, Domaines pseudoconvexes, Tohoku Math. J. 49 (1942), 1552.
  2. P. Lelong, Definition des fonctions plurisousharmoniques, C. R. Acd. Sci. Paris 215 (1942), 398400.
  3. R. E. Greene and H. Wu, -approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Scient. Ec. Norm. Sup. 12 (1979), 4784.
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