# 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

${\displaystyle f\colon G\to {\mathbb {R} }\cup \{-\infty \},}$

with domain ${\displaystyle G\subset {\mathbb {C} }^{n}}$ is called plurisubharmonic if it is upper semi-continuous, and for every complex line

${\displaystyle \{a+bz\mid z\in {\mathbb {C} }\}\subset {\mathbb {C} }^{n}}$ with ${\displaystyle a,b\in {\mathbb {C} }^{n}}$

the function ${\displaystyle z\mapsto f(a+bz)}$ is a subharmonic function on the set

${\displaystyle \{z\in {\mathbb {C} }\mid a+bz\in G\}.}$

In full generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space ${\displaystyle X}$ as follows. An upper semi-continuous function

${\displaystyle f\colon X\to {\mathbb {R} }\cup \{-\infty \}}$

is said to be plurisubharmonic if and only if for any holomorphic map ${\displaystyle \varphi \colon \Delta \to X}$ the function

${\displaystyle f\circ \varphi \colon \Delta \to {\mathbb {R} }\cup \{-\infty \}}$

is subharmonic, where ${\displaystyle \Delta \subset {\mathbb {C} }}$ denotes the unit disk.

### Differentiable plurisubharmonic functions

If ${\displaystyle f}$ is of (differentiability) class ${\displaystyle C^{2}}$, then ${\displaystyle f}$ is plurisubharmonic if and only if the hermitian matrix ${\displaystyle L_{f}=(\lambda _{ij})}$, called Levi matrix, with entries

${\displaystyle \lambda _{ij}={\frac {\partial ^{2}f}{\partial z_{i}\partial {\bar {z}}_{j}}}}$

Equivalently, a ${\displaystyle C^{2}}$-function f is plurisubharmonic if and only if ${\displaystyle {\sqrt {-1}}\partial {\bar {\partial }}f}$ is a positive (1,1)-form.

## Examples

Relation to Kähler manifold: On n-dimensional complex Euclidean space ${\displaystyle \mathbb {C} ^{n}}$ , ${\displaystyle f(z)=|z|^{2}}$ is plurisubharmonic. In fact, ${\displaystyle {\sqrt {-1}}\partial {\overline {\partial }}f}$ is equal to the standard Kähler form on ${\displaystyle \mathbb {C} ^{n}}$ up to constant multiples. More generally, if ${\displaystyle g}$ satisfies

${\displaystyle {\sqrt {-1}}\partial {\overline {\partial }}g=\omega }$

for some Kähler form ${\displaystyle \omega }$, then ${\displaystyle g}$ is plurisubharmonic, which is called Kähler potential.

Relation to Dirac Delta: On 1-dimensional complex Euclidean space ${\displaystyle \mathbb {C} ^{1}}$ , ${\displaystyle u(z)=\log(z)}$ is plurisubharmonic. If ${\displaystyle f}$ is a C-class function with compact support, then Cauchy integral formula says

${\displaystyle f(0)=-{\frac {\sqrt {-1}}{2\pi }}\int _{C}{\frac {\partial f}{\partial {\bar {z}}}}{\frac {dzd{\bar {z}}}{z}}}$

which can be modified to

${\displaystyle {\frac {\sqrt {-1}}{\pi }}\partial {\overline {\partial }}\log |z|=dd^{c}\log |z|}$.

It is nothing but Dirac measure at the origin 0 .

More Examples

• If ${\displaystyle f}$ is an analytic function on an open set, then ${\displaystyle \log |f|}$ is plurisubharmonic on that open set.
• Convex functions are plurisubharmonic
• If ${\displaystyle \Omega }$ is a Domain of Holomorphy then ${\displaystyle -\log(dist(z,\Omega ^{c}))}$ is plurisubharmonic
• Harmonic functions are not necessarily plurisubharmonic

## History

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

## Properties

• if ${\displaystyle f}$ is a plurisubharmonic function and ${\displaystyle c>0}$ a positive real number, then the function ${\displaystyle c\cdot f}$ is plurisubharmonic,
• if ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ are plurisubharmonic functions, then the sum ${\displaystyle f_{1}+f_{2}}$ 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 ${\displaystyle f}$ is plurisubharmonic and ${\displaystyle \phi :\mathbb {R} \to \mathbb {R} }$ a monotonically increasing, convex function then ${\displaystyle \phi \circ f}$ is plurisubharmonic.
• If ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ are plurisubharmonic functions, then the function ${\displaystyle f(x):=\max(f_{1}(x),f_{2}(x))}$ is plurisubharmonic.
• If ${\displaystyle f_{1},f_{2},\dots }$ is a monotonically decreasing sequence of plurisubharmonic functions

then ${\displaystyle f(x):=\lim _{n\to \infty }f_{n}(x)}$ 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.[3]
• The inequality in the usual semi-continuity condition holds as equality, i.e. if ${\displaystyle f}$ is plurisubharmonic then
${\displaystyle \limsup _{x\to x_{0}}f(x)=f(x_{0})}$

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

• Plurisubharmonic functions are subharmonic, for any Kähler metric.
• Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if ${\displaystyle f}$ is plurisubharmonic on the connected open domain ${\displaystyle D}$ and
${\displaystyle \sup _{x\in D}f(x)=f(x_{0})}$

for some point ${\displaystyle x_{0}\in D}$ then ${\displaystyle f}$ is constant.

## Applications

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 ${\displaystyle f:\;M\mapsto {\mathbb {R} }}$ is called exhaustive if the preimage ${\displaystyle f^{-1}(]-\infty ,c])}$ is compact for all ${\displaystyle c\in {\mathbb {R} }}$. A plurisubharmonic function f is called strongly plurisubharmonic if the form ${\displaystyle {\sqrt {-1}}(\partial {\bar {\partial }}f-\omega )}$ is positive, for some Kähler form ${\displaystyle \omega }$ 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.

## References

• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
• Robert C. Gunning. Introduction to Holomorphic Functions in Several Variables, Wadsworth & Brooks/Cole.
• Klimek, Pluripotential Theory, Clarendon Press 1992.