# 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 $f\colon G\to {\mathbb {R} }\cup \{-\infty \},$ with domain $G\subset {\mathbb {C} }^{n}$ is called plurisubharmonic if it is upper semi-continuous, and for every complex line
    $\{a+bz\mid z\in {\mathbb {C} }\}\subset {\mathbb {C} }^{n},$ with $a,b\in {\mathbb {C} }^{n},$
the function $z\mapsto f(a+bz)$ is a subharmonic function on the set
    $\{z\in {\mathbb {C} }\mid a+bz\in G\}.$
In full generality, the notion can be defined on an arbitrary complex manifold or even a complex analytic space $X$ as follows. An upper semi-continuous function $f\colon X\to {\mathbb {R} }\cup \{-\infty \}$ is said to be plurisubharmonic if for any holomorphic map $\varphi \colon \Delta \to X$ the function $f\circ \varphi \colon \Delta \to {\mathbb {R} }\cup \{-\infty \}$ is subharmonic, where $\Delta \subset {\mathbb {C} }$ denotes the unit disk.

### Differentiable plurisubharmonic functions

If $f$ is of (differentiability) class $C^{2}$, then $f$ is plurisubharmonic if and only if the hermitian matrix $L_{f}=(\lambda _{ij})$, called Levi matrix, with entries
    $\lambda _{ij}={\frac {\partial ^{2}f}{\partial z_{i}\partial {\bar {z}}_{j}}}$
is positive semidefinite.
Equivalently, a $C^{2}$-function f is plurisubharmonic if and only if $i\partial {\bar {\partial }}f$ is a positive (1,1)-form.

## Examples

Relation to Kähler manifold: On n-dimensional complex Euclidean space $\mathbb {C} ^{n}$ , $f(z)=|z|^{2}$ is plurisubharmonic. In fact, $i\partial {\overline {\partial }}f$ is equal to the standard Kähler form on $\mathbb {C} ^{n}$ up to constant multiples. More generally, if $g$ satisfies
       $i\partial {\overline {\partial }}g=\omega$
for some Kähler form $\omega$, then $g$ 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 $\mathbb {C} ^{1}$ , $u(z)=\log |z|$ is plurisubharmonic. If $f$ is a C$^∞$-class function with compact support, then Cauchy integral formula says
       $f(0)={\frac {1}{2\pi i}}\int _{D}{\frac {\partial f}{\partial {\bar {z}}}}{\frac {dzd{\bar {z}}}{z}},$
which can be modified to
       ${\frac {i}{\pi }}\partial {\overline {\partial }}\log |z|=dd^{c}\log |z|$.
It is nothing but Dirac measure at the origin 0 .
More Examples
* If $f$ is an analytic function on an open set, then $\log |f|$ is plurisubharmonic on that open set.
* Convex functions are plurisubharmonic.
* If $\Omega$ is a domain of holomorphy then $-\log(dist(z,\Omega ^{c}))$ is plurisubharmonic.

## History

Plurisubharmonic functions were defined in 1942 by Kiyoshi Oka and Pierre Lelong.

## Properties

* The set of plurisubharmonic functions has the following properties like a convex cone:
   * if $f$ is a plurisubharmonic function and $c>0$ a positive real number, then the function $c\cdot f$ is plurisubharmonic,
   * if $f_{1}$ and $f_{2}$ are plurisubharmonic functions, then the sum $f_{1}+f_{2}$ 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 $f$ is plurisubharmonic and $\varphi :\mathbb {R} \to \mathbb {R}$ an increasing convex function then $\varphi \circ f$ is plurisubharmonic. ($\varphi (-\infty )$ is interpreted as $\lim _{x\rightarrow -\infty }\varphi (x)$.)
* If $f_{1}$ and $f_{2}$ are plurisubharmonic functions, then the function $\max(f_{1},f_{2})$ is plurisubharmonic.
* The pointwise limit of a decreasing sequence of plurisubharmonic functions is plurisubharmonic.
* 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.
* The inequality in the usual semi-continuity condition holds as equality, i.e. if $f$ is plurisubharmonic then $\limsup _{x\to x_{0}}f(x)=f(x_{0})$.
* Plurisubharmonic functions are subharmonic, for any Kähler metric.
* Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if $f$ is plurisubharmonic on the domain $D$ and$\sup _{x\in D}f(x)=f(x_{0})$ for some point $x_{0}\in D$ then $f$ is constant.

## Applications

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