# 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

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

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

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

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

$\displaystyle{ \lambda_{ij}=\frac{\partial^2f}{\partial z_i\partial\bar z_j} }$

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

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

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 and Pierre Lelong.

## Properties

• The set of plurisubharmonic functions has the following properties like a convex cone:
• if $\displaystyle{ f }$ is a plurisubharmonic function and $\displaystyle{ c\gt 0 }$ 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.
• 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.

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.