# 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

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

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

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

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

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

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 and only 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

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

is positive semidefinite.

Equivalently, a $C^2$-function f is plurisubharmonic if and only if $\sqrt{-1}\partial\bar\partial f$ 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, $\sqrt{-1}\partial\overline{\partial}f$ is equal to the standard Kähler form on $\mathbb{C}^n$　 up to constant multiplies. More generally, if $g$ satisfies

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

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

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

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

which can be modified to

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

It is nothing but Dirac measure at the origin 0 .

## History

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

## Properties

• 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 $\phi:\mathbb{R}\to\mathbb{R}$ a monotonically increasing, convex function then $\phi\circ f$ is plurisubharmonic.
• If $f_1$ and $f_2$ are plurisubharmonic functions, then the function $f(x):=\max(f_1(x),f_2(x))$ is plurisubharmonic.
• If $f_1,f_2,\dots$ is a monotonically decreasing sequence of plurisubharmonic functions

then $f(x):=\lim_{n\to\infty}f_n(x)$ 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 $f$ is plurisubharmonic then
$\limsup_{x\to x_0}f(x) =f(x_0)$

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

• Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if $f$ is plurisubharmonic on the connected open 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 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 $f:\; M \mapsto {\Bbb R}$ is called exhaustive if the preimage $f^{-1}(]-\infty, c])$ is compact for all $c\in {\Bbb R}$. A plurisubharmonic function f is called strongly plurisubharmonic if the form $\sqrt{-1}(\partial\bar\partial f-\omega)$ 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.

## 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.