In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 used in complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

. On a Kähler manifold

Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

, plurisubharmonic functions form a subset of the subharmonic function

Subharmonic function

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory....

s. However, unlike subharmonic functions (which are defined on a Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

) plurisubharmonic functions can be defined in full generality on complex space

Complex space

In mathematics, n-dimensional complex space is a multi-dimensional generalisation of the complex numbers, which have both real and imaginary parts or dimensions...

s.

Formal definition

A function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

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

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 line
with

the function is a subharmonic function

Subharmonic function

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory....

 on the set


In full generality, the notion can be defined on an arbitrary complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

 or even a complex space

Complex space

In mathematics, n-dimensional complex space is a multi-dimensional generalisation of the complex numbers, which have both real and imaginary parts or dimensions...

  as follows. An upper semi-continuous function

Semi-continuity

In mathematical analysis, semi-continuity is a property of extended real-valued functions that is weaker than continuity...

is said to be plurisubharmonic if and only if for any holomorphic map
the function
is subharmonic

Subharmonic function

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory....

, 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

Positive form

In complex geometry, the term positive formrefers to several classes of real differential formsof Hodge type .- -forms :Real -forms on a complex manifold Mare forms which are of type and real,...

.

History

Plurisubharmonic functions were defined in 1942 by
Kiyoshi Oka

Kiyoshi Oka

was a Japanese mathematician who did fundamental work in the theory of several complex variables. He was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924....

  and Pierre Lelong.

Properties

• The set of plurisubharmonic functions form a convex cone
  Convex cone
  In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...
  
   in the vector space
  Vector space
  A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
  
   of semicontinuous functions, i.e.
• 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.
• 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 is plurisubharmonic and a monotonically increasing, convex function then is plurisubharmonic.
• If and are plurisubharmonic functions, then the function is plurisubharmonic.
• If is a monotonically decreasing sequence of plurisubharmonic functions

then so is .

• Every continuous plurisubharmonic function can be obtained as a limit of monotonically decreasing sequence of smooth plurisubharmonic functions. Moreover, this sequence can be chosen uniformly convergent.
• The inequality in the usual semi-continuity
  Semi-continuity
  In mathematical analysis, semi-continuity is a property of extended real-valued functions that is weaker than continuity...
  
   condition holds as equality, i.e. if is plurisubharmonic then

(see limit superior and limit inferior

Limit superior and limit inferior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence...

 for the definition of lim sup).

• Plurisubharmonic functions are subharmonic
  Subharmonic function
  In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory....
  
  , for any Kähler metric
  Kähler manifold
  In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
  
  .

• Therefore, plurisubharmonic functions satisfy the maximum principle
  Maximum principle
  In mathematics, the maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types. Roughly speaking, it says that the maximum of a function in a domain is to be found on the boundary of that domain...
  
  , i.e. if is plurisubharmonic on the connected
  Connected space
  In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
  
   domain and

for some point then is constant.

Applications

In complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy

Domain of holomorphy

In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a set which is maximal in the sense that there exists a holomorphic function on this set which cannot be extended to a bigger set....

 and Stein manifold

Stein manifold

In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...

s.

Oka theorem

The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by Kiyoshi Oka

Kiyoshi Oka

was a Japanese mathematician who did fundamental work in the theory of several complex variables. He was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924....

 in 1942.

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

Positive form

In complex geometry, the term positive formrefers to several classes of real differential formsof Hodge type .- -forms :Real -forms on a complex manifold Mare forms which are of type and real,...

, for some Kähler form

Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

on M.

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

Stein manifold

In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...

. Conversely, any
Stein manifold

Stein manifold

In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...

admits such a function.