Plurisubharmonic function

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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[edit]

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 + 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[edit]

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[edit]

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[edit]

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

Properties[edit]

  • 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 so is f(x):=\lim_{n\to\infty}f_n(x).

  • 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.[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[edit]

In complex analysis, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy and Stein manifolds.

Oka theorem[edit]

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[edit]

  • 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.

External links[edit]

Notes[edit]

  1. ^ a b K. Oka, Domaines pseudoconvexes, Tohoku Math. J. 49 (1942), 15–52.
  2. ^ P. Lelong, Definition des fonctions plurisousharmoniques, C. R. Acd. Sci. Paris 215 (1942), 398–400.
  3. ^ R. E. Greene and H. Wu, C^\infty-approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Scient. Ec. Norm. Sup. 12 (1979), 47–84.