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 + 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.

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

\limsup_{x\to x_0}f(x) =f(x_0)

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

\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

External links

Notes

  1. 1 2 K. Oka, Domaines pseudoconvexes, Tohoku Math. J. 49 (1942), 1552.
  2. P. Lelong, Definition des fonctions plurisousharmoniques, C. R. Acd. Sci. Paris 215 (1942), 398400.
  3. R. E. Greene and H. Wu, C^\infty-approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Scient. Ec. Norm. Sup. 12 (1979), 4784.
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