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

[math]\displaystyle{ f \colon G \to {\mathbb{R}}\cup\{-\infty\}, }[/math]

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

[math]\displaystyle{ \{ a + b z \mid z \in {\mathbb{C}} \}\subset {\mathbb{C}}^n }[/math] with [math]\displaystyle{ a, b \in {\mathbb{C}}^n }[/math]

the function [math]\displaystyle{ z \mapsto f(a + bz) }[/math] is a subharmonic function on the set

[math]\displaystyle{ \{ z \in {\mathbb{C}} \mid a + b z \in G \}. }[/math]

In full generality, the notion can be defined on an arbitrary complex manifold or even a complex analytic space [math]\displaystyle{ X }[/math] as follows. An upper semi-continuous function

[math]\displaystyle{ f \colon X \to {\mathbb{R}} \cup \{ - \infty \} }[/math]

is said to be plurisubharmonic if and only if for any holomorphic map [math]\displaystyle{ \varphi\colon\Delta\to X }[/math] the function

[math]\displaystyle{ f\circ\varphi \colon \Delta \to {\mathbb{R}} \cup \{ - \infty \} }[/math]

is subharmonic, where [math]\displaystyle{ \Delta\subset{\mathbb{C}} }[/math] denotes the unit disk.

Differentiable plurisubharmonic functions

If [math]\displaystyle{ f }[/math] is of (differentiability) class [math]\displaystyle{ C^2 }[/math], then [math]\displaystyle{ f }[/math] is plurisubharmonic if and only if the hermitian matrix [math]\displaystyle{ L_f=(\lambda_{ij}) }[/math], called Levi matrix, with entries

[math]\displaystyle{ \lambda_{ij}=\frac{\partial^2f}{\partial z_i\partial\bar z_j} }[/math]

is positive semidefinite.

Equivalently, a [math]\displaystyle{ C^2 }[/math]-function f is plurisubharmonic if and only if [math]\displaystyle{ \sqrt{-1}\partial\bar\partial f }[/math] is a positive (1,1)-form.

Examples

Relation to Kähler manifold: On n-dimensional complex Euclidean space [math]\displaystyle{ \mathbb{C}^n }[/math] , [math]\displaystyle{ f(z) = |z|^2 }[/math] is plurisubharmonic. In fact, [math]\displaystyle{ \sqrt{-1}\partial\overline{\partial}f }[/math] is equal to the standard Kähler form on [math]\displaystyle{ \mathbb{C}^n }[/math] up to constant multiples. More generally, if [math]\displaystyle{ g }[/math] satisfies

[math]\displaystyle{ \sqrt{-1}\partial\overline{\partial}g=\omega }[/math]

for some Kähler form [math]\displaystyle{ \omega }[/math], then [math]\displaystyle{ g }[/math] is plurisubharmonic, which is called Kähler potential. These can be readily generated by applying the ddbar lemma to Kähler forms on a Kähler manifold.

Relation to Dirac Delta: On 1-dimensional complex Euclidean space [math]\displaystyle{ \mathbb{C}^1 }[/math] , [math]\displaystyle{ u(z) = \log(z) }[/math] is plurisubharmonic. If [math]\displaystyle{ f }[/math] is a C^∞-class function with compact support, then Cauchy integral formula says

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

which can be modified to

[math]\displaystyle{ \frac{\sqrt{-1}}{\pi}\partial\overline{\partial}\log|z|=dd^c\log|z| }[/math].

It is nothing but Dirac measure at the origin 0 .

More Examples

• If [math]\displaystyle{ f }[/math] is an analytic function on an open set, then [math]\displaystyle{ \log|f| }[/math] is plurisubharmonic on that open set.
• Convex functions are plurisubharmonic
• If [math]\displaystyle{ \Omega }[/math] is a Domain of Holomorphy then [math]\displaystyle{ -\log (dist(z,\Omega^c)) }[/math] 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 [math]\displaystyle{ f }[/math] is a plurisubharmonic function and [math]\displaystyle{ c\gt 0 }[/math] a positive real number, then the function [math]\displaystyle{ c\cdot f }[/math] is plurisubharmonic,
• if [math]\displaystyle{ f_1 }[/math] and [math]\displaystyle{ f_2 }[/math] are plurisubharmonic functions, then the sum [math]\displaystyle{ f_1+f_2 }[/math] 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 [math]\displaystyle{ f }[/math] is plurisubharmonic and [math]\displaystyle{ \phi:\mathbb{R}\to\mathbb{R} }[/math] a monotonically increasing, convex function then [math]\displaystyle{ \phi\circ f }[/math] is plurisubharmonic.
• If [math]\displaystyle{ f_1 }[/math] and [math]\displaystyle{ f_2 }[/math] are plurisubharmonic functions, then the function [math]\displaystyle{ f(x):=\max(f_1(x),f_2(x)) }[/math] is plurisubharmonic.
• If [math]\displaystyle{ f_1,f_2,\dots }[/math] is a monotonically decreasing sequence of plurisubharmonic functions

then [math]\displaystyle{ f(x):=\lim_{n\to\infty}f_n(x) }[/math] 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 [math]\displaystyle{ f }[/math] is plurisubharmonic then

[math]\displaystyle{ \limsup_{x\to x_0}f(x) =f(x_0) }[/math]

(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 [math]\displaystyle{ f }[/math] is plurisubharmonic on the connected open domain [math]\displaystyle{ D }[/math] and

[math]\displaystyle{ \sup_{x\in D}f(x) =f(x_0) }[/math]

for some point [math]\displaystyle{ x_0\in D }[/math] then [math]\displaystyle{ f }[/math] is constant.

Applications

In Several Complex Variables, 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 [math]\displaystyle{ f:\; M \mapsto {\mathbb R} }[/math] is called exhaustive if the preimage [math]\displaystyle{ f^{-1}((-\infty, c]) }[/math] is compact for all [math]\displaystyle{ c\in {\mathbb R} }[/math]. A plurisubharmonic function f is called strongly plurisubharmonic if the form [math]\displaystyle{ \sqrt{-1}(\partial\bar\partial f-\omega) }[/math] is positive, for some Kähler form [math]\displaystyle{ \omega }[/math] 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

• Bremermann, H. J. (1956). "Complex Convexity". Transactions of the American Mathematical Society 82 (1): 17–51. doi:10.1090/S0002-9947-1956-0079100-2. 
• 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.
• Klimek, Pluripotential Theory, Clarendon Press 1992.

External links

• Hazewinkel, Michiel, ed. (2001), "Plurisubharmonic function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p072930