Plurisubharmonic function explained

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 { R }\cup\,with domain G \subset { C }^n is called plurisubharmonic if it is upper semi-continuous, and for every complex line\{ a + b z \mid z \in { C } \}\subset ^n, with a, b \in { C }^n,

the function

z \mapsto f(a + bz) is a subharmonic function on the set\{ z \in { C } \mid a + b z \in G \}.X as follows. An upper semi-continuous function f \colon X \to { R } \cup \ is said to be plurisubharmonic if for any holomorphic map\varphi\colon\Delta\to X the function f\circ\varphi \colon \Delta \to { R } \cup \ is subharmonic, where \Delta\subset{ 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=(λ} _{ij} ) , called Levi matrix, withentriesλ _{ij} = (\partial ^{2f})/(\partial z _{i\partial\bar} z _{j})

is positive semidefinite.

Equivalently, a

C ^{2} -function f is plurisubharmonic if and only if i\partial\bar\partial f is a positive (1,1)-form.

Examples

Relation to Kähler manifold: On n-dimensional complex Euclidean space

C ^{n} , f(z) = |z| ^{2} is plurisubharmonic. In fact, i\partial\overline{\partial}f is equal to the standard Kähler form on C ^{n} up to constant multiples. More generally, if g satisfiesi\partial\overline{\partial}g = \omega for some Kähler form \omega , then g 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

C ^{1} , u(z) = log |z| is plurisubharmonic. If f is a C^∞-class function with compact support, then Cauchy integral formula saysf( 0 ) = (1)/(2 \pi i) \int _{D (\partial f)/(\partial\bar{z)} }\frac,which can be modified to(i)/(\pi) \partial\overline{\partial} log |z| = dd ^{clog|z|} .It is nothing but Dirac measure at the origin 0 .

More Examples

• If

f is an analytic function on an open set, then log |f| is plurisubharmonic on that open set.

• Convex functions are plurisubharmonic.
• If

\Omega is a domain of holomorphy then -log (dist(z,\Omega ^{c))} is plurisubharmonic.

History

Plurisubharmonic functions were defined in 1942 byKiyoshi Oka^[1] and Pierre Lelong.^[2]

Properties

• The set of plurisubharmonic functions has the following properties like a convex cone:

• if

f is a plurisubharmonic function and c> 0 a positive real number, then the function c ⋅ 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 \varphi: R \to R an increasing convex function then \varphi \circ f is plurisubharmonic. (\varphi( -inf ty) is interpreted as \lim _{x} \varphi(x) .)

• If

f _{1} and f _{2} are plurisubharmonic functions, then the function max (f _{1,f} _{2)} is plurisubharmonic.

• The pointwise limit of a decreasing sequence of plurisubharmonic functions is plurisubharmonic.
• Every continuous plurisubharmonic function can be obtained as the limit of a 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)} .

• Plurisubharmonic functions are subharmonic, for any Kähler metric.
• Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if

f is plurisubharmonic on the domain D and\sup _{x\in} f(x) = f(x _{0)} for some point x _{0\in} D then f 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.^[1]

A continuous function

f: M \mapsto { R} is called exhaustive if the preimage f ^{-1} (( -inf ty, c]) is compact for all c\in { R} . A plurisubharmonicfunction f is called strongly plurisubharmonicif the form i(\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, anyStein manifold admits such a function.

References

• 1992976. Complex Convexity. Bremermann. H. J.. Transactions of the American Mathematical Society. 1956. 82. 1. 17–51. 10.1090/S0002-9947-1956-0079100-2. free.
• 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.

Notes

1. note:In the treatise, it is referred to as the pseudoconvex function, but this means the plurisubharmonic function, which is the subject of this page, not the pseudoconvex function of convex analysis.
2. Lelong . P. . Definition des fonctions plurisousharmoniques . C. R. Acad. Sci. Paris . 215 . 398 - 400 . 1942.
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.