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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

A function

f:G→R∪{−∞},{displaystyle fcolon Gto {mathbb {R} }cup {-infty },}

with domain G⊂Cn{displaystyle Gsubset {mathbb {C} }^{n}}
is called plurisubharmonic if it is upper semi-continuous, and for every complex line

{a+bz∣z∈C}⊂Cn{displaystyle {a+bzmid zin {mathbb {C} }}subset {mathbb {C} }^{n}} with a,b∈Cn{displaystyle a,bin {mathbb {C} }^{n}}

the function z↦f(a+bz){displaystyle zmapsto f(a+bz)} is a subharmonic function on the set

{z∈C∣a+bz∈G}.{displaystyle {zin {mathbb {C} }mid a+bzin G}.}

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

f:X→R∪{−∞}{displaystyle fcolon Xto {mathbb {R} }cup {-infty }}

is said to be plurisubharmonic if and only if for any holomorphic map
φ:Δ→X{displaystyle varphi colon Delta to X} the function

f∘φ:Δ→R∪{−∞}{displaystyle fcirc varphi colon Delta to {mathbb {R} }cup {-infty }}

is subharmonic, where Δ⊂C{displaystyle Delta subset {mathbb {C} }} denotes the unit disk.

Differentiable plurisubharmonic functions

If f{displaystyle f} is of (differentiability) class C2{displaystyle C^{2}}, then f{displaystyle f} is plurisubharmonic if and only if the hermitian matrix Lf=(λij){displaystyle L_{f}=(lambda _{ij})}, called Levi matrix, with
entries

λij=∂2f∂zi∂z¯j{displaystyle lambda _{ij}={frac {partial ^{2}f}{partial z_{i}partial {bar {z}}_{j}}}}

is positive semidefinite.

Equivalently, a C2{displaystyle C^{2}}-function f is plurisubharmonic if and only if −1∂∂¯f{displaystyle {sqrt {-1}}partial {bar {partial }}f} is a positive (1,1)-form.

Examples

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

−1∂∂¯g=ω{displaystyle {sqrt {-1}}partial {overline {partial }}g=omega }

for some Kähler form ω{displaystyle omega }, then g{displaystyle g} is plurisubharmonic, which is called Kähler potential.

Relation to Dirac Delta: On 1-dimensional complex Euclidean space C1{displaystyle mathbb {C} ^{1}} , u(z)=log⁡(z){displaystyle u(z)=log(z)} is plurisubharmonic. If f{displaystyle f} is a C^∞-class function with compact support, then Cauchy integral formula says

f(0)=−−12π∫C∂f∂z¯dzdz¯z{displaystyle f(0)=-{frac {sqrt {-1}}{2pi }}int _{C}{frac {partial f}{partial {bar {z}}}}{frac {dzd{bar {z}}}{z}}}

which can be modified to

−1π∂∂¯log⁡|z|=ddclog⁡|z|{displaystyle {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

• The set of plurisubharmonic functions form a convex cone in the vector space of semicontinuous functions, i.e.

• if f{displaystyle f} is a plurisubharmonic function and c>0{displaystyle c>0} a positive real number, then the function c⋅f{displaystyle ccdot f} is plurisubharmonic,


• if f1{displaystyle f_{1}} and f2{displaystyle f_{2}} are plurisubharmonic functions, then the sum f1+f2{displaystyle 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{displaystyle f} is plurisubharmonic and ϕ:R→R{displaystyle phi :mathbb {R} to mathbb {R} } a monotonically increasing, convex function then ϕ∘f{displaystyle phi circ f} is plurisubharmonic.


• If f1{displaystyle f_{1}} and f2{displaystyle f_{2}} are plurisubharmonic functions, then the function f(x):=max(f1(x),f2(x)){displaystyle f(x):=max(f_{1}(x),f_{2}(x))} is plurisubharmonic.


• If f1,f2,…{displaystyle f_{1},f_{2},dots } is a monotonically decreasing sequence of plurisubharmonic functions

then f(x):=limn→∞fn(x){displaystyle f(x):=lim _{nto infty }f_{n}(x)} 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.^[3]
  


• The inequality in the usual semi-continuity condition holds as equality, i.e. if f{displaystyle f} is plurisubharmonic then

lim supx→x0f(x)=f(x0){displaystyle limsup _{xto x_{0}}f(x)=f(x_{0})}

(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 f{displaystyle f} is plurisubharmonic on the connected open domain D{displaystyle D} and

supx∈Df(x)=f(x0){displaystyle sup _{xin D}f(x)=f(x_{0})}

for some point x0∈D{displaystyle x_{0}in D} then f{displaystyle 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↦R{displaystyle f:;Mmapsto {mathbb {R}}}
is called exhaustive if the preimage f−1(]−∞,c]){displaystyle f^{-1}(]-infty ,c])}
is compact for all c∈R{displaystyle cin {mathbb {R}}}. A plurisubharmonic
function f is called strongly plurisubharmonic
if the form −1(∂∂¯f−ω){displaystyle {sqrt {-1}}(partial {bar {partial }}f-omega )}
is positive, for some Kähler form
ω{displaystyle 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

• 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) [1994], "Plurisubharmonic function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  

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