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 f\colon G\to {\mathbb {R} }\cup \{-\infty \},}
with domain G ⊂ C n {\displaystyle G\subset {\mathbb {C} }^{n}} is called plurisubharmonic if it is upper semi-continuous, and for every complex line
{ a + b z ∣ z ∈ C } ⊂ C n , {\displaystyle \{a+bz\mid z\in {\mathbb {C} }\}\subset {\mathbb {C} }^{n},} with a , b ∈ C n , {\displaystyle a,b\in {\mathbb {C} }^{n},}
the function z ↦ f ( a + b z ) {\displaystyle z\mapsto f(a+bz)} is a subharmonic function on the set
{ z ∈ C ∣ a + b z ∈ G } . {\displaystyle \{z\in {\mathbb {C} }\mid a+bz\in 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 f\colon X\to {\mathbb {R} }\cup \{-\infty \}} is said to be plurisubharmonic if for any holomorphic map
φ : Δ → X {\displaystyle \varphi \colon \Delta \to X} the function f ∘ φ : Δ → R ∪ { − ∞ } {\displaystyle f\circ \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 C 2 {\displaystyle C^{2}} , then f {\displaystyle f} is plurisubharmonic if and only if the hermitian matrix L f = ( λ i j ) {\displaystyle L_{f}=(\lambda _{ij})} , called Levi matrix, with entries
λ i j = ∂ 2 f ∂ z i ∂ z ¯ j {\displaystyle \lambda _{ij}={\frac {\partial ^{2}f}{\partial z_{i}\partial {\bar {z}}_{j}}}}
is positive semidefinite. Equivalently, a C 2 {\displaystyle C^{2}} -function f is plurisubharmonic if and only if i ∂ ∂ ¯ f {\displaystyle i\partial {\bar {\partial }}f} is a positive (1,1)-form.
Examples Relation to Kähler manifold: On n-dimensional complex Euclidean space C n {\displaystyle \mathbb {C} ^{n}} , f ( z ) = | z | 2 {\displaystyle f(z)=|z|^{2}} is plurisubharmonic. In fact, i ∂ ∂ ¯ f {\displaystyle i\partial {\overline {\partial }}f} is equal to the standard Kähler form on C n {\displaystyle \mathbb {C} ^{n}} up to constant multiples. More generally, if g {\displaystyle g} satisfies
i ∂ ∂ ¯ g = ω {\displaystyle i\partial {\overline {\partial }}g=\omega }
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