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. 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. A function with domain G ⊂ C n {displaystyle Gsubset {mathbb {C} }^{n}} is called plurisubharmonic if it is upper semi-continuous, and for every complex line the function z ↦ f ( a + b z ) {displaystyle zmapsto f(a+bz)} is a subharmonic function on the set 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 is said to be plurisubharmonic if and only if for any holomorphic map φ : Δ → X {displaystyle varphi colon Delta o X} the function is subharmonic, where Δ ⊂ C {displaystyle Delta subset {mathbb {C} }} denotes the unit disk. 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, withentries