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 with domain is called plurisubharmonic if it is upper semi-continuous, and for every complex line with the function 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 as follows. An upper semi-continuous function is said to be plurisubharmonic if and only if for any holomorphic map the function is subharmonic, where denotes the unit disk. Differentiable plurisubharmonic functions If is of (differentiability) class , then is plurisubharmonic if and only if the hermitian matrix , called Levi matrix, with entries is positive semidefinite. Equivalently, a -function f is plurisubharmonic if and only if is a positive (1,1)-form. Examples Relation to Kähler manifold: On n-dimensional complex Euclidean space , is plurisubharmonic. In fact, is equal to the standard Kähler form on up to constant multiples. More generally, if satisfies for some Kähler form , then 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 , is plurisubharmonic. If is a C∞-class function with compact support, then Cauchy integral formula says which can be modified to . It is nothing but Dirac measure at the origin 0 . More Examples If is an analytic function on an open set, then is plurisubharmonic on that open set. Convex functions are plurisubharmonic If is a Domain of Holomorphy then 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 is a plurisubharmonic function and a positive real number, then the function is plurisubharmonic, if and are plurisubharmonic functions, then the sum 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 is plurisubharmonic and a monotonically increasing,