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. 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. 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. 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 Plurisubharmonic functions were defined in 1942 by Kiyoshi Oka and Pierre Lelong.