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
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 an increasing convex function then is plurisubharmonic. ( is interpreted as .)
If and are plurisubharmonic functions, then the function is plurisubharmonic.
The pointwise limit of a decreasing sequence of plurisubharmonic functions is plurisubharmonic.
Every continuous plurisubharmonic function can be obtained as the limit of a 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 is plurisubharmonic then .
The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by Kiyoshi Oka in 1942.[1]
A continuous function
is called exhaustive if the preimage
is compact for all . A plurisubharmonic
function f is called strongly plurisubharmonic
if the form
is positive, for some Kähler form 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.