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 ![{\displaystyle a,b\in {\mathbb {C} }^{n},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
the function
is a subharmonic function on the set
![{\displaystyle \{z\in {\mathbb {C} }\mid a+bz\in G\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 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
![{\displaystyle \lambda _{ij}={\frac {\partial ^{2}f}{\partial z_{i}\partial {\bar {z}}_{j}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
![{\displaystyle i\partial {\overline {\partial }}g=\omega }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
![{\displaystyle f(0)={\frac {1}{2\pi i}}\int _{D}{\frac {\partial f}{\partial {\bar {z}}}}{\frac {dzd{\bar {z}}}{z}},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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.
History
Plurisubharmonic functions were defined in 1942 byKiyoshi Oka[1] and Pierre Lelong.[2]
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 increasing, convex function then
is plurisubharmonic. - If
and
are plurisubharmonic functions, then the function
is plurisubharmonic. - If
is a decreasing sequence of plurisubharmonic functions then its pointwise limit 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
. - Plurisubharmonic functions are subharmonic, for any Kähler metric.
- Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if
is plurisubharmonic on the domain
and
for some point
then
is constant.
Applications
In several complex variables, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy and Stein manifolds.
Oka theorem
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 plurisubharmonicif 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, anyStein manifold admits such a function.
References
- Bremermann, H. J. (1956). "Complex Convexity". Transactions of the American Mathematical Society. 82 (1): 17–51. doi:10.1090/S0002-9947-1956-0079100-2. JSTOR 1992976.
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
- Robert C. Gunning. Introduction to Holomorphic Functions in Several Variables, Wadsworth & Brooks/Cole.
- Klimek, Pluripotential Theory, Clarendon Press 1992.
External links
Notes
- ^ a b Oka, Kiyoshi (1942), "Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes", Tohoku Mathematical Journal, First Series, 49: 15–52, ISSN 0040-8735, Zbl 0060.24006 note:In the treatise, it is referred to as the pseudoconvex function, but this means the plurisubharmonic function, which is the subject of this page, not the pseudoconvex function of convex analysis.Bremermann (1956)
- ^ Lelong, P. (1942). "Definition des fonctions plurisousharmoniques". C. R. Acad. Sci. Paris. 215: 398–400.
- ^ R. E. Greene and H. Wu,
-approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Scient. Ec. Norm. Sup. 12 (1979), 47–84.