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.
the function is a subharmonic function on the set
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
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.
which can be modified to
It is nothing but Dirac measure at the origin 0 .
- 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
- The set of plurisubharmonic functions form a convex cone in the vector space of semicontinuous functions, i.e.
- 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, convex function then is plurisubharmonic.
- If and are plurisubharmonic functions, then the function is plurisubharmonic.
- If is a monotonically decreasing sequence of plurisubharmonic functions
then is plurisubharmonic.
- Every continuous plurisubharmonic function can be obtained as the limit of a monotonically decreasing sequence of smooth plurisubharmonic functions. Moreover, this sequence can be chosen uniformly convergent.
- The inequality in the usual semi-continuity condition holds as equality, i.e. if is plurisubharmonic then
(see limit superior and limit inferior for the definition of lim sup).
- Plurisubharmonic functions are subharmonic, for any Kähler metric.
- Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if is plurisubharmonic on the connected open domain and
for some point then is constant.
- 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.
- Hazewinkel, Michiel, ed. (2001) , "Plurisubharmonic function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- K. Oka, Domaines pseudoconvexes, Tohoku Math. J. 49 (1942), 15–52.
- P. Lelong, Definition des fonctions plurisousharmoniques, C. R. Acd. Sci. Paris 215 (1942), 398–400.
- R. E. Greene and H. Wu, -approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Scient. Ec. Norm. Sup. 12 (1979), 47–84.