In mathematics, in the area of

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...

, a pluripolar set is the analog of a

polar set In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X lying in the dual space X^. The bipolar of a subset is the polar of A^, but li ...

for

plurisubharmonic function In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of function (mathematics), functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subs ...

Definition

Let

G \subset ^n

and let

f \colon G \to  \cup \

be a

which is not identically

-\infty

. The set :

:= \

is called a ''complete pluripolar set''. A ''pluripolar set'' is any subset of a complete pluripolar set. Pluripolar sets are of

Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, o ...

at most

2n-2

and have zero

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...

. If

f

is a

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

then

\log ,  f ,

is a plurisubharmonic function. The zero set of

f

is then a pluripolar set.

References

*Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992. {{PlanetMath attribution, id=6021, title=pluripolar set Potential theory

Definition

See also

References