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 lies ...

for

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 ...

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, of ...

at most

2n-2

and have zero Lebesgue measure. If

f

is a holomorphic function 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