Pluripolar set

In mathematics, in the area of potential theory, a pluripolar set is the analog of a polar set for plurisubharmonic functions.

Definition

Let ${\displaystyle G\subset {\mathbb {C} }^{n}}$ and let ${\displaystyle f\colon G\to {\mathbb {R} }\cup \{-\infty \}}$ be a plurisubharmonic function which is not identically ${\displaystyle -\infty }$. The set

${\displaystyle {\mathcal {P}}:=\{z\in G\mid f(z)=-\infty \}}$

is called a complete pluripolar set. A pluripolar set is any subset of a complete pluripolar set. Pluripolar sets are of Hausdorff dimension at most ${\displaystyle 2n-2}$ and have zero Lebesgue measure.^[1]

If ${\displaystyle f}$ is a holomorphic function then ${\displaystyle \log |f|}$ is a plurisubharmonic function. The zero set of ${\displaystyle f}$ is then a pluripolar set if ${\displaystyle f}$ is not the zero function.

See also

• Skoda-El Mir theorem