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.

See also

• Skoda-El Mir theorem

References

1. Sibony, Nessim; Schleicher, Dierk; Cuong, Dinh Tien; Brunella, Marco; Bedford, Eric; Abate, Marco (2010). Gentili, Graziano; Patrizio, Giorgio; Guenot, Jacques (eds.). Holomorphic Dynamical Systems: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 7-12, 2008. Springer Science & Business Media. p. 275. ISBN 978-3-642-13170-7.

• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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