# Examples of pluripolar sets

I have a very basic question on pluripolar sets. First remind their definition. Let $\Omega\subset \mathbb{C}^n$ be a domain. A subset $E\subset \Omega$ is called pluripolar if there exists a plurisubharmonic function $f\colon \Omega\to \mathbb{R}$, not identically $-\infty$, such that $E\subset \{f=-\infty\}$.

Question. What are the examples of pluripolar sets? For example, when a closed real analytic (or smooth) submanifold of $\Omega$ is pluripolar?

Remark. According to Wikipedia, pluripolar sets have Hausdorff dimension at most $2n-2$. Also the zero set of a holomorphic function is pluripolar.

Pluripolarity is a quite subtle property.

E. Bedford characterized pluripolar real-analytic submanifolds.

Added: Bedford (in Lelong-Skoda seminar, 1981) called a real submanifold $M$ of $\mathbb{C}^n$ a generating at a point $p$ if the tangent space $T_p M$ is not contained in a proper complex-linear subspace of $\mathbb{C}^n$. It is easy to see that a generating at any point submanifold is not pluripolar. Bedford prove that a real-analytic nowhere generating submanifold is pluripolar.

For example, all real analytic curves in $\mathbb{C}^n$ are pluripolar for $n>1$.

In opposite direction, it's useful to know examples of sets that are not pluripolar. There is a very surprising example due to K. Diederich and J. E. Fornæss (1982) of a non-pluripolar $C^\infty$ smooth curve in $\mathbb{C}^2$. This construction can be extended to $\mathbb{C}^n$.

• Can you state please Bedford's result? – makt Jan 28 '16 at 15:41
• @sva See the edit. – Oleg Eroshkin Jan 28 '16 at 17:47

As you said, analytic sets (other than the whole space) are pluripolar. By playing with subharmnic functions of the form $\sum u_j$ you can arrange some more complicated sets which are neither open nor closed (but the set where a plurisubharmonic function is $-\infty$ must be $G_\delta$). So you can have a dense pluripolar set, for example. Other non-analytic examples can be made by taking subsets of pluripolar sets. Pluripolar sets in (complex) dimension $1$ are easier to visualize: they are just very small, and sufficient conditions for a set to be pluripolar can be given in terms of Haurdorff-like measure terms (but only sufficient or necessary, there is a theorem that says that a complete characterization in metric terms is impossible). Then you can take products of these one dimensional sets times $C^k$.

But there is the following remarkable theorem:

If a pluripolar set in $C^{n+1}$ is a graph of a continuous function $f:C^n\to C$, then this function is analytic (thus the set is also analytic).

Nikolay Shcherbina, Pluripolar graphs are holomorphic. Acta Math. 194 (2005), no. 2, 203–216.