Pluripolar hulls and convergence sets

Abstract

The pluripolar hull of a pluripolar set E in ${\mathbb{P}}^n$ is the intersection of all complete pluripolar sets in ${\mathbb{P}}^n$ that contain $E$. We prove that the pluripolar hull of each compact pluripolar set in ${\mathbb{P}}^n$ is $F_{\sigma }$. The convergence set of a divergent formal power series $f(z_0,\dots ,z_n)$ is the set of all "directions" $\xi \in {\mathbb{P}}^n$ along which $f$ is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in ${\mathbb{P}}^n$ is the convergence set of some divergent series $f$. The convergence sets on $\Gamma :=\{[1:z:\psi (z)]:z\in \mathbb{C}\}\subset {\mathbb{C}}^2\subset {\mathbb{P}}^2$, where $\psi$ is a transcendental entire holomorphic function, are also studied and we obtain that a subset on $\Gamma$ is a convergence set in ${\mathbb{P}}^2$ if and only if it is a countable union of compact projectively convex sets, and hence the union of a countable collection of convergence sets on $\Gamma$ is a convergence set.

24 pages

References in corpus (1)

• A note on holomorphic extensions