Pluripolar hulls and convergence sets

TL;DR

This paper investigates the properties of pluripolar hulls in complex projective space and characterizes convergence sets of divergent series, establishing their structure and relation to pluripolar sets.

Contribution

It proves that pluripolar hulls of compact sets are $F_\sigma$, and characterizes convergence sets as unions of pluripolar hulls and countable unions of convex sets.

Findings

Pluripolar hulls of compact sets are $F_\sigma$ sets.

Union of pluripolar hulls of countable collections forms a convergence set.

Convergence sets on certain curves are countable unions of convex 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.