Pluripolar Hulls and Convergence Sets

Abstract

The pluripolar hull of a pluripolar set E in ℙn is the intersection of all complete pluripolar sets in ℙn that contain E. We prove that the pluripolar hull of each compact pluripolar set in ℙn is Fσ. The convergence set of a divergent formal power series f(z0, … , zn) is the set of all “directions” ξ ∈ ℙn along which f is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in ℙn is the convergence set of some divergent series f. The convergence sets on Γ := {[1 : z : ψ (z)] : z ∈ ℂ} ⊂ ℂ2 ⊂ ℙ2, where ψ is a transcendental entire holomorphic function, are also studied and we obtain that a subset on Γ is a convergence set in ℙ2 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 Γ is a convergence set.