Pluripolar hulls and fine analytic structure

TL;DR

This paper explores the relationship between pluripolar hulls and fine analytic structures, demonstrating that for any non-polar subset of the complex plane, a pluripolar set can be constructed with a hull that lacks fine analytic structure but projects onto the entire plane.

Contribution

It proves the existence of pluripolar sets with hulls lacking fine analytic structure, extending understanding of pluripolar hulls in complex analysis.

Findings

Constructed pluripolar sets with specific hull properties

Showed hulls can lack fine analytic structure

Proved projection covers entire complex plane

Abstract

We discuss the relation between pluripolar hulls and fine analytic structure. Our main result is the following. For each non polar subset $S$ of the complex plane $\mathbb C$ we prove that there exists a pluripolar set $E \subset (S \times \mathbb C)$ with the property that the pluripolar hull of $E$ relative to $\mathbb C^2$ contains no fine analytic structure and its projection onto the first coordinate plane equals $\mathbb C$.