Approximation of polynomial hulls by analytic varieties: A counterexample

TL;DR

This paper presents a counterexample in complex analysis, showing that certain polynomial hulls cannot be approximated by analytic varieties with boundary conditions, by constructing specific sets and sequences of Poletsky discs.

Contribution

It constructs a connected, compact set in ^2 with points in its polynomial hull that cannot be approximated by analytic sets with boundary, providing explicit sequences of Poletsky discs.

Findings

Counterexample to polynomial hull approximation by analytic varieties

Explicit construction of Poletsky discs for points outside the set

Computation of weak limits of Green currents under these discs

Abstract

We construct a connected, compact set $K \subset \mathbb{C}^2$ with the following property: there exist points $p \in \hat{K} \setminus K$ such that there does not exist a sequence $\{A_\nu\}$ of analytic sets $A_\nu \subset\subset \mathbb{C}^2$ with boundary satisfying $p \in A_\nu$ for every $\nu \in \mathbb{N}$ and $\lim_{\nu\to\infty} bA_\nu \subset K$. For every point in $\hat{K} \setminus K$, we explicitly construct a sequence of Poletsky discs, and we compute the weak limit of the pushforwards of the Green current under these discs.