No topological condition implies equality of polynomial and rational hulls

TL;DR

The paper demonstrates that topological properties alone do not determine the equality of polynomial and rational hulls, providing counterexamples and confirming equality in specific complex manifold cases.

Contribution

It constructs sets homeomorphic to any compact set that are rationally but not polynomially convex, and proves equality of hulls for certain complex surfaces and manifolds.

Findings

Existence of sets homeomorphic to any compact set with rational but not polynomial convexity

Equality of polynomial and rational hulls for specific complex surfaces without analytic discs

Confirmation of hull equality for certain manifolds with nontrivial polynomial hulls

Abstract

It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any compact subset $K$ of a Euclidean space, there exists a set $X$, in some ${\mathbb C}^N$, that is homeomorphic to $K$ and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in ${\mathbb C}^3$ constructed by Izzo and Stout, whose polynomial hulls are nontrivial but contain no analytic discs, the polynomial and rational hulls coincide, thereby answering a question of Gupta. Equality of polynomial and rational hulls is shown also for $m$-dimensional manifolds ($m\geq 2$) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold.