A compact set that is locally holomorphically convex but not holomorphically convex

Michael Benton Freeman and Reese Harvey

Vol. 48 (1973), No. 1, 77–81

DOI: 10.2140/pjm.1973.48.77

Abstract

It is shown that a certain simple imbedding T of the ordinary two-dimensional torus in C² contains a polynomially convex compact T-neighborhood of each of its points, but T is not holomorphically convex in even the weakest presently accepted sense. This example illustrates some of the limitations of a theory of lower dimensional sets in Cⁿ. In particular, it shows the difficulty of developing a theory based on local information.

Mathematical Subject Classification 2000

Primary: 32E30

Secondary: 32E20

Milestones

Received: 22 June 1972

Published: 1 September 1973