Vol. 48, No. 1, 1973

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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

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
Received: 22 June 1972
Published: 1 September 1973
Michael Benton Freeman
Reese Harvey