×

Large polynomial hulls with no analytic structure. (English) Zbl 0885.32010

Ancona, V. (ed.) et al., Complex analysis and geometry. Proceedings of the CIRM conferences on complex analysis and geometry XII, Trento, Italy, June 5–9, 1995 and vector bundles on Fano threefolds II, Trento, Italy, December 2–5, 1995. Bonn: Longman. Pitman Res. Notes Math. Ser. 366, 119-122 (1997).
Let \(X\) be a compact set in \(\mathbb{C}^2\) and \(\widehat X\) denote the polynomial hull of \(X;\) i.e., the set of points \(x\in \mathbb{C}^2\) satisfying the inequality \(|P(x)|\leq\max_X|P|\) for each holomorphic polynomial hull. The set \(X\) is called polynomial convex if it coincides with its polynomial hull.
The goal of this note is to give a version of the Stolzenberg construction [G. Stolzenberg, J. Math. Mech. 12, 103-111 (1963; Zbl 0113.29101)], which allows the authors to construct hulls which are rather large (e.g., having positive measure) but which have no analytic structure. Let \(K\) be a compact, polynomially convex subset of the unit ball \(B\) in \(\mathbb{C}^2.\) Then there exists a compact set \(X\) in the boundary of the ball \(B\) such that \(K\subset\widehat X\) and \(\widehat X\setminus K\) has no analytic structure. In particular, if \(K\) itself contains no analytic structure, then \(\widehat X\) has no analytic structure. Taking \(K\) to be a product of two Jordan arcs having positive measure in \(\mathbb{C},\) the following corollary is obtained. There exists a compact set \(X\) in the boundary of the unit ball \(B\) in \(\mathbb{C}^2\) whose polynomial hull \(\widehat X\) has no analytic structure but is of positive Lebesgue measure.
For the entire collection see [Zbl 0869.00034].

MSC:

32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables

Citations:

Zbl 0113.29101
PDFBibTeX XMLCite