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Polynomial hulls and analytic discs. (English) Zbl 1382.32008

The author constructs a compact connected subset \(K\subset \mathbb C^n\) whose polynomial hull \(\hat K\) is not covered by analytic discs with boundaries in small neighborhoods of \(K\). This answers a question raised in [the reviewer and F. Forstnerič, Ill. J. Math. 56, No. 1, 53–65 (2012; Zbl 1311.32013)].
More precisely, the author defines the disc hull \(\hat K_{\text{disc}}\) of a compact subset \(K\subset \mathbb C^n\) as the set of all points \(z\in \mathbb C^n\) such that for any \(\epsilon>0\) there is a holomorphic map \(f: \mathbb D\to \mathbb C^n\) continuous up to the boundary \(\mathbb T\) with \(f(0)=z\) and such that its boundary \(f(\mathbb T)\) lies in the \(\epsilon\)-neighborhood of \(K\). He shows that for any compact set \(K\) in the boundary of the unit ball \(\mathbb B^n\subset\mathbb C^n\), with finitely many connected components, there is a connected compact set \(L\subset \mathbb C^n\) such that \(L\cap \bar{\mathbb B}^n=K\), \(\hat L=L\cup \hat K\) and \(\hat L_{\text{disc}}=L\cup \hat K_{\text{disc}}\).
Note that P. Ahern and W. Rudin [Contemp. Math. 137, 1–27 (1992; Zbl 0769.32004)] defined the disc hull differently: the disc hull \(\mathcal{D}(K)\) of a compact subset \(K\subset \mathbb C^n\) is the union of all bounded holomorphic maps \(f: \mathbb D\to \mathbb C^n\) such that \(f^*(z)\in K\) for almost every \(z\in \mathbb T\). It is not difficult to see that \(\hat K_{\text{disc}}\subset \mathcal{D}(K)\) but the two definitions do not coincide in general.

MSC:

32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32T05 Domains of holomorphy
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