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The Lu Qi-Keng conjecture fails generically. (English) Zbl 0857.32010

A domain in \(\mathbb{C}^n\) is a Lu Qi-Keng domain if its Bergman kernel function has no zero. An open set in \(\mathbb{C}^n\) is a Lu Qi-Keng set if all its connected components are Lu Qi-Keng domains. The importance of this concept lies on the fact that the vanishing of the Bergman kernel function obstructs the global definition of the Bergman representative coordinates. This has motivated Lu Qi-Keng [Q.-K. Lu, Chinese Math. 8, 283-289 (1966) \(\equiv\) Acta Math. Sinica 16, 269-281 (1966; Zbl 0173.33001)] to raise the by now famous Lu Qi-Keng conjecture which, in effect, conjectures that most domains with some additional geometric properties are Lu Qi-Keng domains. Of course, the notion of a Lu Qi-Keng domain is a biholomorphically invariant concept. In \(\mathbb{C}^1\), a domain \(\Omega \neq \mathbb{C}^1\) is a Lu Qi-Keng domain if and only if \(\Omega\) is simply connected (i.e. if and only if \(\Omega\) if biholomorphically equivalent to a disk) [see N. Sinta and A. Yamada, Proc. Am. Math. Soc. 59, 222-224 (1976; Zbl 0333.30004)]. That no analogous topological characterization of Lu Qi-Keng domains can hold in \(\mathbb{C}^n\), \(n>1\), was shown by the author in Proc. Am. Math. Soc. 97, 374-375 (1986; Zbl 0596.32032). In the present note the author shows that the bounded Lu Qi-Keng domains of holomorphy in \(\mathbb{C}^n\) form a nowhere dense subset, with respect to a suitable topology defined by a variant of the Hausdorff distance on bounded open sets, of all bounded domains of holomorphy. The author concludes the paper with some remaining open questions. In particular, one would like to settle the question whether every bounded convex domain in \(\mathbb{C}^n\) is a Lu Qi-Keng domain.

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32D05 Domains of holomorphy
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