Counterexample to the Lu Qi-Keng conjecture. (English) Zbl 0596.32032

In 1966, Lu Qi-Keng [Chinese Math. 8, 283-298 (1966); translation from Acta Math. Sinica 16, 269-281 (1966; Zbl 0173.330)] raised the question whether for every domain in \({\mathbb{C}}^ n\) the Bergman kernel function is zero-free (it is for the unit ball). The affirmative answer has come to be known as the Lu Qi-Keng conjecture. It was soon realized that the conjecture fails for annuli in the plane [M. Skwarczynski: Proc. Am. Math. Soc. 22, 305-310 (1969; Zbl 0182.110), P. Rosenthal, Proc. Am. Math. Soc. 21, 33-35 (1969; Zbl 0172.101)], and these counterexamples can be lifted to higher dimensions [R. E. Greene and S. G. Krantz: Ann. Math. Stud. 100, 179-198 (1981; Zbl 0483.32014)] However, the conjecture remained open for topologically trivial domains in higher dimensions. In the note under review, the author settles it in the negative. Specifically, he proves the following:
Theorem: There exists a smooth, bounded, strongly pseudoconvex domain in \({\mathbb{C}}^ 2\) whose closure is diffeomorphic to the ball and whose Bergman kernel function has zeroes.
It should be pointed out that the domains obtained are remarkably simple, and the vanishing of the kernel function is not to be considered a pathology.
At this time, it seems to be open what happens for convex domains.
Reviewer: E.Straube


32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32T99 Pseudoconvex domains
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