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The Lu Qi-Keng conjecture fails generically

Author: Harold P. Boas
Journal: Proc. Amer. Math. Soc. 124 (1996), 2021-2027
MSC (1991): Primary 32H10
MathSciNet review: 1317032
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Abstract: The bounded domains of holomorphy in ${\mathbf {C}} ^n$ whose Bergman kernel functions are zero-free form a nowhere dense subset (with respect to a variant of the Hausdorff distance) of all bounded domains of holomorphy.

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  • 1. H. P. Boas, Counterexample to the Lu Qi-Keng conjecture, Proc. Amer. Math. Soc. 97 (1986), 374--375. MR 87i:32035
  • 2. H. P. Boas and E. J. Straube, Equivalence of regularity for the Bergman projection and the $\overline \partial $-Neumann operator, manuscripta math. 67 (1990), 25--33. MR 90k:32057
  • 3. Yakov Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, International J. Math. 1 (1990), no. 1, 29--46. MR 91k:32012
  • 4. John E. Fornæss and Bill Zame, Runge exhaustions of domains in ${\mathbf {C}} ^n$, Math. Z. 194 (1987), 1--5. MR 88c:32026
  • 5. John Erik Fornæss and Edgar Lee Stout, Spreading polydiscs on complex manifolds, Amer. J. Math. 99 (1977), no. 5, 933--960. MR 57:10009
  • 6. R. E. Greene and Steven G. Krantz, Stability properties of the Bergman kernel and curvature properties of bounded domains, in Recent Developments in Several Complex Variables, Princeton Univ. Press, Princeton, NJ, 1981, 179--198. MR 83d:32023
  • 7. Lars Hörmander, $L^2$ estimates and existence theorems for the $\overline \partial$-operator, Acta Math. 113 (1965) 89--152. MR 31:3691
  • 8. M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter, Berlin, New York, 1993. MR 94k:32039
  • 9. J. J. Kohn, Global regularity for $\overline \partial$ on weakly pseudo-convex manifolds, Trans. Amer. Math. Soc. 181 (1973), 273--292. MR 49:9442
  • 10. Steven G. Krantz, Function Theory of Several Complex Variables, second edition, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1992. MR 93c:32001
  • 11. Lu Qi-Keng, On Kaehler manifolds with constant curvature, Chinese Math. 8 (1966), 283--298; English translation of Acta Math. Sinica 16 (1966), 269--281. MR 34:6806
  • 12. I. Ramadanov, Sur une propriété de la fonction de Bergman, C. R. Acad. Bulgare Sci. 20 (1967), 759--762. MR 37:1632
  • 13. I.-P. Ramadanov, Some applications of the Bergman kernel to geometrical theory of functions, Complex Analysis, Banach Center Publications, Vol. 11, PWN---Polish Scientific Publishers, Warsaw, 1983, 275--286. MR 85h:32040
  • 14. N. V. Shcherbina, On fibering into analytic curves of the common boundary of two domains of holomorphy, Math. USSR Izvestiya 21 (1983), no. 2, 399--413; English translation of Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1106--1123. MR 84b:32018
  • 15. M. Skwarczynski, Biholomorphic invariants related to the Bergman function, Dissertationes Math. 173 (1980). MR 82e:32038
  • 16. N. Suita and A. Yamada, On the Lu Qi-Keng conjecture, Proc. Amer. Math. Soc. 59 (1976), 222-224. MR 54:13142

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Additional Information

Harold P. Boas
Affiliation: Department of Mathematics Texas A&M University College Station Texas 77843–3368

Received by editor(s): December 10, 1994
Additional Notes: This research was partially supported by NSF grant number DMS-9203514.
Communicated by: Eric Bedford
Article copyright: © Copyright 1996 American Mathematical Society

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