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Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

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Abstract

We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \({\mathbb C^n}\) for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szegö kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.

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References

  1. Attioui A.: Version réelle de la conjecture de Ramadanov. Ann. Sci. école Norm. Sup. 29(4), 273–285 (1996)

    MATH  MathSciNet  Google Scholar 

  2. Axler S., Conway J.B., McDonald G.: Toeplitz operators on Bergman spaces. Canad. J. Math. 34, 466–483 (1982)

    MATH  MathSciNet  Google Scholar 

  3. Beals M., Fefferman C., Grossmann R.: Strictly pseudoconvex domains in \({\mathbb C^n}\) . Bull. Am. Math. Soc. (N.S.) 8, 125–322 (1983)

    Article  Google Scholar 

  4. Berezin, F.A.: Quantization in complex symmetric spaces. Izvestiya Akad. Nauk SSSR Ser. Mat. 39, 363–402 (1975) (English translation, Math. USSR—Izvestiya 9, 341–379) (1975)

    Google Scholar 

  5. Boichu D., Cœ uré G.: Sur le noyau de Bergman des domaines de Reinhardt. Invent. Math. 72, 131–152 (1983)

    Article  MathSciNet  Google Scholar 

  6. Bott R., Tu L.W.: Differential forms in algebraic topology. Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)

    Google Scholar 

  7. Boutet de Monvel, L.: Le noyau de Bergman en dimension 2, Séminaire sur les équations aux Dérivées Partielles 1987–1988, Exp. no. XXII, p. 13. école Polytech., Palaiseau (1988)

  8. Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. Journées: équations aux Dérivées Partielles de Rennes (1975) Soc. Math. France, Paris, 1976, pp. 123–164. Astérisque, No. 34–35

  9. Burns D., Shnider S.: Spherical hypersurfaces in complex manifolds. Invent. Math. 33, 223–246 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chern S.-S., Ji S.: On the Riemann mapping theorem. Ann. Math. 144, 421–439 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Faraut J., Korányi A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 64–89 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fefferman C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Inv. Math. 26, 1–65 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  13. Graham C.R.: Higher asymptotics of the complex Monge-Ampére equation. Compos. Math. 64, 133–155 (1987)

    MATH  Google Scholar 

  14. Graham, C.R.: Scalar boundary invariants and the Bergman kernel. In: Complex Analysis II (College Park, Maryland 1985–86). Lecture Notes in Math., vol. 1276, pp. 108–135. Springer, Berlin (1987)

  15. Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  16. Helgason S.: Differential Geometry. Lie Groups and Symmetric Spaces. Academic Press, New York (1978)

    MATH  Google Scholar 

  17. Hirachi, K.: Scalar pseudo-Hermitian invariants and the Szegö kernel on three-dimensional CR manifolds. In: Complex Geometry (Osaka 1990). Lect. Notes Pure Appl. Math., vol. 143, pp. 67–76. Dekker, New York (1993)

  18. Hirachi K.: The second variation of the Bergman kernel of ellipsoids. Osaka J. Math. 30, 457–473 (1993)

    MATH  MathSciNet  Google Scholar 

  19. Hirachi, K., Komatsu, G.: Local Sobolev-Bergman kernels of strictly pseudoconvex domains. In: Analysis and geometry in several complex variables (Katata, 1997). Trends Math, pp. 63–96. Birkhäuser, Boston (1999).

  20. Kobayashi S.: Geometry of bounded domains. Trans. Am. Math. Soc. 92, 267–290 (1959)

    Article  MATH  Google Scholar 

  21. Komuro M.: On Atiyah-Patodi-Singer η-invariant for S 1-bundles over Riemann surfaces. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30, 525–548 (1984)

    MATH  MathSciNet  Google Scholar 

  22. Loos O.: Bounded Symmetric Domains and Jordan Pairs. University of California, Irvine (1977)

    Google Scholar 

  23. Lu Z., Tian G.: The log term of the Szegö kernel. Duke Math. J. 125, 351–387 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Milnor J., Stasheff J.: Characteristic classes. Annals of Math. Studies. Princeton University Press, New Jersey (1974)

    Google Scholar 

  25. Nakazawa N.: Asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains in \({\mathbb C^2}\) . Proc. Jpn. Acad. Ser. A Math. Sci. 66, 39–41 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ramadanov I.P.: A characterization of the balls in \({\mathbb C^n}\) by means of the Bergman kernel. C. R. Acad. Bulgare Sci. 34, 927–929 (1981)

    MATH  MathSciNet  Google Scholar 

  27. Zhang G.: Berezin transform on compact Hermitian symmetric spaces. Manuscr. Math. 97, 371–388 (1998)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Institute, Silesian University, Na Rybníčku 1, 74601, Opava, Czech Republic

    Miroslav Engliš

  2. Mathematics Institute, Academy of Sciences, Žitná 25, 11567, Prague 1, Czech Republic

    Miroslav Engliš

  3. Department of Mathematical Sciences, Chalmers University of Technology, Göteborg, Sweden

    Genkai Zhang

  4. Department of Mathematical Sciences, Göteborg University, 412 96, Göteborg, Sweden

    Genkai Zhang

Authors
  1. Miroslav Engliš
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  2. Genkai Zhang
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Correspondence to Genkai Zhang.

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Research supported by the Swedish Science Council (VR), GA ČR grant 201/06/0128, and Czech Ministry of Education research plan MSM4781305904.

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Engliš, M., Zhang, G. Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces. Math. Z. 264, 901–912 (2010). https://doi.org/10.1007/s00209-009-0495-x

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