Geometriae Dedicata

, Volume 161, Issue 1, pp 119–128 | Cite as

Berezin quantization of homogeneous bounded domains

Original Paper

Abstract

We prove that a homogeneous bounded domain admits a Berezin quantization.

Keywords

Kähler metrics Berezin quantization Bounded homogeneous domain Calabi’s diastasis function 

Mathematics Subject Classification (2000)

53D05 53C55 58F06 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arezzo C., Loi A.: Quantization of Kähler manifolds and the asymptotic expansion of Tian–Yau–Zelditch. J. Geom. Phys. 47, 87–99 (2003)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arezzo C., Loi A.: Moment maps, scalar curvature and quantization of Kähler manifolds. Comm. Math. Phys. 246, 543–549 (2004)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Auslander L., Kostant B.: Polarization and unitary representations of solvable Lie groups. Invent. Math. 14, 255–354 (1971)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Berezin F.A.: Quantization. Math. USSR Izvestiya 8, 1109–1163 (1974) MR 52:16404MATHCrossRefGoogle Scholar
  5. 5.
    Calabi E.: Isometric imbeddings of complex manifolds. Ann. Math. 58, 1–23 (1953)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds I: geometric interpretation of Berezin’s quantization. JGP 7, 45–62 (1990)MathSciNetMATHGoogle Scholar
  7. 7.
    Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds II. Trans. Am. Math. Soc. 337, 73–98 (1993)MathSciNetMATHGoogle Scholar
  8. 8.
    Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds III. Lett. Math. Phys. 30, 291–305 (1994)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds IV. Lett. Math. Phys. 34, 159–168 (1995)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Di Scala, A.J., Ishi, H., Loi, A.: Kaehler immersions of homogeneous Kaehler manifolds into complex space forms. Asian J. Math. (To appear)Google Scholar
  11. 11.
    Donaldson S.: Scalar curvature and projective embeddings, I. J. Diff. Geom. 59, 479–522 (2001)MathSciNetMATHGoogle Scholar
  12. 12.
    Dorfmeister J.: Simply transitive groups and Kähler structures on homogeneous Siegel domains. Trans. Am. Math. Soc. 288, 293–305 (1985)MathSciNetMATHGoogle Scholar
  13. 13.
    Engliš M.: Berezin quantization and reproducing kernels on complex domains. Trans. Am. Math. Soc. 348, 411–479 (1996)MATHCrossRefGoogle Scholar
  14. 14.
    Gindikin S.G.: Analysis in homogeneous domains. Russ. Math. Surv. 19, 1–89 (1964)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gramchev T., Loi A.: TYZ expansion for the Kepler manifold. Comm. Math. Phys. 289, 825–840 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Greco A., Loi A.: Radial balanced metrics on the unit disk. J. Geom. Phys. 60, 53–59 (2010)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Ishi H.: Representations of the affine transformation groups acting simply transitively on Siegel domains. J. Funct. Anal. 167(2), 425–462 (1999)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Ishi, H.: Unitary holomorphic multiplier representations over a homogeneous bounded domain. Adv. Pure Appl. Math. (To appear)Google Scholar
  19. 19.
    Ji S.: Inequality for distortion function of invertible shaves on Abelian varieties. Duke Math. J. 58, 657–667 (1989)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kempf, G.R.: Metrics on invertible shaves on abelian varieties. Topics in algebraic geometry (Guanajuato, 1989): Aportationes Mat. Notas Investigacion 5, Soc. Mat. Mexicana, Mexico, pp. 107–108 (1992)Google Scholar
  21. 21.
    Loi A.: Regular quantizations of K manifolds and constant scalar curvature metrics. J. Geom. Phys. 53, 354–364 (2005)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Loi A.: Calabi’s diastasis function for Hermitian symmetric spaces. Diff. Geom. Appl. 24, 311–319 (2006)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Loi A., Zedda M.: K-Einstein submanifolds of the infinite dimensional projective space. Math. Ann. 350, 145–154 (2011)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Loi, A., Zedda, M.: Balanced metrics on Cartan and Cartan-Hartogs domain. Math. Zeitschrift (2011). (To appear)Google Scholar
  25. 25.
    Loi A., Zedda M.: Balanced metrics on Hartogs domains. Abh. Math. Sem. Univ. Hamburg 81(1), 69–77 (2011)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Nomura T.: Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains. J. Lie Theory 11, 185–206 (2001)MathSciNetMATHGoogle Scholar
  27. 27.
    Piatetskii-Shapiro I.I.: Automorphic Functions and the Geometry of Classical Domains. Gordon and Breach, New York, NY (1969)Google Scholar
  28. 28.
    Rawnsley J.: Coherent states and K manifolds. Q. J. Math. Oxf 2(28), 403–415 (1977)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rossi H., Vergne M.: Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group. J. Funct. Anal. 13, 324–389 (1973)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Zhang S.: Heights and reductions of semi-stable varieties. Comp. Math. 104, 77–105 (1996)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di CagliariCagliariItaly

Personalised recommendations