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On the coefficients of TYZ expansion of locally Hermitian symmetric spaces

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Abstract

In this paper we address the problem of studying those Kähler manifolds whose first two coefficients of the associated TYZ expansion vanish and we prove that for a locally Hermitian symmetric space this happens only in the flat case. We also prove that there exist nonflat locally Hermitian symmetric spaces where all the odd coefficients vanish.

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References

  1. Abreu M.: Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds. Ann. Glob. Anal. Geom. 41, 209–239 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arezzo C., Loi A.: Quantization of Kähler manifolds and the asymptotic expansion of Tian–Yau–Zelditch. J. Geom. Phys. 47, 87–99 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arezzo C., Loi A.: Moment maps, scalar curvature and quantization of Kähler manifolds. Commun. Math. Phys. 243, 543–559 (2004)

    Article  MathSciNet  Google Scholar 

  4. Arezzo C., Loi A., Zuddas F.: Szegö kernel, regular quantizations and spherical CR-structures. Math. Z. 275, 1207–1216 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berezin, F.A.: Quantization. Izv. Akad. Nauk SSSR Ser. Mat. 38, 1116–1175 (1974) (Russian)

  6. Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds I: geometric interpretation of Berezin’s quantization. JGP. 7, 45–62 (1990)

    MATH  MathSciNet  Google Scholar 

  7. Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds II. Trans. Am. Math. Soc. 337, 73–98 (1993)

    MATH  MathSciNet  Google Scholar 

  8. Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds III. Lett. Math. Phys. 30, 291–305 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds IV. Lett. Math. Phys. 34, 159–168 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Calabi E.: Isometric imbeddings of complex manifolds. Ann. Math. 58, 1–23 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  11. Di Scala A.J., Loi A.: Symplectic duality of symmetric spaces. Adv. Math. 217, 2336–2352 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Donaldson S.: Scalar Curvature and Projective Embeddings, I. J. Diff. Geom. 59, 479–522 (2001)

    MATH  MathSciNet  Google Scholar 

  13. Donaldson S.: Scalar curvature and projective embeddings, II. Q. J. Math. 56, 345–356 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Engliš M.: Berezin quantization and reproducing kernels on complex domains. Trans. Am. Math. Soc. 348(2), 411–479 (1996)

    Article  MATH  Google Scholar 

  15. Engliš M.: A Forelli-Rudin construction and asymptotics of weighted Bergman kernels. J. Funct. Anal. 177(2), 257–281 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Engliš M.: The asymptotics of a Laplace integral on a Kähler manifold. J. Reine Angew. Math. 528, 1–39 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Engliš M., Zhang G.: Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces. Math. Z. 264(4), 901–912 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Feng Z., Tu Z.: On canonical metrics on Cartan-Hartogs domains. Math. Zeit. 278(1-2), 301–320 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kempf G.R.: Metric on invertible sheaves on abelian varieties. Topics in algebraic geometry, Guanajuato (1989)

    Google Scholar 

  20. Ji S.: Inequality for distortion function of invertible sheaves on Abelian varieties. Duke Math. J. 58, 657–667 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  21. LeBrun C.: Scalar flat Kähler metrics on blown-up ruled surfaces. J. Reine Angew. Math. 420, 161–177 (1991)

    MATH  MathSciNet  Google Scholar 

  22. Loi A.: The Tian–Yau–Zelditch asymptotic expansion for real analytic Kähler metrics. Int. J. Geom. Methods Mod. Phys. 1, 253–263 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Loi A.: A Laplace integral, the T-Y-Z expansion and Berezin’s transform on a Kähler manifold. Int. J. Geom. Meth. Mod. Phys. 2, 359–371 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Loi A.: Regular quantizations of Kähler manifolds and constant scalar curvature metrics. J. Geom. Phys. 53, 354–364 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Loi A.: Calabi’s diastasis function for Hermitian symmetric spaces. Diff. Geom. Appl. 24, 311–319 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Loi A., Gramchev T.: TYZ expansion for the Kepler manifold. Comm. Math. Phys. 289, 825–840 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Loi A., Mossa R.: Berezin quantization of homogeneous bounded domains. Geom. Dedicata 161, 119–128 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  28. Loi A., Mossa R.: The diastatic exponential of a symmetric space. Math. Z. 268(3–4), 1057–1068 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Loi A., Zedda M.: Kähler–Einstein submanifolds of the infinite dimensional projective space. Math. Ann. 350(1), 145–154 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Loi A., Zedda M.: Balanced metrics on Cartan and Cartan–Hartogs domains. Math. Z. 270(3-4), 1077–1087 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Loi A., Zedda M., Zuddas F.: Some remarks on the Kähler geometry of the Taub-NUT metrics. Ann. Glob. Anal. Geom. 41(4), 515–533 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lu Z.: On the lower terms of the asymptotic expansion of Tian–Yau–Zelditch. Am. J. Math. 122, 235–273 (2000)

    Article  MATH  Google Scholar 

  33. Lu Z., Tian G.: The log term of Szegö Kernel. Duke Math. J. 125(2), 351–387 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Ma X., Marinescu G.: Holomorphic morse inequalities and Bergman kernels. Progress in Mathematics, Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  35. Rawnsley J.: Coherent states and Kähler manifolds. Q. J. Math. Oxford (2) 28,403–415 (1977)

  36. Simanca S.R.: Kähler metrics of constant scalar curvature on bundles over ℂP n-1. Math. Ann. 291, 239–246 (1991)

  37. Xu H.: A closed formula for the asymptotic expansion of the Bergman kernel. Commun. Math. Phys. 314(3), 555585 (2012)

    Article  Google Scholar 

  38. Zedda, M.: Canonical metrics on Cartan–Hartogs domains. Int. J. Geom. Methods Mod. Phys. 9(1), 1250011, 13 pp. (2012)

  39. Zelditch S.: Szegö Kernels and a Theorem of Tian. Int. Math. Res. Not. 6,317–331 (1998)

  40. Zhang S.: Heights and reductions of semi-stable varieties. Comp. Math. 104,77–105 (1996)

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale, 72, 09124, Cagliari, Italy

    Andrea Loi

  2. Dipartimento di Matematica “G. Peano” Università di Torino, Turin, Italy

    Michela Zedda

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  1. Andrea Loi
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  2. Michela Zedda
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Correspondence to Andrea Loi.

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Loi, A., Zedda, M. On the coefficients of TYZ expansion of locally Hermitian symmetric spaces. manuscripta math. 148, 303–315 (2015). https://doi.org/10.1007/s00229-015-0746-6

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  • DOI: https://doi.org/10.1007/s00229-015-0746-6

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