Skip to main content
SpringerLink
Log in

Asymptotics of G-equivariant Szegő kernels

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Let \((X, T^{1,0}X)\) be a compact connected orientable CR manifold of dimension \(2n+1\) with non-degenerate Levi curvature. Assume that X admits a connected compact Lie group G action. Under certain natural assumptions about the group G action, we define G-equivariant Szegő kernels and establish the associated Boutet de Monvel–Sjöstrand type theorems. When X admits also a transversal CR \(S^1\) action, we study the asymptotics of Fourier components of G-equivariant Szegő kernels with respect to the \(S^1\) action.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boutet de Monvel, L., Sjöstrand, J.: Sur la singularitédes noyaux de Bergman et de Szegö, Astérisque,34–35, 123–164 (1976) MR0590106, Zbl 0344.32010

  2. Boutet de Monvel, L., Guillemin, V.: The spectral theory of Toeplitz operators, Ann. Math. Stud., vol 99, Princeton Univ. Press, Princeton, NJ, (1981)

  3. D. Catlin, The Bergman kernel and a theorem of Tian, in Analysis and geometry in several complex variables (Katata, 1997),1–23, Trends Math., Birkhäuser, Boston, 1999, MR1699887, Zbl0941.32002

  4. Engliš, M.: Weighted Bergman kernels and quantization. Comm. Math. Phys. 227(2), 211–241 (2002). (MR1903645, Zbl 1010.32002)

    Article  MathSciNet  Google Scholar 

  5. Fritsch, K., Herrmann, H., Hsiao, C.-Y.: \(G\)-equivariant embedding theorems for CR manifolds of high codimension, arXiv: 1810.09629, to appear in Mich. Math. J

  6. Guillemin, V.: Star products on compact pre-quantizable symplectic manifolds. Lett. Math. Phys. 35(1), 85–89 (1995)

    Article  MathSciNet  Google Scholar 

  7. Herrmann, H., Hsiao, C.-Y., Li, X.: Szegő kernels and equivariant embedding theorems for CR manifolds, arXiv: 1710.04910, to appear in Math. Res. Lett

  8. Hörmander, L.: The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, (2003)

  9. Hsiao, C.-Y.: Projections in several complex variables, Mém. Soc.Math. France, Nouv. Sér. 123 (2010), 131 p, MR2780123,Zbl 1229.32002

  10. Hsiao, C.-Y., Huang, R.-T.: \(G\)-invariant Szegő kernel asymptotics and CR reduction. Calc. Var. Partial Differential Equations 60(1), 48 (2021)

    Article  Google Scholar 

  11. Hsiao, C.-Y., Ma, X., Marinescu, G.: Geometric quantization on CR manifolds, arXiv: 1906.05627

  12. Hsiao, C.-Y., Marinescu, G.: Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles. Comm. Anal. Geom. 22(1), 1–108 (2014)

    Article  MathSciNet  Google Scholar 

  13. Hsiao, C.-Y., Marinescu, G.: On the singularities of the Szegő projections on lower energy forms. J. Differential Geom. 107(1), 83–155 (2017)

    Article  MathSciNet  Google Scholar 

  14. Kohn, J.J.: The range of the tangential Cauchy–Riemann operator. Duke Math. J. 53(2), 307–562 (1986)

    Article  MathSciNet  Google Scholar 

  15. Ma, X., Marinescu, G.: The first coefficients of the asymptotic expansion of the Bergman kernel of the \(spin^c\) Dirac operator. Internat. J. Math. 17(6), 737–759 (2006)

    Article  MathSciNet  Google Scholar 

  16. Ma, X., Marinescu, G.: Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics. Birkhäuser Verlag, Basel (2007)

    MATH  Google Scholar 

  17. Ma, X., Marinescu, G.: Generalized Bergman kernels on symplectic manifolds. Adv. Math. 217(4), 1756–1815 (2008)

    Article  MathSciNet  Google Scholar 

  18. Ma, X.: Geometric quantization on Kähler and symplectic manifolds, International Congress of Mathematicians, vol. II, Hyderabad, India, August 19-27, 785–810 (2010)

  19. Ma, X., Zhang, W.: Geometric quantization for proper moment maps: the Vergne conjecture. Acta Math. 212(1), 11–57 (2014)

    Article  MathSciNet  Google Scholar 

  20. Melin, A., Sjöstrand, J.: Fourier integral operators with complex-valued phase functions, Springer. Lecture Notes Math. 459, 120–223 (1975)

    Article  Google Scholar 

  21. Rossi, H.: Attaching analytic spaces to an analytic space along a pseudoconcave boundary, Proc. Conf. Complex. Manifolds (Minneapolis), Springer–Verlag, New York, pp. 242–256 (1965)

  22. Shen, W.-C.: Asymptotics of torus equivariant Szegő kernel on a compact CR manifold, Bull. Inst. Math. Acad. Sin. (N.S.) 14(3), 331-383 (2019)

  23. Shiffman, B., Zelditch, S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Comm. Math. Phys. 200, 661–683 (1999)

    Article  MathSciNet  Google Scholar 

  24. Taylor, M.E.: Fourier series on compact Lie groups. Proc. Amer. Math. Soc. 19(5), 1103–1105 (1968)

    Article  MathSciNet  Google Scholar 

  25. Zelditch, S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Central University, Chung-Li, Taoyuan, 32001, Taiwan

    Rung-Tzung Huang

  2. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, Guangdong, China

    Guokuan Shao

Authors
  1. Rung-Tzung Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guokuan Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guokuan Shao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was supported by Taiwan Ministry of Science and Technology projects 107-2115-M-008-007-MY2 and 109-2115-M-008-007-MY2. The second author was supported by National Natural Science Foundation of China (Grant No.12001549), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110250) and Science and Technology Projects in Guangzhou (Grant No. 201904010436).

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, RT., Shao, G. Asymptotics of G-equivariant Szegő kernels. Ann Glob Anal Geom 61, 869–893 (2022). https://doi.org/10.1007/s10455-022-09838-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-022-09838-0

Keywords

Mathematics Subject Classification

Search

Navigation