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Szegő kernel expansion and equivariant embedding of CR manifolds with circle action

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Abstract

Let X be a compact strongly pseudoconvex CR manifold with a transversal CR \(S^1\)-action. In this paper, we establish the asymptotic expansion of Szegő kernels of positive Fourier components, and by using the asymptotics, we show that X can be equivariant CR embedded into some \(\mathbb {C}^N\) equipped with a simple \(S^1\)-action. An equivariant embedding of quasi-regular Sasakian manifold is also derived.

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References

  1. Baouendi, M.-S., Rothschild, L.-P., Treves, F.: CR structures with group action and extendability of CR functions. Invent. Math. 83, 359–396 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Burns, D.M.: Global behavior of some tangential Cauchy–Riemann equations. In: Partial Differential Equations and Geometry (Proceedings of Conference, Park City, 1977). Lecture Notes in Pure and Applied Mathematics, vol. 48, pp. 51–56. Dekker, New York (1979)

  3. Burns, D.-M., Epstein, C.-L.: Embeddability for three-dimensional CR manifolds. J. Am. Math. Soc. 3(4), 809–841 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blair, D.-E.: Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics, vol. 509. Springer, Berlin (1976)

    Book  Google Scholar 

  5. Boutet de Monvel, L.: Intégration des équations de Cauchy–Riemann induites formelles. In: Séminaire Goulaouic-Lions-Schwartz 1974–1975; Équations aux derivées partielles linéaires et non linéaires, Centre Math., École Polytech., Paris, Exp. no. 9, pp. 13 (1975)

  6. Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2006)

    Google Scholar 

  7. Boyer, C.P., Galicki, K., Simanca, S.: Canonical Sasakian metrics. Commun. Math. Phys. 279, 705C733 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegő. Astérisque 34–35, 123–164 (1976)

    MATH  Google Scholar 

  9. Chanillo, S., Chiu, H.-L., Yang, P.: Embeddability for three-dimensional Cauchy–Riemann manifolds and CR Yamabi invariants. Duke Math. J. 161(15), 2909–2921 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chanillo, S., Chiu, H.-L., Yang, P.: Embedded three-dimensional CR manifolds and the non-negativity of Paneitz operators. Contemp. Math. 599, 65–82 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cheng, J.-H., Hsiao, C.-Y., Tsai, I.-H.: Index theorems on CR manifolds with \(S^1\) actions. arXiv:1511.00063

  12. Epstein, C.L.: CR-structures on three dimensional circle bundles. Invent. Math. 109, 351–403 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Geiges, H.: Normal contact structures on 3-manifolds. Tohoku Math. J. 49, 415–422 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hörmander, L.: The Analysis of Linear Partial Differential Operators. I, Classics in Mathematics. Springer, Berlin (2003)

    Book  Google Scholar 

  16. Hsiao, C.-Y.: Projections in several complex variables. Mem. Soc. Math. France Nouv. Ser. 123, 1–131 (2010)

  17. Hsiao, C.-Y.: Bergman kernel asymptotics and a pure analytic proof of the kodaira embedding theorem. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, KT., Shcherbina, N. (eds.) Complex Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol. 144, pp. 161–173. Springer, Tokyo (2015)

  18. Hsiao, C.-Y.: Existence of CR sections for high power of semi-positive generalized Sasakian CR line bundles over generalized Sasakian CR manifolds. Ann. Glob. Anal. Geom. 47(1), 13–62 (2015). doi:10.1007/s10455-014-9434-0

    Article  MathSciNet  MATH  Google Scholar 

  19. Hsiao, C.-Y.: Szegő kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds. Memories of American Mathematical Society. arXiv:1401.6647

  20. Hsiao, C.-Y., Li, X.: Szegö kernel asymptotics and Morse inequalities on CR manifolds with \(S^1\) action. arXiv:1502.02365

  21. Hsiao, C.-Y., Li, X.: Morse inequalities for Fourier components of Kohn Rossi cohomology of CR manifolds with \(S^1\) action. Math. Z. 284(1–2), 441–468 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hsiao, C.-Y., Li, X., Marinescu, G.: Equivariant Kodaira embedding of CR manifolds with circle action. arXiv:1603.08872

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Hsiao, C.-Y., Marinescu, G.: On the singularities of the Szegő projections on lower energy forms. J. Differ. Geom. arXiv:1407.6305

  25. Lempert, L.: On three dimensional Cauchy–Riemann manifolds. J. Am. Math. Soc. 5(4), 923–969 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Marinescu, G., Yeganefar, N.: Embeddability of some strongly pseudoconvex manifolds. Trans. Am. Math. Soc. 359, 4757–4771 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Melin, A., Sjöstrand, J.: Fourier Integral Operators with Complex-Valued Phase Functions. Springer Lecture Notes in Mathematics, vol. 459, pp. 120–223. Springer, Berlin (1975)

  28. Ornea, L., Verbitsky, M.: Embeddings of compact Sasakian manifolds. Math. Res. Lett. 14(4), 703–710 (2007)

  29. Ornea, L., Verbitsky, M.: Sasakian structures on CR-manifolds. Geom. Dedicata 125(1), 159–173 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rossi, H.: Attaching analytic spaces to an analytic space along a pseudoconvex boundary. In: Aeppli A., Calabi E., Rohrl R. (eds.) Proceedings of Conference on Complex Manifolds, pp. 242–256. Springer, Berlin (1965)

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Acknowledgements

The authors would like to thank Professor Paul Yang for his interest in this work. The third-named author would like to thank the Institute of Mathematics, University of Cologne, for the hospitality during his visit. We also thank the referee for many detailed remarks that have helped to improve the presentation.

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Authors and Affiliations

  1. Mathematical Institute, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Hendrik Herrmann

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, China

    Xiaoshan Li

  3. Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No.1, Sec.4, Roosevelt Road, Taipei, 10617, Taiwan

    Chin-Yu Hsiao & Xiaoshan Li

Authors
  1. Hendrik Herrmann
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  2. Chin-Yu Hsiao
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  3. Xiaoshan Li
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Correspondence to Xiaoshan Li.

Additional information

Hendrik Herrmann was supported by the Graduiertenkolleg 1269: “Global Structures in Geometry and Analysis” and by the Institute of Mathematica, Academia Sinica. Chin-Yu Hsiao was partially supported by Taiwan Ministry of Science of Technology Project 104-2628-M-001-003-MY2 and the Golden-Jade fellowship of Kenda Foundation. Xiaoshan Li was supported by NSFC No. 11501422, Postdoctoral Science Foundation of China 2015M570660 and Central university research Fund 2042015kf0049.

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Herrmann, H., Hsiao, CY. & Li, X. Szegő kernel expansion and equivariant embedding of CR manifolds with circle action. Ann Glob Anal Geom 52, 313–340 (2017). https://doi.org/10.1007/s10455-017-9559-z

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  • DOI: https://doi.org/10.1007/s10455-017-9559-z

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