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A direct approach to Bergman kernel asymptotics for positive line bundles

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Arkiv för Matematik

Abstract

We give an elementary proof of the existence of an asymptotic expansion in powers of k of the Bergman kernel associated to L k, where L is a positive line bundle over a compact complex manifold. We also give an algorithm for computing the coefficients in the expansion.

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References

  1. Berman, R., Bergman kernels and local holomorphic Morse inequalities, Math. Z. 248 (2004), 325–344.

    Article  MATH  MathSciNet  Google Scholar 

  2. Berman, R., Bergman kernels and equilibrium measures for line bundles over projective manifolds, Preprint, 2007. arXiv:0710.4375.

  3. Berndtsson, B. and Andersson, M., Henkin–Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble) 32:3 (1982), v–vi, 91–110.

    Google Scholar 

  4. Boutet de Monvel, L. and Sjöstrand, J., Sur la singularité des noyaux de Bergman et de Szegő, in Journées: Équations aux dérivées partielles de Rennes (1975), Astérisque 3435, pp. 123–164, Soc. Math. France, Paris, 1976.

  5. Catlin, D., The Bergman kernel and a theorem of Tian, in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math., pp. 1–23, Birkhäuser, Boston, MA, 1999.

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Grigis, A. and Sjöstrand, J., Microlocal Analysis for Differential Operators, London Mathematical Society Lecture Note Series 196, Cambridge University Press, Cambridge, 1994.

    MATH  Google Scholar 

  8. Hörmander, L., Fourier integral operators. I, Acta Math. 127 (1971), 79–183.

    Article  MATH  MathSciNet  Google Scholar 

  9. Hörmander, L., The Analysis of Linear Partial Differential Operators. III, Grundlehren der Mathematischen Wissenschaften 274, Springer, Berlin–Heidelberg, 1985.

  10. Keller, J., Asymptotique du noyau de Bergman généralisé sur une varieté de Kähler, ouverte, Preprint, Toulouse, 2004.

  11. Lu, Z., On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch, Amer. J. Math. 122 (2000), 235–273.

    MATH  MathSciNet  Google Scholar 

  12. Melin, A. and Sjöstrand, J., Determinants of pseudodifferential operators and complex deformations of phase space, Methods Appl. Anal. 9 (2002), 177–237.

    MATH  MathSciNet  Google Scholar 

  13. Sjöstrand, J., Singularités Analytiques Microlocales, Astérisque 95, Soc. Math. France, Paris, 1982.

    Google Scholar 

  14. Tian, G., On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), 99–130.

    MATH  MathSciNet  Google Scholar 

  15. Zelditch, S., Szegő kernels and a theorem of Tian, Int. Math. Res. Notices 1998 (1998), 317–331.

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Correspondence to Bo Berndtsson.

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Berman, R., Berndtsson, B. & Sjöstrand, J. A direct approach to Bergman kernel asymptotics for positive line bundles. Ark Mat 46, 197–217 (2008). https://doi.org/10.1007/s11512-008-0077-x

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  • DOI: https://doi.org/10.1007/s11512-008-0077-x

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