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Abstract Bergman kernel expansion and its applications

Authors: Chiung-ju Liu and Zhiqin Lu
Journal: Trans. Amer. Math. Soc. 368 (2016), 1467-1495
MSC (2010): Primary 32Q15; Secondary 53A30
Published electronically: July 9, 2015
MathSciNet review: 3430370
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Abstract: We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method actually provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial.

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  • [1] Robert Berman, Bo Berndtsson, and Johannes Sjöstrand, A direct approach to Bergman kernel asymptotics for positive line bundles, Ark. Mat. 46 (2008), no. 2, 197-217. MR 2430724 (2009k:58050),
  • [2] S. Bochner, Curvature in Hermitian metric, Bull. Amer. Math. Soc. 53 (1947), 179-195. MR 0019983 (8,490d)
  • [3] L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegő, Journées: Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris, 1976, pp. 123-164. Astérisque, No. 34-35 (French). MR 0590106 (58 #28684)
  • [4] David Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997) Trends Math., Birkhäuser Boston, Boston, MA, 1999, pp. 1-23. MR 1699887 (2000e:32001)
  • [5] Xianzhe Dai, Kefeng Liu, and Xiaonan Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), no. 1, 1-41. MR 2215454 (2007k:58043)
  • [6] Jean-Pierre Demailly, Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 93-114. MR 1128538 (93b:32048)
  • [7] S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479-522. MR 1916953 (2003j:32030)
  • [8] S. K. Donaldson, Stability, birational transformations and the Kahler-Einstein problem, Surveys in differential geometry. Vol. XVII, Surv. Differ. Geom., vol. 17, Int. Press, Boston, MA, 2012, pp. 203-228. MR 3076062,
  • [9] Miroslav Engliš, The asymptotics of a Laplace integral on a Kähler manifold, J. Reine Angew. Math. 528 (2000), 1-39. MR 1801656 (2002j:32038),
  • [10] Charles Fefferman, Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), no. 2, 131-262. MR 526424 (80j:32035),
  • [11] Chiung-ju Liu, The asymptotic Tian-Yau-Zelditch expansion on Riemann surfaces with constant curvature, Taiwanese J. Math. 14 (2010), no. 4, 1665-1675. MR 2663940 (2011m:32029)
  • [12] Chiung-ju Liu and Zhiqin Lu, Uniform Asymptotic Expansion on Riemann Surfaces. To appear in Proceedings in honor of D. Phong's 60th Birthday.
  • [13] Zhiqin Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235-273. MR 1749048 (2002d:32034)
  • [14] Zhiqin Lu and Gang Tian, The log term of the Szegő kernel, Duke Math. J. 125 (2004), no. 2, 351-387. MR 2096677 (2006e:32026),
  • [15] Zhiqin Lu and Bernard Shiffman, Asymptotic expansion of the off-diagonal Bergman kernel on compact Kähler manifolds. To appear in J. Geom. Anal., arXiv, 1301.2166.
  • [16] Xiaonan Ma and George Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007. MR 2339952 (2008g:32030)
  • [17] Wei-Dong Ruan, Canonical coordinates and Bergmann [Bergman] metrics, Comm. Anal. Geom. 6 (1998), no. 3, 589-631. MR 1638878 (2000a:32050)
  • [18] Bernard Shiffman, Uniformly bounded orthonormal sections of positive line bundles on complex manifolds. To appear in Proceedings in honor of D. Phong's 60th Birthday.
  • [19] Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99-130. MR 1064867 (91j:32031)
  • [20] Hao Xu, A closed formula for the asymptotic expansion of the Bergman kernel, Comm. Math. Phys. 314 (2012), no. 3, 555-585. MR 2964769,
  • [21] Steve Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 6 (1998), 317-331. MR 1616718 (99g:32055),

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Additional Information

Chiung-ju Liu
Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan 106

Zhiqin Lu
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875

Keywords: Szeg\H{o} kernel, asymptotic expansion, ample line bundle
Received by editor(s): May 15, 2014
Received by editor(s) in revised form: November 11, 2014
Published electronically: July 9, 2015
Additional Notes: The first author was supported by NSC grant 982115M002007 in Taiwan. The second author was partially supported by NSF grant DMS-12-06748.
Article copyright: © Copyright 2015 American Mathematical Society

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