Abstract Bergman kernel expansion and its applications

Liu, Chiung-ju; Lu, Zhiqin

doi:10.1090/tran/6621

Abstract Bergman kernel expansion and its applications
HTML articles powered by AMS MathViewer

by Chiung-ju Liu and Zhiqin Lu PDF

Trans. Amer. Math. Soc. 368 (2016), 1467-1495 Request permission

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.

References

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, DOI 10.1007/s11512-008-0077-x
S. Bochner, Curvature in Hermitian metric, Bull. Amer. Math. Soc. 53 (1947), 179–195. MR 19983, DOI 10.1090/S0002-9904-1947-08778-4
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), Astérisque, No. 34-35, Soc. Math. France, Paris, 1976, pp. 123–164 (French). MR 0590106
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
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
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, DOI 10.4310/pamq.2011.v7.n4.a6
S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479–522. MR 1916953, DOI 10.4310/jdg/1090349449
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, DOI 10.4310/SDG.2012.v17.n1.a5
Miroslav Engliš, The asymptotics of a Laplace integral on a Kähler manifold, J. Reine Angew. Math. 528 (2000), 1–39. MR 1801656, DOI 10.1515/crll.2000.090
Charles Fefferman, Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), no. 2, 131–262. MR 526424, DOI 10.1016/0001-8708(79)90025-2
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, DOI 10.11650/twjm/1500405976
Chiung-ju Liu and Zhiqin Lu, Uniform Asymptotic Expansion on Riemann Surfaces. To appear in Proceedings in honor of D. Phong’s 60th Birthday.
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, DOI 10.1353/ajm.2000.0013
Zhiqin Lu and Gang Tian, The log term of the Szegő kernel, Duke Math. J. 125 (2004), no. 2, 351–387. MR 2096677, DOI 10.1215/S0012-7094-04-12526-6
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.
Xiaonan Ma and George Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007. MR 2339952, DOI 10.1007/978-3-7643-8115-8
Wei-Dong Ruan, Canonical coordinates and Bergmann [Bergman] metrics, Comm. Anal. Geom. 6 (1998), no. 3, 589–631. MR 1638878, DOI 10.4310/CAG.1998.v6.n3.a5
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.
Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99–130. MR 1064867
Hao Xu, A closed formula for the asymptotic expansion of the Bergman kernel, Comm. Math. Phys. 314 (2012), no. 3, 555–585. MR 2964769, DOI 10.1007/s00220-012-1531-y
Steve Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 6 (1998), 317–331. MR 1616718, DOI 10.1155/S107379289800021X