Abstract
In this chapter, we establish the asymptotic expansion of the Bergman kernel associated to high tensor powers of a positive line bundle on a compact complex manifold. Thanks to the spectral gap property of the Kodaira Laplacian, Theorem 1.5.5, we can use the finite propagation speed of solutions of hyperbolic equations, (Theorem D.2.1), to localize our problem to a problem on ℝ2n. Comparing with Section 1.6, the key point here is that we need to extend the connection of the line bundle L such that its curvature becomes uniformly positive on ℝ2n. Then we still have the spectral gap property on ℝ2n. Thus we can instead study the Bergman kernel on ℝ2n (cf. (4.1.27)), and use various resolvent representations (4.1.59), (4.2.22) of the Bergman kernel on ℝ2n. We conclude our results by employing functional analysis resolvent techniques.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
4.3 Bibliographic notes
R. Berman, B. Berndtsson, and J. Sjöstrand, Asymptotics of Bergman kernels, Preprint available at arXiv:math.CV/0506367, 2005.
J.-M. Bismut, Equivariant immersions and Quillen metrics, J. Differential Geom. 41 (1995), no. 1, 53–157. §11
J.-M. Bismut and G. Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math. (1991), no. 74, ii+298 pp. (1992). §11
J.-M. Bismut and E. Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), 355–367.
P. Bleher, B. Shiffman, and S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), no. 2, 351–395.
Th. Bouche, Convergence de la métrique de Fubini-Study d’un fibré linéare positif, Ann. Inst. Fourier (Grenoble) 40 (1990), 117–130.
L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Journées: Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris, 1976, pp. 123–164. Astérisque, No. 34–35.
D. 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.
L. Charles, Berezin-Toeplitz operators, a semi-classical approach, Comm. Math. Phys. 239 (2003), 1–28.
X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), no. 1, 1–41; announced in C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 193–198. §5
S.K. Donaldson, Planck’s constant in complex and almost-complex geometry, XIIIth International Congress on Mathematical Physics (London, 2000), Int. Press, Boston, MA, 2001, pp. 63–72.
S.K. Donaldson, Some numerical results in complex differential geometry, (2006), math.DG/0512625.
C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65.
L. Hörmander, An introduction to complex analysis in several variables, 1966, third ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990.
A.V. Karabegov and M. Schlichenmaier, Identification of Berezin-Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49–76.
K. Liu and X. Ma, A remark on’ some numerical results in complex differential geometry’, Math. Res. Lett. 14 (2007), 165–171.
Z. Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235–273.
Z. Lu and G. Tian, The log term of the Szegõ kernel, Duke Math. J. 125 (2004), no. 2, 351–387.
X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Acad. Sci. Paris 339 (2004), no. 7, 493–498, The full version: math.DG/0411559, Adv. in Math. §3.4
X. Ma and W. Zhang, Bergman kernels and symplectic reduction, C. R. Math. Acad. Sci. Paris 341 (2005), 297–302, see also Toeplitz quantization and symplectic reduction, Nankai Tracts in Mathematics Vol. 10, World Scientific, 2006, 343–349. The full version: math.DG/0607605. §3.3
X. Ma and W. Zhang, Superconnection and family Bergman kernels, C. R. Math. Acad. Sci. Paris 344 (2007), 41–44.
W.-D. Ruan, Canonical coordinates and Bergmann metrics, Comm. Anal. Geom. 6 (1998), no. 3, 589–631.
B. Shiffman and S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002), 181–222.
G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), 99–130.
X. Wang, Canonical metrics on stable vector bundles, Comm. Anal. Geom. 13 (2005), no. 2, 253–285.
S.-T. Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1–2, 109–158.
S.-T. Yau, Perspectives on geometric analysis, Survey in Differential Geometry, X (2006), 275–379.
S. Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices (1998), no. 6, 317–331.
Rights and permissions
Copyright information
© 2007 Birkhäuser Verlag AG
About this chapter
Cite this chapter
(2007). Asymptotic Expansion of the Bergman Kernel. In: Holomorphic Morse Inequalities and Bergman Kernels. Progress in Mathematics, vol 254. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8115-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8115-8_5
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8096-0
Online ISBN: 978-3-7643-8115-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)