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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Asymptotic Expansion of the Bergman Kernel
Erschienen in: Holomorphic Morse Inequalities and Bergman Kernels
Verlag: Birkhäuser Basel
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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 ℝ
2
n
. 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 ℝ
2
n
. Then we still have the spectral gap property on ℝ
2
n
. Thus we can instead study the Bergman kernel on ℝ
2
n
(cf. (4.1.27)), and use various resolvent representations (4.1.59), (4.2.22) of the Bergman kernel on ℝ
2
n
. We conclude our results by employing functional analysis resolvent techniques.