Asymptotic Expansion of the Bergman Kernel
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.
KeywordsAsymptotic Expansion Line Bundle Dirac Operator Heat Kernel Formal Power Series
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4.3 Bibliographic notes
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