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A direct approach to Bergman kernel asymptotics for positive line bundles. (English) Zbl 1161.32001

The authors give a new, direct, and elementary construction of the asymptotic expansion with respect to \(k\) of the Bergman kernel associated to a high power \(L^k\) of a positive line bundle \(L\) over a compact complex manifold. Previously, S. Zelditch [Int. Math. Res. Not. 1998, No. 6, 317–331 (1998; Zbl 0922.58082)] and D. Catlin [Trends in Mathematics, 1–23 (1999; Zbl 0941.32002)] obtained such an asymptotic expansion by applying work of L. Boutet de Monvel and J. Sjöstrand [Astérisque 34–35, 123–164 (1976; Zbl 0344.32010)] concerning asymptotic expansions of kernels on strictly pseudoconvex domains. This article uses instead some ideas from a book of the third author [Astérisque 95, 1–166 (1982; Zbl 0524.35007)] to compute local asymptotic Bergman kernels on small coordinate patches. The positivity hypothesis implies that the global Bergman kernel is asymptotically equal to the local kernels.

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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References:

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