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Asymptotic Expansion of the Off-Diagonal Bergman Kernel on Compact Kähler Manifolds

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Abstract

We compute the first four coefficients of the asymptotic off-diagonal expansion in Shiffman and Zelditch (J. Reine Angew. Math. 544:181–222, 2002) of the Bergman kernel for the N-th power of a positive line bundle on a compact Kähler manifold, and we show that the coefficient b 1 of the N −1/2 term vanishes when we use a K-frame. We also show that all the coefficients of the expansion are polynomials in the K-coordinates and the covariant derivatives of the curvature and are homogeneous with respect to the weight w in Lu (Am. J. Math. 122(2):235–273, 2000).

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Notes

  1. The curvature tensor itself is regarded as the 0-th covariant derivative. We use this convention throughout the paper.

  2. The Kähler form we use is π −1 times the Kähler form in [13]. Therefore, the expansion (35) differs from the one in [13] by a factor π −m.

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Authors and Affiliations

  1. Department of Mathematics, UC Irvine, Irvine, CA, 92697, USA

    Zhiqin Lu

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Bernard Shiffman

Authors
  1. Zhiqin Lu
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  2. Bernard Shiffman
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Correspondence to Bernard Shiffman.

Additional information

Communicated by Marco M. Peloso.

Research of the first author is partially supported by NSF grant DMS-1206748. Research of the second author is partially supported by NSF grant DMS-1201372.

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Lu, Z., Shiffman, B. Asymptotic Expansion of the Off-Diagonal Bergman Kernel on Compact Kähler Manifolds. J Geom Anal 25, 761–782 (2015). https://doi.org/10.1007/s12220-013-9445-2

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