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Complex Manifolds

Ed. by Fino, Anna Maria

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CiteScore 2018: 0.64

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Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

Robert Xin Dong
Published Online: 2017-02-08 | DOI: https://doi.org/10.1515/coma-2017-0002


We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.

Keywords: variation of Bergman kernel; degeneration of hyperelliptic curve; node; cusp


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About the article

Published Online: 2017-02-08

Published in Print: 2017-02-23

Citation Information: Complex Manifolds, Volume 4, Issue 1, Pages 7–15, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2017-0002.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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