Abstract
For a Legendre family of elliptic curves, the two-term asymptotic expansion of the relative Bergman kernel metric near the degenerate boundary is obtained by an approach based on the Taylor series of Abelian differentials and Riemann periods. Namely, the curvature form has hyperbolic growth in the transversal direction with an explicit second term at the node. For another nodal degenerate family of elliptic curves, the result turns out to be the same. But for two cusp cases, it is either trivial with a constant period or reducible to the Legendre family case. The proofs do not depend on special elliptic functions, and work also for higher genus cases. In the last part, we discuss invariant properties on curves.
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Acknowledgements
This paper is dedicated to Professor Kang-Tae Kim on the occasion of his sixtieth birthday. The author sincerely thanks Professor Takeo Ohsawa for his patient guidance and Professor Tomoyuki Hisamoto for bringing attention the book [9]. This work is supported by the Ideas Plus grant 0001/ID3/2014/63 of the Polish Ministry of Science and Higher Education, KAKENHI and the Grant-in-Aid for JSPS Fellows (No. 15J05093).
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Dong, R.X. (2018). Boundary Asymptotics of the Relative Bergman Kernel Metric for Elliptic Curves IV: Taylor Series. In: Byun, J., Cho, H., Kim, S., Lee, KH., Park, JD. (eds) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol 246. Springer, Singapore. https://doi.org/10.1007/978-981-13-1672-2_10
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DOI: https://doi.org/10.1007/978-981-13-1672-2_10
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