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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