Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

Abstract

Abstract: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.

Boundary asymptotics of the relative Bergman kernel metric for elliptic curves

“…Our results on the Jacobian varieties are stated in Section 10. An announcement of this paper for curves appeared in [24].…”