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.
@article{bwmeta1.element.doi-10_1515_coma-2017-0002,
author = {Robert Xin Dong},
title = {Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves},
journal = {Complex Manifolds},
volume = {4},
year = {2017},
pages = {7-15},
zbl = {06695097},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_coma-2017-0002}
}
Robert Xin Dong. Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves. Complex Manifolds, Tome 4 (2017) pp. 7-15. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_coma-2017-0002/