Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

Robert Xin Dong. "Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves." Complex Manifolds 4.1 (2017): 7-15. <http://eudml.org/doc/287969>.

@article{RobertXinDong2017,
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.},
author = {Robert Xin Dong},
journal = {Complex Manifolds},
keywords = {variation of Bergman kernel; degeneration of hyperelliptic curve; node; cusp; variation of the Bergman kernel; elliptic curves; hyperelliptic curves},
language = {eng},
number = {1},
pages = {7-15},
title = {Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves},
url = {http://eudml.org/doc/287969},
volume = {4},
year = {2017},
}

TY - JOUR
AU - Robert Xin Dong
TI - Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves
JO - Complex Manifolds
PY - 2017
VL - 4
IS - 1
SP - 7
EP - 15
AB - 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.
LA - eng
KW - variation of Bergman kernel; degeneration of hyperelliptic curve; node; cusp; variation of the Bergman kernel; elliptic curves; hyperelliptic curves
UR - http://eudml.org/doc/287969
ER -