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Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

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Dong R. X.

Complex Manifolds

2017

tom 4

nr 1

7-15

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.

Ograniczanie wyników

1 Complex Manifolds

1 Dong R. X.

1 2017

Wyniki wyszukiwania

Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves