Boundary asymptotics of the relative Bergman kernel metric for curves
Boundary asymptotics of the relative Bergman kernel metric for curves
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DOI：
10.1007/s00526-022-02347-9
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发表时间：
2020-05
影响因子：
2.1
通讯作者：
R. X. Dong
中科院分区：
文献类型：
--
作者：
R. X. Dong
研究了退化超椭圆黎曼曲面的全纯族及其Jacobi簇上的相对Bergman核度量的性质。在一个节点或尖点附近，我们得到精确的渐近公式与显式系数。一般情况下，尖点族上的Bergman核并不总是收敛于极限曲面正则部分上的Bergman核，这与节点族的情况不同。事实证明，奇异性和复杂结构的信息有助于伯格曼核的各种渐近行为。我们的方法涉及到经典的泰勒展开的阿贝尔微分和周期矩阵。
We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit coefficients. In general the Bergman kernels on a given cuspidal family do not always converge to that on the regular part of the limiting surface, which is different from the nodal case. It turns out that information on both the singularity and the complex structure contributes to various asymptotic behaviors of the Bergman kernel. Our method involves the classical Taylor expansion for Abelian differentials and period matrices.