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ベルグマン核とその境界挙動

伯格曼核及其边界行为

基本信息

批准号：
15J05093
负责人：
董欣
金额：
$ 1.22万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2015
资助国家：
日本
起止时间：
2015-04-24 至 2017-03-31
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15J05093/
关键词：
Bergman kernel degeneration of curve node cusp

项目摘要

The Bergman kernel on a complex manifold is a canonical volume depending on the complex structure. I study the Bergman kernel and its variations (in particular its asymptotic behaviors) at degeneration in a quantitative way. In general, the curvature semi-positivities characterize certain convexities and are associated with L2 estimates and extensions. At least 3 approaches work for this problem: elliptic function, Taylor expansion and pinching coordinate.Let p be a polynomial of degree >=2 with roots of distinct values different from t, 0. Locally on a cuspidal family of hyperelliptic curves, its Bergman kernel function as t tends to 0 becomes small. Also, the second term is harmonic in t and doesn't necessarily possess a positive coefficient. Moreover, the Jacobian varieties remain being manifolds (i.e., non-degenerate), as t tends to 0.For distinct a, b, t in C-{0}, we consider another family of genus two curves. Then, both coefficients of the first two terms depend only on the information away from the cusp, which is not the case for the previous case. For the Jacobian varieties, the curvature form of the relative Bergman kernel has hyperbolic growth again.

复流形上的Bergman核是依赖于复结构的正则体积。我研究的伯格曼核及其变化（特别是其渐近行为）退化的定量方法。通常，曲率半正性刻画了某些凸性，并与L2估计和扩张相关联。至少有三种方法可以解决这个问题：椭圆函数，泰勒展开和拼挤坐标。设p是一个次数>=2的多项式，其根的不同值不同于t，0。局部地在尖点超椭圆曲线族上，其Bergman核函数随着t趋于0而变小。此外，第二项在t中是调和的，不一定具有正系数。此外，雅可比变量仍然是流形（即，对于C-{0}中不同的a，B，t，我们考虑了另一类亏格为2的曲线.然后，前两项的系数都只依赖于远离尖点的信息，这与前一种情况不同。对于雅可比簇，相对Bergman核的曲率形式再次具有双曲增长。