ベルグマン核とその境界挙動

复杂流形上的Bergman核是一个依赖于复杂结构的正则体积。我研究伯格曼核和它的变化（特别是它的渐近行为）在退化以定量的方式。一般来说，曲率半正性具有一定的凸性，并且与L2估计和扩展有关。至少有三种方法可以解决这个问题：椭圆函数、泰勒展开和捏紧坐标。设p为次为>=2的多项式，其根的不同值不同于t， 0。在超椭圆曲线的锯齿族上，当t趋于0时，其Bergman核函数变小。同样，第二项在t中是调和的，并不一定是正系数。而且，当t趋于0时，雅可比矩阵仍然是流形（即不退化）。对于C-{0}中不同的a， b, t，我们考虑另一类双属曲线。然后，前两项的两个系数只依赖于远离尖端的信息，这与前一种情况不同。对于雅可比矩阵，相对伯格曼核的曲率形式再次具有双曲增长。

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.