ベルグマン核とその境界挙動

复流形上的Bergman核是依赖于复流形结构的标准体积。我定量地研究了Bergman核及其在退化时的变化(特别是它的渐近行为)。一般而言，曲率半正性刻画了某些凸性，并与L2估计和扩张有关。至少有三种方法可以解决这个问题：椭圆函数、泰勒展开式和拼接坐标。设p是&gt；=2次多项式，根的值不同于t，0。在超椭圆曲线族的局部上，当t趋于0时，其Bergman核函数变小。另外，第二项是t中的调和项，不一定是正系数。此外，当t趋于0时，雅可比簇仍然是流形(即非退化的)，对于C-{0}中不同的a，b，t，我们考虑另一族亏格为2的曲线族。然后，前两项的两个系数只依赖于远离尖点的信息，这与前一种情况不同。对于雅可比变种，相对Bergman核的曲率形式又具有双曲线增长。

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.