ベルグマン核とその境界挙動

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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.