2016 Fiscal Year Annual Research Report
| Project/Area Number |
15J05093
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| Research Institution | Nagoya University |
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Principal Investigator |
董 欣 名古屋大学, 多元数理科学研究科, 特別研究員(DC2)
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| Project Period (FY) |
2015-04-24 – 2017-03-31
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| Keywords | Bergman kernel |
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| Outline of Annual Research Achievements |
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.
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| Research Progress Status |
28年度が最終年度であるため、記入しない。
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| Strategy for Future Research Activity |
28年度が最終年度であるため、記入しない。
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Research Products
(10 results)
