| 研究実績の概要 |
The Bergman kernel on each complex manifold is a canonical volume-form determined by the complex structure and we study the variation of Bergman kernels at degeneration. For a Legendre family of elliptic curves, using special elliptic functions I showed that the horizontal Levi form of the relative Bergman kernel metric is strictly positive everywhere inside the moduli space, blows up and has hyperbolic growth near the node. By analyzing the period of curves, I later gave the above results an alternative proof, which involves precise coefficients but does not depend on special elliptic functions. For other families of elliptic curves degenerating to nodes and cusps, it is more or less trivial.
For a holomorphic family of hyperelliptic curves and their Jacobians, asymptotic formulas of the relative Bergman kernel metrics near the degenerate boundaries could also be obtained. The curvature form induces an incomplete metric which is different from genus one curves. In the special case of a family of genus two curves degenerating to a node, we determine the precise coefficients. Habermann-Jost had obtained using the pinching-coordinate method the asymptotic results for the Bergman kernel on general curves near nodal singularities. Nevertheless, my results for hyperelliptic curves are based on a different method and therefore are more precise in the sense that the coefficients are explicitly written down, especially if one wants to know how the given complex structures relate to these coefficients, which usually indicate the geometry of base varieties and their singularities.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
There is a quick progress in my research on Bergman kernel on algebraic curves. However, our result coincides with the result of Habermann-Jost in some cases. Nevertheless, I could consider other types of singularities where Bergman kernels degenerate. In particular, the higher dimensional cases and the pluri-canonical cases seem far away from being solved.
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| 今後の研究の推進方策 |
As I mentioned, my results for hyperelliptic curves are more precise in the sense that the coefficients are explicitly written down, especially if one wants to know how the given complex structures relate to these coefficients, which usually indicate the geometry of base varieties and their singularities. I would first continue this direction by writing the results down carefully in detail.
Then I attempt to work for the cusp case by making an analogue of the pinching coordinate , which is useful for general curves. After that I will move to higher dimensional cases, followed by the Jacobians of general curves. Another closely related direction is the pluri-canonical cases which is far away from being solved. I will attack this direction by stating with Takayama's paper.
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