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Applications of algebro-analytic method in algebraic geometry

代数解析方法在代数几何中的应用

基本信息

批准号：
17540023
负责人：
SAITO Morihiko
金额：
$ 1.34万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

项目摘要

Concerning the injectivity of the refined cycle maps that was mentioned in the research proposal, no good results were obtained so that the target of the research was changed to Murre's conjecture on the Chow- Kunneth decomposition. We proved this in the case the level of the cohomology is at most 1, and generalized it to the case of a morphism of smooth varieties whose singular points are discrete and whose target is a curve.We found a relatively simple method to compute the b-function of a hyperplane arrangement by combining a generalization of Malgrange's theorem with a solution of Aomoto conjecture due to Esnault, Schechtman, Viehweg. Using this, we can calculate by hand the b-function in the case the dimension is 3 and the degree is at most 8. As for the multiplier ideals of a hyperplane arrangement, we proved a conjecture of Mustata that the jumping coefficients are combinatorial invariants.We showed with Dimca, Maisonobe and Torrelli that the multiplier ideals of a hypersurface are compatible with the restriction to a noncharacteristic hyperplane section, and deduced some results on the spectrum.After discussions with Hanamura on the algebraic cycle classes in intersection cohomology, we proved a relation between the intersection cohomology and the weight filtration of the cohomology, generalizing a result of Weber.We showed with Dimca that the Hodge and pole order filtrations on the local cohomology differ for sufficiently general projective hypersurfaces, and proved a conjecture of Wotzlaw related to this in the case there are not so many singular points.

关于研究提案中提到的精化循环图的内射性，由于没有得到好的结果，因此研究目标改为穆雷关于Chow-Kunneth分解的猜想。我们在上同调的级数至多为1的情况下证明了这一点，并将其推广到奇点离散、目标为曲线的光滑簇射的情况。通过将马尔格朗日定理的推广与Esnault、Schechtman、Viehweg提出的青本猜想的解相结合，我们找到了一种相对简单的方法来计算超平面排列的b函数。利用这一点，我们可以手工计算维数为 3、次数最多为 8 的情况下的 b 函数。对于超平面排列的乘子理想，我们证明了 Mustata 的猜想，即跳跃系数是组合不变量。我们与 Dimca、Maisonobe 和 Torrelli 一起证明了超曲面的乘子理想与非特征性的限制相容。超平面截面，并推导出谱上的一些结果。在与 Hanamura 讨论交上同调中的代数环类之后，我们证明了交上同调与上同调的权重过滤之间的关系，推广了 Weber 的结果。我们与 Dimca 一起证明了对于足够一般的射影超曲面，局部上同调上的 Hodge 和极阶过滤是不同的，并且在没有那么多奇点的情况下证明了与此相关的沃茨瓦夫猜想。