Applications of algebro-analytic method in algebraic geometry

关于研究方案中提到的精化循环映射的内射性问题，由于没有得到很好的结果，所以研究的目标改为Murre关于Chow-Kunneth分解的猜想。我们在上同调的水平至多为1的情况下证明了这一点，并将其推广到奇点离散且目标为曲线的光滑簇的态射的情况.我们将Malgrange定理的推广与Esnault，Schechtman，Viehweg的Aomoto猜想的解结合起来，找到了计算超平面排列的b-函数的一个比较简单的方法.使用这个，我们可以手工计算维数为3且次数至多为8的情况下的b函数。关于超平面排列的乘子理想，我们证明了Mustata关于跳跃系数是组合不变量的猜想，与Dimca，Maisonobe和Torrelli证明了超曲面的乘子理想与非特征超平面截面的限制相容，并导出了谱的一些结果，在与Hanamura讨论了交上同调的代数圈类之后，证明了交上同调与上同调的权滤子之间的关系，推广了Weber的一个结果，与Dimca一起证明了对于充分一般的射影超曲面，局部上同调的Hodge滤子与极序滤子是不同的，并在奇点不多的情况下证明了Wotzlaw的一个与此有关的猜想.

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.