Applications of algebro-analytic method in algebraic geometry

关于研究计划中提到的精细化周期图的注入性，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。我们在上同调的阶不超过1的情况下证明了这一点，并将其推广到奇异点为离散且目标为曲线的光滑变体的态射的情况。将Malgrange定理的推广与Esnault， Schechtman， Viehweg的Aomoto猜想的解相结合，找到了一种计算超平面排列的b函数的相对简单的方法。利用这种方法，我们可以手工计算维数为3，度最多为8的情况下的b函数。对于超平面排列的乘子理想，证明了Mustata关于跳跃系数是组合不变量的一个猜想。我们用Dimca， Maisonobe和Torrelli证明了超曲面的乘法器理想与非特征超平面截面的限制是相容的，并在谱上推导了一些结果。在与Hanamura讨论了交上同调中的代数循环类之后，我们证明了交上同调与上同调的权过滤之间的关系，推广了Weber的一个结果。我们用Dimca证明了对于足够一般的射影超曲面，在局部上同调上的Hodge滤波和极点阶滤波是不同的，并证明了在没有那么多奇异点的情况下与此相关的一个Wotzlaw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.