绝对集与分解定理
Absolute sets and the Decomposition Theorem
摘要：
我们给出了一个框架，从自然函子类别之间产生可构造的功能，而不需要一个态射的模空间模型函子。我们表明使用黎曼-希尔伯特对应，任何自然（派生）函子上的可构造层光滑复代数簇可以用来构造一种特殊的可构造集，称为绝对集，推广的概念介绍了辛普森在模的存在。我们推测局部系统的绝对集满足一个“特殊变种包”，其中是Manin-Mumford、Mordell-Lang和André-Oort猜想的类似物。假设几何层的分解定理成立，该猜想给出了所有半单反常层的分解定理的一个简单证明。我们证明了猜想的秩一的情况下，在这种情况下，封闭的绝对集是有限工会的挠平移仿射环面。这将作者关于上同调跳跃基因座的结构结果推广到任何其他自然跳跃基因座。例如，交叉上同调和Leray滤子的跳跃轨迹。我们还表明，勒雷谱序列的开放嵌入在一个良好的紧化退化的所有秩一的本地系统在通常的页面，而不仅仅是酉本地系统。
英文摘要：
We give a framework to produce constructible functions from natural functors between categories, without need of a morphism of moduli spaces to model the functor. We show using the Riemann-Hilbert correspondence that any natural (derived) functor on constructible sheaves on smooth complex algebraic varieties can be used to construct a special kind of constructible sets, called absolute sets, generalizing a notion introduced by Simpson in presence of moduli. We conjecture that the absolute sets of local systems satisfy a "special varieties package", among which is an analog of the Manin-Mumford, Mordell-Lang, and André-Oort conjectures. The conjecture gives a simple proof of the Decomposition Theorem for all semi-simple perverse sheaves, assuming the Decomposition Theorem for the geometric ones. We prove the conjecture in the rank one case by showing that the closed absolute sets in this case are finite unions of torsion-translated affine tori. This extends a structure result of the authors for cohomology jump loci to any other natural jump loci. For example, to jump loci of intersection cohomology and Leray filtrations. We also show that the Leray spectral sequence for the open embedding in a good compactification degenerates for all rank one local systems at the usual page, not just for unitary local systems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
