D-模上的滤子与Bernstein-Sato多项式根的重数
Filtrations on D-modules and multiplicities of roots of Bernstein-Sato polynomials
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中文总结 AI 辅助
本文关联Bernstein-Sato多项式根的重数与奇点不变量，引入模b-函数并给出其算法，解决Gelfand问题的渐近解，还将结果应用于Torelli问题并给出肯定答案。
中文摘要 AI 辅助
本文将全纯函数f对应的Bernstein-Sato型多项式的重数与若干奇点不变量关联起来。首先，我们引入了特定的“模b-函数”，证明其可刻画正则全纯D-模沿f的局部化上的权滤子，并提供了计算它们的算法。其次，我们证明这些模b-函数可通过幂b-函数（即关于f的幂的b-函数）的根的重数来近似。此外，我们对由这类重数的某一和给出的元素的Hodge水平给出了一个精确上界。接下来，我们通过确定一个明确的阈值，给出了Gelfand问题的有效渐近解，该阈值之后，b_f(s)根的每个整数平移都是f的阿基米德ζ函数的极点。我们还证明这些极点的阶等于对数单值算子的幂零指数，且该指数可进一步表示为幂b-函数根的重数的极限。我们定义了若干滤子，并将它们与权滤子和Hodge滤子关联起来，基于此，我们留下了一些开放问题，当f具有齐次孤立奇点、超平面构形或球形簇上的半不变量时，我们对这些问题给出了肯定解答。我们将结果应用于若干场景，包括对Torelli问题给出肯定回答：若超曲面具有对数典范奇点，则1/f属于f的超曲面的交复形当且仅当-1是b_f(s)的单根。
英文摘要
In this paper, we relate multiplicities of Bernstein--Sato-type polynomials with respect to a holomorphic function f to several singularity invariants. First, we introduce certain ``mod'' b-functions and show that they characterize the weight filtration on the localization of a simple regular holonomic D-module along f, and we provide an algorithm to compute them. Second, we show that they can be approximated by the multiplicities of roots of power b-functions (the b-functions with respect to powers of f). Further, we give a sharp upper bound for the Hodge level of elements given by a certain sum of such multiplicities. Next, we give an effective asymptotic solution to the Gelfand problem by determining an explicit threshold after which every integer shift of a root of b_f(s) is a pole of the Archimedean zeta function of f. We also show that the order of these poles is equal to the nilpotency index of the logarithmic monodromy operator, which we further express as the limit of the multiplicities of roots of power b-functions. We define several filtrations, relating them to the weight and Hodge filtrations, based upon which we leave some open questions that we address in the affirmative in the case when f has a homogeneous isolated singularity, or it is a hyperplane arrangement, or it is a semi-invariant on a spherical variety. We give several immediate applications to our results, including a positive answer to a question of Torelli assuming the hypersurface has log canonical singularities: 1/f lies in the intersection complex of the hypersurface of f if and only if -1 is a simple root of b_f(s).
