论文标题
超平面布置满足（UN）扭曲对数比较定理，应用于$ \ Mathscr {d} _ {x} $ - 模块
Hyperplane Arrangements Satisfy (un)Twisted Logarithmic Comparison Theorems, Applications to $\mathscr{D}_{X}$-modules
论文作者
论文摘要
对于减少的超平面排列，我们证明了分析性扭曲的对数比较定理，在定义扭曲的重量上受到轻度组合算术条件。这给出了扭曲的对数DE RHAM复合物与扭曲的Meromorormormorormormormorormormormorormormormorormormorormormorormormorormormorormormorormormormoright。后者计算了该安排补充的共同体，该系数与相应等级的一个局部系统的系数。我们还证明了代数变体（当布置是中心的时），以及分析和代数（无键）对数比较定理。最后一项积极解决了terao的旧猜想。我们还证明：每个非平凡等级都可以通过这些扭曲的对数比较定理来计算补体上的一个局部系统；这些计算是明确的有限维线性代数。最后，我们给出了一些$ \ mathscr {d} _ {x} $ - 模块应用程序：例如，我们对多元伯恩斯坦（SATE）的组成的一个组成部分进行了彻底的限制，即与安排任意分解相关的理想。界限对应于（在单变量的情况下，给出了M. Saito的独立证明）。
For a reduced hyperplane arrangement we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement's complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite dimensional linear algebra. Finally, we give some $\mathscr{D}_{X}$-module applications: for example, we give a sharp restriction on the codimension one components of the multivariate Bernstein--Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito's result that the roots of the Bernstein--Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$
