The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
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超平面排列的 Milnor 纤维化：从模共振到代数单向性
DOI：
10.1112/plms.12027
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发表时间：
2014
影响因子：
1.8
通讯作者：
Alexander I. Suciu
中科院分区：
文献类型：
--
作者：
S. Papadima;Alexander I. Suciu
排列理论中的一个中心问题是确定作用在复超平面排列A的Milnor纤维的同调群Hq（F（A），C）上的代数单值的特征多项式Δq是否由交格L（A）确定。在简单的组合条件下，我们证明了对应于素数p的a次幂的某些特征值的Δ1的因子的重数等于Aomoto-Betti数βp（A），而β p（A）又是从L（A）中提取的.当A定义了一组只有二重点和三重点的射影直线时，这就导致了代数单值性的组合公式。为了获得这些结果，我们涉及网的基础拟阵的A共振品种的积极特征。利用网的模不变量，我们找到了拟阵的一个新的可实现性障碍（C上），并估计了拟阵A的第一复共振簇的本质分支的个数。我们的方法还揭示了模共振与SL 2（C）-表示变体的几何结构的一个相当意外的联系，这是由Maurer-Cartan方程控制的。
A central question in arrangement theory is to determine whether the characteristic polynomial Δq of the algebraic monodromy acting on the homology group Hq(F(A),C) of the Milnor fiber of a complex hyperplane arrangement A is determined by the intersection lattice L(A) . Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers βp(A) , which in turn are extracted from L(A) . When A defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of A to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over C ) for matroids, and estimate the number of essential components in the first complex resonance variety of A . Our approach also reveals a rather unexpected connection of modular resonance with the geometry of SL2(C) ‐representation varieties, which are governed by the Maurer–Cartan equation.