自由阿贝尔覆盖的特征簇与贝蒂数
Characteristic varieties and Betti numbers of free abelian covers
摘要：
有限胞腔复形X的正则\Z^r -覆盖由H ^1（X，\Q）中r-平面的格拉斯曼数参数化.在这种变化中移动，并记录当贝蒂数b_1，.，b_i是Grassmannian的某些子集Ω^i_r（X）的有限剖分。我们在这里提出了一种方法，基本上可以追溯到德怀尔和弗里德，用于计算这些集合的跳跃轨迹的同源性与系数在秩1本地系统的X。利用这些跳跃轨迹的指数切锥，我们证明了每个Ω-不变量都包含在与H ^1（X，\Q）中线性子空间的排列相关联的舒伯特簇的并的补中。当X的特征簇是平移环面的并集时，该理论可以变得非常明确。但即使在这种情况下，Ω-不变量也不一定是开的，即使X是光滑复射影簇也不一定。作为应用，我们讨论了几类群的几何有限性。
英文摘要：
The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves out certain subsets Ω^i_r(X) of the Grassmannian. We present here a method, essentially going back to Dwyer and Fried, for computing these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. Using the exponential tangent cones to these jump loci, we show that each Ω-invariant is contained in the complement of a union of Schubert varieties associated to an arrangement of linear subspaces in H^1(X,\Q). The theory can be made very explicit in the case when the characteristic varieties of X are unions of translated tori. But even in this setting, the Ω-invariants are not necessarily open, not even when X is a smooth complex projective variety. As an application, we discuss the geometric finiteness properties of some classes of groups.
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