On the fundamental group of an abelian cover
On the fundamental group of an abelian cover
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关于阿贝尔覆盖的基本群
DOI：
10.1142/s0129167x9500033x
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发表时间：
1993
影响因子：
0.6
通讯作者：
F. Tovena
中科院分区：
文献类型：
--
作者：
R. Pardini;F. Tovena
设X， Y为维数n≥2的光滑复射影变数，设f: Y→X为完全分叉的阿贝尔覆盖。假设f的分支除数的分量是充裕的。则映射f*: π1(Y)→π1(X)是满射，并得到一个中心扩展：\begin{equation} 0 \to K \to \pi_1 (Y)\to \pi_1(X) \to 1 \end{equation}，其中K是有限群。本文给出了在伽罗瓦群的作用下，核K和(1)的上同调类c(f)∈H2（π1(X)， K）是如何用f的分支因子和$f_*{\mathcal O}_Y$的特征束的分量的陈氏类来计算的。利用这一结果，对于任意整数m>0，我们构造了m个变量X1，…，Xm，尽管它们具有相同的数值不变量，并且它们被实现为具有相同伽罗瓦群、分支轨迹和惯性子群的相同射影变量X的覆盖，但其中没有两个是同胚的。
Let X, Y be smooth complex projective varieties of dimension n≥2 and let f: Y→X be a totally ramified abelian cover. Assume that the components of the branch divisor of f are ample. Then the map f*: π1(Y)→π1(X) is surjective and gives rise to a central extension: \begin{equation} 0 \to K \to \pi_1 (Y)\to \pi_1(X) \to 1 \end{equation} where K is a finite group. Here we show how the kernel K and the cohomology class c(f) ∈ H2(π1(X), K) of (1) can be computed in terms of the Chern classes of the components of the branch divisor of f and of the eigensheaves of $f_*{\mathcal O}_Y$ under the action of the Galois group. Using this result, for any integer m>0, we construct m varieties X1,…, Xm no two of which are homeomorphic, even though they have the same numerical invariants and they are realized as covers of the same projective variety X with the same Galois group, branch locus and inertia subgroups.