超曲面补的高阶Alexander不变量
Higher-order Alexander Invariants of Hypersurface Complements
摘要：
我们定义了高阶Alexander模$A_{n,i}(\mathcal{U})$和高阶度$δ_{n,i}(\mathcal{U})$，它们是复超曲面补$\mathcal{U}$的不变量.这些不变量来自超曲面补的某些可解覆盖的同调的模结构。这种不变量最初是由T. Cochran在[1]和S. Harvey在[8]中提出的，并用于研究纽结和三维流形.本文推广了C. Leidy和L. Maxim [22]从平面曲线补到高维超曲面补。Zebriki观察到奇异复平面曲线上奇点的位置会影响曲线的拓扑结构。我关于高阶超曲面补的结果也表明，全局拓扑是由局部拓扑控制的。特别地，超曲面补的高阶度由局部链路对的高阶度的线性组合界定。
英文摘要：
We define the higher-order Alexander modules $A_{n,i}(\mathcal{U})$ and higher-order degrees $δ_{n,i}(\mathcal{U})$ which are invariants of a complex hypersurface complement $\mathcal{U}$. These invariants come from the module structure of the homology of certain solvable covers of the hypersurface complement. Such invariants were originally developed by T. Cochran in [1] and S. Harvey in [8], and were used to study knots and 3-manifolds. In this paper, I generalize the result proved by C. Leidy and L. Maxim [22] from the plane curve complements to higher-dimensional hypersurface complements. Zariski observed that the position of singularities on a singular complex plane curve affects the topology of the curve. My results on higher-order degrees of hypersurface complements also show that global topology is controlled by the local topologies. In particular, the higher-order degrees of the hypersurface complement are bounded by a linear combination of the higher-order degrees of the local link pairs.
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