Nearby cycles and Alexander modules of hypersurface complements

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• 作者：Yongqiang Liu ^{liuyq@mail.ustc.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 32S20 ; 32S25 ; 32S35 ; 32S55 ; 32S60 ; secondary ; 14J70 ; 14F17
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：19 March 2016
• 年：2016
• 卷：291
• 期：Complete
• 页码：330-361
• 全文大小：577 K

文摘

Let f:Cⁿ⁺¹→C$f:{\mathbb{C}}^{n+1}\to \mathbb{C}$ be a polynomial map, which is transversal at infinity. Using Sabbah's specialization complex, we give a new description of the Alexander modules of the hypersurface complement Cⁿ⁺¹∖f⁻¹(0)${\mathbb{C}}^{n+1}\setminus f^{-1}(0)$, and obtain a general divisibility result for the associated Alexander polynomials. As a byproduct, we prove a conjecture of Maxim on the decomposition of the Cappell–Shaneson peripheral complex of the hypersurface. Moreover, as an application, we use nearby cycles to recover the mixed Hodge structure on the torsion Alexander modules, as defined by Dimca and Libgober. We also explore the relation between the generic fibre of f and the nearby cycles.