Nearby cycles and Alexander modules of hypersurface complements

DOI10.1016/J.AIM.2015.10.032zbMATH Open1351.32040arXiv1405.2343OpenAlexW2300741015MaRDI QIDQ2634784

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Publication date: 18 February 2016

Published in: (Search for Journal in Brave)

Abstract: Let $f:CNightarrowC$ 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 $CNsetminus{f}^{-1}(0)$, 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.

Full work available at URL: https://arxiv.org/abs/1405.2343

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