Reidemeister torsion, peripheral complex and Alexander polynomials of hypersurface complements

Liu, Yongqiang¹ ; Maxim, Laurenţiu²

¹School of Mathematical Sciences, University of Science and Technology of China, No 96, JinZhai Road, Hefei, 230026, China
²Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Office 713, Madison, WI 53706-1388, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 5, pp. 2755-2785

Résumé

Let f : ℂn+1 → ℂ be a polynomial that is transversal (or regular) at infinity. Let U = ℂn+1 ∖ f−1(0) be the corresponding affine hypersurface complement. By using the peripheral complex associated to f, we give several estimates for the (infinite cyclic) Alexander polynomials of U induced by f, and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of U.

Détail

DOI : 10.2140/agt.2015.15.2757

Classification : 32S25, 32S55, 32S60
Keywords: Reidemeister torsion, Sabbah specialization complex, nearby cycles, peripheral complex, hypersurface complement, Milnor fibre, non-isolated singularities, Alexander polynomial, boundary manifold, mixed Hodge structure

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