Divisibility Results For Alexander Type Invariants Of Hypersurface Complements

The aim of this paper is to give the divisibility results for the Alexander type in-variants of hypersurface complements.Assume that f:Cn+lâ†'C is transversal at infinity. Set F0=f-1(0) and U= Cn+1\F0. In Chapter 2 and 3, we study the classical Alexander modules of hypersurface complement U and its boundary manifold, in particular, the divisibility results.In Chapter 2, we realize the Alexander modules by the Sabbah specialization com-plex, which is closely related to the nearby cycles of f. The nearby cycles provide a good way to glue all the local singular information together:the stalks of the nearby cy-cles are isomorphic to the cohomology groups of the local Milnor fibre. Therefore, this new approach reveals the relation between the Alexander modules and the singularities of the affine hypersurface F0, and provides a new general divisibility result. Moreover, as nearby cycle is an important ingredient in the theory of mixed Hodge modules, we can use nearby cycles to obtain a mixed Hodge structure on the Alexander modules.In Chapter 3, we provide a generalization to the case of hypersurface with non-isolated singularities of the Cogolludo-Florens identity for Alexander polynomials. Our main tool will be the Cappell-Shaneson peripheral complex associated to f. In more detail, we give a new description of the peripheral complex, from which we deduce several error estimates for the Alexander polynomials of the complement. Moreover, by exploiting the relation between the Alexander polynomials and Reidemeister torsion, we show how these estimates can be further refined by using the intersection form appearing in the duality for Reidemeister torsion.Our new description of the peripheral complex can also be used to show that the peripheral complex underlies an algebraic mixed Hodge module. In particular, after explaining the relation between the peripheral complex and the boundary manifold of the complement U, we obtain mixed Hodge structures on the Alexander modules of this boundary manifold.Choose hypersurface V (?) CPn+1, and set M*= CPn+1\V.In Chapter 4, we give divisibility results for the (global) Alexander varieties (or characteristic varieties) of hypersurface complements M* expressed in terms of the lo-cal Alexander varieties (or local characteristic varieties) at points along one of the irre-ducible components of the hypersurface V. These divisibility results are very general, in the sense that no additional assumptions on the hypersurace V are needed (we drop the assumption that f is transversal at infinity). As an application, we recast old and obtain new finiteness and divisibility results for the classical (infinite cyclic) Alexan-der modules of complex hypersurface complements. Moreover, using Suciu’s notion of locally straight spaces, we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties.