Hypersurface Complements, Alexander Modules and Monodromy

Abstract

We consider an arbitrary polynomial map $f:{\mathbb C}^{n+1}\to {\mathbb C} $ and we study the Alexander invariants of ${\mathbb C}^{n+1}\setminus X$ for any fiber $X$ of $f$. The article has two major messages. First, the most important qualitative properties of the Alexander modules are completely independent of the behaviour of $f$ at infinity, or about the special fibers. Second, all the Alexander invariants of all the fibers of the polynomial $f$ are closely related to the monodromy representation of $f$. In fact, all the torsion parts of the Alexander modules (associated with all the possible fibers) can be obtained by factorization of a unique universal Alexander module, which is constructed from the monodromy representation. Additionally, the article extends some results of A. Libgober about Alexander modules of hypersurface complements.