Reidemeister torsion, peripheral complex and Alexander polynomials of hypersurface complements

Abstract

Let $f:{ℂ}^{n+1}\to ℂ$ be a polynomial that is transversal (or regular) at infinity. Let $\mathcal{U}={ℂ}^{n+1}\setminus f^{-1}\left(0\right)$ 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 $\mathcal{U}$ 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 $\mathcal{U}$.