Milnor and Tjurina numbers for a hypersurface germ with isolated singularity

Assume that $f:(\mathbb{C}^n,0) \to (\mathbb{C},0)$ is an analytic function germ at the origin with only isolated singularity. Let $\mu$ and $\tau$ be the corresponding Milnor and Tjurina numbers. We show that $\dfrac{\mu}{\tau} \leq n$. As an application, we give a lower bound for the Tjurina number in terms of $n$ and the multiplicity of $f$ at the origin.Comment: 5 pages. A remark is added to explain the recent result for isolated plane curve case due to A. Dimca and G.-M. Greuel. Some typos fixe

Reidemeister Torsion, Peripheral Complex, and Alexander Polynomials of Hypersurface Complements

Let f:\CN \rightarrow \C be a polynomial, which is transversal (or regular) at infinity. Let \U=\CN\setminus 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.Comment: comments are very welcom

Characteristic Varieties of Hypersurface Complements

We give divisibility results for the (global) characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface. As an application, we recast old and obtain new finiteness and divisibility results for the classical (infinite cyclic) Alexander modules of complex hypersurface complements. Moreover, for the special case of hyperplane arrangements, we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties.Comment: v2: much of the paper has been re-written, including a more detailed introduction and updated reference

Spectral pairs, Alexander modules, and boundary manifolds

Let f: \CN \rightarrow \C be a reduced polynomial map, with $D=f^{-1}(0)$, \U=\CN \setminus D and boundary manifold M=\partial \U. Assume that $f$ is transversal at infinity and $D$ has only isolated singularities. Then the only interesting non-trivial Alexander modules of \U and resp. $M$ appear in the middle degree $n$. We revisit the mixed Hodge structures on these Alexander modules and study their associated spectral pairs (or equivariant mixed Hodge numbers). We obtain upper bounds for the spectral pairs of the $n$-th Alexander module of \U, which can be viewed as a Hodge-theoretic refinement of Libgober's divisibility result for the corresponding Alexander polynomials. For the boundary manifold $M$, we show that the spectral pairs associated to the non-unipotent part of the $n$-th Alexander module of $M$ can be computed in terms of local contributions (coming from the singularities of $D$) and contributions from "infinity".Comment: comments are very welcom