1,571 research outputs found
Milnor and Tjurina numbers for a hypersurface germ with isolated singularity
Assume that is an analytic function
germ at the origin with only isolated singularity. Let and be the
corresponding Milnor and Tjurina numbers. We show that . As an application, we give a lower bound for the Tjurina number in terms of
and the multiplicity of 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
, we give several estimates for the (infinite cyclic) Alexander polynomials
of \U induced by , 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 ,
\U=\CN \setminus D and boundary manifold M=\partial \U. Assume that is
transversal at infinity and has only isolated singularities. Then the only
interesting non-trivial Alexander modules of \U and resp. appear in the
middle degree . 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 -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 , we show that the spectral pairs associated to the
non-unipotent part of the -th Alexander module of can be computed in
terms of local contributions (coming from the singularities of ) and
contributions from "infinity".Comment: comments are very welcom
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