# Intersection homology and Alexander modules of hypersurface complements

@article{Maxim2004IntersectionHA, title={Intersection homology and Alexander modules of hypersurface complements}, author={Laurenţiu Maxim}, journal={Commentarii Mathematici Helvetici}, year={2004}, volume={81}, pages={123-155} }

Let $V$ be a degree $d$, reduced hypersurface in $\mathbb{CP}^{n+1}$, $n \geq 1$, and fix a generic hyperplane, $H$. Denote by $\mathcal{U}$ the (affine) hypersurface complement, $\mathbb{CP}^{n+1}-V \cup H$, and let $\mathcal{U}^c$ be the infinite cyclic covering of $\mathcal{U}$ corresponding to the kernel of the total linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules $H_i(\mathcal{U}^c;\mathbb{Q})$ of the hypersurface…

## 48 Citations

On Betti Numbers of Milnor Fiber of Hyperplane Arrangements

- Mathematics
- 2015

Let $\mathcal{A}$ be a central hyperplane arrangement in $\mathbb{C}^{n+1}$ and $H_i,i=1,2,...,d$ be the defining equations of the hyperplanes of $\mathcal{A}$. Let $f=\prod_i H_i$. There is a global…

Spectral pairs, Alexander modules, and boundary manifolds

- Mathematics
- 2016

Let $$n>0$$n>0 and $$f: {\mathbb {C}}^{n+1}\rightarrow {\mathbb {C}}$$f:Cn+1→C be a reduced polynomial map, with $$D=f^{-1}(0)$$D=f-1(0), $${\mathcal {U}}={\mathbb {C}}^{n+1}{\setminus } D$$U=Cn+1\D…

The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties

- Mathematics
- 2016

Given any connected topological space X, assume that there exists an epimorphism $$\phi {:}\; \pi _1(X) \rightarrow {\mathbb {Z}}$$ϕ:π1(X)→Z. The deck transformation group $${\mathbb {Z}}$$Z acts on…

Higher-order Alexander Invariants of Hypersurface Complements

- Mathematics
- 2015

We define the higher-order Alexander modules $A_{n,i}(\mathcal{U})$ and higher-order degrees $\delta_{n,i}(\mathcal{U})$ which are invariants of a complex hypersurface complement $\mathcal{U}$. These…

Reidemeister torsion, peripheral complex and Alexander polynomials of hypersurface complements

- Mathematics
- 2015

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…

L2-Betti Numbers of Hypersurface Complements

- Mathematics
- 2014

In \cite{DJL07} it was shown that if $\scra$ is an affine hyperplane arrangement in $\C^n$, then at most one of the $L^2$--Betti numbers $b_i^{(2)}(\C^n\sm \scra,\id)$ is non--zero. In this note we…

Motivic zeta functions and infinite cyclic covers

- Mathematics
- 2017

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are…

Nearby cycles and Alexander modules of hypersurface complements

- Mathematics
- 2016

Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

- Mathematics
- 2021

In previous work jointly with Geske, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules, which generalizes the MHS on the cohomology of the Milnor…

## References

SHOWING 1-10 OF 65 REFERENCES

Hypersurface Complements, Alexander Modules and Monodromy

- Mathematics
- 2002

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…

Homotopy groups of the complements to singular hypersurfaces, II

- Mathematics
- 1992

The homotopy group $\pi_{n-k} ({\bf C}^{n+1}-V)$ where $V$ is a hypersurface with a singular locus of dimension $k$ and good behavior at infinity is described using generic pencils. This is analogous…

Milnor fibers and higher homotopy groups of arrangements

- Mathematics
- 2001

We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts…

Singular spaces, characteristic classes, and intersection homology

- Mathematics
- 1991

global topological invariant to be studied will be the L-classes Li(X) E Hi(X; Q), X a stratified pseudomanifold with even-codimension strata. For manifolds, these characteristic classes are the…

Intersection homology II

- Mathematics
- 1983

In [19, 20] we introduced topological invariants IH~,(X) called intersection homology groups for the study of singular spaces X. These groups depend on the choice of a perversity p: a perversity is a…

ALEXANDER POLYNOMIALS OF PLANE ALGEBRAIC CURVES

- Mathematics
- 1994

The author studies the fundamental group of the complement of an algebraic curve defined by an equation . Let be the morphism defined by the equation . The main result is that if the generic fiber is…

Singularities and Topology of Hypersurfaces

- Mathematics
- 1992

This book systematically presents a large number of basic results on the topology of complex algebraic varieties using the information on the local topology and geometry of a singularity. These…