Intersection homology and Alexander modules of hypersurface complements
By Laurentiu G. Maxim. 2004
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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 complement and show that, if $i \leq n$, these are torsion over the ring of rational Laurent polynomials. We also obtain obstructions on the associated global polynomials: their zeros are roots of unity of order $d$ and are entirely determined by the local topological information encoded by the link pairs of singular strata of a stratification of the pair $(\mathbb{CP}^{n+1},V)$. As an application, we give obstructions on the eigenvalues of monodromy operators associated to
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/c3b2b9ff1107c14474ce9cda9bbffa51700c257d
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Singular Intersection Homology (2020, background): “Further work on intersection spaces, including extending the construction to more stratified spaces, developing connections to de Rham and L(2) cohomologies, and studying applications to algebraic varieties can be found in [25, 17, 24, 19, 18, 162, 21, 141, 140, 22].”