Related papers: Complements to ample divisors and Singularities
We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin's vanishing theorem for perverse sheaves.…
This is an introduction to topology of complement to plane curves and hypersurfaces in the projective space and is based on the lectures given in Lumini in February and in ICTP (Trieste) in August of 2005. We discuss key problems concerning…
We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend classical local-to-global divisibility results to the twisted setting.…
We survey various Alexander-type invariants of plane curve complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to complex plane curves. Also included are some new…
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 invariants come from the module…
The paper introduces a number of new techniques to handle minimal hyersurface singularities. In particular, they allow to extend the obstruction theory for postive scalr curvature to any dimension.
We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and higher-order degrees $\delta_n(C)$ which are invariants of the algebraic planar curve $C$. These come from analyzing the module structure of the homology of certain…
The dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed…
In the study of normal surface singularities the relation between analytical and topological properties and invariants of the singularity is a very rich problem. This relation is particularly close for surface singularities constructed from…
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 major messages. First, the most important…
We give a short introduction to the theory of twisted Alexander polynomials of a 3--manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted…
We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We…
We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify…
It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in ${\mathbb C}^2,0$}.…
We study the topology of the complements of discriminants of simple real boundary singularities by counting the connected components of these sets and assigning to them certain topological characteristics. Results of this paper serve as a…
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…
We study complements of hypersurfaces in schemes with respect to the property being affine.
We introduce modular inequalities for complements of plane curves, based on a Combinatorial Aomoto complex construction associated with the weak combinatorial type of a curve. We use this as a tool to investigate twisted Alexander…
Let X be a complex smooth quasi-projective variety with an epimorphism $\nu \colon \pi_1(X)\twoheadrightarrow \mathbb{Z}^n$. We survey recent developments about the asymptotic behaviour of Betti numbers with any field coefficients and the…
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…