We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and higher-order degrees $δ_n(C)$ which are invariants of the algebraic planar curve $C$. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve $C$. These invariants are in the spirit of those developed by T. Cochran in \cite{C} and S. Harvey in \cite{H} and \cite{Har}, which were used to study knots, 3-manifolds, and finitely presented groups, respectively. We show that for curves in general position at infinity, the higher-order degrees are finite. This provides new obstructions on the type of groups that can arise as fundamental groups of complements to affine curves in general position at infinity.
We relate the geometry of the resonance varieties associated to a commutative differential graded algebra model of a space to the finiteness properties of the completions of its Alexander-type invariants. We also describe in simple algebraic terms the non-translated components of the degree-one characteristic varieties for a class of non-proper complex manifolds.
We construct a bigraded spectral sequence from the gl(0)-homology to knot Floer homology. This spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules.
This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.
The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper we characterize the vanishing of such invariants for transversal unions of plane curves $C'$ and $C''$ in terms of the finiteness, the vanishing properties of the invariants of $C'$ and $C''$, and whether they are irreducible or not. As a consequence, we characterize which of these types of curves have trivial multivariable Alexander polynomial in terms of their defining equations. Our results impose obstructions on the class of groups that can be realized as fundamental groups of comp…
To any complex algebraic variety endowed with a morphism to a complex affine torus we associate multivariable cohomological Alexander modules, and define natural mixed Hodge structures on their maximal Artinian submodules. The key ingredients of our construction are Gabber-Loeser's Mellin transformation and Hain-Zucker's work on unipotent variations of mixed Hodge structures. As applications, we prove the quasi-unipotence of monodromy, we obtain upper bounds on the sizes of the Jordan blocks of monodromy, and we explore the change in the Alexander modules after removing fibers of the map. We a…
Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let $U$ be a smooth connected complex algebraic variety and let $f\colon U\to \mathbb{C}^*$ be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of $\mathbb{C}^*$ by $f$ gives rise to an infinite cyclic cover $U^f$ of $U$. The action of the deck group $\mathbb{Z}$ on $U^f$ induces a $\mathbb{Q…
We consider a braid $β$ which acts on a punctured plane. Then we construct a local system on this plane and find a homology cicle $D$ in its symmetric power such that $D\cdot β(D)$ coincides with the Alexander polynomial of the plate closure of $β$.
We revisit generic vanishing results for perverse sheaves with any field coefficients on a complex semi-abelian variety, and indicate several topological applications. In particular, we obtain finiteness properties for the integral Alexander modules of complex algebraic varieties mapping to semi-abelian varieties. Similar results were recently derived by the authors by using Morse-theoretic arguments.
In this paper we discuss some properties of fundamental groups and Alexander polynomials of plane curves. We discuss the relationship of the non-triviality of Alexander polynomials and the notion of (nearly) freeness for irreducible plane curves. We reprove and restate in modern terms a somewhat forgotten result of Zariski. Finally, we describe some topological properties of curves with abelian fundamental group.
We introduce a new invariant of tangles along with an algebraic framework in which to understand it. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of the computationally efficient members of the family of Alexander invariants, it is the most meaningful. These are lecture notes for talks given by the first author, written and completed by the second. The talks, with handouts and videos, are available at http://www.math.toronto.edu/drorbn/Talks/Regina-1206/. See also further comments at http://www.math.tor…
We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We show that the I-adic completion of the Alexander invariant of a 1-formal group G is determined solely by the cup-product map in low degrees. It follows that the germs at the origin of the characteristic and resonance varieties of G are analytically isomorphic; in particular, the tangent cone to V_k(G) at 1 equals R_k(G). This provides new obstructions to 1-formality. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerf…
We extend results of Cogolludo-Agustin and Libgober relating the Alexander polynomial of a plane curve $C$ with the Mordell--Weil rank of certain isotrivial families of jacobians over $\mathbf{P}^2$ of discriminant $C$. In the second part we introduce a height pairing on the $(2,3,6)$ quasi-toric decompositions of a plane curve. We use this pairing and the results in the first part of the paper to construct a pair of degree 12 curves with 30 cusps and Alexander polynomial $t^2-t+1$, but with distinct height pairing. We use the height pairing to show that these curves from a Zariski pair.
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 computations of higher-order degrees of curves, which are invariants defined in a previous paper of the authors.
On the Alexander polynomials of conic-line arrangements
Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学）；Simion Stoilow Institute of Mathematics（西蒙·斯托伊洛瓦数学研究所）；University of the National Education Commission Krakow（克拉科夫国民教育委员会大学）；Ovidius University（奥维迪乌斯大学）
We study finiteness (and vanishing) properties of the higher order degrees associated to complements of complex affine plane curves with mild singularities at infinity. Our results impose new obstructions on the class of groups that can be realized as fundamental groups of affine plane curve complements. We also clarify the relationship between the higher order degrees and the multivariable Alexander polynomial of a non-irreducible plane curve.
We generalize and complete some of Maxim's recent results on Alexander invariants of a polynomial transversal to the hyperplane at infinity. Roughly speaking, and surprisingly, such a polynomial behaves both topologically and algebraically (e.g. in terms of the variation of MHS on the cohomology of its smooth fibers), like a homogeneous polynomial.
