Ştefan Papadima

Topology and geometry of cohomology jump loci

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.

Non-abelian cohomology jump loci from an analytic viewpoint

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

UNIVERSAL REPRESENTATIONS OF BRAID AND BRAID-PERMUTATION GROUPS

Drinfeld used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We view braid groups as subgroups of braid-permutation groups. We construct a family of universal representations of braid-permutation groups, without using associators. All representations in the family are faithful, defined over

The universal finite-type invariant for braids, with integer coefficients

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Flat connections and resonance varieties: From rank one to higher ranks

by simple explicit formulae. We show that they give universal Vassiliev-type invariants for braid-permutation groups.

Non-finiteness properties of fundamental groups of smooth projective varieties

Abstract We give a simple, explicit construction of a universal finite-type invariant for braids, which is multiplicative, at the associated graded level. Our approach is valid for arbitrary ring coefficients. A key ingredient is provided by the properties of fundamental groups of fiber-type arrangements. We examine the relation with the Vassiliev theory for links.

On the monodromy action on Milnor fibers of graphic arrangements

Given a finitely-generated group

Deformations of hypersolvable arrangements

pi

On the geometry and topology of partial configuration spaces of Riemann surfaces

and a linear algebraic group

Non-Abelian Resonance: Product and Coproduct Formulas

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