Around the Tangent Cone Theorem

Abstract

A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several classes of examples from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their Milnor fibers, configuration spaces, and elliptic arrangements.

Cohomology jump loci of 3-manifolds

“…The following basic relationship between the characteristic and resonance varieties was established by Libgober in [25] in the case when X is a finite CW-complex and i is arbitrary; a similar proof works in the generality that we work in here (see [10,45] for an even more general setup).…”

“…All these notions admit 'partial' versions: e.g., for a fixed q ě 1, we may speak of a q-finite q-model pA, dq for X, in which case the sets R i k pAq are Zariski closed for all i ď q. For more details on all this, we refer to [10,11,13,27,45] and references therein.…”

“…, δ 0 q " H i pAq; thus, the point 0 P H 1 pAq belongs to R i k pAq if and only if b i ě k. In particular, since A is connected, we have that R 0 1 pAq " t0u. In general, the resonance varieties of a cdga may not be invariant under scalar multiplication, see [11,27,45]. Nevertheless, when the differential of A is zero (that is, A is simply a cga), the varieties R i k pAq are homogeneous subsets of H 1 pAq " A 1 .…”

The homotopy type of elliptic arrangements

“…b k = 0. Suciu in [Suc16] has studied a larger class of elliptic arrangements allowing translations of any element c k ∈ E (we consider only translations with c k = b k +ωb k for some b k ∈ S 1 ). In [Bib16], Bibby has studied central elliptic arrangements.…”

Cohomology of abelian arrangements

“…In [Dup15], Dupont uses our decomposition of the Leray spectral sequence in Lemma 3.1 to show that all toric arrangements are formal. In [Suc15], Suciu uses the model given in Theorem 4.1 to study resonance varieties and formality of elliptic arrangements.…”