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Combinatorial covers and vanishing of cohomology. (English) Zbl 1407.55008

The authors develop a Mayer-Vietoris spectral sequence for cohomology of local systems over spaces equipped with certain types of combinatorial stratifications. This is applied in several settings: complements of arrangements of hyperplanes in \({\mathbb{CP}}^n\); complements of elliptic arrangements, that is, arrangements of kernels of homomorphisms \(E^n \to E\), \(E\) an elliptic curve; and toric complexes: unions of coordinate subtori of \((S^1)^n\) determined by simplicial complexes. The purpose is to establish general conditions under which the local system cohomology vanishes except in a single degree. The spectral sequence applies to local systems of modules over arbitrary fields or the integers, with no assumption of finite generation or other restrictions. In particular it applies with group ring coefficients, in which case the vanishing results are used to establish duality and abelian duality properties. These in turn have consequences for propagation of resonance, a nesting phenomenon for cohomology jump loci treated in a subsequent paper by the same authors, see [Sel. Math., New Ser. 23, No. 4, 2331–2367 (2017; Zbl 1381.55005)].
Roughly speaking, a combinatorial cover of a space \(X\) is a cover of \(X\) and an auxiliary ranked poset \(P\), with an order-preserving map from the nerve of the cover to \(P\) that respects homotopy types in a certain sense. Given such data, and a locally constant sheaf \({\mathcal F}\) of modules over \(X\), the authors build a spectral sequence abutting to \(H^\cdot(X,{\mathcal F})\) whose \(E_2\) term is described in terms of cohomology of local systems over the order complexes of intervals in \(P\). Such covers are constructed for \(X\) the complement of a complex projective hyperplane arrangement using the DeConcini-Procesi wonderful model, with \(P\) being the poset of nested sets. Certain free abelian subgroups of \(\pi_1(X)\) are identified, coming from centers of local fundamental groups at strata corresponding to nested sets. For cohomology vanishing the operative assumptions are that \({\mathcal F}\) comes from a module which is maximal Cohen-Macaulay over the (Laurent polynomial) group rings of these subgroups – this replaces the condition on nonvanishing residues that appears in earlier results – and that the strata are Stein manifolds. With these assumptions the spectral sequence implies the vanishing results for hyperplane arrangements, which are then used to establish related vanishing results for complements of elliptic arrangements, implying in particular that they are duality and abelian duality spaces. Consequently the pure braid group of an elliptic curve is a duality and abelian duality group. The methods also apply to show that a toric complex is an abelian duality space if and only if the associated simplicial complex is Cohen-Macaulay.
Serious readers are advised of a smattering of occasionally confusing typos and notational inconsistencies.

MSC:

55T99 Spectral sequences in algebraic topology
14F17 Vanishing theorems in algebraic geometry
32S22 Relations with arrangements of hyperplanes
55N25 Homology with local coefficients, equivariant cohomology
16G50 Cohen-Macaulay modules in associative algebras
16S34 Group rings

Citations:

Zbl 1381.55005
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References:

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