Topology and geometry of cohomology jump loci

By Alexandru Dimca, Stefan Papadima, Alexander I. Suciu. 2009

132 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e435d205344d3229e9e4d6139b171face73a7ec8

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Koszul modules and Green’s conjecture (2018, background): “ancevarieties(andoftheassociatedcohomologyjumpingloci)inHodgetheoryhas beeninitiatedbyGreenandLazarsfeld[15],andithasbeenactively pursuedinthetheoryofhyperplane arrangements and topology. We refer to [6], [24] and the references therein for an overview, and for the 4 M. APRODU, G. FARKAS, S¸. PAPADI”
Characteristic varieties of quasi-projective manifolds and orbifolds (2010, methodology): “In this way, we recover and extend some of the properties found by Dimca [27; 26; 25], Dimca and Maxim [28], and Dimca, Papadima and Suciu [30; 29] using Arapura’s Theorem.”
Bernstein-Sato ideals and local systems (2012, unknown): “[19, 20], Arapura [1, 2], Simpson [42], Budur [9], Libgober [27], Dimca-PapadimaSuciu [17], Popa-Schnell [37], Dimca-Papadima [16].”
The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy (2014, background): “…manifestation, Milnor’s construction associates to each homogeneous polynomial Q ∈ C[z1, . . . , z ] a smooth fibration over C∗, by restricting the polynomial map Q : C → C to the complement of the zero-set of Q. Received 3 December 2015; revised 17 December 2016; published online 6 March 2017.”