Purity, Formality, and Arrangement Complements

Dupont, Clément

doi:10.1093/imrn/rnv260

Cited by 20 publications

(24 citation statements)

References 22 publications

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“…-Using étale cohomology with coefficients in Q instead of Z , we can also prove formality (without bound) of cohomology of complements of hyperplane and toric arrangements with Q -coefficients. This gives an alternative proof of the results of Brieskorn and Dupont (proved respectively in [3] and [14]). Let us mention however that the étale cohomology method only yields formality if the arrangement is defined over a p-adic field.…”

Section: Annales De L'institut Fouriersupporting

confidence: 53%

“…It should be noted that the approach outlined in this section is dependent on Conjecture 4.1. However the result is true regardless of this conjecture as was proved, in various degrees of generality, in [7,12,14]. It should also be noted that if instead the variety X has the property that H k (X, R) is a pure Hodge structure of weight αk for all k, where α is a fixed non-zero rational number, then, using Theorem 3.1, we also get a formality result that recovers the one of [7].…”

Section: Hodge Theorymentioning

confidence: 92%

“…The results we get are thus less general than those of Brieskorn and Dupont. If instead of étale cohomology we use Hodge theory as in subsection 4.1 we can completely recover the results of [3] and [14].…”

Section: Annales De L'institut Fouriermentioning

confidence: 94%

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Homotopy transfer and formality

Drummond-Cole¹,

Horel²

2022

Annales De l'Institut Fourier

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In this paper, we prove that, given appropriate hypotheses, nformality of a differential graded algebraic structure is equivalent to the existence of a chain-level automorphism lifting a degree twisting isomorphism relative to a unit of order greater than n. A similar result with slightly different hypothesis was proved by Joana Cirici and the second author. We use the homotopy transfer theorem and an explicit inductive procedure in order to kill the higher operations. As an application of our result, we prove formality with coefficients in the p-adic integers of certain dg-algebras coming from hyperplane and toric arrangements and configuration spaces.Résumé. -Dans cet article nous montrons que, sous certaines hypothèses, la n-formalité d'une structure algébrique différentielle graduée est équivalente à l'existence d'un automorphisme au niveau des chaînes relevant un isomorphisme gradué tordant relatif à une unité d'ordre plus grand que n. Un résultat similaire sous des hypothèse légèrement différentes avait été prouvé par Joana Cirici et le second auteur. Nous utilisons le théorème de transfert homotopique et une procédure récursive explicite pour tuer les opérations supérieures. Comme application de ce résultat, nous prouvons la formalité à coefficients dans les entiers p-adiques de certaines dg-algèbres venant d'arrangements d'hyperplans ou d'arrangements toriques ainsi que des espaces de configurations.

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Section: Annales De L'institut Fouriersupporting

confidence: 53%

Section: Hodge Theorymentioning

confidence: 92%

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Homotopy transfer and formality

Drummond-Cole¹,

Horel²

2022

Annales De l'Institut Fourier

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“…In Section 4, we prove that the MHS on Tors R H j (U f ; k) 1 does not depend on finite covers of U . Finally, in Section 5, we prove Corollary 1.7 using results of Dupont [9] and Budur-Liu-Wang [3], discuss the formula for the dimension of Tors R H j (U f ; k) 1 (Remark 5.12) in the cases of Remark 1.8, and talk about the MHS on Alexander modules of hyperplane arrangement complements in more detail.…”

Section: Indeed Ifmentioning

confidence: 99%

Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

Elduque¹,

Cueto²

2021

Preprint

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In previous work jointly with Geske, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this paper, we give a positive answer to that question, which implies that the direct sum decomposition of the torsion part of Alexander modules into generalized eigenspaces is in fact a decomposition of MHS. We also show that the MHS on the generalized eigenspace of eigenvalue 1 can be constructed without passing to a suitable finite cover (as is the case for the MHS on the torsion part of the Alexander modules), and compute it under some purity assumptions on the base space.

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“…A deeper inspection of the Salvetti's complex was made in [5] (see also the erratum); this leads to a presentation of the cohomology module with integer coefficients. The wonderful model was described in [8,9,21] and the formality was proven in [13].…”

Section: Introductionmentioning

confidence: 99%

Combinatorics of toric arrangements

Pagaria

2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.

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Purity, Formality, and Arrangement Complements

Cited by 20 publications

References 22 publications

Homotopy transfer and formality

Homotopy transfer and formality

Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

Combinatorics of toric arrangements

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