The Orlik-Solomon model for hypersurface arrangements
[Le modèle d’Orlik-Solomon pour les arrangements d’hypersurfaces]
Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2507-2545.

Nous mettons au point un modèle pour la cohomologie du complémentaire d’un arrangement d’hypersurfaces dans une variété complexe projective lisse. Cela généralise le cas des diviseurs à croisements normaux, découvert par P. Deligne dans le cadre de la théorie de Hodge mixte des variétés complexes lisses. Notre modèle est une version globale de l’algèbre d’Orlik-Solomon, qui calcule la cohomologie du complémentaire d’une union d’hyperplans dans un espace affine. L’outil principal est le complexe des formes logarithmiques le long d’un arrangement d’hypersurfaces, et sa filtration par le poids. Nous étudions aussi des liens avec les compactifications magnifiques et les espaces de configuration de points sur des courbes.

We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the configuration spaces of points on curves are also studied.

DOI : https://doi.org/10.5802/aif.2994
Classification : 14C30,  14F05,  14F25,  52C35
Mots clés : arrangements, théorie de Hodge mixte, formes logarithmiques, espaces de configuration
@article{AIF_2015__65_6_2507_0,
     author = {Dupont, Cl\'ement},
     title = {The Orlik-Solomon model for hypersurface arrangements},
     journal = {Annales de l'Institut Fourier},
     pages = {2507--2545},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2994},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2994/}
}
Dupont, Clément. The Orlik-Solomon model for hypersurface arrangements. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2507-2545. doi : 10.5802/aif.2994. http://archive.numdam.org/articles/10.5802/aif.2994/

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