The Orlik-Solomon model for hypersurface arrangements

Dupont, Clément

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.

[1] Aluffi, Paolo Chern classes of free hypersurface arrangements, J. Singul., Volume 5 (2012), pp. 19-32 | MR | Zbl

[2] Arnolʼd, V. I. The cohomology ring of the group of dyed braids, Mat. Zametki, Volume 5 (1969), pp. 227-231 | MR | Zbl

[3] Bibby, Christin Cohomology of abelian arrangements (2013) (http://arxiv.org/abs/1310.4866) | MR

[4] Bibby, Christin; Hilburn, Justin Quadratic-linear duality and rational homotopy theory of chordal arrangements (2014) (http://arxiv.org/abs/1409.6748)

[5] Bloch, S. Motives, the fundamental group, and graphs (2012) (preprint)

[6] Brieskorn, Egbert Sur les groupes de tresses [d’après V. I. Arnol${}^{\text{'}}$d], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Springer, Berlin, 1973, p. 21-44. Lecture Notes in Math., Vol. 317 | Numdam | MR | Zbl

[7] Catanese, Fabrizio; Hoşten, Serkan; Khetan, Amit; Sturmfels, Bernd The maximum likelihood degree, Amer. J. Math., Volume 128 (2006) no. 3, pp. 671-697 | MR | Zbl

[8] De Concini, C.; Procesi, C. Wonderful models of subspace arrangements, Selecta Math. (N.S.), Volume 1 (1995) no. 3, pp. 459-494 | DOI | MR | Zbl

[9] Deligne, Pierre Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971) no. 40, pp. 5-57 | Numdam | MR | Zbl

[10] Deligne, Pierre Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974) no. 44, pp. 5-77 | Numdam | MR | Zbl

[11] Dolgachev, Igor V. Logarithmic sheaves attached to arrangements of hyperplanes, J. Math. Kyoto Univ., Volume 47 (2007) no. 1, pp. 35-64 | MR | Zbl

[12] Fulton, William; MacPherson, Robert A compactification of configuration spaces, Ann. of Math. (2), Volume 139 (1994) no. 1, pp. 183-225 | DOI | MR | Zbl

[13] Getzler, E. Resolving mixed Hodge modules on configuration spaces, Duke Math. J., Volume 96 (1999) no. 1, pp. 175-203 | DOI | MR | Zbl

[14] Hu, Yi A compactification of open varieties, Trans. Amer. Math. Soc., Volume 355 (2003) no. 12, pp. 4737-4753 | DOI | MR | Zbl

[15] Kříž, Igor On the rational homotopy type of configuration spaces, Ann. of Math. (2), Volume 139 (1994) no. 2, pp. 227-237 | DOI | MR | Zbl

[16] Leray, Jean Le calcul différentiel et intégral sur une variété analytique complexe. (Problème de Cauchy. III), Bull. Soc. Math. France, Volume 87 (1959), pp. 81-180 | Numdam | MR | Zbl

[17] Li, Li Wonderful compactification of an arrangement of subvarieties, Michigan Math. J., Volume 58 (2009) no. 2, pp. 535-563 | DOI | MR | Zbl

[18] Looijenga, Eduard Cohomology of ${ℳ}_3$ and ${ℳ}_3^1$, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) (Contemp. Math.), Volume 150, Amer. Math. Soc., Providence, RI, 1993, pp. 205-228 | DOI | MR | Zbl

[19] Morgan, John W. The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. (1978) no. 48, pp. 137-204 | Numdam | MR | Zbl

[20] Orlik, Peter; Solomon, Louis Combinatorics and topology of complements of hyperplanes, Invent. Math., Volume 56 (1980) no. 2, pp. 167-189 | DOI | MR | Zbl

[21] Orlik, Peter; Terao, Hiroaki Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 300, Springer-Verlag, Berlin, 1992, pp. xviii+325 | DOI | MR | Zbl

[22] Peters, Chris A. M.; Steenbrink, Joseph H. M. Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 52, Springer-Verlag, Berlin, 2008, pp. xiv+470 | MR | Zbl

[23] Saito, Kyoji Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 27 (1980) no. 2, pp. 265-291 | MR | Zbl

[24] Terao, Hiroaki Forms with logarithmic pole and the filtration by the order of the pole, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo (1978), pp. 673-685 | MR | Zbl

[25] Totaro, Burt Configuration spaces of algebraic varieties, Topology, Volume 35 (1996) no. 4, pp. 1057-1067 | DOI | MR | Zbl

[26] Voisin, Claire Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002, pp. x+322 (Translated from the French original by Leila Schneps) | DOI | MR | Zbl

[27] Yuzvinskiĭ, S. Orlik-Solomon algebras in algebra and topology, Uspekhi Mat. Nauk, Volume 56 (2001) no. 2(338), pp. 87-166 | DOI | MR | Zbl