BGG resolutions via configuration spaces

Falk, Michael; Schechtman, Vadim; Varchenko, Alexander

doi:10.5802/jep.9

BGG resolutions via configuration spaces
[Résolutions BGG via les espaces de configurations]

Falk, Michael ¹ ; Schechtman, Vadim ² ; Varchenko, Alexander ³

¹ Department of Mathematics and Statistics, Northern Arizona University Flagstaff, AZ 86011, USA
² Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 Route de Narbonne, 31062 Toulouse, France
³ Department of Mathematics, University of North Carolina at Chapel Hill Chapel Hill, NC 27599-3250, USA

Journal de l’École polytechnique — Mathématiques, Tome 1 (2014), pp. 225-245.

Résumé
Abstract

Nous étudions les éclatements d’espaces de configuration. Ces espaces ont une structure de variété que nous appelons d’Orlik-Solomon ; elle permet de calculer la cohomologie d’intersection de certaines connexions plates avec singularités logarithmiques à l’aide de complexes de formes logarithmiques du type d’Aomoto. En utilisant cette construction, nous donnons une réalisation géométrique de la résolution de Bernstein–Gelfand–Gelfand pour ${𝔰𝔩}_{2}$ comme un complexe d’Aomoto.

We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik–Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the ${𝔰𝔩}_{2}$ Bernstein–Gelfand–Gelfand resolution as an Aomoto complex.

DOI : 10.5802/jep.9

Classification : 55R80, 17B10, 32S22, 17B55
Keywords: Configuration space, normal-crossing divisor, resolution, residue, local system, cohomology, Orlik-Solomon algebra, Aomoto complex, BGG resolution
Mot clés : Espace de configuration, diviseur à croisements normaux, résolution, résidu, système local, cohomologie, algèbre d’Orlik-Solomon, complexe d’Aomoto, résolution BGG

Affiliations des auteurs :

Falk, Michael ¹ ; Schechtman, Vadim ² ; Varchenko, Alexander ³

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Falk, Michael; Schechtman, Vadim; Varchenko, Alexander. BGG resolutions via configuration spaces. Journal de l’École polytechnique — Mathématiques, Tome 1 (2014), pp. 225-245. doi : 10.5802/jep.9. http://www.numdam.org/articles/10.5802/jep.9/

Bibliographie
Cité par

[AV12] Arinkin, D.; Varchenko, A.; Bjorner, A.; Cohen, F.; De Concini, C. C. Procesi; Salvetti, M. Intersection cohomology of a rank one local system on the complement of a hyperplane-like divisor, Configuration spaces. Geometry, combinatorics and topology (Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series), Volume 14, Edizioni della Normale, Pisa, 2012, pp. 49-53 (arXiv:1106.5732) | MR | Zbl

[BB81] Beilinson, A.; Bernstein, J. Localisation de $𝔤$ -modules, C. R. Acad. Sci. Paris Sér. I Math., Volume 292 (1981) no. 1, pp. 15-18 | MR | Zbl

[BFS98] Bezrukavnikov, R.; Finkelberg, M.; Schechtman, V. Factorizable sheaves and quantum groups, Lect. Notes in Math., 1691, Springer-Verlag, Berlin, 1998, pp. x+287 | MR | Zbl

[BG92] Beilinson, A.; Ginzburg, V. Infinitesimal structure of moduli spaces of $G$ -bundles, Internat. Math. Res. Notices (1992) no. 4, pp. 63-74 | DOI | MR | Zbl

[BGG75] Bernstein, I. N.; Gelfand, I. M.; Gelfand, S. I. Differential operators on the base affine space and a study of $𝔤$ -modules, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 21-64 | MR | Zbl

[DCP95] 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

[Dim92] Dimca, A. Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992, pp. xvi+263 | DOI | MR | Zbl

[ESV92] Esnault, H.; Schechtman, V.; Viehweg, E. Cohomology of local systems on the complement of hyperplanes, Invent. Math., Volume 109 (1992) no. 3, pp. 557-561 Erratum: Ibid. 112 (1993), p. 447 | DOI | MR | Zbl

[Kem78] Kempf, G. The Grothendieck-Cousin complex of an induced representation, Adv. in Math., Volume 29 (1978) no. 3, pp. 310-396 | DOI | MR | Zbl

[KS97] Khoroshkin, S.; Schechtman, V. Factorizable $𝒟$ -modules, Math. Res. Lett., Volume 4 (1997) no. 2-3, pp. 239-257 | DOI | MR | Zbl

[KV06] Khoroshkin, S.; Varchenko, A. Quiver $𝒟$ -modules and homology of local systems over an arrangement of hyperplanes, IMRP Int. Math. Res. Pap. (2006) (Art. ID 69590) | MR | Zbl

[OT92] Orlik, P.; Terao, H. Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, 300, Springer-Verlag, Berlin, 1992, pp. xviii+325 | DOI | MR | Zbl

[STV95] Schechtman, V.; Terao, H.; Varchenko, A. Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, J. Pure Appl. Algebra, Volume 100 (1995) no. 1-3, pp. 93-102 | DOI | MR | Zbl

[SV91] Schechtman, V.; Varchenko, A. Arrangements of hyperplanes and Lie algebra homology, Invent. Math., Volume 106 (1991) no. 1, pp. 139-194 | DOI | MR | Zbl

[Var95] Varchenko, A. Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Advanced Series in Mathematical Physics, 21, World Scientific Publishing Co., Inc., River Edge, NJ, 1995, pp. x+371 | DOI | MR | Zbl

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