BGG resolutions via configuration spaces

Michael Falk[1]; Vadim Schechtman[2]; Alexander Varchenko[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 (2014)

  • Volume: 1, page 225-245
  • ISSN: 2270-518X

Abstract

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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.

How to cite

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Falk, Michael, Schechtman, Vadim, and Varchenko, Alexander. "BGG resolutions via configuration spaces." Journal de l’École polytechnique — Mathématiques 1 (2014): 225-245. <http://eudml.org/doc/275461>.

@article{Falk2014,
abstract = {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 $\mathfrak\{sl\}_2$ Bernstein–Gelfand–Gelfand resolution as an Aomoto complex.},
affiliation = {Department of Mathematics and Statistics, Northern Arizona University Flagstaff, AZ 86011, USA; Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 Route de Narbonne, 31062 Toulouse, France; Department of Mathematics, University of North Carolina at Chapel Hill Chapel Hill, NC 27599-3250, USA},
author = {Falk, Michael, Schechtman, Vadim, Varchenko, Alexander},
journal = {Journal de l’École polytechnique — Mathématiques},
keywords = {Configuration space; normal-crossing divisor; resolution; residue; local system; cohomology; Orlik-Solomon algebra; Aomoto complex; BGG resolution; configuration space},
language = {eng},
pages = {225-245},
publisher = {École polytechnique},
title = {BGG resolutions via configuration spaces},
url = {http://eudml.org/doc/275461},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Falk, Michael
AU - Schechtman, Vadim
AU - Varchenko, Alexander
TI - BGG resolutions via configuration spaces
JO - Journal de l’École polytechnique — Mathématiques
PY - 2014
PB - École polytechnique
VL - 1
SP - 225
EP - 245
AB - 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 $\mathfrak{sl}_2$ Bernstein–Gelfand–Gelfand resolution as an Aomoto complex.
LA - eng
KW - Configuration space; normal-crossing divisor; resolution; residue; local system; cohomology; Orlik-Solomon algebra; Aomoto complex; BGG resolution; configuration space
UR - http://eudml.org/doc/275461
ER -

References

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  11. S. Khoroshkin, A. Varchenko, Quiver 𝒟 -modules and homology of local systems over an arrangement of hyperplanes, IMRP Int. Math. Res. Pap. (2006) Zbl1116.32005MR2282180
  12. P. Orlik, H. Terao, Arrangements of hyperplanes, 300 (1992), Springer-Verlag, Berlin Zbl0757.55001MR1217488
  13. V. Schechtman, H. Terao, A. Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, J. Pure Appl. Algebra 100 (1995), 93-102 Zbl0849.32025MR1344845
  14. V. Schechtman, A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194 Zbl0754.17024MR1123378
  15. A. Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, 21 (1995), World Scientific Publishing Co., Inc., River Edge, NJ Zbl0951.33001MR1384760

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