Abstract
We introduce a basis of the Orlik–Solomon algebra labeled by chambers, the so called chamber basis. We consider structure constants of the Orlik–Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen’s minimal complex from the Aomoto complex.
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Cohen, D. C., Cohomology and intersection cohomology of complex hyperplane arrangements, Adv. Math. 97 (1993), 231–266.
Cohen, D. C. and Orlik, P., Arrangements and local systems, Math. Res. Lett. 7 (2000), 299–316.
Cohen, D. C. and Suciu, A. I., Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), 33–53.
De Concini, C. and Procesi, C., Nested sets and Jeffrey–Kirwan residues, in Geometric Methods in Algebra and Number Theory, Progr. Math. 235, pp. 139–149, Birkhäuser, Boston, MA, 2005.
Denham, G., The Orlik–Solomon complex and Milnor fibre homology, Topology Appl. 118 (2002), 45–63.
Dimca, A. and Papadima, S., Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math. 158 (2003), 473–507.
Esnault, H., Schechtman, V. and Viehweg, E., Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), 557–561.
Jewell, K. and Orlik, P., Geometric relationship between cohomology of the complement of real and complexified arrangements, Topology Appl. 118 (2002), 113–129.
Kohno, T., Homology of a local system on the complement of hyperplanes, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 144–147.
Orlik, P. and Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189.
Orlik, P. and Terao, H., Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften 300, Springer, Berlin–Heidelberg, 1992.
Papadima, S. and Suciu, A. I., Higher homotopy groups of complements of complex hyperplane arrangements, Adv. Math. 165 (2002), 71–100.
Papadima, S. and Suciu, A. I., The spectral sequence of an equivariant chain complex and homology with local coefficients, Preprint, 2007. arXiv:0708.4262.
Randell, R., Morse theory, Milnor fibers and minimality of hyperplane arrangements, Proc. Amer. Math. Soc. 130:9 (2002), 2737–2743.
Salvetti, M., Topology of the complement of real hyperplanes in C N, Invent. Math. 88 (1987), 603–618.
Salvetti, M. and Settepanella, S., Combinatorial Morse theory and minimality of hyperplane arrangements, Geom. Topol. 11 (2007), 1733–1766.
Schechtman, V., Terao, H. and Varchenko, A., Local systems over complements of hyperplanes and the Kac–Kazhdan conditions for singular vectors, J. Pure Appl. Algebra 100 (1995), 93–102.
Suciu, A. I., Translated tori in the characteristic varieties of complex hyperplane arrangements, Topology Appl. 118 (2002), 209–223.
Varchenko, A. and Gel′fand, I. M., Heaviside functions of a configuration of hyperplanes, Funktsional. Anal. i Prilozhen. 21 (1987), 1–18 (Russian). English transl.: Functional Anal. Appl. 21 (1987), 255–270.
Whitehead, G. W., Elements of Homotopy Theory, Graduate Texts in Mathematics 61, Springer, New York, 1978.
Yoshinaga, M., Hyperplane arrangements and Lefschetz’s hyperplane section theorem, Kodai Math. J. 30 (2007), 157–194.
Zaslavsky, T., Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 154 (1975).
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Yoshinaga, M. The chamber basis of the Orlik–Solomon algebra and Aomoto complex. Ark Mat 47, 393–407 (2009). https://doi.org/10.1007/s11512-008-0085-x
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DOI: https://doi.org/10.1007/s11512-008-0085-x