Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$\beta \,\mathbf {nbc}$-bases for cohomology of local systems
on hyperplane complements

Authors: Michael Falk and Hiroaki Terao
Journal: Trans. Amer. Math. Soc. 349 (1997), 189-202
MSC (1991): Primary 52B30
MathSciNet review: 1401770
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes ${\mathcal A} $. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement ${\mathcal A} $. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of ${\mathcal A} $. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.

References [Enhancements On Off] (What's this?)

  • 1. Kazuhiko Aomoto, Les équations aux différences linéaires et les intégrales des fonctions multiformes, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 271–297 (French). MR 0508176
    Kazuhiko Aomoto, Une correction et un complément à l’article: “Les équations aux différences linéaires et les intégrales des fonctions multiformes” [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 271–297; MR 58 #22688c], J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), no. 3, 519–523 (French). MR 560011
  • 2. Kazuhiko Aomoto, On vanishing of cohomology attached to certain many valued meromorphic functions, J. Math. Soc. Japan 27 (1975), 248–255. MR 0440065
  • 3. Aomoto, K., Kita, M.: Hypergemetric functions (in Japanese), Springer-Verlag, Tokyo, 1994
  • 4. Anders Björner, On the homology of geometric lattices, Algebra Universalis 14 (1982), no. 1, 107–128. MR 634422, 10.1007/BF02483913
  • 5. Anders Björner, The homology and shellability of matroids and geometric lattices, Matroid applications, Encyclopedia Math. Appl., vol. 40, Cambridge Univ. Press, Cambridge, 1992, pp. 226–283. MR 1165544, 10.1017/CBO9780511662041.008
  • 6. Anders Björner and Günter M. Ziegler, Broken circuit complexes: factorizations and generalizations, J. Combin. Theory Ser. B 51 (1991), no. 1, 96–126. MR 1088629, 10.1016/0095-8956(91)90008-8
  • 7. Brylawski, T., Varchenko, A.: The determinant formula for a matroid bilinear form, Advances in Math., to appear.
  • 8. Henry H. Crapo, A higher invariant for matroids, J. Combinatorial Theory 2 (1967), 406–417. MR 0215744
  • 9. Douai, A., Terao, H.: The determinant of a hypergeometric period matrix, preprint, 1996.
  • 10. Hélène Esnault, Vadim Schechtman, and Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), no. 3, 557–561. MR 1176205, 10.1007/BF01232040
    H. Esnault, V. Schechtman, and E. Viehweg, Erratum: “Cohomology of local systems on the complement of hyperplanes” [Invent. Math. 109 (1992), no. 3, 557–561; MR1176205 (93g:32051)], Invent. Math. 112 (1993), no. 2, 447. MR 1213111, 10.1007/BF01232443
  • 11. I. M. Gel′fand, General theory of hypergeometric functions, Dokl. Akad. Nauk SSSR 288 (1986), no. 1, 14–18 (Russian). MR 841131
  • 12. Roger Godement, Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). MR 0102797
  • 13. Kanarek, H.: Monodromy hypergeometric functions arising from arrangements of hyperplanes, Ph.D. thesis, University of Essen, 1996.
  • 14. Kita, M.: Private communication.
  • 15. Toshitake Kohno, Homology of a local system on the complement of hyperplanes, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 4, 144–147. MR 846350
  • 16. Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488
  • 17. Schechtman, V. V., Terao, H., Varchenko, A. N.: Local systems over complements of hyperplanes and the Kac-Kazhdan condition for singular vectors, J. Pure Appl. Algebra 100 ``Physics and Algebra" (1995), 93-102.
  • 18. Vadim V. Schechtman and Alexander N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), no. 1, 139–194. MR 1123378, 10.1007/BF01243909
  • 19. Varchenko, A.N.: The Euler beta-function, the Vandermonde determinant, Legendre's equation, and critical values of linear functions on a configuration of hyperplanes. I. Math. USSR Izvestiya, 35 (1990), 543-571, II. Math. USSR Izvestiya, 36 (1991), 155-167. MR 91c:32031a,b (of Russian original)
  • 20. Varchenko, A.N.: Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Advanced Series in Mathematical Physics - Vol. 21, World Scientific Publishers, 1995. CMP 96:11
  • 21. Michelle L. Wachs and James W. Walker, On geometric semilattices, Order 2 (1986), no. 4, 367–385. MR 838021, 10.1007/BF00367425
  • 22. Yuzvinsky, S.: Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm. Algebra 23 (1995), 5339-5354. CMP 96:05
  • 23. Günter M. Ziegler, Matroid shellability, 𝛽-systems, and affine hyperplane arrangements, J. Algebraic Combin. 1 (1992), no. 3, 283–300. MR 1194080, 10.1023/A:1022492019120

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 52B30

Retrieve articles in all journals with MSC (1991): 52B30

Additional Information

Michael Falk
Affiliation: Department of Mathematics, University of Wisconsin - Madison, Madison, Wisconsin 53704

Hiroaki Terao
Affiliation: Department of Mathematics, University of Wisconsin - Madison, Madison, Wisconsin 53704

Received by editor(s): April 30, 1995
Additional Notes: The first author was partially supported by a Northern Arizona University Organized Research Grant
Dedicated: In memory of Michitake Kita
Article copyright: © Copyright 1997 American Mathematical Society