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
DOI: https://doi.org/10.1090/S0002-9947-97-01844-8
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. Aomoto, K.: Les équations aux différences linéaires et les intégrales des fonctions multiformes, J. Fac. Sci. Univ. Tokyo 22 (1975) 271-297; 26 (1979) 519-523. MR 58:22688c; MR 82h:32013
  • 2. Aomoto, K.: On vanishing of cohomology attached to certain many valued meromorphic functions, J. Math. Soc. Japan 27 (1975), 248-255 MR 55:12946
  • 3. Aomoto, K., Kita, M.: Hypergemetric functions (in Japanese), Springer-Verlag, Tokyo, 1994
  • 4. Björner, A.: On the homology of geometric lattices, Algebra Universalis 14 (1982), 107-128. MR 83d:05029
  • 5. Björner, A.: Homology and shellability of matroids and geometric lattices, in Matroid Applications (ed. N. White), Cambridge University Press 1992, pp. 226-283. MR 94a:52030
  • 6. Björner, A., Ziegler, G.: Broken circuit complexes: Factorizations and generalizations, J. Combin. Theory Ser. B 51 (1991), 96-126. MR 92b:52027
  • 7. Brylawski, T., Varchenko, A.: The determinant formula for a matroid bilinear form, Advances in Math., to appear.
  • 8. Crapo, H.: A higher invariant for matroids, J. of Combinatorial Theory 2 (1967) 406-417. MR 35:6579
  • 9. Douai, A., Terao, H.: The determinant of a hypergeometric period matrix, preprint, 1996.
  • 10. Esnault, H., Schechtman, V., Viehweg, E.: Cohomology of local systems of the complement of hyperplanes, Invent. Math. 109 (1992), 557-561; Erratum, ibid. 112 (1993) 447. MR 93g:32051; MR 94b:32061
  • 11. Gelfand, I.M.: General theory of hypergeometric functions, Soviet Math. Dokl. 33 (1986), 573-577. MR 87h:22012 (of Russian original)
  • 12. Godement, R.: Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958. MR 21:1583
  • 13. Kanarek, H.: Monodromy hypergeometric functions arising from arrangements of hyperplanes, Ph.D. thesis, University of Essen, 1996.
  • 14. Kita, M.: Private communication.
  • 15. Kohno, T.: Homology of a local system on the complement of hyperplanes, Proc. Japan Acad. Ser. A 62 (1986), 144-147. MR 87i:32019
  • 16. Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Math. Wiss. 300, Springer Verlag, 1992. MR 94e:52014
  • 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. Schechtman, V. V., Varchenko, A. N.: Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194. MR 93b:17067
  • 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. Wachs, M. L., Walker, J.W.: On geometric semilattices, Order, 2 (1986), 367-385. MR 87f:06004
  • 22. Yuzvinsky, S.: Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm. Algebra 23 (1995), 5339-5354. CMP 96:05
  • 23. Ziegler, G.: Matroid shellability, $\beta $-systems, and affine arrangements, J. Algebraic Combinatorics, 1 (1992), 283-300. MR 93j:52022

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
Email: mjf@odin.math.nau.edu

Hiroaki Terao
Affiliation: Department of Mathematics, University of Wisconsin - Madison, Madison, Wisconsin 53704
Email: terao@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01844-8
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

American Mathematical Society