Annali di Matematica Pura ed Applicata

, Volume 159, Issue 1, pp 341–355 | Cite as

On local systems over complements to arrangements of hyperplanes associated to grassmann strata

  • M. Salvetti
  • M. C. Prati
Article

Summary

In the first two parts we recall the construction of generalized hypergeometric functions and of the cellular complex homotopy equivalent to the complement of a family of hyperplanes in CN. In the third part we find a generalization of some results in [2] about the homology of local systems on an affine space less some hyperplanes. Our method is based on [7] and it gives also informations about the cellular complex there constructed. In the last part explicit bases for the only non-vanishing homology group are described in terms of the cells of the above mentioned complex. The configurations of hyperplanes which we examine are those giving fundamental strata in the grassmanian ([2], [3]) and strata in G3,n allowing also triple points.

Keywords

Local System Triple Point Hypergeometric Function Homology Group Affine Space 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I. M. Gel'fand,The general theory of hypergeometric functions, Sov. Math. Dokl.,33, n. 3 (1986), pp. 573–577.Google Scholar
  2. [2]
    I. M. Gel'fand -V. A. Vasilev -A. V. Zelewinski,General hypergeometric functions on complex grassmanians, Funk. Anal. Appl.,21 (1987), pp. 19–31.Google Scholar
  3. [3]
    I. M. Gel'fand -R. M. Goreski -R. D. MacPherson -V. V. Serganova,Combinatorial Geometries, Convex Polyhedra, and Schubert Cells, Adv. in Math.,63 (1987), pp. 301–316.Google Scholar
  4. [4]
    A. Hattori, Topology of CN minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. Tokyo,22 (1975), pp. 248–255.Google Scholar
  5. [5]
    T. Khono,Homology of a local system on the complement of hyperplanes, Proc. Jap. Acad.,62, Ser. A (1986), pp. 144–147.Google Scholar
  6. [6]
    P.Orlik,Introduction to Arrangements, CBMS Regional Conference Series, 1989.Google Scholar
  7. [7]
    M. Salvetti, Topology of the complement of real hyperplanes in CN, Inv. Math.,88 (1987), pp. 603–618.Google Scholar
  8. [8]
    M. Salvetti,Arrangements of lines and monodromy of plane curves, Comp. Math.,68 (1988), pp. 103–122.Google Scholar
  9. [9]
    M. Salvetti, On the homotopy type of the complement to an arrangement of lines in C2, Boll. U.M.I. (7), 2-A (1988), pp. 337–344.Google Scholar
  10. [10]
    G. Whitehead,Elements of Homotopy Theory, Graduate Texts in Math.,61, Springer, Berlin, 1978.Google Scholar
  11. [11]
    T.Zawlaski,Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math.,154 (1975).Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1991

Authors and Affiliations

  • M. Salvetti
    • 1
  • M. C. Prati
    • 2
  1. 1.Dipartimento di MatematicaUniversità di PisaPisa
  2. 2.I.N.F.N.Scuola Normale SuperiorePisa

Personalised recommendations