A vanishing result for local system cohomology

by Alexandru Dimca

Let A be a hyperplane arrangement in the complex projective space ℙ^{n} or the affine space ℂ^{n} and let us denote by M(A) its complement.

In the first talk I have explained how the cohomology algebra of M(A) can be computed from a combinatorial invariant associated to the hyperplane arrangement A, namely the lattice of all non empty intersections of hyperplanes in A ordered by reverse inclusion. This result was originated by Arnold, proved by Brieskorn and is now most conveniently expressed in terms of Orlik-Solomon algebras, see [OT1] for all the details.

The second talk was devoted to the fundamental group of the complement M(A), in particular to results by Zarisky - van Kampen on how to obtain presentations of this group (for the best results in this direction see the papers by Randell and Arvola on the reference list in [OT1]) and the results of Brieskorn-Deligne giving conditions when such a complement M(A) is an aspherical space, i.e. all the higher homotopy groups of M(A) vanish. We have also described recent results by S. Papadima, A. Suciu and myself on how to determine the first nonzero higher homotopy group of M(A) when the space M(A) is not aspherical, see [DS] .

The third talk described results on the cohomology of M(A) with local coefficients, in particular the results from [ESV] and [STV] which are well presented in the recent book [OT2]. We have also explained how the use of perverse sheaves, see [BBD], [KS], can lead to improvements of results by Libgober [L] and Massey [M] on the Milnor fibers of arrangements, see also [D], [CS] as general references on this subject.

References

[BBD] A. Beilinson, J. Bernstein and P. Deligne : Faisceaux Pervers, Astérisque 100, Soc. Math. France 1983.

[CS] D. Cohen and A. Suciu: On Milnor fibrations of arrangements, J. London Math. Soc. 51 (1995), 105-119.

[D] A. Dimca : Singularities and topology of hypersurfaces, Universitext Springer,1992.

[DS] A. Dimca and S. Papadima: Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements, Annals of Math. (to appear)

[ESV] H. Esnault, V. Schechtman and E. Viehweg: Cohomology of local systems on the complement of hyperplanes, Invent.math. 109(1992),557-561; Erratum,ibid.112(1993),447.

[KS] M. Kashiwara and P. Schapira : Sheaves on Manifolds, Grund der math. Wiss. 292, Springer Verlag 1990.

[L] A. Libgober: Eigenvalues for the monodromy of the Milnor fibers of arrangements, Trends in Singularities, Eds. A. Libgober, M. Tibar, Trends in Maths, Birkhäuser, 2002, pp. 141-150.

[M] D. Massey: Perversity, duality and arrangements in ℂ^{3}, Topologyand its Appl.73(1996),169-179.

[OT1] P. Orlik and H. Terao: Arrangements of hyperplanes, Springer 1992.

[OT2] P. Orlik and H. Terao: Arrangements and Hypergeometric Integrals, MSJ Memoirs, Volume 9, Tokyo, 2001.

[STV] V. Schechtman, H. Terao and A. Varchenko: Local systems over complementsof hyperplanes and the Kas-Kazhdan conditions for singular vectors, JPAA 100(1995),93-102.