Morse Inequalities for Arrangements

✍ Scribed by Daniel C. Cohen

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 138 KB
• Volume: 134
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1694

✦ Synopsis

Let A=[H 1 , ..., H n ] be an arrangement of complex hyperplanes, and let L be a local system of coefficients on the complement M of A. The cohomology of M with coefficients in L arises in a number of contexts representations of braid groups, generalized hypergeometric functions, Knizhnik Zamolodchikov equations, etc. and has been the subject of considerable recent interest; see for instance [Ko, SV, AK, Va, CS], and see [OT] as a general reference for arrangements. In this brief note, we settle a question raised by Aomoto and Kita concerning the ranks of the cohomology groups H k (M; L) in the case where L is a complex local system of rank one.

Let :=(: 1 , ..., : n ) # C n be a collection of ``weights.'' Associated to :, we have a representation =\ : : ? 1 (M) Ä C* given by ( g j )=exp(&2?i: j ) for any meridian g j about the hyperplane H j of A, and a local system of coefficients L=L : on M. If : satisfies certain genericity conditions, the cohomology, H*(M; L), of M with coefficients in L may be computed using the Orlik Solomon algebra of A; see [ESV, STV]. This leads to results such as the following.

Proposition (Aomoto and Kita [AK, Proposition 2.13.2]). For almost all weights :, we have

(1)

Aomoto and Kita subsequently remark that it is not known if the above inequality holds for all :.

Our purpose is to point out that the inequality (1) does indeed hold for any local system on the complement of any arrangement. Let L be an arbitrary complex rank one local system on the complement M of an Article No. AI971694 43 0001-8708Â98 25.00