Abstract |
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A subspace arrangement A in ℂ<sup>m</sup> is a finite set {x <inf>0</inf>, , x<inf>n</inf> }of vector subspaces. The complement space M(A) is ℂ<sup>m</sup> \∪<inf>εA</inf>A x. When each subspace is an hyperplane, it is also known as an arrangement of hyperplanes. In that case, it is known that the Poincaré polynomials of M(A) is connected to the Poincaré polynomials of the complements of the deleted arrangement A′ = A \{x<inf>0</inf>} and of the restricted arrangement A″ = {x<inf>0</inf> ∩ y|y ε A′} by the nice formula Poin(M(A),t) = Poin(M(A′),t) + t Poin(M(″),t). In this paper, we prove that for a subspace arrangement, there is a long exact sequence in cohomology which connects M(A) to M(A′) and M(A″). Using it, we can extend the above formula to arrangements with a geometric lattice, and to some other specific arrangements. |