Recursion for Poincaré polynomials of subspace arrangements

Abstract

A subspace arrangement $\mathcal{A}$ in $\mathbb{C}^m$ is a finite set $\{x_0, \dots, x_n \}$ of vector subspaces. The complement space $M(\mathcal{A})$ is $\mathbb{C}^m \setminus \bigcup_{x \in \mathcal{A}} x$. 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 $M(\mathcal{A})$ is connected to the Poincaré polynomials of the complements of the deleted arrangement $\mathcal{A}' = \mathcal{A} \setminus \{x_0\}$ and of the restricted arrangement $\mathcal{A}'' = \{ x_0 \cap y \st y \in \mathcal{A}' \}$ by the nice formula \[ Poin(M(\mathcal{A}),t) = Poin(M(\mathcal{A}'),t) + tPoin(M(\mathcal{A}''),t). \] In this paper, we prove that for a subspace arrangement, there is a long exact sequence in cohomology which connects $M(\mathcal{A})$ to $M(\mathcal{A}')$ and $M(\mathcal{A}'')$. Using it, we can extend the above formula to arrangements with a geometric lattice, and to some other specific arrangements.