Recursion for Poincaré polynomials of subspace arrangements (Q2430151)

Suppose \(\mathcal A\) is an arrangement of \({\mathbb C}\)-linear subspaces of \({\mathbb C}^m\). The Poincaré polynomial of \({\mathcal A}\) is \(P({\mathcal A},t)=\sum_{q\geq 0} \dim H^q(M,{\mathbb Q})\,t^q\), where \(M={\mathbb C}^m - \bigcup_{x \in {\mathcal A}} x\) is the complement of \({\mathcal A}\). Given \(x_0 \in {\mathcal A}\) one constructs two smaller arrangements, the deletion \({\mathcal A}'={\mathcal A}-\{x_0\}\) and restriction \({\mathcal A}''=\{x \cap x_0 \mid x \in {\mathcal A}'\}\), with complements \(M'\) and \(M''\) respectively. The author proves the identity \[ P({\mathcal A},t)=P({\mathcal A}',t)+t^{2c -1}P({\mathcal A}'',t) \] where \(c=\operatorname{codim}(x_0)\), under the assumption that the intersection lattice \(L\) of \({\mathcal A}\) is geometric. This generalizes a well-known identity for arrangements of hyperplanes, originally proved in the reviewer's dissertation. Here the argument is based on a combinatorially-defined rational model \(D({\mathcal A})\) for \(M\) constructed by \textit{E. M. Feichtner} and \textit{S. Yuzvinsky} [Zap. Nauchn. Semin. POMI 326, 235--247 (2005); translated in J. Math. Sci., New York 140, No.~3, 472--479 (2007; Zbl 1084.52519)]. The author constructs a short exact sequence \[ 0 \to D({\mathcal A}') \to D({\mathcal A}) \to D({\mathcal A}'') \to 0 \] that is of independent interest, as it relates the rational homotopy types of \(M\), \(M'\), and \(M''\). The induced long exact sequence in cohomology \[ \cdots \to H^q(M',{\mathbb Q}) \to H^q(M,{\mathbb Q}) \to H^{q-c}(M',{\mathbb Q}) \to \cdots \] can be derived more easily using the long exact sequence of the pair \((M',M)\), applying the Thom Isomorphism Theorem to a tubular neighborhood of \(M''\) in \(M'\). The recursive identity holds if and only if \(H^q(M',{\mathbb Q})\) injects into \(H^q(M,{\mathbb Q})\) for each \(q\geq 0\). This is established using a calculation of \(P({\mathcal A},t)\) due to \textit{P. Deligne, M. Goresky} and \textit{R. MacPherson} [Mich. Math. J. 48, Spec. Vol., 121--136 (2000; Zbl 1083.14504)] and classical deletion-restriction identities for Möbius functions of geometric lattices. The recursive identity may fail to hold if \(L\) is not geometric. The reviewer notes that the vanishing of \(\chi(M)\) (Corollary 3.7) is an immediate consequence of the fact that \(M\) supports a free \({\mathbb C}^*\)-action, which implies \(\chi(M)\) is a multiple of \(\chi({\mathbb C}^*)\) .