Rigidity of differentiable structure for new class of line arrangements (Q871732)

An arrangement of hyperplanes is a finite collection of \(C\)-linear subspaces of codimension one in a complex vector space \(C^\ell\). For such an arrangement, \(\mathcal A\), there is a natural projective arrangement \(\mathcal A^*\) of hyperplanes in \(CP^{\ell-1}\) associated to it. Let \(M(\mathcal A) = C^\ell - \bigcup_{H \in\mathcal A} H\) and \(M(\mathcal A^*) = CP^{\ell -1} -\bigcup_{H^* \in \mathcal A^*} H^*\). One of the central topics in the theory of arrangements is to find connections between the topology or differentiable structure of \(M(\mathcal A)\) (or \(M(A^*)\)) and the combinatorial geometry of \(\mathcal A\). A partial solution to this problem was given by \textit{T. Jiang} and \textit{S. S.-T. Yau} [Compos. Math. 92, No. 2, 133--155 (1994; Zbl 0828.57018)]. In the paper under review, the authors introduce a new class of simple arrangements in \(CP^2\). This class of simple arrangements is much larger than the class of nice arrangements studied in the previous paper. It is proven that any two simple arrangements with the same underlying matriod have diffeomorphic complements.