The diffeomorphic types of the complements of arrangements in \(\mathbb C\mathbb P^{3}\). II.
An arrangement of hyperplanes \(\mathcal{A}^*\) in \(\mathbb{CP}^n\) is a finite collection of hyperplanes of dimension \(n - 1\) in \(\mathbb{CP}^n\). Associated with \(\mathcal{A}^*\) is an open real \(2n\)-manifold, the complement \(M(\mathcal{A}^*) = \mathbb{CP}^n - \bigcup_{H^* \in \mathcal{A}^*} H^*\). One of the central problems in this area is to decide to what extent the topology or differentiable structure of \(M(\mathcal{A}^*)\) is determined by the combinatorial geometry of \(\mathcal{A}^*\) and vice versa. It is well-known that the combinatorial data of \(\mathcal{A}^*\) are coded by \(L(\mathcal{A}^*)\) which is the set of all intersections of elements of \(\mathcal{A}^*\) partially ordered by reverse inclusion. The authors introduced in the first part [J. Math. Soc. Japan 59, No. 2, 423--447 (2007; Zbl 1140.14032)] the concept of point arrangements of hyperplanes in \(\mathbb{CP}^3\) and proved the following theorem. Theorem 1.1 Let \(\mathcal{A}^*\) be a nice point arrangement of hyperplanes in \(\mathbb{CP}^3\). The moduli space of \(\mathcal{A}^*\) with fixed combinatorics \(L(\mathcal{A}^*)\) is connected. In the paper under review, the authors generalize the above theorem to a class of nice arrangement of hyperplanes in \(\mathbb{CP}^3\). In general, for any arrangement \(\mathcal{A}^*\) of hyperplanes in \(\mathbb{CP}^3\), one introduces a soul \(\mathcal{G}(\mathcal{A}^*)\) which is a pseudo-complex completely determined by the combinatoric data of the arrangement. If the soul consists of \(\mathcal{G}(1)\) (a set of lines or 1-simplices) and \(\mathcal{G}(2)\) ( a set of planes or 2-simplices), then the arrangement is called a line arrangement. A line arrangement is called a nice arrangement if after removing disjoint stars of \(\mathcal{G}\), the remaining pseudo-complex contains no loop. The following is proved: Theorem A. Let \(\mathcal{A}_0^*\) and \(\mathcal{A}_1^*\) be two nice arrangements of hyperplanes in \(\mathbb{CP}^3\). If \(L(\mathcal{A}_0^*)\) and \(L(\mathcal{A}^*_1)\) are isomorphic, then \(M(\mathcal{A}_0^*)\) and \(M(\mathcal{A}_1^*)\) are diffeomorphic to each other. In the course of proving theorem A, one proved the following Theorem. Theorem B. Let \(\mathcal{A}^*\) be a nice arrangement of hyperplanes in \(\mathbb{CP}^3\). The moduli space of \(\mathcal{A}^*\) with fixed combinatorics \(L(\mathcal{A}^*)\) is connected.
differential and topological structure of complement of arrangement
scientific article; zbMATH DE number 5323357
scientific article