The diffeomorphic types of the complements of arrangements in \(\mathbb{C}\mathbb{P}^3\). I: Point arrangements (Q2370107)

An arrangement of hyperplanes \(\mathcal A^{\ast}\) in \(\mathbb C\mathbb P^{n}\) is a finite collection of hyperplanes of dimension \(n-1\) in \(\mathbb C\mathbb P^{n}\). Associated with \(\mathcal A^{\ast}\) is an open real \(2n\)-manifold, the complement \(M(\mathcal A^{\ast}) = \mathbb C\mathbb P^{n}\setminus\displaystyle\bigcup_{H^{\ast}\in\mathcal A^{\ast}}H^{\ast}\). The combinatorial data of \(\mathcal A^{\ast}\) is coded by \(L(\mathcal A^{\ast})\), which is the set of all intersections of elements of \(\mathcal A^{\ast}\) partially ordered by reverse inclusion. For any arrangement \(\mathcal A^{\ast}\) in \(\mathbb C\mathbb P^{3}\), the authors introduce a soul \(\mathcal G(\mathcal A^{\ast})\) which is a pseudo-complex completely determined by the combinatoric data of the arrangement. If the soul consists of \(\mathcal G(0)\) (a set of points or 0-simplices) and \(\mathcal G(2)\) (a set of planes or 2-simplices), then the arrangement is called point arrangement. A point arrangement is called a nice arrangement if after removing disjoint stars of \(\mathcal G\), the remaining pseudo-complex contains no loop. The main results of the paper are the following. Theorem A. Let \(\mathcal A^{\ast}_{0}\) and \(\mathcal A^{\ast}_{1}\) be two nice point arrangements of hyperplanes in \(\mathbb C\mathbb P^{3}\). If \(L(\mathcal A^{\ast}_{0})\) and \(L(\mathcal A^{\ast}_{1})\) are isomorphic, then \(M(\mathcal A^{\ast}_{0})\) and \(M(\mathcal A^{\ast}_{1})\) are diffeomorphic to each other. Theorem B. Let \(\mathcal A^{\ast}\) be a nice point arrangement of hyperplanes in \(\mathbb C\mathbb P^{3}\). The moduli space of \(\mathcal A^{\ast}\) with fixed combinatories \(L(\mathcal A^{\ast})\) is connected.