The diffeomorphic types of the complements of arrangements in \(\mathbb C\mathbb P^{3}\). II.
type: Item
instance of: scholarly article
title: The diffeomorphic types of the complements of arrangements in \(\mathbb C\mathbb P^{3}\). II.
zbMATH Open document ID: 1192.14042
author: Shaobo Wang
author: Stephen Yau
published in: Science in China. Series A
publication date: 2008-09-09T00:00:00Z
review text: 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(\ma…
reviewed by: Vasile Brînzănescu
Mathematics Subject Classification ID: 14N20
Mathematics Subject Classification ID: 32S22
Mathematics Subject Classification ID: 57R50
zbMATH DE Number: 5323357
zbMATH Keywords: arrangement of hyperplanes
zbMATH Keywords: soul of arrangement
zbMATH Keywords: combinatorial geometry of arrangement
zbMATH Keywords: differential and topological structure of complement of arrangement
MaRDI profile type: Publication
full work available at URL: s11425-007-0174-5
OpenAlex ID: W2040443024
cites work: Q3473822
cites work: On the algebra associated with a geometric lattice
cites work: Homotopy types of line arrangements
cites work: Topological invariance of intersection lattices of arrangements in 𝐂𝐏²
cites work: Q4225589
cites work: Orlik-Solomon algebras and Tutte polynomials
cites work: Topology and combinatorics of real line arrangements
cites work: Lattice-Isotopic Arrangements are Topologically Isomorphic
cites work: On the homotopy theory of arrangements. II
cites work: Q4301626
cites work: Rigidity of differentiable structure for new class of line arrangements
cites work: The diffeomorphic types of the complements of arrangements in \(\mathbb{C}\mathbb{P}^3\). I: Point arrangements
cites work: The fundamental group of the complement of an arrangement of complex hyperplanes
cites work: Complexified real arrangements of hyperplanes
cites work: Arrangements and cohomology
cites work: Q3043303
cites work: Q3142351
cites work: Q3209735
cites work: Combinatorics and topology of complements of hyperplanes
cites work: Unitary reflection groups and cohomology
cites work: The fundamental group of the complement of a union of complex hyperplanes: Correction
DOI: 10.1007/S11425-007-0174-5
description: scientific article; zbMATH DE number 5323357