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Diffeomorphic types of the complements of arrangements of hyperplanes. (English) Zbl 0828.57018
A fundamental open problem in the theory of complex hyperplane arrangements has been the conjecture that the homotopy or topological type of the complement of a finite collection of hyperplanes is a function only of the underlying matroid or intersection lattice [P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss. 300 (1992; Zbl 0757.55001)]. This problem has now been resolved in the negative in unpublished work of Grigory Rybnikov [On the fundamental group of a complex hyperplane arrangement (preprint, 1993)]. In the paper under review, the authors provide a sufficient condition for the conjecture to hold. Given an arrangement $${\mathcal A}$$ of lines in $$CP^2$$, construct the graph $$G({\mathcal A})$$ whose vertices are the points of multiplicity greater than two, with two vertices adjacent when there is a line of $${\mathcal A}$$ containing them. The star of a vertex is the set of lines containing it (which may contain several collinear edges). An arrangement is called “nice” if there is a set of pairwise disjoint stars in $$G({\mathcal A})$$ whose complement is a forest. In this case it is shown that for any arrangement $${\mathcal A}'$$ with the same intersection lattice as $${\mathcal A}$$, the arrangements along the segment from $${\mathcal A}$$ to $${\mathcal A}'$$ have constant intersection lattice. In other words, the realization space of the matroid of $${\mathcal A}$$ is convex. Then the lattice isotopy theorem of R. Randell [Proc. Am. Math. Soc. 107, 555-559 (1989; Zbl 0681.57016)] implies that $${\mathcal A}$$ and $${\mathcal A}'$$ have diffeomorphic complements.