Rigidity of differentiable structure for new class of line arrangements

An arrangement of hyperplanes is a finite collection of {\bf C}-linear subspaces of codimension one in a complex vector space ${\bf C}^l$. For such an arrangement ${\cal A}$, there is a natural projective arrangement ${\cal A}^*$ of hyperplanes in ${\bf CP}^{l-1}$ associated to it. Let $M({\cal A})={\bf C}^l- \bigcup_{H \in {\cal A}} H$ and $M({\cal A}^*)= {\bf CP}^{l-1}- \bigcup_{H^* \in {\cal A}^*}H^*$. One of central topics in the theory of arrangements is to find connections between the topology or differentiable structure of $M({\cal A})$ (or $M({\cal A}^*)$) and the combinatorial geometry of ${\cal A}$. A partial solution to this problem was given by Jiang and Yau [6]. Specially, they showed that for a class of nice arrangement in ${\bf CP}^2$, the diffeomorphic types of the complements are combinatorial in nature. In this paper, we introduce a new class of simple arrangements in ${\bf CP}^2$. This class of simple arrangements is much larger than the class of nice arrangements. We prove that for this new class of simple arrangements, the diffeomorphic types of the complements are still combinatorial in nature. In fact, the moduli space of simple arrangements with fixed combinatorial data is connected.

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Published 1 January 2005