Wonderful Models of Subspace Arrangements

0 Introduction In this paper we describe, for any given nite family of sub-spaces of a vector space or for linear subspaces in aane or projective space, a smooth model, proper over the given space, in which the complement of these subspaces is unchanged but the family of subspaces is replaced by a divisor with normal crossings. This model can be described explicitly in a combinatorial way and also by an explicit sequence of blow ups. It is suitable for several computations as cohomology or rational homotopy and it is in some sense minimal. The motivation stems from our attempt to understand Drinfeld's construction (cf. Dr2]) of special solutions of the Khniznik-Zamolodchikov equation (cf. K-Z]) with some prescribed asymptotic behavior and its consequences for some universal constructions associated to braiding: universal unipotent monodromy representations of braid groups, the construction of a universal Vassiliev invariant for knots, braided categories etc.. The K-Z connection is a special at meromorphic connection on C n with simple poles on a family of hyperplanes. It turns out that the prescription of the asymptotic behavior for such connections is controlled by the geometry of a suitable modiication of C n in which the union of the polar hyperplanes is replaced by a divisor with normal crossings. In the process of developing this geometry we realized that our constructions could be developed more generally for subspace arrangements and became aware of the paper of Fulton-MacPherson F-M] in which a Hironaka model is described for the complement of the big diagonal in the power of a smooth variety X. It became clear to us that our techniques were quite similar to theirs and so we adopted their notation of nested set in the appropriate general form. Although we work in a linear subspaces setting it is clear that the methods are essentially local and one can recover their results from our analysis applied to certain special conngurations of subspaces. In fact the theory can be applied whenever we have a subvariety of a smooth variety which locally (in the etale topology) appears as a union of subspaces. We have decided to present the results relative to the holonomy computations in a separate paper \Hyperplane arrangements and holonomy" since the material in the present paper is quite independent from the problem which initially motivated our research. This paper is entirely dedicated to the combinatorial and geometric constructions arising from …