Nonpositive curvature of blow-ups

Consider the following situation: M C is a complex manifold of complex dimension n, and D C is a union of smooth complex codimension-one submanifolds (i.e., D C is a smooth divisor). Examples of this situation include: (1) arrangements of projective hyperplanes in C P n , as well as various blow-ups of such arrangements along intersections of hyperplanes, (2) nonsingular toric varieties (where D C is the complement of the (C) n-orbit), and (3) certain compactiications of point-conngurations in C P 1 (where D C is the complement of the nondegenerate conngurations). In such examples there is often a \real version" of (M C ; D C) which we will denote by (M; D). By this we mean that M is the xed point set of a smooth involution on M C which is locally isomorphic to complex conjugation on C n and that D = D C \ M. Thus, M is a smooth n-manifold and D is a union of codimension-one smooth submanifolds. Our primary interest in this paper is the geometry and topology of the pair (M; D). The examples in which we are interested will have the features discussed in (A), (B), and (C) below. (A) Cellulations by polytopes. The divisor D cuts M into regions, called chambers, which are combinatorially equivalent to convex polytopes. In this case, we say D gives a cellulation of M. In addition, D will be locally isomorphic to an arrangement of hyperplanes. (If D has the last property, then each dual cell in M will be a \zonotope".) (B) The associated cellulation by cubes. If the above cellulation is by simple polytopes (an n-dimensional polytope is simple if n-edges meet at each vertex), then so is its dual cellulation. Moreover, these cellulations will have a common subdivision by cubes. (C) Nonpositive curvature and asphericity. Any cubical cell complex K has a natural piecewise-Euclidean structure in which each cell is identiied with a regular Euclidean cube of some xed size (say, of edge length 1). This then deenes a \length metric" on each component of K: the length of a linear path in some cell is its Euclidean length, and the distance between two points in K is the innmum of the lengths of all piecewise linear paths between them. It follows that any two points in the same path component of K can be connected by a geodesic segment …