  We find an explicit combinatorial gradient vector field on the well known
complex S (Salvetti complex) which models the complement to an arrangement of
complexified hyperplanes. The argument uses a total ordering on the facets of
the stratification of R^n associated to the arrangement, which is induced by a
generic system of polar coordinates. We give a combinatorial description of the
singular facets, finding also an algebraic complex which computes local
homology. We also give a precise construction in the case of the braid
arrangement.
