  Let X be a projective scheme carrying a circle action S with isolated fixed
points. We associate a simplicial complex Delta(X,S) of "closure chains" using
a refinement of its Morse/Bialynicki-Birula decomposition. If this
decomposition is a stratification (e.g. when X is a flag manifold), then
Delta(X,S) is just the order complex of the poset of fixed points. For X a
toric variety, Delta(X,S) is a triangulation of the moment polytope. We compute
some other examples, including a Bott-Samelson manifold and the punctual
Hilbert scheme of 4 points in the plane.
  Summing over the facets of Delta(X,S), we obtain a positive formula for the
Duistermaat-Heckman measure on the moment polytope of X, defined for any torus
action extending S. We explain how, through brutal use of partial fractions,
this can be extended to an AB/BV-type formula for integrating general classes.
Throughout we work with equivariant Chow groups, and do not make any smoothness
requirements on X.
