A COMPACTLY SUPPORTED FORMULA FOR EQUIVARIANT LOCALIZATION and SIMPLICIAL COMPLEXES OF BIAŁYNICKI-BIRULA DECOMPOSITIONS

The Duistermaat-Heckman formula for their induced measure on a moment polytope is nowadays seen as the Fourier transformof the Atiyah-Bott localization formula, applied to the T -equivariant Liouville class. From this formula one does not see directly that the measure is positive, nor that it vanishes outside the moment polytope. In [Knutson99] we gave a formula for the Duistermaat-Heckman measure whose terms are all positive and compactly supported, using a Morse decomposition. Its derivation required that the stable and unstable Morse strata intersect transversely. In this paper, we remove this very restrictive condition, at the cost of working with an “iterated” Morse (or Białynicki-Birula) decomposition. This leads in a natural way to a simplicial complex of “closure chains”, which in the toric variety case is just a pulling triangulation of the moment polytope. To handle the singularities of the closed strata we restrict to the projective algebraic setting. Conversely, this allows us to work from the beginning with singular projective schemes over algebraically closed ground fields.