A compactly supported formula for equivariant localization, and, simplicial complexes of Bialynicki-Birula decompositions

TL;DR

This paper introduces a new combinatorial complex associated with circle actions on projective schemes, providing a compact formula for the Duistermaat-Heckman measure and extending to integrals of classes without smoothness assumptions.

Contribution

It constructs a simplicial complex of closure chains refining Bialynicki-Birula decompositions and derives a positive formula for the Duistermaat-Heckman measure applicable to various schemes.

Findings

The complex simplifies the computation of the Duistermaat-Heckman measure.

Provides explicit examples including flag manifolds and Hilbert schemes.

Extends formulas to non-smooth schemes using equivariant Chow groups.

Abstract

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.