The Duistermaat-Heckman formula and Chern-Schwartz-MacPherson classes

TL;DR

This paper links the Duistermaat-Heckman formula with Chern-Schwartz-MacPherson classes, providing new interpretations and extensions within symplectic geometry and algebraic topology.

Contribution

It introduces a novel interpretation of Duistermaat-Heckman measure terms as measures of cycles using Ginzburg's classes, extending the classical formula.

Findings

Individual cone terms as measures of cycles in T^*M

Extension of the Duistermaat-Heckman formula to include Brianchon-Gram theorem

Connection between symplectic geometry and algebraic topology via Chern-Schwartz-MacPherson classes

Abstract

Let M be a smooth complex projective variety, bearing a K\"ahler symplectic form \omega and a Hamiltonian action of a torus T, with finitely many fixed points M^T. One standard form of the Duistermaat-Heckman theorem gives a formula for M's Duistermaat-Heckman measure DH_T(M,\omega) as an alternating sum of projections of cones, with overall direction determined by a Morse decomposition of M. Using Victor Ginzburg's construction of Chern-Schwartz-MacPherson classes, we show that these individual cone terms can themselves be interpreted as Duistermaat-Heckman measures of cycles in T^*M. (This has a similar goal to the symplectic cobordism approach of Viktor Ginzburg, Guillemin, and Karshon.) Our approach also suggests extensions of the formula, including the Brianchon-Gram theorem.