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

Let M be a smooth complex projective variety, bearing a Kähler symplectic form ωand 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,ω) 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.

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