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

2022/02/03 by Allen Knutson, Knutson, Allen
Mathematics · #53D20 14F10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2202.01774

openalex publication_date 2022/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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