Poisson–de Rham homology of hypertoric varieties and nilpotent cones

• Nicholas Proudfoot ^[1] ; Travis Schedler ^[2]
  1. [1] University of Oregon
  2. [2] University of Texas
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 23, Nº. 1, 2017, págs. 179-202
• Idioma: inglés
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• Resumen

  • Abstract We prove a conjecture of Etingof and the second author for hypertoric varieties that the Poisson–de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson–de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson–de Rham–Poincaré polynomial and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham (J Algebra 242(1):160–175, 2001). We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.