Poisson–de Rham homology of hypertoric varieties and nilpotent cones

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–Poincare 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.