Poisson-de Rham homology of hypertoric varieties and nilpotent cones

TL;DR

This paper proves a conjecture relating Poisson-de Rham homology and de Rham cohomology for hypertoric varieties and more general symplectic resolutions, providing explicit computations and conjectural formulas involving matroid invariants.

Contribution

It establishes the isomorphism between Poisson-de Rham homology and de Rham cohomology for hypertoric cones and extends the result to a broader class of symplectic resolutions, with explicit polynomial computations.

Findings

Poisson-de Rham homology of hypertoric cones matches de Rham cohomology of their resolutions.

Computed the 2-variable Poisson-de Rham-Poincare polynomial for hypertoric varieties.

Connected the polynomial to Tutte polynomials and Kostka polynomials, with conjectures for 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. 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.