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–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

Equivariant Slices for Symplectic Cones

“…Therefore, if we assume nothing but the fact that Y is a symplectic leaf closed under the C × -action and that X has symplectic singularities, we know that the LHS of (45) must be a weakly equivariant local system on C × · y (at this point, we only need symplectic singularities to guarantee that M(X) is holonomic, which more generally follows if X has finitely many symplectic leaves). We can describe this local system using an ordinary Darboux-Weinstein slice S, i.e., such that X y ∼ =Ŷ y× S. Then, similarly to [PS14,§5], the grading on HP 0 (S) is given by an arbitrary vector field η such that [η, π S ] = −kπ S (using now the fact that all Poisson vector fields are Hamiltonian, from Theorem 42). Such a vector field can be obtained from the Euler vector field Eu X of X by the projection π S :X y → S, namely η = (π S ) * (Eu X | {0}×S ).…”

“…The impetus for this work was [PS14], and I thank Nick Proudfoot for his collaboration there. This work was partially supported by NSF grant DMS-1406553.…”

Derived Equivalences for Symplectic Reflection Algebras

“…Since ρ is a Poisson morphism, dim L ′ = dim ρ(L ′ ) implies that there is an open subset L 0 ⊂ L ′ that is an unramified cover of an open leaf L in ρ(L ′ ). By the work of Namikawa, [N4] (the case of open leaf) and of Proudfoot and Schedler, the proof of [PS,Proposition 3.1] (the general case), the algebraic fundamental group of L is finite. It follows that the algebraic fundamental group of L 0 is finite.…”

Quiver theories and Hilbert series of classical Slodowy intersections

“…15 5 Unitary Quivers for A 4 Slodowy Intersections. 16 6 A Series 3d Mirror Symmetry 20 7 BCD Series Multi-flavoured Quiver Types 21 8 O-USp Quivers for B 1 ∼ = C 1 Slodowy Intersections. 23 9 O-USp Quivers for B 2 ∼ = C 2 Slodowy Intersections.…”

“…There is no settled terminology: in[4] these are termed "Slodowy varieties"; in[3] and[5] "intersections"; in[6] "nilpotent Slodowy slices"; in[7] "S3-varieties". Since the spaces are not generally nilpotent with respect to their symmetry group, F ⊆ G, and are always intersections (labelled by a pair of orbits), we prefer "Slodowy intersections".…”