Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities

TL;DR

This paper computes the zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities, confirming a conjecture in the Kleinian case and providing new results for elliptic cases.

Contribution

It explicitly calculates the zeroth Poisson homology for symmetric powers of these singularities, confirming a conjecture for Kleinian surfaces and extending results to elliptic cases.

Findings

Confirmed Alev's conjecture for Kleinian surfaces.

Computed Hochschild homology for symmetric powers of elliptic algebras.

Degeneration of Brylinski spectral sequence in the Kleinian case.

Abstract

Let X be a surface with an isolated singularity at the origin, given by the equation Q(x,y,z)=0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = C^2/G for G < SL(2,C) finite. Let Y be the n-th symmetric power of X. We compute the zeroth Poisson homology of Y, as a graded vector space with respect to the weight grading. In the Kleinian case, this confirms a conjecture of Alev, that the zeroth Poisson homology of the n-th symmetric power of C^2/G is isomorphic to the zeroth Hochschild homology of the n-th symmetric power of the algebra of G-invariant differential operators on C. That is, the Brylinski spectral sequence degenerates in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, for the parameter equal to all but countably many points of the elliptic curve.