Poisson traces in positive characteristic

TL;DR

This paper investigates Poisson traces in positive characteristic, providing explicit computations for various singularities and quotient spaces, revealing infinite-dimensionality in general but finite-dimensional cases under specific conditions.

Contribution

It offers the first complete calculations of Poisson trace spaces for certain singularities in positive characteristic, extending the understanding of D-modules in this setting.

Findings

Computed HP_0(A) for large p in specific singularities

Established finite-dimensionality conditions in characteristic zero

Provided explicit Hilbert series for various cases

Abstract

We study Poisson traces of the structure algebra A of an affine Poisson variety X defined over a field of characteristic p. According to arXiv:0908.3868v4, the dual space HP_0(A) to the space of Poisson traces arises as the space of coinvariants associated to a certain D-module M(X) on X. If X has finitely many symplectic leaves and the ground field has characteristic zero, then M(X) is holonomic, and thus HP_0(A) is finite dimensional. However, in characteristic p, the dimension of HP_0(A) is typically infinite. Our main results are complete computations of HP_0(A) for sufficiently large p when X is 1) a quasi-homogeneous isolated surface singularity in the three-dimensional space, 2) a quotient singularity V/G, for a symplectic vector space V by a finite subgroup G in Sp(V), and 3) a symmetric power of a symplectic vector space or a Kleinian singularity. In each case, there is a finite nonnegative grading, and we compute explicitly the Hilbert series. The proofs are based on the theory of D-modules in positive characteristic.