Vanishing cycles on Poisson varieties

TL;DR

This paper extends previous work to compute characteristic cycles of intersection cohomology sheaves on certain Poisson varieties, proposing a conjecture linking hyperbolic stalks and microlocalization at fixed points.

Contribution

It introduces new computations of characteristic cycles on Poisson varieties and formulates a conjecture connecting hyperbolic stalks with microlocalization.

Findings

Computed characteristic cycles on slices in the double affine Grassmannian

Computed characteristic cycles on hypertoric varieties

Proposed a conjecture relating hyperbolic stalks and microlocalization

Abstract

We extend slightly the results of Evens-Mirkovi\'c, and "compute" the characteristic cycles of Intersection Cohomology sheaves on the transversal slices in the double affine Grassmannian and on the hypertoric varieties. We propose a conjecture relating the hyperbolic stalks and the microlocalization at a torus-fixed point in a Poisson variety.