Equivariant slices for symplectic cones

TL;DR

This paper develops a C*-equivariant version of the Darboux-Weinstein decomposition for Poisson cones, with applications to symplectic resolutions, algebraic geometry, and noncommutative algebra, including examples and open questions.

Contribution

It introduces a new C*-equivariant decomposition theory for Poisson cones, establishing existence, uniqueness, and applications to quantization and examples.

Findings

Established basic results on existence and uniqueness of C*-equivariant slices.

Studied examples including quotient singularities and hypertoric varieties.

Explored applications to noncommutative algebra and posed open questions.

Abstract

The Darboux-Weinstein decomposition is a central result in the theory of Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the point and a formal slice. Recently, conical symplectic resolutions, and more generally, Poisson cones, have been very actively studied in representation theory and algebraic geometry. This motivates asking for a C*-equivariant version of the Darboux-Weinstein decomposition. In this paper, we develop such a theory, prove basic results on their existence and uniqueness, study examples (quotient singularities and hypertoric varieties), and applications to noncommutative algebra (their quantization). We also pose some natural questions on existence and quantization of C*-actions on slices to conical symplectic leaves.