Hamiltonian reductions as affine closures of cotangent bundles

TL;DR

This paper demonstrates that Hamiltonian reductions of certain affine G-varieties are affine closures of cotangent bundles, with implications for symplectic singularities, resolutions, and algebraic differential operators.

Contribution

It establishes the structure of Hamiltonian reductions as affine closures of cotangent bundles for 2-large actions, and provides conditions for the non-existence of symplectic resolutions.

Findings

Hamiltonian reductions are affine closures of cotangent bundles.

Conditions for the non-existence of symplectic resolutions are given.

Applications include proofs of theorems on differential operators and symplectic structures.

Abstract

Let $Y$ be an irreducible non-singular affine $G$-variety with a $2$-large action. We show that the Hamiltonian reduction $T^*Y/\!\!/\!\!/G$ is a symplectic variety with terminal singularities, isomorphic to the affine closure of $T^*Z_{\text{reg}}$ where $Z:=Y/\!/G$. Furthermore, we provide sufficient conditions for the non-existence of a symplectic resolution for such varieties. These results yield three main applications: (i) providing a short proof of G. Schwarz's theorem on the graded surjectivity of the push-forward map $\mathcal{D}(Y)^G \to \mathcal{D}(Z)$; (ii) establishing the surjectivity of the symbol map on $Z$; and (iii) confirming the non-linear analog of a conjecture of Kaledin--Lehn--Sorger for $2$-large actions.