Hamiltonian reductions as affine closures of cotangent bundles

Abstract

Let $Y$ be an irreducible non-singular affine $G$-variety with a $2$-large action. We show that the Hamiltonian reduction $T^{\ast }Y///G$ is a symplectic variety with terminal singularities, isomorphic to the affine closure of $T^{\ast }{Z}_{\text{reg}}$ 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.

22 pages. Any comments are welcome. v2: add connection to a conjecture of Kaledin--Lehn--Sorger