Hamiltonian reductions as affine closures of cotangent bundles
Baohua Fu, Jie Liu
TL;DR
The paper proves that for an irreducible non-singular affine $G$-variety $Y$ with a $2$-large action, the Hamiltonian reduction $T^*Y/\!/G$ is a symplectic variety with terminal singularities and equals the affine closure of $T^*(Y/\!/G)_{\mathrm{reg}}$, thereby connecting cotangent-bundle geometry to Hamiltonian reduction. It develops a Hamiltonian slice framework to analyze isotropy-stratifications of reductions and studies factoriality and (F)PIG conditions, deriving criteria under which $T^*Y/\!/G$ is factorial or $\mathbb{Q}$-factorial and when it admits no symplectic resolution. The paper further applies these results to differential operators on orbit spaces, giving a short proof of Schwarz's graded surjectivity theorem and establishing the surjectivity of the symbol map for $Z=Y/\!/G$. Together, these results unify two robust constructions of symplectic varieties, provide precise factoriality criteria, and yield concrete consequences for $D$-modules and algebraic symplectic geometry of orbit spaces.
Abstract
For an irreducible non-singular affine $G$-variety $Y$ whose action is $2$-large, we prove that the Hamiltonian reduction $T^*Y/\!\!/\!\!/G$ is a symplectic variety with terminal singularities, isomorphic to the affine closure of $T^*Z_{\text{reg}}$ for $Z:=Y/\!/G$. As applications, we provide a short proof of G. Schwarz's theorem on the graded surjectivity of the push-forward map $\mathcal{D}(Y)^G\rightarrow \mathcal{D}(Z)$, and we establish the surjectivity of the symbol map on $Z$.