Hamiltonian reductions as affine closures of cotangent bundles

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.

💡 Research Summary

The paper studies Hamiltonian reductions of cotangent bundles of affine varieties equipped with a reductive group action. Let (Y) be an irreducible smooth affine (G)-variety whose action is “2‑large” (a condition satisfied by almost all representations). The cotangent bundle (T^{*}Y) carries a natural Hamiltonian (G)-action with moment map (\mu). The authors consider the Hamiltonian reduction \