Canonical torus action on symplectic singularities
Yoshinori Namikawa, Yuji Odaka
TL;DR
The work proves that any symplectic singularity arising from a smoothable polarized projective symplectic variety admits a canonical local torus action, realized through a conical affine symplectic cone with a hyperKähler cone metric and a canonical $\mathbb{R}_{>0}$-rescaling. The authors bridge Donaldson–Sun theory of local metric tangent cones with Poisson deformation theory to construct scale-up deformations $X\rightsquigarrow W\rightsquigarrow C$, extend holomorphic forms, and deduce a germ-level isomorphism $(X,x)\cong (C,0)$ via formal completion and Artin approximation. A key technical outcome is establishing $W=C$ as affine conical symplectic varieties under mild hypotheses, enabling a canonical $T=(\mathbb{G}_m)^r$-action on the germ with $r\ge1$, and, in favorable scenarios, $r=1$. The results conditionally strengthen Kaledin’s conjecture and connect with hyperKähler reduction theory, including potential applications to Nakajima quiver varieties and HK quotients. The approach yields a metric-geometry–algebraic framework that illuminates the local structure of singular symplectic spaces and their deformation theory.
Abstract
We show that any symplectic singularity lying on a smoothable projective symplectic variety locally admits a good action of an algebraic torus of dimension $r \geq 1$, which is canonical. Under mild assumptions, $r=1$ is also confirmed. In particular, it admits (canonical) good $\mathbb{C}^*$-action. This proves Kaledin's conjecture conditionally but in a substantially stronger form. Our key idea is to use Donaldson-Sun theory on local Kahler metrics in complex differential geometry to connect with the theory of Poisson deformations of symplectic varieties. For general symplectic singularities, we prove the same assertion -- namely, the existence of a canonical (local) torus action -- assuming that the Donaldson-Sun theory extends to such singularities along with suitable singular (hyper)Kahler metrics. Conversely, our results can be also used to study local behaviour of such metrics around the germ. For instance, we show that such singular hyperKahler metric around isolated singularity is close to a metric cone in a polynomial order, and satisfies $r=1$ i.e., has a good canonical (local) $\mathbb{C}^*$-action, as the complexification of the cone metric rescaling. Our theory also fits well to singularities on many hyperKahler reductions.