Canonical torus action on symplectic singularities

Abstract

We show that any symplectic singularity lying on a smoothable projective symplectic variety locally admits a good action of $(\mathbb{C}^*)^r$, which is canonical. Under mild assumptions, we actually prove such singularity germ is the cone vertex over a contact orbifold with weak K\"ahler-Einstein metric, forcing $r=1$. In particular, it admits a (canonical) good $\mathbb{C}^*$-action, which also extends to (canonical) actions of $\mathbb{H}^*\supset SU(2)$. These settle Kaledin's conjecture conditionally but in a substantially stronger form by establishing the canonicity, the extensibility of the action, for instance. Our key idea is to use the Donaldson-Sun theory on local K\"ahler metrics in complex differential geometry to connect with the theory of Poisson deformations of symplectic varieties. For general symplectic singularities, we prove the same assertions, assuming that the Donaldson-Sun theory extends to such singularities along with suitable singular (hyper)K\"ahler metrics. Conversely, our results can also be used to study the local behavior of such metrics around the germ.