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 an algebraic torus of dimension $r \geq 1$, which is canonical. In particular, it admits a 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.