Towards a characterization of toric hyperk\"{a}hler varieties among symplectic singularities II

This is a continuation of arXiv: 2408.03012. We answer affirmatively Question 5.10 posed in the previous article. More precisely, let $(X,\omega )$ be a conical symplectic variety of dimension $2n$ with $wt(\omega )=2$, which has a projective symplectic resolution. Assume that $X$ admits an effective Hamiltonian action of an $n$-dimensional algebraic torus $T^n$, compatible with the conical ${\mathbf{C}}^{\ast }$-action. Then we prove that there is a $T^n$-equivariant algebraic isomorphism $(X,\omega )\cong (Y(A,0),{\omega }_{Y(A,0)})$ for a toric hyperkahler variety $Y(A,0)$ with $A$ unimodular.