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

TL;DR

This paper proves that certain conical symplectic varieties with torus actions are isomorphic to toric hyperk"ahler varieties, confirming a specific conjecture about their structure.

Contribution

It establishes an isomorphism between a class of symplectic varieties with torus actions and toric hyperk"ahler varieties, confirming a key conjecture.

Findings

Affirmative answer to Question 5.10 from previous work.

Proves the isomorphism for varieties with specific symplectic and torus action properties.

Shows the toric hyperk"ahler variety is characterized by the given conditions.

Abstract

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 $\mathbf{C}^*$-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.