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

TL;DR

This paper characterizes toric hyperk"ahler varieties among symplectic singularities by showing that certain conical symplectic varieties with torus actions are isomorphic to these hyperk"ahler varieties, under specific conditions.

Contribution

It proves that conical symplectic varieties with torus actions and projective symplectic resolutions are essentially toric hyperk"ahler varieties with unimodular matrices.

Findings

Such varieties are isomorphic to toric hyperk"ahler varieties.

The isomorphism respects the symplectic form and moment maps.

Centers of the varieties are mapped to each other.

Abstract

Let $(X, \omega)$ be a conical symplectic variety of dimension $2n$ 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. A typical example of $X$ is a toric hyperkahler variety $Y(A,0)$. In this article, we prove that this property characterizes $Y(A,0)$ with $A$ unimodular. More precisely, if $(X, \omega)$ is such a conical symplectic variety, then there is a $T^n$-equivariant (complex analytic) isomorphism $\varphi: (X, \omega) \to (Y(A,0), \omega_{Y(A,0)})$ under which both moment maps are identified. Moreover $\varphi$ sends the center $0_X$ of $X$ to the center $0_{Y(A,0)}$ of $Y(A,0)$.