Noncommutative projective partial resolutions and quiver varieties

Let $\mathrm{\Gamma }\in {\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ be a finite subgroup. We introduce a class of projective noncommutative surfaces ${\mathbb{P}}_I^2$, indexed by a set of irreducible $\mathrm{\Gamma }$-representations. Extending the action of $\mathrm{\Gamma }$ from ${\mathbb{C}}^2$ to ${\mathbb{P}}^2$, we show that these surfaces generalise both $[{\mathbb{P}}^2\mathrm{/}\mathrm{\Gamma }]$ and ${\mathbb{P}}^2\mathrm{/}\mathrm{\Gamma }$. We prove that isomorphism classes of framed torsion-free sheaves on any ${\mathbb{P}}_I^2$ carry a canonical bijection to the closed points of appropriate Nakajima quiver varieties. In particular, we provide geometric interpretations for a class of Nakajima quiver varieties using noncommutative geometry. Our results partially generalise several previous results on such quiver varieties.