Abstract
For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/\Gamma$ as a Nakajima quiver variety for the framed McKay quiver of $\Gamma$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of stability parameter.
| Original language | English |
|---|---|
| Pages (from-to) | 680-704 |
| Number of pages | 24 |
| Journal | Algebraic Geometry |
| Volume | 8 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - 30 Nov 2021 |
Acknowledgements
We thank Gwyn Bellamy, Ben Davison and Hiraku Nakajima for helpful discussions. We are also grateful to the anonymous referee whose helpful comments enabled us to bypass a case by-case analysis of ADE diagrams in the situation of primary interest where our quiver variety admits a morphism to Hilb[n] C2 , and who identified a gap in an earlier, more complicated construction of our ne moduli spaces, leading to a substantial simpli cation.Funding
While working on this project, S.G. was supported by an Aker Scholarship, whereas A.Gy. and B.Sz. were supported by EPSRC grant EP/R045038/1.
| Funders | Funder number |
|---|---|
| Engineering and Physical Sciences Research Council | EP/R045038/1 |
Keywords
- math.AG
- math.RT
