Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 099, 21 pages      arXiv:2106.10115      https://doi.org/10.3842/SIGMA.2021.099
Contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday

Quot Schemes for Kleinian Orbifolds

Alastair Craw a, Søren Gammelgaard b, Ádám Gyenge c and Balázs Szendrői b
a) Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
b) Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
c) Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, 1053, Budapest, Hungary

Received June 29, 2021, in final form November 03, 2021; Published online November 10, 2021; Theorem 1.1 corrected December 06, 2021

Abstract
For a finite subgroup $\Gamma\subset {\rm SL}(2,{\mathbb C})$, we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold $\big[{\mathbb C}^2\!/\Gamma\big]$. We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of $\Gamma$, taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our work [Algebr. Geom. 8 (2021), 680-704] on the Hilbert scheme of points on ${\mathbb C}^2/\Gamma$; we present arguments that completely bypass the ADE classification.

Key words: Quot scheme; quiver variety; Kleinian orbifold; preprojective algebra; cornering.

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