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Quiver varieties and a noncommutative P²

Baranovsky, V. and Ginzburg, V. and Kuznetsov, A. (2002) Quiver varieties and a noncommutative P². Compositio Mathematica, 134 (3). pp. 283-318. ISSN 0010-437X. doi:10.1023/A:1020930501291.

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To any finite group Γ ⊂ SL₂(ℂ) and each element t in the center of the group algebra Of Γ we associate a category, Coh(ℙ²_(Γ, τ),ℙ¹). It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh(ℙ²_(Γ, τ),ℙ¹) should be thought of as the category of coherent sheaves on a ‘noncommutative projective space’, ℙ²_(Γ, τ), equipped with a framing at ℙ¹, the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh(ℙ²_(Γ, τ),ℙ¹) and the Nakajima quiver variety arising from G via the McKay correspondence. We apply the above isomorphism to deduce a generalization of the Crawley-Boevey and Holland conjecture, saying that the moduli space of ‘rank 1’ projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for Γ = {1}, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, A₁, by points of the Calogero– Moser space, due to Cannings and Holland and Berest and Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on ℙ²_(Γ, τ). It is totally different from the one used by Berest and Wilson, involving τ-functions.

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Additional Information:© 2002 Kluwer Academic Publishers. Received: 4 April 2001; accepted in final form: 20 August 2001. Published online by Cambridge University Press 01 Jun 2005.
Subject Keywords:noncommutative geometry; quiver varieties; McKay correspondence
Issue or Number:3
Classification Code:MSC 2000: 14D20 (14A22, 16S38)
Record Number:CaltechAUTHORS:BARcm02
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Official Citation:Baranovsky, V., Ginzburg, V., & Kuznetsov, A. (2002). Quiver Varieties and a Noncommutative P2. Compositio Mathematica, 134(3), 283-318. doi:10.1023/A:1020930501291
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2951
Deposited By: Tony Diaz
Deposited On:08 May 2006
Last Modified:08 Nov 2021 19:52

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