2001/03/12 by Vladimir Baranovsky, Victor Ginzburg, Baranovsky, Vladimir +3
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #math.AG #math.QA
paper · pdf · doi:10.48550/arxiv.math/0103068
36pp, Crawley-Boevey & Holland conjecture generalized and proved
arxiv created 2001/07/06 · arxiv updated 2009/11/30
To any finite group G in SL2(C), and each `t' in the center of the group algebra of G, we associate a category, Coht. 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 Coht should be thought of as the category of coherent sheaves on a `noncommutative projective 2-space', equipped with a framing at the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coht and the Nakajima quiver variety arising from G via the McKay correspodence. We apply the above isomorphism to deduce generalized Crawley-Boevey & 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 G=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-Holland and Berest-Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on the noncommutative projective 2-space. It is totally different from the one used by Berest-Wilson, involving τ-functions.