2005-10-14

Farkhod Eshmatov, Cornell University

DG models and Nakajima quiver varieties

Associated to each (extended) Dynkin diagram there is a family of noncommutative algebras which deforms the coordinate ring of the Kleinian singularity corresponding to that diagram. These algebras were defined by W. Crawley-Boevey and M. Holland in 1998, who also suggested a conjectural decription of projective modules over these algebras in terms of Nakajima quiver varieties. In 2002, Baranovski, Ginzburg and Kuznetsov proved the Crawley-Boevey-Holland conjecture using the methods of noncommutative projective geometry. In this talk we will present a refined (G-equivariant) version of this conjecture and give a new proof of the Baranovski-Ginzburg-Kuznetsov result using some simple ideas from homotopical algebra. In the case of cyclic groups our proof leads to a completely explicit description of ideals of the Crawley-Boevey-Holland algebras.