Relative critical loci, quiver moduli, and new lagrangian subvarieties

The preprojective algebra of a quiver naturally appears when computing the cotangent to the quiver moduli, via the moment map. When considering the derived setting, it is replaced by its differential graded (dg) variant, introduced by Ginzburg. This construction can be generalized using potentials, so that one retrieves critical loci when considering moduli of perfect modules. Our idea is to consider some relative, or constrained critical loci, deformations of the above, and study Calabi--Yau structures on the underlying relative versions of Ginzburg's dg algebras. It yields for instance some new lagrangian subvarieties of the Hilbert schemes of points on the plane.

This report is a joint work with Damien Calaque and Sarah Scherotzke.