Francesco Sala, Kavli Institute for the Physics and the Mathematics of the Universe
“Moduli spaces and stacks of sheaves on resolutions of toric singularities”
Moduli spaces of torsion free sheaves on surfaces play a preeminent role in geometric rep- resentation theory. For example, thanks to the works of Nakajima, Grojnowski, Vasserot, Schiffmann, etc, the (equivariant) cohomology of Hilbert schemes of points on the complex plane C2 admits an action of the Heisenberg algebra (and also of certain bigger algebras, such as the Yangians). More recently, algebraic stacks of torsion sheaves on surfaces have also started to play an important role in this area: for example, Schiffmann and Vasserot defined an associative algebra structure ( la Hall) on the (equivariant) K-theory/cohomology of the algebraic stack of zero-dimensional sheaves on C2, that acts on the (equivariant) K- theory/cohomology of Hilbert schemes of points on the same surface (action that generalises the one defined by Nakajima, etc.). Thus one can state the following paradigm: there exist algebras associated with the K-theory/cohomology of algebraic stacks of torsion sheaves on a surface, which act on the K-theory/cohomology of moduli spaces of torsion free sheaves on the same surface.
In the present talk, I will discuss this paradigm in the case where the surface is the minimal resolution of a toric singularity: this case is more interesting than the one on C2, because one can consider also torsion sheaves supported on the exceptional curves of the resolution.
MC 5413