The geometric McKay correspondence for quantum plane invariants

Sue Sierra (University of Edinburgh)

Wednesday 10th December 16:00-17:00
Maths 311B

Abstract

Let k be a field, q a nonzero scalar, and let A be the quantum plane k_q[x,y] , which satisfies the relation xy=qyx.. Let n \geq 2,  let the cyclic group  C_n act on A  with respective weights 1 and -1,  and consider the ring of invariants  $A^{C_n}$ and the associated category $\mathcal X = \operatorname{mod}-A^{C_n}$. The category $\mathcal X$ is singular, that is $A^{C_n}$ has infinite global dimension. It is well-known that the smash product  $A \hash C_n$ can be considered as a resolution of $\mathcal X$, and taking q -> 1  we obtain the standard  "noncommutative crepant resolution" of the cyclic quotient singularity $X = \mathbb A^2/C_n$. We can thus view $A \hash C_n$ as a  “noncommutative algebraic   resolution” of  $\mathcal X$. Of course $X$ also has a geometric (crepant) resolution of singularities, and our work is motivated by the question of whether $\mathcal X$  has a “noncommutative geometric resolution”.

We answer this question positively: we deform the the minimal (geometric) resolution  $\widetilde{X}$ of  $X$ to obtain a category $\widetilde{\mathcal X}$ which we show is derived equivalent to $A \hash C_n$, together with a “morphism” $g: \widetilde{\mathcal X} \to \mathcal X$. We obtain  $\widetilde{\mathcal X}$ in two ways: from an $ \mathbb N$-graded ring, and as a noncommutative toric variety. We further study the exceptional locus of $g$ and show that its intersection theory gives an  $A_{n-1}$ graph, giving a noncommutative version of the geometric McKay correspondence. 

We will begin the talk by reviewing the commutative McKay correspondence for Kleinian singularities and will assume minimal algebraic geometry.  

This is joint work with Simon Crawford.

Add to your calendar

Download event information as iCalendar file (only this event)