Counting curves in determinantal varieties and a connection to quiver mutations

Suppose X is a smooth projective variety, E and F are vector bundles on X, and M: E —> F is a map of vector bundles.  More concretely, M defines a family of matrices {M_x}, parametrized by the points x of the variety X.  For a positive integer k, we can define the kth determinantal variety of M to be the locus of points x in X for which the linear map M_x has rank at most k.  Such varieties give some of the simplest examples of subvarieties of X which are not complete intersections.  

Determinantal varieties are almost always singular, however there are two natural desingularizations, called the PAX and PAXYmodels, defined using basic concepts from linear algebra.  It is natural to ask what the relationship is between these two resolutions: How do their cohomology rings compare?  their derived categories?  etc… 

In this talk I will describe a beautiful correspondence, conjectured by physicists, between the number of curves in each of the two resolutions.  I will give a sketch of the proof and connect this correspondence to mutations of quiver varieties.  This is based on joint work with Nathan Priddis and Yaoxiong Wen.