Curious Kernels for V.G.I.T.

Robert Vandermolen – University of South Carolina

The study of Derived Categories, first introduced by Grothendieck and his student Verdier, has come a long a way in describing deep connections between algebraic geometry, commutative algebra, representation theory, number theory, symplectic geometry, and theoretical physics. Despite that a lot of work has been done with derived categories there are still many open questions and interesting conjectures. One example is from the mid 90’s and early 2000’s where Bondal, Orlov and (independently) Kawamata conjectured that if two smooth complex varieties are related by a flop then they should be derived equivalent. It has been shown by Reid and others that flops can be equivalently explained by wall crossings from variations of geometric invariant theory. Using this approach Ballard, Diemer, and Favero have suggested a Kernel as a candidate to realize this conjectured derived equivalence. In recent joint work with Ballard, Chidambaram, Favero, and McFaddin, we have shown that a generalization of this kernel realizes the derived equivalence of the Grassmann Flop, which was first introduced by Donovan and Segal. In this talk we will examine some of the curious algebraic and geometric properties of a brand new generalization which describes these previously studied kernels and may describe many more wall crossings.