Dave Anderson: Operational K-theory

Given any covariant "homology" theory on algebraic varieties, the bivariant machinery of Fulton and MacPherson constructs a "operational" bivariant theory, which formally includes a contravariant "cohomology" component. Taking the homology theory to be Chow homology, this is how the Chow cohomology of singular varieties is defined. In this talk, I will describe joint work with Sam Payne in which we study the operational K-theory associated to the K-homology of coherent sheaves. Remarkably, despite its formal definition, the operational theory has many properties which make it easier to understand than the K-theory of vector bundles or perfect complexes. This is illustrated most vividly by singular toric varieties, where relatively little is known about K-theory of vector bundles, while the operational equivariant K-theory has a simple description in terms of the fan, directly generalizing the smooth case.