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. ## TalkDownload video. |

Yale 2013 >