The operator dbar in holomorphic K-theory

Abstract

The holomorphic (or semi-topological) K-theory of a smooth projective variety sits between the algebraic K-theory of the variety and the topological K-theory of the underlying topological space. We describe how to define a family of dbar operators on holomorphic K-theory in a manner analogous to Atiyah's construction of a family of elliptic operators in topological K-theory. In the process, we prove a result akin to Bott periodicity for holomorphic mapping spaces. These results first appeared in the author's Stanford University Ph.D. thesis under the direction of Ralph Cohen.