Homology, Homotopy and Applications
Volume 8 (2006)
The operator dbar in holomorphic $K$-theory
Pages: 187 – 210
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.
holomorphic $K$-theory, dbar operator, Bott periodicity
2010 Mathematics Subject Classification