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.
Homology, Homotopy and Applications, Vol. 8 (2006), No. 1, pp.187-210.
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