Homology, Homotopy and Applications

Volume 8 (2006)

Number 1

The operator dbar in holomorphic $K$-theory

Pages: 187 – 210

DOI: https://dx.doi.org/10.4310/HHA.2006.v8.n1.a6


Dana Powell Roland (Mathematics Department, Merrimack College, North Andover, Massachusetts, U.S.A.)


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

19L47, 55P91

Published 1 January 2006