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For a complex manifold $X$ one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain $K$-theory (I do not know whether the $K$-groups given by the "...

Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. ...