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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 "standard" constructions are isomorphic in this setting) and $\Lambda$-operations for them.

My question is: did anyone study these $K$-groups and $\gamma$-filtration for them; are there any methods to compute them (at least, for some examples) if $X$ is compact but not algebraic? Is there a significant distinction from the properties of algebraic $K$-theory of smooth complex varieties?

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    $\begingroup$ Have you looked at Exposés IV and V in SGA 6? The whole theory is made in the general framework of a ringed topos. This should give you at least all the formal part, though it probably doesn't say much about computing concrete examples. $\endgroup$
    – abx
    Mar 6, 2015 at 9:03
  • $\begingroup$ I recently asked the same question to some algebraic geometer and he told me that there is natural map from this K-theory to the usual one, that this map has a huge kernel (continuous families in the kernel) and that surjectivity would be a consequence of the Hodge conjecture. (This is just what I got from hearsay and possibly I got it wrong.) $\endgroup$
    – ThiKu
    Mar 6, 2015 at 17:16

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