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Mikhail Bondarko
Mikhail Bondarko
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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?

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?

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?

Source Link
Mikhail Bondarko
Mikhail Bondarko
  • 16.9k
  • 4
  • 34
  • 97

K-theory of coherent sheaves on complex manifolds: references and gamma-filtration?

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?