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Let $A$ be a smooth affine variety of dimension $n$. Are there any facts known on $K_{\ast}(A)$ and its $\gamma$-filtration which do not hold for $K_*(V)$ for an arbitrary smooth $V$ (of the same dimension)?

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I suppose you mean geometric K-theory? On the other hand, are you working over $\mathbb{R}$ or $\mathbb{C}$? If the former, Nash has proved that every connected compact smooth manifold is diffeomorphic to a connected component of a real algebraic variety, so refinements in the algebraic case would surprise me. But if you mean complex affine varieties... !? – some guy on the street Sep 9 2011 at 17:16
I meant algebraic $K$-theory. Actually, I need $K$-theory with compact support.:) – Mikhail Bondarko Sep 9 2011 at 19:08

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