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)?

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 streetSep 9 '11 at 17:16

I meant algebraic $K$-theory. Actually, I need $K$-theory with compact support.:)
–
Mikhail BondarkoSep 9 '11 at 19:08