Working over $\mathbb{C}$, the Barth-Larsen results tell us a lot about the ordinary cohomology of non-singular varieties of small codimension in projective space. For example if $X \subseteq \mathbb{P^n}$ has codimension $l$ then the restriction map: $$ H^i(\mathbb{P^n}, \mathbb{Z}) \to H^i(X, \mathbb{Z}) $$ is an isomporphism for $i \le n-2l$.
My question is
What can be said about the topological $K$-theory of such an $X$?
Of course the $K$-theory and ordinary cohomology of $X$ are closely related: if $X$ is torsion-free then they are rationally isomorphic. Better still, if $$K(X) = K_{(0)}(X) \supseteq K_{(1)}(X) \supseteq K_{(2)}(X) \supseteq \cdots $$ is the topological filtration of $K(X)$, then the associated graded ring of this filtration is the even part of the integral ordinary cohomology. So technically, we can use this to translate the Barth-Larsen result above into a statement in the language of $K$-theory, but I was hoping for a statement a bit more "native" to $K$-theory. Is anyone aware of such a statement?
My motivation for this question relates to Hartshorne's complete intersection conjecture, stated in his influential 1974 paper [1]. Much of Hartshorne's motivation for this conjecture stems from the fact that the topology of non-singular varieties of small codimension is identical to that of complete intersections, at least as measured by the fundamental group and ordinary cohomology. So it seems natural to wonder about what $K$-theory has to say about this (partly as I have some vague reasons for suspecting it might have something to say).
[1] Robin Hartshorne, Varieties of small codimension in projective space. Bull. Amer. Math. Soc., 80:1017—1032, 1974.
