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Let $X$ be a smooth complete intersection in $\mathbb{P}^n$. I am searching for literature on the K-theory for $X$? I guess the K-theory is known...

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Do you mean topological $K$-theory, or the algebraic one? The first one is known. The algebraic one is essentially equivalent to the Chow ring, which is far from being understood. – abx Dec 21 '13 at 9:19
I mean the algebraic one. So what is known for $K_0(X)$? – Aleksa Dec 21 '13 at 9:20
Essentially nothing. Please ask a precise question. – abx Dec 21 '13 at 9:36
For example under what kind of assumptions on $X$ $K_0(X)$ is finitely generated ? Maybe for complete intersections of genus zero... – Aleksa Dec 21 '13 at 9:38
What is genus 0?? This happens for degree 1 and 2 and for cubic surfaces. That's all. – abx Dec 21 '13 at 9:52

As suggested by Daniel Loughran, I will try to answer the question whether $K_0(X)$ is finitely generated. First, the Chern character from $K_0(X)$ to the Chow ring $CH(X)$ is an isomorphism modulo torsion: see for instance Fulton, Intersection Theory, Cor. 18.3.2. Thus we can transfer the question to $CH(X)$, and use this question. So, at least conjecturally, for a complete intersection $X$ the group $K_0(X)$ is finitely generated iff $h^{p,q}(X)=0$ for $p\neq q$. This holds if and only if $X$ is a quadric, a cubic surface, or an even-dimensional intersection of 2 quadrics (see M. Rapoport, Complément à l'article de P. Deligne "La conjecture de Weil pour les surfaces K3", Inv. math. 15 (1972), 227-236).

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