K-theory of cones of smooth varieties

Abstract

Let R be the homogeneous coordinate ring of a smooth projective variety X over a field k of characteristic 0. We calculate the K-theory of R in terms of the geometry of the projective embedding of X. In particular, if X is a curve then we calculate K0(R) and K1(R), and prove that K−1(R) = ⊕H1 (C, O(n)). The formula for K0(R) involves the Zariski cohomology of twisted K¨ahler differentials on the variety.