𝐾-theory of cones of smooth varieties

Abstract

Let R R be the homogeneous coordinate ring of a smooth projective variety X X over a field k k of characteristic 0. We calculate the K K -theory of R R in terms of the geometry of the projective embedding of X X . In particular, if X X is a curve, then we calculate K 0 ( R ) K_0(R) and K 1 ( R ) K_1(R) , and prove that K − 1 ( R ) = ⨁ H 1 ( C , O ( n ) ) K_{-1}(R)=\bigoplus H^1(C,\mathcal {O}(n)) . The formula for K 0 ( R ) K_0(R) involves the Zariski cohomology of twisted Kähler differentials on the variety.