Let X be an integral n-dimensional variety over a field k of characteristic zero, regular in codimension 1 and with singular locus Z. We establish a right exact sequence, coming from the Brown-Gersten spectral sequence, that computes K1-n(X) from KH1-n(X) and NK1-n(X). We then compute each of these pieces separately, and then analyze the map NK1-n(X) → K1-n(X).
We show that the KH1-n(X) contribution almost has a geometric structure. When k is algebraically closed, X is projective, and Z is either smooth over k or of codimension greater than 2, we prove that there is a 1-motive ML → G] over k, and a map α : G(k) → KH1-n(X) whose kernel and cokernel are finitely generated. Thus the k-points G(k) of the group scheme G approximates KH1-n(X) up to some finitely generated abelian groups. Furthermore, when n = 3, the sequence L(k) → G(k) → KH-2(X) is exact. In addition, M is computable, as under Deligne's equivalence between torsion-free 1-motives and torsion-free mixed Hodge structures of type {(0,0),(0,1),(1,0),(1,1)} such that Gr1WH polarizable, the free complex 1-motive (M ×k ©)fr is the 1-motive that corresponds to the unique largest such H coming from the weight 2 part W2Hn(X(©), Z) of the nth cohomology group Hn(X(©), Z).
When X is not projective, the result still holds, except that the 1-motive M comes from W2Hn(Y(©), Z) where Y is an algebraic compactification of X. Furthermore, the non-lattice parts of the M we get, and hence the map α, are independent of the choice of compactification.
For the NK1-n(X) contribution, when Z is an isolated singularity, we show that K1-n(X) is an extension of KH1-n(X) by the cdh-cohomology group Hn>/italic>-1cdh(U;O), where U is any open affine neighborhood of Z. Furthermore, Hn>/italic>-1cdh(U;O) is a finite-dimensional k-vector space, whose dimension is the Du Bois invariant b0,n-1 of the isolated singularity Z.
All in all, we have a full computation of K-2(X) when X is three-dimensional over an algebraically closed field and has only isolated singularities.
