Let $X$ be an $n$-dimensional variety over a field $k$ of characteristic zero, regular in codimension 1 with singular locus $Z$. In this paper we study the negative $K$-theory of $X$, showing that when $Z$ is sufficiently nice, $K_{ 1-n}(X)$ is an extension of $KH_{ 1-n}(X)$ by a finite dimensional vector space, which we compute explicitly. We also show that $KH_{ 1-n}(X)$ almost has a geometric structure. Specifically, we give an explicit 1-motive $[L \rightarrow G]$ and a map $G(k) \rightarrow KH_{ 1-n}(X)$ whose kernel and cokernel are finitely generated abelian groups.
Categories: math.KT math.AG