On the Negative $K$-theory of Singular Varieties

Abstract

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 $K{H}_{1-n}(X)$ by a finite dimensional vector space, which we compute explicitly. We also show that $K{H}_{1-n}(X)$ almost has a geometric structure. Specifically, we give an explicit 1-motive $[L\to G]$ and a map $G(k)\to K{H}_{1-n}(X)$ whose kernel and cokernel are finitely generated abelian groups.

24 pages

References in corpus (2)

• A note on the dual complex associated to a resolution of singularities
• Dual graphs of exceptional divisors