On the Negative $K$-theory of Singular Varieties

TL;DR

This paper investigates the negative K-theory of singular varieties over characteristic zero fields, providing explicit computations and geometric structures for certain K-groups related to the singular locus.

Contribution

It explicitly computes the negative K-theory groups for varieties with controlled singularities and introduces a geometric 1-motive structure for these groups.

Findings

$K_{1-n}(X)$ is an extension of $KH_{1-n}(X)$ by a finite dimensional vector space.

$KH_{1-n}(X)$ admits an explicit 1-motive structure.

The kernel and cokernel of the map from $G(k)$ to $KH_{1-n}(X)$ are finitely generated abelian groups.

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 $KH_{1-n}(X)$ 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.