Zero cycles with modulus and zero cycles on singular varieties

TL;DR

This paper explores the structure of zero cycles with modulus on smooth varieties, establishing a decomposition of the cohomological Chow group, constructing an Albanese variety with modulus, and proving torsion properties and injectivity results.

Contribution

It introduces a canonical decomposition of 0-cycle Chow groups with modulus, constructs an Albanese variety with modulus, and proves the Roitman torsion theorem for these cycles.

Findings

Decomposition of cohomological Chow group of 0-cycles on the double of X along D.

Construction of an Albanese variety with modulus as a universal regular quotient.

Proof that CH_0(X|D) is torsion-free and injects into K_0(X,D) for affine X.

Abstract

Given a smooth variety $X$ and an effective Cartier divisor $D \subset X$, we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles ${\rm CH}_0(X)$ and the Chow group of 0-cycles with modulus ${\rm CH}_0(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of ${\rm CH}_0(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that ${\rm CH}_0(X|D)$ is torsion-free and there is an injective cycle class map ${\rm CH}_0(X|D) \hookrightarrow K_0(X,D)$ if $X$ is affine. For a smooth affine surface $X$, this is strengthened to show that $K_0(X,D)$ is an extension of ${\rm CH}_1(X|D)$ by ${\rm CH}_0(X|D)$.