Remarks on approximate decompositions of the diagonal

TL;DR

This paper explores the relationship between different essential CH_0-dimensions of complex varieties with trivial 0-cycle groups, providing conditions under which these dimensions coincide.

Contribution

It establishes necessary and sufficient conditions for varieties with trivial 0-cycle groups and essential CH_0-dimension ≤ 2 to actually have essential CH_0-dimension 0.

Findings

Identifies conditions linking essential CH_0-dimension 2 and 0

Provides criteria for varieties to have trivial essential CH_0-dimension

Advances understanding of decompositions of the diagonal in algebraic geometry

Abstract

In this paper, we investigate, for varieties over $\mathbb C$ with trivial group of $0$-cycles, the gap between essential $\mathrm{CH}_0$-dimension $2$ and essential $\mathrm{CH}_0$-dimension $0$. In particular, we present sufficient (and necessary) conditions for a variety with trivial group of $0$-cycles and essential $\mathrm{CH}_0$-dimension $\leq 2$ to have, in fact, essential $\mathrm{CH}_0$-dimension $0$.