$\mathbb{A}^1$-equivalence of zero cycles on surfaces

TL;DR

This paper investigates $ ext{A}^1$-equivalence classes of zero cycles on open complex algebraic surfaces, proving a logarithmic version of Mumford's theorem and confirming log Bloch's conjecture for certain surfaces.

Contribution

It introduces a logarithmic framework for zero cycles and proves key conjectures for quasiprojective surfaces with negative log Kodaira dimension.

Findings

Logarithmic Mumford's theorem established.

Log Bloch's conjecture verified for surfaces with log Kodaira dimension -∞.

Advances understanding of zero cycles in open algebraic surfaces.

Abstract

In this paper, we study $\mathbb{A}^1$-equivalence classes of zero cycles on open complex algebraic surfaces. We prove the logarithmic version of Mumford's theorem on zero cycles and prove that log Bloch's conjecture holds for quasiprojective surfaces with log Kodaira dimension $-\infty$.