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

TL;DR

This paper proves the log Bloch conjecture for certain open smooth complex surfaces by leveraging recent advances in mixed motives, confirming the conjecture for all $Q$-homology planes and some non-log general type surfaces.

Contribution

It establishes the log Bloch conjecture for open surfaces based on the validity of the classical Bloch conjecture for their compactifications, using mixed motives theory.

Findings

Log Bloch conjecture holds for all $Q$-homology planes.

Validates the conjecture for open surfaces not of log general type.

Connects the conjecture's validity to the classical Bloch conjecture via mixed motives.

Abstract

Using recent developments in the theory of mixed motives, we prove that the log Bloch conjecture holds for an open smooth complex surface if the Bloch conjecture holds for its compactification. This verifies the log Bloch conjecture for all $\mathbb{Q}$-homology planes and for open smooth surfaces which are not of log general type.