Bloch's conjecture for Catanese and Barlow surfaces

TL;DR

This paper proves that the zero-cycle Chow group of Catanese surfaces is trivial, extending the result to Barlow surfaces, and discusses conditional implications for low degree K3 surfaces under the variational Hodge conjecture.

Contribution

It establishes the triviality of the CH_0 group for Catanese and Barlow surfaces and explores conditional applications to K3 surfaces assuming the variational Hodge conjecture.

Findings

CH_0 group of Catanese surfaces is Z

CH_0 group of Barlow surfaces is Z

Conditional results for K3 surfaces under Hodge conjecture

Abstract

Catanese surfaces are regular surfaces of general type with $p_g=0$. They specialize to double covers of Barlow surfaces. We prove that the $CH_0$ group of a Catanese surface is equal to $\mathbb{Z}$, which implies the same result for the Barlow surfaces. We will also give a conditional application (more precisely, assuming the variational Hodge conjecture) of the same method to the Chow motive of low degree $K3$ surfaces.