Algebraic cycles and Todorov surfaces

Abstract

Motivated by the Bloch–Beilinson conjectures, Voisin has formulated a conjecture about $0$-cycles on self-products of surfaces of geometric genus one. We verify Voisin’s conjecture for the family of Todorov surfaces with $K^2=2$ and fundamental group $\mathbb{Z}/2\mathbb{Z}$. As a by-product, we prove that certain Todorov surfaces have finite-dimensional motive.