Algebraic cycles and Todorov surfaces

TL;DR

This paper verifies Voisin's conjecture for specific Todorov surfaces with geometric genus one, demonstrating their finite-dimensional motives and advancing understanding of algebraic cycles in complex surfaces.

Contribution

It confirms Voisin's conjecture for a family of Todorov surfaces with particular invariants and shows these surfaces have finite-dimensional motives, linking conjectures in algebraic geometry.

Findings

Voisin's conjecture verified for Todorov surfaces with $K^2=2$ and fundamental group $ ext{Z}/2 ext{Z}$

Proved certain Todorov surfaces possess finite-dimensional motives

Advances understanding of algebraic cycles and motives in complex 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.