Bloch-type conjectures and an example of a threefold of general type

TL;DR

This paper explores the relationship between Hodge structures and Chow groups through Bloch-type conjectures, extending Voisin's method to demonstrate a threefold of general type satisfying these conjectures.

Contribution

It introduces an extension of Voisin's method to verify Bloch's generalized conjecture for a specific threefold of general type.

Findings

Demonstrated a threefold of general type with small Chow groups

Extended Voisin's method to broader classes of varieties

Provided evidence supporting the link between Hodge structures and Chow groups

Abstract

The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that "small Hodge diamonds" go with "small Chow groups". Voisin's method (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A threefold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture.