Bloch's Conjecture and Chow Motives

TL;DR

This paper proves Bloch's conjecture for certain complex surfaces by linking it to properties of homologically trivial correspondences, showing the conjecture holds if and only if these correspondences vanish, with implications for the structure of the Chow motive.

Contribution

It establishes an equivalence between Bloch's conjecture and the vanishing of homologically trivial idempotent correspondences for complex surfaces with no nontrivial 2-forms.

Findings

Bloch's conjecture holds if and only if homologically trivial idempotents vanish.

The cube of the ideal of homologically trivial correspondences is zero under these conditions.

The results apply to surfaces not of general type.

Abstract

Let X be a complex surface with no nontrivial 2-forms. Then we show that Bloch's conjecture is true (i.e. the Albanese map in this case is injective) if and only if any homologically trivial idempotent in the ring of correspondences vanishes. Furthermore the cube of the ideal of homologically trivial correspondences is zero if these equivalent conditions are satisfied (e.g. if X is not of general type).