On a multiplicative version of Bloch's conjecture

TL;DR

This paper explores a partial converse to a known decomposition theorem, establishing that under certain conditions, the decomposition of the top-degree cohomology implies a Chow group decomposition for specific varieties.

Contribution

It proves a weak version of the converse of a decomposition theorem for varieties of dimension up to 5 with finite-dimensional motive and satisfying the Lefschetz standard conjecture.

Findings

The weak converse holds for certain varieties of dimension ≤ 5.

The proof uses Vial's refined Chow-Kunneth decomposition.

Results connect Chow group decompositions with cohomology decompositions.

Abstract

A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove that (a weak version of) the converse holds for varieties of dimension at most 5 that have finite-dimensional motive and satisfy the Lefschetz standard conjecture. The proof is based on Vial's construction of a refined Chow-Kunneth decomposition for these varieties.