On a multiplicative version of Mumford's theorem

TL;DR

This paper extends Mumford's theorem to higher codimension Chow groups under the assumption of the Lefschetz standard conjecture, linking algebraic cycles to cohomological decompositions.

Contribution

It generalizes a known decomposition result from 0-cycles to arbitrary codimension Chow groups assuming the Lefschetz standard conjecture.

Findings

Proves a decomposition theorem for Chow groups of arbitrary codimension.

Establishes a connection between Chow group decompositions and cohomology under the conjecture.

Extends Mumford's theorem to a broader class of algebraic cycles.

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 a similar statement for Chow groups of arbitrary codimension, provided the variety satisfies the Lefschetz standard conjecture.