  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.
