Rational Lawson–Massey brackets with nonzero quotient class

Determine whether a smooth complex projective variety admits a defined triple Lawson–Massey product with rational coefficients whose quotient class is nonzero.

Background

The paper constructs integral triple Lawson–Massey brackets on smooth complex projective varieties whose quotient classes are nonzero but whose images under the Lawson cycle map vanish in the corresponding singular-homology Massey quotient. These examples are torsion phenomena: the selected values have exact order two, so they disappear after rationalization.

The paper observes that every defined rational singular Massey product on a smooth complex projective variety vanishes by Kähler formality. Consequently, a rational Lawson–Massey bracket with a nonzero quotient class would provide a secondary Lawson invariant invisible to rational singular homology. The constructions in the paper do not resolve whether such a rational example exists.