Rational points versus zero-cycles of degree one over arbitrary fields

Determine whether, for every projective homogeneous variety over an arbitrary field, the existence of a zero-cycle of degree one is equivalent to the existence of a rational point.

Background

The paper’s Tate trace analysis assumes that every projective homogeneous variety has a rational point if and only if it has a zero-cycle of degree one. This equivalence is known in several important cases, including spinor and orthogonal groups of quadratic forms, special and projective linear groups of central simple algebras, and generally over number fields. Establishing whether the equivalence holds over arbitrary fields would clarify the scope of the paper’s method for recovering integral Tate traces from Bialynicki–Birula–Brosnan motivic decompositions.