Examples of non-algebraic classes in the Brown-Peterson tower

For every $n\ge 0$, we construct classes in the Brown-Peterson cohomology $BP⟨n⟩$ of smooth projective complex algebraic varieties which are not in the image of the cycle map from the corresponding motivic Brown-Peterson cohomology. This generalizes the examples of Atiyah and Hirzebruch to all finite levels in the Brown-Peterson tower.