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

TL;DR

This paper constructs specific classes in the Brown-Peterson cohomology of smooth projective complex varieties that are not detected by the cycle map, extending previous examples to all levels of the Brown-Peterson tower.

Contribution

It provides the first examples of non-algebraic classes in all finite levels of the Brown-Peterson tower, generalizing Atiyah-Hirzebruch examples.

Findings

Constructed classes in $BP\langle n \rangle$ not in the cycle map image

Extended Atiyah-Hirzebruch examples to all levels of the tower

Demonstrated non-algebraic classes in complex algebraic varieties

Abstract

For every $n\ge 0$, we construct classes in the Brown-Peterson cohomology $BP\langle n \rangle$ 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.