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

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HHHW03 - Derived algebraic geometry and chromatic homotopy theory

It is a classical problem in algebraic geometry to decide whether a class in the singular cohomology of a smooth complex variety X is algebraic, that is if it can be realized as the fundamental class of an algebraic subvariety of X. One can ask a similar question for motivic spectra: Given a motivic spectrum E, which classes in the topological E-cohomology of X come from motivic classes. I would like to discuss this question and examples of non-algebraic classes for the tower of Brown-Peterson spectra.

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