We know from the work of Kontsevich, for example, that birational Calabi-Yau complex varieties have the same Hodge numbers. I want to understand to what extent the equivalence of cohomological invariants fails for two such varieties. Precisely, I am looking for examples of birational Calabi-Yau varieties over any field that have some non-isomorphic cohomological invariants (Chow groups, Hodge structures, K-groups, cohomology groups, etc.)
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4$\begingroup$ I believe that the result you attribute to Kontsevich was proved earlier by Chi-Lung Wang following ideas of Batyrev. Anyway, the product (i.e., the ring structure) on cohomology is not always invariant under K-equivalence. The Crepant Transformation Conjecture of Ruan predicts how the ring structure behaves, after extending from cohomology to quantum cohomology and then allowing an analytic continuation. $\endgroup$– Jason StarrCommented Dec 28, 2017 at 16:17
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3$\begingroup$ Typo correction: "Chi-Lung Wang" --> "Chin-Lung Wang". Sorry about that. $\endgroup$– Jason StarrCommented Dec 28, 2017 at 16:50
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Take a look at arXiv:math/0703315. It gives an explicit pair of birational Calabi-Yau threefolds which are cohomologically non-isomorphic.
