We know from the work of Kontsevich, for example, that birational CalabiYau 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 CalabiYau varieties over any field that have some nonisomorphic cohomological invariants (Chow groups, Hodge structures, Kgroups, cohomology groups, etc.)
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4$\begingroup$ I believe that the result you attribute to Kontsevich was proved earlier by ChiLung Wang following ideas of Batyrev. Anyway, the product (i.e., the ring structure) on cohomology is not always invariant under Kequivalence. 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

3$\begingroup$ Typo correction: "ChiLung Wang" > "ChinLung 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 CalabiYau threefolds which are cohomologically nonisomorphic.