TITLE: Birational Calabi-Yau varieties with non-isomorphic cohomological invariants
QUESTION [6 upvotes]: 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.)

REPLY [2 votes]: Take a look at arXiv:math/0703315.
It gives an explicit pair of birational Calabi-Yau threefolds which are cohomologically non-isomorphic.