6
$\begingroup$

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.)

$\endgroup$
2
  • 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 Starr Dec 28 '17 at 16:17
  • 3
    $\begingroup$ Typo correction: "Chi-Lung Wang" --> "Chin-Lung Wang". Sorry about that. $\endgroup$ – Jason Starr Dec 28 '17 at 16:50
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.