Isotropic Chow groups conjecture over flexible fields

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Let kk be a field, and let Ch⁡k/k∗\operatorname{Ch}^*_{k/k} denote the isotropic Chow groups with Fp\mathbb{F}_p-coefficients, obtained from Chow groups by quotienting by classes coming from pp-anisotropic varieties. Let Ch⁡Num∗\operatorname{Ch}^*_{Num} denote the corresponding numerical Chow groups. Isotropic Chow groups conjecture. If kk is flexible, then

Ch⁡k/k∗=Ch⁡Num∗.\operatorname{Ch}^*_{k/k}=\operatorname{Ch}^*_{Num}.

This is the central conjecture concerning pure isotropic motives and predicts that, over flexible fields, isotropic cycles are exactly the numerically nontrivial cycles. The source gives no resolution status.

References

Primary source

Alexander Vishik, “On isotropic and numerical equivalence of cycles”, arXiv:2201.02984 (2022).

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