Isotropic Chow groups conjecture with finite coefficients

Let kk be a flexible field and let XX be a smooth projective variety over kk. Write CHk/k(X,Z/pm)\operatorname{CH}_{k/k}(X,\mathbb{Z}/p^m) for the isotropic Chow group with Z/pm\mathbb{Z}/p^m-coefficients, and let CHNum(X,Z/pm)\operatorname{CH}_{Num}(X,\mathbb{Z}/p^m) be the quotient by the kernel of the numerical degree pairing. Isotropic Chow groups conjecture with finite coefficients. If kk is flexible, then for all mm,

CHk/k(X,Z/pm)=CHNum(X,Z/pm).\operatorname{CH}_{k/k}(X,\mathbb{Z}/p^m)=\operatorname{CH}_{Num}(X,\mathbb{Z}/p^m).

This extends the conjecture for Fp\mathbb{F}_p-coefficients to all pp-primary finite coefficients. The source gives no resolution status.

Sources & referencesView supporting material

Primary source

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

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