Equality of the generator lattice with the numerical-cycle lattice

Determine whether the lattice generated by the maximal totally isotropic linear subspaces of an even-dimensional Hermitian variety equals the full group of codimension-middle numerical cycles, namely whether $\Lambda=\mathrm N^{r-1}(X)$ for $X=(2r-1,q^2)$ over an algebraic closure of $F_p$.

Background

For the Hermitian variety X=(2r−1,q2)X=(2r-1,q^2), the paper defines Λ\Lambda as the lattice generated by the classes of the maximal totally isotropic linear subspaces, or generators, inside the group Nr−1(X)\mathrm N^{r-1}(X) of codimension-(r−1)(r-1) cycles modulo numerical equivalence. Previous work shows that the quotient Nr−1(X)/Λ\mathrm N^{r-1}(X)/\Lambda is a finite p-group.

The paper proves p-adic bounds on this quotient, including that qr−1Nr−1(X)⊆Λq^{r-1}\mathrm N^{r-1}(X)\subseteq\Lambda, but does not determine whether the quotient is always trivial. Thus the unresolved issue is whether the generators form the entire numerical-cycle lattice.

References

By , $\mathrm N{r-1}(X)/\Lambda$ is a finite $p$-group, and whether it vanishes is open in general.

— An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with $p$-adic applications  (2610.01731 - Harashita, 1 Oct 2026) in Section 1, subsection “The lattice of generators, $p$-adically”