For every nonsingular Hermitian variety X of dimension 2m over a finite field, let h denote the hyperplane class and let N^m(X)_R be the space of codimension-m algebraic cycles modulo numerical equivalence, tensored with R. On the primitive subspace N^m(X)_R^prim=alpha∈ N^m(X)_R:alpha· h^m=0, the intersection form satisfies (-1)^m(alpha·alpha)>0 for every nonzero alpha. Equivalently, the intersection form is definite on the primitive middle-dimensional algebraic classes.

Grothendieck’s standard conjecture of Hodge type for Hermitian varieties

For every nonsingular Hermitian variety $X$ of dimension $2m$ over a finite field, let $h$ denote the hyperplane class and let $N^m(X)_{\mathbb{R}}$ be the space of codimension- $m$ algebraic cycles modulo numerical equivalence, tensored with $\mathbb{R}$ . On the primitive subspace $N^m(X)_{\mathbb{R}}^{\mathrm{prim}}=\{\alpha\in N^m(X)_{\mathbb{R}}:\alpha\cdot h^m=0\}$ , the intersection form satisfies $(-1)^m(\alpha\cdot\alpha)>0$ for every nonzero $\alpha$ . Equivalently, the intersection form is definite on the primitive middle-dimensional algebraic classes.

References

Primary source

arXiv

Additional references

An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with p-adic applications — arXiv — Shushi Harashita

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed progress

A new preprint claims a rigorous advance for Hermitian varieties, but it addresses only a special case and does not settle Grothendieck’s broader conjecture.

The problem concerns Grothendieck’s standard conjecture of Hodge type, restricted to Hermitian varieties. The reported work gives a proof for this class together with an arithmetic refinement, rather than resolving the conjecture for arbitrary varieties.

Known results

No additional classical partial results were identified in the retrieved sources.

October 2026 special-case proof

Shushi Harashita’s preprint claims an elementary dual-polar-graph argument proving positivity of the primitive intersection form for Hermitian varieties and establishing divisibility bounds for its $p$ -adic elementary divisors. The result has consequences for algebraic cycles and finite covers, but its mathematical validity is unverified in the retrieved material.

Current status (as of October 2026): A special case for Hermitian varieties is claimed in a preprint, while the general standard conjecture remains open.

Sources

No solutions have been posted yet.