Grothendieck's standard conjecture of Hodge type

About 19 years old · traced to 2007

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field, and let $\ell \in A^1_{\operatorname{num}}(X)$ be an ample class.

Grothendieck's standard conjecture of Hodge type. For any $i \le d/2$ , the symmetric bilinear form

(x, y) \mapsto (-1)^i\deg(\ell^{d - 2i} xy)

is positive definite on the kernel of multiplication by $\ell^{d - 2i + 1}$ .

This is the standard conjecture of Hodge type for numerical Chow groups. The paper verifies new cases using its Kähler-package theorem, while the conjecture remains open for the projectivization of a toric vector bundle over a field of positive characteristic.

References

Primary source

Matt Larson and Ethan Partida, “Hodge theory for combinatorial projective bundles”, arXiv:2604.21925 (2026).

Additional references

13 papers in this index state this conjecture (2007–2026). The statement above is taken from the most recent of them; the others are arXiv:2511.02026, arXiv:2510.21562, arXiv:2210.14001, arXiv:2111.10276, arXiv:1903.01244, arXiv:1806.03216, arXiv:1802.06244, arXiv:1204.2165, arXiv:1110.3505, arXiv:1002.5011, arXiv:0907.4046, arXiv:0709.3040.

Posted 1 month ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

Recent papers establish the conjecture in several important families, but the general statement remains open.

Grothendieck proposed the conjecture in 1968; it asserts positivity of a natural intersection pairing on numerical algebraic cycles. The general case, including projectivizations of toric vector bundles in positive characteristic, remains unresolved.

Known results

The conjecture is known for complex varieties and for surfaces.
Ancona (2018) proved it for abelian fourfolds in characteristic $p$ .
Later work proves it for powers of abelian threefolds and for powers of simple abelian varieties of prime dimension.
For every prime $p$ and $g \geq 4$ , infinitely many abelian varieties of dimension $g$ over $\overline{\mathbb{F}}_{p}$ satisfy it.

2026 advances

In April 2026, Larson and Partida reported new cases for projective bundles, using a Kähler-package theorem; their paper explicitly leaves the positive-characteristic toric-vector-bundle case open. On October 1, 2026, Harashita reported an elementary proof for Hermitian varieties with additional $p$ -adic applications. These are substantial special cases, not a solution of the general conjecture.

Current status (as of October 2026): The conjecture is established in several geometric families, but remains open in general, including the stated toric-vector-bundle case.

Sources

No solutions have been posted yet.