For integers m≥ 1 and n≥ 1, let X_m^n⊂ P^n+1_C be the Fermat hypersurface x_0^m+x_1^m+…+x_n+1^m=0. The Hodge conjecture for Fermat varieties asserts that, for every codimension p with 0≤ p≤ n, every rational Hodge class is algebraic: H^2p(X_m^n,Q)cap H^p,p(X_m^n)=operatornameIm!(operatornameclcolon CH^p(X_m^n)otimesQlongrightarrow H^2p(X_m^n,Q)). Equivalently, every class in H^2p(X_m^n,Q) of Hodge type (p,p) is a rational linear combination of cohomology classes of algebraic cycles of codimension p.

Hodge conjecture for Fermat varieties

For integers $m\ge 1$ and $n\ge 1$ , let $X_m^n\subset \mathbb{P}^{n+1}_{\mathbb{C}}$ be the Fermat hypersurface $x_0^m+x_1^m+\cdots+x_{n+1}^m=0$ . The Hodge conjecture for Fermat varieties asserts that, for every codimension $p$ with $0\le p\le n$ , every rational Hodge class is algebraic:

H^{2p}(X_m^n,\mathbb{Q})\cap H^{p,p}(X_m^n)=\operatorname{Im}\!\left(\operatorname{cl}\colon \mathrm{CH}^p(X_m^n)\otimes\mathbb{Q}\longrightarrow H^{2p}(X_m^n,\mathbb{Q})\right).

Equivalently, every class in $H^{2p}(X_m^n,\mathbb{Q})$ of Hodge type $(p,p)$ is a rational linear combination of cohomology classes of algebraic cycles of codimension $p$ .

References

Primary source

arXiv

Additional references

Lengths of Hodge characters in Fermat varieties — arXiv — Lucas Martelotte, Maximiliano Miranda, Hossein Movasati, Roberto Villaflor

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A September 2026 paper claims a major expansion of the cases known for Fermat varieties, but the full conjecture remains open.

The problem asks whether the Hodge conjecture holds for Fermat varieties. The latest report claims a length-reduction method proving it in every dimension across a substantially larger degree range.

Known results

Shioda: all dimensions for degrees $m<21$ .
Shioda and Ran: all dimensions for prime degree and $m=4$ .
da Silva: all Fermat fourfolds $X_m^4$ when $m$ is coprime to $6$ .
Aoki: all dimensions for $m=p^2$ with $p$ prime.

September 2026 length-reduction advance

Lucas Martelotte, Maximiliano Miranda, Hossein Movasati, and Roberto Villaflor report that a length-reduction method proves the conjecture in every dimension for the stated degree range. The result is an arXiv claim and is not independently verified here; the all-degree problem and three exceptional degrees remain open.

Current status (as of September 2026): A substantial degree range is claimed to be settled in every dimension, while the all-degree conjecture and three exceptional degrees remain open.

Sources

No solutions have been posted yet.