For every integer m≥ 1, let X_m^4=x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0⊂P^5 be the Fermat fourfold. The Hodge conjecture asserts that every rational Hodge class of codimension 2 on X_m^4 is algebraic; equivalently, the cycle-class map operatornameCH^2(X_m^4)otimes_ZQlongrightarrow H^4(X_m^4,Q)cap H^2,2(X_m^4) is surjective.

Hodge conjecture for Fermat fourfolds

The problem asks whether the relevant cohomology classes on Fermat fourfolds come from algebraic cycles. The new claim concerns a large but finite family, not the conjecture for all Fermat fourfolds.

Known results

Shioda's method proves the conjecture for every Fermat fourfold of degree $m$ coprime to $6$ .
The conjecture is known for Fermat varieties of degree $m \le 20$ , and for degrees $21$ and $27$ in every dimension.
The Fermat sextic fourfold is known to satisfy the conjecture.
A computer implementation confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds; this is a different, stronger statement.

August 2026 finite-family claim

Rifat Jumagulov's preprint claims the ordinary Hodge conjecture for Fermat fourfolds of odd degree at most $199$ . It reports a census of $78{,}299$ Galois-orbit representatives, geometric closure criteria, and machine-checked witnesses. The scan found no independent verification or reported objection.

Current status (as of August 2026): Earlier special cases are established, while Jumagulov's extension through odd degree $199$ remains an unverified preprint claim and the general Fermat-fourfold problem remains open.

Hodge conjecture for Fermat fourfolds

Primary source

Additional references

Progress summary

Solutions 0