The Hodge conjecture for Fermat varieties of degree 35

Let X^n_m ⊂ P^{n+1} be the Fermat variety of dimension n and degree m. We give a proof, independent of Kang's theorem on the generalized Hodge conjecture for Fermat threefolds and fourfolds, of the Hodge conjecture for X^n_m for every n when m = 2^a 3^b 5^c 7^d is not divisible by 420. For cd = 0 this is due to Aoki. Martelotte, Miranda, Movasati and Villaflor, in work posted shortly before the first version of this paper, independently proved the Hodge conjecture for all Fermat varieties of degree less than 65 other than 44, 51 and 52, among them the degree 35, by a different method which rests on Kang's theorem; combined with Aoki's reduction, their result also covers the other degrees above. By Aoki's work on the gap group, what is missing beyond Aoki's results is the algebraicity of the Hodge classes attached to one exceptional character of X^6_35. We obtain it from a coniveau statement: the 24-dimensional rational sub-Hodge structure of level one of H^3(X^3_35, Q) attached to the character (1,2,16,21,30) of μ_35^5 is supported on a divisor. This is a case of the generalized Hodge conjecture which is included in the statement of Kang's theorem; our proof of it is explicit and does not use that theorem. It uses an explicit one-parameter family of curves on X^3_35, a residue formula for the derivative of the Abel–Jacobi map along the family, and a computer-verified interval-arithmetic certificate at one point of the family. For the ten degrees m = 35k, 1 ≤ k ≤ 10, the reduction to this one class is verified by an exact finite computation which does not use Aoki's structure theorem. As consequences we obtain the Hodge conjecture for products of Fermat varieties of one such degree and for abelian varieties of Fermat type of these degrees, among them the Jacobian of the curve y^2 = x^35 − 1. Finally, for m = 420 we show that the Hodge characters in the one remaining class of the gap group have dimension at least 14, a bound which is attained, and deduce the Hodge conjecture for X^n_420 for n ≤ 12 (for n ≤ 4 this is also included in the statement of Kang's theorem). Preprint, not yet peer reviewed. The repository contains the paper (LaTeX source and PDF) and the verification scripts with their expected output.