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The Hodge conjecture for the sextic fourfolds of an eight-dimensional family of K3 surfaces with real multiplication

Year: 2026
Type: preprint
Abstract: In an earlier note we wrote down an explicit eight-dimensional family of K3 surfaces with real multiplication by ℚ(√5): double planes w² = F for a family of plane sextics F, with the Hodge conjecture for S × S for the very general member S. Here we treat the sextic fourfolds X = {F(x₀,x₁,x₂) + F(y₀,y₁,y₂) = 0} in ℙ⁵ attached to the same family. For very general F in the family we prove the Hodge conjecture for X, and we show that its rational Hodge classes of degree four form a space of dimension 9, two more than for a very general plane sextic. Besides the square of the hyperplane class and the six planes y = ηx (η⁶ = −1) contained in X, the Hodge classes come from Hodge classes on S × S; the two additional dimensions are due to the Néron–Severi group and the real multiplication of S, and their algebraicity comes from the earlier note. Two cubic surfaces in X give, together with the square of the hyperplane class and the planes, explicit cycles for eight of the nine dimensions. By the construction of Shioda and Katsura the problem reduces to Hodge classes in tensor products of eigenspaces of the sextic surface w⁶ + F = 0. The main point is the upper bound. For two of the eigenspace pairs the first-order bound of Carlson and Griffiths is 22 instead of 1, and 18 of the 21 extra dimensions come from a single class of type (1,1) that satisfies the first-order conditions. A second-order computation, in terms of endomorphisms that preserve the holomorphic forms and carried out with Griffiths' reduction of pole order, shows that this class does not stay of type (1,1) and gives the bound 1. We found no earlier statement of the Hodge conjecture for these fourfolds beyond the Fermat case, and no earlier use of second-order conditions to bound the Hodge classes of a very general member. The bounds are computed at an explicit member of the family, exactly over ℚ and modulo two primes, with two independent implementations. Code and data are at github.com/sallerk/HC-k3-rm5-fourfold.
DOI: 10.5281/zenodo.23325021
Language: English
Open Access status: green
OpenAlex ID: W7221536070
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