math.AG — Algebraic Geometry

The Chaise Longue Theorem: Conjecture 1.2 of Degtyarev–Shimada for the Fermat varieties of every degree in every even dimension, and the integral Hodge conjecture for those whose degree is prime, equal to 4, or prime to (n+1)!

Contributed by Rafael Amichis Luengo

Let $X \subset \mathbb{P}^{n+1}$ be the complex Fermat variety of degree $m \ge 3$ and even dimension $n = 2k$, and let $L(X) \subseteq H_n(X; \mathbb{Z})$ be the subgroup generated by the classes of its standard $k$-dimensional linear subspaces. Degtyarev and Shimada conjectured that $H_n(X; \mathbb{Z})/L(X)$ is torsion free [DS, Conjecture 1.2]. We prove this conjecture for every degree $m \ge 3$ in every even dimension, modulo two topological results used by [DS]: Pham's theorem on the homology of the affine Fermat variety, and their intersection numbers. Its algebraic form — the quotient $\mathbb{Z}[G]/(\psi_J)$ of the group ring of [DS] is free abelian of rank $m^{2k+1} - Q_k(m)$ — is proved with no topology, and it has been verified for every degree by a complete proof in the proof assistant Lean 4 with the Mathlib library, with no gap and no axiom beyond the three standard ones. The proof reduces the conjecture, prime by prime, to dimension counts in truncated polynomial rings, proved by peeling off one variable at a time. Consequences: the integral Hodge conjecture for $X$ whenever $m$ is prime, or $m = 4$, or $m$ is prime to $(n+1)!$, with the integral Hodge classes of middle degree generated by the standard linear subspaces — in particular for the Fermat quartic of every even dimension; and the primitivity of the sublattices spanned by the linear subspaces that [DS] exhibit on the partial Fermat varieties, which [DS] proved conditionally on the conjecture. This work has not been refereed by a human expert.