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)!
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…
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