On the Hodge conjecture for Fermat varieties

Abstract

LaTeX source (amsart, figures as PNG), compiled PDF and verification code for the paper On the Hodge conjecture for Fermat varieties by Deep Bhattacharjee. The paper proves the Hodge conjecture for the Fermat varieties of degree 114 in every dimension, and more generally of every degree 2a3b19c with a ≤ 1, together with their products and the abelian varieties of Fermat type of these degrees. For degrees divisible by 57 it is the first proof that does not rest on a statement of Kang whose proof has a gap; for 114, 171, 342, 513 and the other multiples of 57 other than 57 itself, the theorem had not been stated before. Aoki's reduction leaves one class in degree 114, which is the join of two characters of the Fermat threefold of degree 114. Each of the two corresponding pieces of H3 is shown to be carried by curves, through families of curves on the threefold and a residue formula for the derivative of their Abel–Jacobi maps, extending Peterson's method for degree 35. One residue is computed by hand; the other is certified to be nonzero by interval arithmetic (Arb), the only computer-assisted step of the paper. The paper also proves the Hodge conjecture for Fermat fourfolds of every odd degree, determines which combinations of the standard statements around the Hodge conjecture imply it, proves the conjecture for every Fermat variety under the recently claimed Hodge conjecture for CM abelian varieties, and extracts from Schoen's cycles one subvariety whose semiregularity would settle the split Weil families over Q(√−3) in every dimension. The Hodge conjecture itself is not proved. paper/ holds the source and paper/main.pdf; scripts/build_paper.sh builds the PDF, a tex.zip and an arXiv tarball that compiles with pdflatex alone. verification/ re-checks the finite steps in Python, Julia, C, Lean 4 (core and Mathlib) and Macaulay2, holds the interval certificate for the degree-114 family (python-flint), and checks every reference online. Author contact: d.bhattacharjee@erl-forschung.de, itsdeep@live.com. Affiliation: Formerly, Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha 751030, India.