Residual Characters of Odd-Degree Fermat Fourfolds
Year: 2026
Type: preprint
Abstract: For a Fermat fourfold of odd degree, the rational Hodge classes in middle degree are spanned by character blocks indexed by six-entry Hodge characters. Two direct constructions make such a block algebraic: a contraction of two surface characters, which includes the case of an inverse pair, and an exact 5-standard tuple of Aoki. We call the remaining characters *residual*. We prove that the residual characters at every odd degree are exactly the inflations of six explicit seeds in degrees 33, 39, 45 and 105. The proof is uniform in the degree and uses no computer enumeration. Maximal-order Fourier kernels, a classification of surface characters at odd conductor and an analysis of four points of a five-progression reduce an arbitrary primitive residual conductor to the degrees 15, 21, 33, 39, 45 and 105, which are then settled by the same character equations. An exhaustive census through degree 599, and enumerations at the 103 degrees up to 2001 not covered by the earlier subfamily theorems, give an independent check. Since the seeds are algebraic, the classification yields a proof of the Hodge conjecture for these fourfolds that does not use Kang's refined motivic dimension estimates. That conclusion is also covered by the written proof of Bhattacharjee [5], and the six seeds were first isolated by Jumagulov [8] through degree 199; the new content is the exact positive classification at every odd degree. We also record the all-dimensional consequences for the prime supports $\{3,11\}$ and $\{3,13\}$, and give an explicit family of Fermat characters on which the tensor filtration identity used in Kang's product argument fails. Archival copy of version 4 of this paper, first made public on rStation on 5 November 2026 (rStation hodge_conjecture paper 104, v4; version 1 posted 4 November 2026). paper.md is the manuscript source as posted there (SHA-256 cac9f460be44dd1abe019c6eee5b42d898df8acad458e704a084bec91155cadd); the PDF is typeset from it with the posted title and abstract. supplement.zip contains the reproduction bundles (SHA-256 68e062729bbe49f0e92b435d0f4394fc4f156a82b9b8bc9aa1f503a386d91657). This paper follows the author's preprint The Hodge Conjecture for Odd-Degree Fermat Fourfolds, public on 8 September 2026 (ResearchGate, DOI 10.13140/RG.2.2.26853.77288). The work was produced largely with AI systems and is released for public human verification; it has not been checked by a human expert.
Author: Li, Hongmin
Institutions: Institute of Science Tokyo, The University of Tokyo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225595359