Lengths of Hodge characters in Fermat varieties
Abstract
We revisit the Hodge conjecture for Fermat (and weighted Fermat) varieties. We review the reduction of the Hodge conjecture in terms of Hodge characters introduced by Shioda and its further reduction in terms of the formal module of tuples introduced by Aoki. Following Aoki, we introduce several lengths on this formal module, and by means of Kang's theorem on the Hodge conjecture for Fermat fourfolds we reduce the conjecture to bound the lengths of exceptional tuples by 6. By applying length reduction algorithms we prove the Hodge conjecture for all Fermat varieties (of any dimension) of degree less than 65 and different from 44, 51 and 52. The main novelty of our length reduction method is the introduction of a lift operation on the degree of the tuple.
Related
- The Hodge conjecture for Fermat fourfolds of degree at most 100, and the semi-standard obstruction at degree 70
- Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties
- Quotients of Calabi-Yau Generalized Fermat manifolds and Log Enriques varieties
- On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type of dimension divisible by four
- Generic vanishing for Dolbeault cohomology of mixed Hodge modules on abelian varieties
BibTeX
@misc{miranda2026lengths,
title = {{Lengths of Hodge characters in Fermat varieties}},
author = {Maximiliano Miranda and Hossein Movasati and Lucas Rufino and Roberto Villaflor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27301},
eprint = {2609.27301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27301},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lengths of Hodge characters in Fermat varieties
AU - Maximiliano Miranda
AU - Hossein Movasati
AU - Lucas Rufino
AU - Roberto Villaflor
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.27301
UR - https://arxiv.org/abs/2609.27301
AB - We revisit the Hodge conjecture for Fermat (and weighted Fermat) varieties. We review the reduction of the Hodge conjecture in terms of Hodge characters introduced by Shioda and its further reduction in terms of the formal module of tuples introduced by Aoki. Following Aoki, we introduce several lengths on this formal module, and by means of Kang's theorem on the Hodge conjecture for Fermat fourfolds we reduce the conjecture to bound the lengths of exceptional tuples by 6. By applying length reduction algorithms we prove the Hodge conjecture for all Fermat varieties (of any dimension) of degree less than 65 and different from 44, 51 and 52. The main novelty of our length reduction method is the introduction of a lift operation on the degree of the tuple.
ER -