Fermat 簇中 Hodge 特征的长度
Lengths of Hodge characters in Fermat varieties
AI总结：
本文通过引入长度缩减算法和提升操作，将Fermat簇的Hodge猜想简化为限制例外元组长度，并证明了度数小于65（除44、51、52）的所有Fermat簇的猜想。
AI中文摘要：
我们重新审视 Fermat（以及加权 Fermat）簇的 Hodge 猜想。我们回顾了由 Shioda 引入的用 Hodge 特征来简化 Hodge 猜想的方法，以及由 Aoki 引入的用元组的正式模来进一步简化该猜想的方法。遵循 Aoki，我们在该正式模上引入了若干长度，并借助 Kang 关于 Fermat 四维簇的 Hodge 猜想定理，将猜想简化为将例外元组的长度限制为 6。通过应用长度缩减算法，我们证明了所有度数小于 65 且不等于 44、51 和 52 的 Fermat 簇（任意维数）的 Hodge 猜想。我们长度缩减方法的主要新颖之处在于引入了对元组度数的提升操作。
英文摘要：
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.
