The Apéry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau–Ginzburg (LG) models — and thus, in particular, as periods. We also construct an Apéry motive, whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard–Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG-models mirror to several Fano threefolds. By describing the “elementary” Apéry numbers in terms of regulators of higher cycles (i.e., algebraic $K$-theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG-models are modular families of $K3$ surfaces, and the distinction between multiples of $\zeta (2)$ and $\zeta (3)$ (or ${(2\pi \mathbf{i})}^3$) translates ultimately into one between algebraic $K_1$ and $K_3$ of the family.

中文翻译：

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Apéry 扩展

Fano 簇的Apéry 数是其量子微分方程的渐近不变量。在本文中，我们启动了一个计划，将这些不变量展示为（镜像）相关兰道-金茨堡（LG）模型上更高循环的限制扩展类，因此，特别是作为周期。我们还构造了一个Apéry 动机，其混合 Hodge 结构被显示为分解定理的应用，以包含所讨论的限制扩展类。使用由高阶正规函数满足的非齐次皮卡德-福克斯方程的新技术成果，我们通过对 LG 模型镜像到多个 Fano 三重的详细计算来说明这一建议。通过用更高周期的调节器来描述“基本”阿佩里数（即代数$K$-理论/动机上同调类），我们获得了对其算术性质的令人满意的解释。事实上，在每种情况下，LG 型号都是模块化系列$K3$表面，以及倍数之间的区别$\delta （2）$和$\delta （3）$（或者${（2\pi \mathbf{我}）}^3$) 最终转化为代数之间的一种$K_1$和$K_3$家庭的。