论文标题
Apéry扩展
Apéry extensions
论文作者
论文摘要
FANO品种的Apéry数量是其量子微分方程的渐近不变。在本文中，我们启动了一个计划，将这些不变性显示为（镜像）在相关的Landau-Ginzburg模型上限制更高周期的扩展类别 - 因此，尤其是作为时期。我们还构建了一个``apéry动机''，其混合杂货结构显示为分解定理的应用，以包含所讨论的限制扩展类。 使用较高的正常功能满足的不均匀的Picard-fuchs方程中的新技术结果，我们用LG模型镜像的详细计算说明了该提案，向几个Fano三倍。通过描述``基本''apéry数字在较高周期的监管因子方面（即代数$ k $ - 理论/动机共同体学类），我们可以获得对其算术属性的令人满意的解释。实际上，在每种情况下，LG模型是$ k3 $表面的模块化家族，$ζ（2）$和$ζ（2）$和$ζ（3）$（或$（2π\ Mathbf {i} 3 $）之间的区别最终转化为一个代数$ k_1 $和$ k_3 $ of Family of Family of Family的一个。
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 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 $ζ(2)$ and $ζ(3)$ (or $(2π\mathbf{i})^3$) translates ultimately into one between algebraic $K_1$ and $K_3$ of the family.
