Apéry extensions
Apéry extensions
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Apéry 扩展
DOI：
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Tokio Sasaki
中科院分区：
文献类型：
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作者：
V. Golyshev;M. Kerr;Tokio Sasaki
Fano簇的Apéry数是其量子微分方程的渐近不变量。在本文中，我们发起了一个计划，以展示这些不变量作为（镜像）限制扩展类的相关朗道-金兹伯格（LG）模型上的更高的周期-因此，特别是，作为周期。我们还构造了一个Apéry动机，其混合霍奇结构，作为分解定理的应用，包含问题中的极限扩张类。利用一个关于高阶正规函数所满足的非齐次Picard-Fuchs方程的新技术结果，我们通过对LG模型镜像到几个Fano三重的详细计算来说明这一建议。通过描述“初等”Apéry数的更高周期的调节器（即，代数K$K$-理论/动机上同调类），我们得到了令人满意的解释，他们的算术性质。事实上，在每种情况下，LG‐模型都是K3 $K3 $曲面的模族，并且在n（2）$zeta（2）$和n（3）$zeta（3）$的倍数（或（2πi）3$（2 pi mathbf {i}）^3$）之间的区别最终转化为族的代数K1 $K_1 $和K3 $K_3 $之间的区别。
The Apéry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau–Ginzburg (LG) models — and thus, in particular, as periods. We also construct an Apéry motive, whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard–Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG‐models mirror to several Fano threefolds. By describing the “elementary” Apéry numbers in terms of regulators of higher cycles (i.e., algebraic K$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$K3$ surfaces, and the distinction between multiples of ζ(2)$zeta (2)$ and ζ(3)$zeta (3)$ (or (2πi)3$(2pi mathbf {i})^3$ ) translates ultimately into one between algebraic K1$K_1$ and K3$K_3$ of the family.
单能扩张和微分方程（仿布洛赫·弗拉森科）
影响因子：
1.9
作者：
Kerr, Matt
通讯作者：
Kerr, Matt