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| Main Authors: | , , |
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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2009.14762 |
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| _version_ | 1866911780277583872 |
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| author | Golyshev, Vasily Kerr, Matt Sasaki, Tokio |
| author_facet | Golyshev, Vasily Kerr, Matt Sasaki, Tokio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_14762 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Apéry extensions Golyshev, Vasily Kerr, Matt Sasaki, Tokio Algebraic Geometry Number Theory 11G42, 14D07, 19F27 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. |
| title | Apéry extensions |
| topic | Algebraic Geometry Number Theory 11G42, 14D07, 19F27 |
| url | https://arxiv.org/abs/2009.14762 |