Apéry extensions

The Ap & eacute;ry 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 Ap & eacute;ry 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 Ap & eacute;ry 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 i)(3)) translates ultimately into one between algebraic K-1 and K-3 of the family.

Automated summary

The paper introduces a program to exhibit the Apéry numbers of Fano varieties as limiting extension classes of higher cycles on associated Landau-Ginzburg models. The authors also construct an Apéry motive and illustrate their proposal with detailed calculations for LG models mirror to several Fano threefolds. The paper explains the arithmetic properties of elementary Apéry numbers by describing them in terms of regulators of higher cycles.