Affiliation:
1. ICTP Math Section Trieste Italy
2. Algebra and Number Theory Lab Institute for Information Transmission Problems Moscow Russia
3. Department of Mathematics Washington University in St. Louis St. Louis Missouri USA
4. Department of Mathematics University of Miami Coral Gables Florida USA
Abstract
AbstractThe 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 ‐theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG‐models are modular families of surfaces, and the distinction between multiples of and (or ) translates ultimately into one between algebraic and of the family.
Funder
Simons Foundation
National Science Foundation
Reference42 articles.
1. Hypergeometric functions and rings generated by monomials
2. P.Belmans S.Galkin andS.Mukhopadhyay Graph potentials and symplectic geometry of moduli spaces of vector bundles arXiv:2206.11584.
3. A family of K3$K3$ surfaces and ζ(3)$\zeta (3)$;Beukers F.;J. Reine Angew. Math.,1984
4. Algebraic cycles and higher K-theory
5. The moving lemma for higher Chow groups;Bloch S.;J. Algebraic Geom.,1994