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Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 1, Pages 27–54 (Mi izv8343)  

This article is cited in 2 scientific papers (total in 2 papers)

Proof of the gamma conjecture for Fano 3-folds of Picard rank 1

V. V. Golysheva, D. Zagierbc

a Institute for Information Trnsmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
b Max Planck Institute for Mathematics
c International Centre for Theoretical Physics

Abstract: We verify the (first) gamma conjecture, which relates the gamma class of a Fano variety to the asymptotics at infinity of the Frobenius solutions of its associated quantum differential equation, for all 17 of the deformation classes of Fano 3-folds of rank 1. This involves computing the corresponding limits (‘Frobenius limits’) for the Picard–Fuchs differential equations of Apéry type associated by mirror symmetry with the Fano families, and is achieved using two methods, one combinatorial and the other using the modular properties of the differential equations. The gamma conjecture for Fano 3-folds always contains a rational multiple of the number $\zeta(3)$. We present numerical evidence suggesting that higher Frobenius limits of Apéry-like differential equations may be related to multiple zeta values.

Keywords: gamma class, gamma conjecture, Picard–Fuchs equation, Fano 3-fold.

Funding Agency Grant Number
Russian Science Foundation 14-50-00150
The work of the first author was supported by the Russian Science Foundation under grant no. 14-50-00150 at the Institute for Information Transmission Problems.


DOI: https://doi.org/10.4213/im8343

Full text: PDF file (694 kB)
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English version:
Izvestiya: Mathematics, 2016, 80:1, 24–49

Bibliographic databases:

UDC: 512.776+515.178.1+517.926.4
MSC: 11B33, 11F37, 14J45, 14J81, 14N35
Received: 25.01.2015
Revised: 09.06.2015

Citation: V. V. Golyshev, D. Zagier, “Proof of the gamma conjecture for Fano 3-folds of Picard rank 1”, Izv. RAN. Ser. Mat., 80:1 (2016), 27–54; Izv. Math., 80:1 (2016), 24–49

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    This publication is cited in the following articles:
    1. Jie Zhou, “GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities”, SIGMA, 13 (2017), 030, 32 pp.  mathnet  crossref
    2. McCarthy D., Osburn R., Straub A., “Sequences, Modular Forms and Cellular Integrals”, Math. Proc. Camb. Philos. Soc., 168:2 (2020), PII S0305004118000774, 379–404  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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