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

摘要

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 Apery 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 Apery-like differential equations may be related to multiple zeta values.