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

*(English. Russian original)*Zbl 1369.14054
Izv. Math. 80, No. 1, 24-49 (2016); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 80, No. 1, 27-54 (2016).

Let \(X\) be a Fano variety, and let \(A_X\in H^*(X)\) be the principal asymptotic class of the quantum differential equation associated with \(X\). Let \(\hat{\Gamma}(X)\in H^*(X)\) be the gamma class of \(X\). The gamma conjecture is the validity of the equality \(A_X=\hat{\Gamma}_X\). For a Fano 3-fold of Picard rank \(1\), the modified gamma class is given by a formula involving the first Chern class of \(X\), and a rational multiple of \(\zeta(3)\).

The paper under review verifies the gamma conjecture for each of the 17 deformation classes in Iskovskikh’s classification of smooth Fano 3-folds of Picard rank 1. The conjecture 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 deformation classes of Fano 3-folds of Picard rank 1.

The gamma conjecture may be regarded as a mirror symmetry for these Fano 3-folds. Mirror symmetry predicts that with each of the 17 families of Fano varieties, there should be associated a family \({\mathcal{E}}\) of \(K3\) surfaces over \({\mathbb{P}}^1\) in such a way that the “quantum differential equation” on the Fano 3-fold (the A-model side) is the Laplace transform of the Picard-Fuchs differential equation satisfied by the periods of \({\mathcal{E}}\) (the B-model side). The family \({\mathcal{E}}\) is a family of Kuga-Sato type whose base space is the modular curve \(X^*_0(N)\) classifying the unordered pairs (\(E, E^{\prime})\) of \(N\)-isogenous elliptic curves for some \(N\), and the fiber is the smooth resolution of the quotient of \(E\times E^{\prime}\) by \((-1)\), with Picard number \(19\). Their Picard-Fuchs differential equations have modular parametrizations.

The proof 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 is combinatorial, and the other uses modular properties of the differential equations.

Further, numerical evidence is presented suggesting that higher Frobenius limits of Apéry-like differential equations may be related to multiple zeta values.

The paper under review verifies the gamma conjecture for each of the 17 deformation classes in Iskovskikh’s classification of smooth Fano 3-folds of Picard rank 1. The conjecture 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 deformation classes of Fano 3-folds of Picard rank 1.

The gamma conjecture may be regarded as a mirror symmetry for these Fano 3-folds. Mirror symmetry predicts that with each of the 17 families of Fano varieties, there should be associated a family \({\mathcal{E}}\) of \(K3\) surfaces over \({\mathbb{P}}^1\) in such a way that the “quantum differential equation” on the Fano 3-fold (the A-model side) is the Laplace transform of the Picard-Fuchs differential equation satisfied by the periods of \({\mathcal{E}}\) (the B-model side). The family \({\mathcal{E}}\) is a family of Kuga-Sato type whose base space is the modular curve \(X^*_0(N)\) classifying the unordered pairs (\(E, E^{\prime})\) of \(N\)-isogenous elliptic curves for some \(N\), and the fiber is the smooth resolution of the quotient of \(E\times E^{\prime}\) by \((-1)\), with Picard number \(19\). Their Picard-Fuchs differential equations have modular parametrizations.

The proof 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 is combinatorial, and the other uses modular properties of the differential equations.

Further, numerical evidence is presented suggesting that higher Frobenius limits of Apéry-like differential equations may be related to multiple zeta values.

Reviewer: Noriko Yui (Kingston)