The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian \(G(2,7)\).

*(English)*Zbl 0974.14026Let \(V = \wedge^2({\mathbb C}^7)\) be the linear space of skew-symmetric \(7\times 7\)-matrices and \(Z \subset {\mathbb P}(V)\cong {\mathbb P}^{20}\) be the variety parametrizing rank 4 matrices. It is defined by the Pfaffians of the diagonal \(6\times 6\)-minors. Its degree is 14 and dimension 17. The intersection of \(Z\) with a general linear space \(A\) of dimension 6 is a Calabi-Yau 3-fold \(X_A\) with Hodge numbers \(h^{11} = 1\) and \(h^{12} = 50\).

The author considers a pencil of 6-dimensional spaces \(L(\mu,\nu)\) of matrices

\(A(\mu,\nu) = (x_{i+j}y_{i-j})_{i,j\in {\mathbb Z}_7}\), where \(y_0 = y_{3} = y_4 = 0\), \(y_1 = y_6 = \mu\), \(y_2 = y_5 = \nu\).

The 3-folds \(X_{\mu,\nu} = Z\cap L(\mu,\nu)\) have 56 nodes and are invariant with respect to the natural action of the group \(H = L(\mu,\nu)\) defined by \(x_k\to e^{2\pi ik/7}x_{k}\). After taking the quotient and resolving the singularities one obtains a 1-dimensional family of Calabi-Yau 3-folds with \(h^{11} = 50\), \(h^{12} = 1\).

There are two points on the base of the family corresponding to the degeneration with maximal monodromy. Conjecturally, we obtain two mirror families. One corresponds to the original family of \(X_A\) with general \(A\). Another one corresponds to the family of Calabi-Yau 3-folds \(Y_{B}\) obtained as linear sections \(B\cap G(2,7)\), where \(\dim B = 13\), of the Grassmannian \(G(2,7)\subset {\mathbb P}^{20}\). The author computes the Picard-Fuchs equations at each degeneracy point. The predicted number of rational curves on the mirrors coincides with the actual one for some small degrees.

The author considers a pencil of 6-dimensional spaces \(L(\mu,\nu)\) of matrices

\(A(\mu,\nu) = (x_{i+j}y_{i-j})_{i,j\in {\mathbb Z}_7}\), where \(y_0 = y_{3} = y_4 = 0\), \(y_1 = y_6 = \mu\), \(y_2 = y_5 = \nu\).

The 3-folds \(X_{\mu,\nu} = Z\cap L(\mu,\nu)\) have 56 nodes and are invariant with respect to the natural action of the group \(H = L(\mu,\nu)\) defined by \(x_k\to e^{2\pi ik/7}x_{k}\). After taking the quotient and resolving the singularities one obtains a 1-dimensional family of Calabi-Yau 3-folds with \(h^{11} = 50\), \(h^{12} = 1\).

There are two points on the base of the family corresponding to the degeneration with maximal monodromy. Conjecturally, we obtain two mirror families. One corresponds to the original family of \(X_A\) with general \(A\). Another one corresponds to the family of Calabi-Yau 3-folds \(Y_{B}\) obtained as linear sections \(B\cap G(2,7)\), where \(\dim B = 13\), of the Grassmannian \(G(2,7)\subset {\mathbb P}^{20}\). The author computes the Picard-Fuchs equations at each degeneracy point. The predicted number of rational curves on the mirrors coincides with the actual one for some small degrees.

Reviewer: Igor V.Dolgachev (Ann Arbor)

##### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14J30 | \(3\)-folds |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |