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The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian $$G(2,7)$$. (English) Zbl 0974.14026
Let $$V = \wedge^2({\mathbb C}^7)$$ 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.

##### 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)
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