Abstract. We formulate a generalization of Givental^Kim's quantum hyperplane principle.This is applied to compute the quantum cohomology of a Calabi^Yau 3-fold de¢ned as the rank 4 locus of a general skew-symmetric 7 Â 7 matrix with coef¢cients in P 6 . The computation veri¢es the mirror symmetry predictions of R(dland [25]. Mathematics Subject Classi¢cations (2000). 4N10, 14J30, 14M12.Key words. Pfa¤an, Calabi^Yau, mirror symmetry, quantum cohomology. IntroductionThe rank 4 degeneracy locus of a general skew-symmetric 7 Â 7-matrix with Gy P 6 1-coef¢cients de¢nes a noncomplete intersection Calabi^Yau 3-fold M 3 with h 1Y1 1. We recall some results of R(dland [25] on the mirror symmetry of M 3 : a potential mirror family W q is constructed as (a resolution of) the orbifold M 3 q aZ 7 , where M 3 q is a one-parameter family of invariants of a natural Z 7 -action on the space of all skew-symmetric 7 Â 7-matrices. It is shown that the Hodge diamond of W q mirrors the one of M 3 . Further, at a point of maximal unipotent monodromy*, the Picard^Fuchs operator for the periods is computed to be (with D qdadq):