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A generalized construction of mirror manifolds. (English) Zbl 0842.32023
Yau, Shing-Tung (ed.), Essays on mirror manifolds. Cambridge, MA: International Press. 388-407 (1992).
B. R. Greene and M. R. Plesser [Nuclear Phys. B 338, No. 1, 15-37 (1990)] showed explicitly how to construct a mirror Calabi-Yau for a hypersurface in weighted projective space which is of Fermat type. This is done by factoring out by a subgroup of the (finite) symmetry group and resolving singularities. In this paper the authors give a very natural extension to certain hypersurfaces of non-Fermat type. They require only that the number of monomials be equal to the number of variables (five in the case of hypersurfaces of weighted projective space); they associate to such a polynomial a matrix \(P = (p_{ij})\),where \(p_{ij}=\) power of the \(i\)th variable in the \(j\)th monomial. As they show, the transposed matrix \(^TP\) determines again a Calabi-Yau hypersurface, and the finite group of symmetries \(G\) of the original hypersurface \(X\) again acts on this transpose \(^TX\). Then they find a subgroup \(H \subset G\) of this symmetry group, such that a candidate for the mirror \(X^m\) of \(X\) is given by \(^TX/HS\) (with singularities resolved). By candidate we mean that the authors have checked that the Hodge diamond are mirrors, but not that the Yukawa couplings coincide. Since for a Fermat polynomial the transpose coincides with \(P\), this construction reduces in this case to that of Greene and Plesser [loc. cit.].
These considerations are applied to give an example of a (candidate for a) mirror for one of the manifolds listed in a paper by P. Candelas, M. Lynker and R. Schimmrigk [Nuclear Phys. B 341, No. 2, 383-402 (1990)] which is listed without a mirror. This is also applied to describe a mirror for the “\(D^k\), \(N = 2\) superconformal minimal model” (where the superpotential is given by the polynomial of \(D_k\) type \(x^{k-1}_1 + x_1 x^2_2\)).
For the entire collection see [Zbl 0816.00010].

32J17 Compact complex \(3\)-folds
32G81 Applications of deformations of analytic structures to the sciences
14J30 \(3\)-folds