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Mirror principle. I. (English) Zbl 0953.14026
This paper deals with a complete proof of the computation of the number \(n_d\) of rational curves of degree \(d\) on a general quintic in the 4-dimensional complex projective space \({\mathbb P}^4\), for any \(d\). In an earlier paper [P. Candelas, X. de la Ossa, P. Green and L. Parkes, Nucl. Phys. B 359 21-47 (1991)] there was conjectured a formula giving a generating function for the numbers \(n_d\) explicitly in terms of some hypergeometric functions. This computation is based on the existence of the so-called “mirror manifolds”. B. Greene and R. Plesser [Nucl. Phys. B 338, 15-37 (1990)] proved the existence of mirror manifolds for a particular case of Calabi-Yau threefolds and this class contains the general quintic.
The paper under review contains the first complete proof of the generating formula for the numbers \(n_d\) and the authors say that their “mirror principle” can be applied to compute other multiplicative equivariant characteristic classes on stable map moduli space. They announce the now published paper: “Mirror principle. II” [Asian J. Math. 3, No. 1, 109-146 (1999)] which contains the case of toric varieties and homogeneous manifolds.

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles
14J30 \(3\)-folds
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