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Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard- Fuchs equations. (English) Zbl 0789.14005
This paper computes the Yukawa-potential \(\kappa^{(\lambda)}_{sss}= \int_{W_ \lambda} \omega \wedge {d^ 3 \omega \over d \lambda^ 3}\) associated to the family of mirrors \((W_ \lambda)_{\lambda \in \mathbb{C}}\) of the Calabi-Yau complete intersections \(V_ \lambda \subset \mathbb{P}^ 5_ \mathbb{C}\) of two cubic equations: \(x^ 3_ 1+ x^ 3_ 2+ x^ 3_ 3-\lambda x_ 4x_ 5 x_ 6=0=x^ 3_ 4+ x^ 3_ 5+x^ 3_ 6-\lambda x_ 1x_ 2x_ 3\). In this case the \(W_ \lambda\) are the smooth models of quotients of \(V_ \lambda\) by a certain abelian subgroup \(G\) of order 81 of \(\mathbb{P}\text{Gl}(5,\mathbb{C})\).
The Picard-Fuchs equation of the family is determined: it is a generalized hypergeometric equation with parameters \(({1\over 3},{1\over 3},{2\over 3},{2\over 3})\) (relative to \(z= \lambda^{-6})\). The solutions of that equation then determine \(\kappa_{sss}\). Putting \(s(z)=(F_ 1/F_ 0)(z)\) and \(q=\exp(s(z))\), where \(F_ 0\) and \(F_ 1\) are appropriate solutions, one also has: \(\kappa_{sss}=9+ \sum (n_ dd^ 3q^ d/(1-q^ d))\).
It is predicted that \(n_ d\) should be the number of rational curves of degree \(d\) on a generic deformation of \(V_ \lambda\). This is checked here for \(d=1\). Moreover, the numbers \(n_ d\) \((d\leq 10)\) are also computed in the case of complete intersections in \(V_{a,b,c,d} \subset \mathbb{P}^{N-1}_{(\mathbb{C})}\), where \(N= a+b+c+d\), for \((a,b,c,d)=(1,1,2,4)\), (1,2,2,3) and (2,2,2,2) assuming that the Picard- Fuchs equation is still hypergeometric of appropriate parameter in that cases.

MSC:
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14M10 Complete intersections
14J35 \(4\)-folds
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