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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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