The Picard-fuchs Equation of a Family of Calabi-yau Threefolds without Maximal Unipotent Monodromy

Recently J.C. Rohde constructed families of Calabi-Yau threefolds parametrised by Shimura varieties. The points corresponding to threefolds with CM are dense in the Shimura variety and, moreover, the families do not have boundary points with maximal unipotent monodromy. Both aspects are of interest for Mirror Symmetry. In this paper we discuss one of Rohde’s examples in detail and we explicitly give the Picard-Fuchs equation for this one dimensional family. In this note we work out an example of J.C. Rohde of a one dimensional family of Calabi-Yau threefolds Xλ (with h (Xλ) = 1), parametrised by a Shimura variety, such that the associated Picard-Fuchs equation has no maximal unipotent monodromy. Actually, as already pointed out by Rohde, the Picard-Fuchs equation of degree four reduces to two differential equations of degree two. In section 2.5 we give these two equations, which are hypergeometric differential equations, explicitly. This is of some interest for Mirror Symmetry, which suggests that a family like the Xλ should be the Mirror of another family of Calabi-Yau threefolds Yμ. A particular solution of the Picard-Fuchs equation, chosen using the maximal unipotent monodromy, should have a Taylor expansion whose d-th coefficient is related to the number of rational curves of degree d on a general Yμ (more precisely, it is related to a certain Gromow-Witten invariant of Yμ). In the absence of maximal unipotent monodromy, the recipe for identifying the solution of the Picard-Fuchs equation should be modified. It might however be the case that the family Xλ is not the Mirror of any family of Calabi-Yau’s, which would also be of some interest. We follow the approach indicated by Rohde in [R]. The main point is a good understanding of a family of K3 surfaces which is used in the construction of the Xλ’s. This leads to an explicit description of H(Xλ,Q) in terms of H (Cλ,Q) for a genus two curve Cλ. The only thing to do then is to recall a classical result on the Picard-Fuchs equations of the Cλ. In section 2.4 we briefly comment on the Calabi-Yau threefolds Xλ with complex multiplication (CM), in particular we observe that these are related to elliptic curves with CM. It is conjectured that Calabi-Yau threefolds with CM are related to Rational Conformal Field Theories (RCFT) and these should be easier to understand then the more general Conformal Field Theories (cf. [GV]). 1. The one parameter family of Calabi-Yau threefolds.