We formulate general conjectures about the relationship between the A-model connection on the cohomology of a d-dimensional Calabi-Yau complete intersection V of r hypersurfaces V 1 ,. .. , V r in a toric variety P Σ and the system of differential operators annihilating the special generalized hypergeometric function Φ 0 depending on the fan Σ. In this… (More)
The main part of this paper is a big table (see Appendix A) containing what we believe to be a complete list of all fourth order equations of Calabi–Yau type known so far. In the text preceding the tables we explain what a differential equation of Calabi–Yau type is and we briefly discuss how we found these equations. We also describe an electronic version… (More)
In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural de-generation of Grassmannians G(k, n) to some Gorenstein toric Fano varieties P (k,… (More)
We provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of generalized hypergeometric equations of order 4 and 5. As application, we present several examples of algebraic transformations of Calabi– Yau differential equations.
We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.
The concept of Apéry limit for second and third order differential equations is extended to fourth and fifth order equations, mainly of Calabi–Yau type. For those equations obtained from Hadamard products of second and third order equations we can prove that the limits are determined in terms of the factors by a certain formula. Otherwise the limits are… (More)
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F
Euler number of the compactified Jacobian and multiplicity of rational curves. Abstract In this paper we show that the Euler number of the compactified Jacobian of a rational curve C with locally planar singularities is equal to the multiplicity of the δ-constant stratum in the base of a semi-universal deformation of C. In particular, the multiplicity… (More)
Let A be a principally polarized abelian variety of dimension four and let Θ ⊂ A be a symmetric theta-divisor, which we assume to be smooth. Using the Hodge structure on H 3 (Θ) we associate to A two abelian subvarieties J(K) ⊂ J(H) of the intermediate jacobian J(Θ) of Θ of dimensions five and nine respectively. We show that J(H) is generated by the image… (More)