Fibration structure in toric hypersurface Calabi-Yau threefolds

We present two methods for studying fibrations of Calabi-Yau manifolds embedded in toric varieties described by single weight systems. We analyse 184,026 such spaces and identify among them 124,701 which are K3 fibrations. As some of the weights give rise to two or three distinct types of fibrations, the total number we find is 167,406. With our methods one can also study elliptic fibrations of 3-folds and K3 surfaces. We also calculate the Hodge numbers of the 3-folds obtain...

We present an algorithm for efficiently exploring inequivalent Calabi-Yau threefold hypersurfaces in toric varieties. A direct enumeration of fine, regular, star triangulations (FRSTs) of polytopes in the Kreuzer-Skarke database is foreseeably impossible due to the large count of distinct FRSTs. Moreover, such an enumeration is needlessly redundant because many such triangulations have the same restrictions to 2-faces and hence, by Wall's theorem, lead to equivalent Calabi-Ya...

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic-K3 fibrations whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fa...

We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of "tops" in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs ($h^{1, 1},h^{2, 1}$) with $h^{1,1...

The number of rational points in toric data are given for two-parameter Calabi-Yau $n$-folds as toric hypersurfaces over finite fields $\mathbb F_p$ . We find that the fundamental period is equal to the number of rational points of the Calabi-Yau $n$-folds in zeroth order $p$-adic expansion. By analyzing the solution set of the GKZ-system given by the enhanced polyhedron, we deduce that under type II/F-theory duality the 3D and 4D Calabi-Yau manifolds have the same number of ...

One may construct a large class of Calabi-Yau varieties by taking anticanonical hypersurfaces in toric varieties obtained from reflexive polytopes. If the intersection of a reflexive polytope with a hyperplane through the origin yields a lower-dimensional reflexive polytope, then the corresponding Calabi-Yau varieties are fibered by lower-dimensional Calabi-Yau varieties. A top generalizes the idea of splitting a reflexive polytope into two pieces. In contrast to the classifi...

This paper is a continuation of our paper math.AG/0205321 where we have built a combinatorial model for the torus fibrations of Calabi-Yau toric hypersurfaces. This part addresses the connection between the model torus fibration and the complex and K\"ahler geometry of the hypersurfaces.

In this expository note, we review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice of the Picard lattice. We then show how to combine information about the Picard group of a toric ambient space with data about automorphisms of the toric variety to identify families of K3 surfaces with high Picard rank.

Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. ...

We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a special Lagrangian torus fibration in the Strominger-Yau-Zaslow conjecture. We study the topological torus fibration in the large complex structure limit and show that it coincides with our combinatorial model.