Fibration structure in toric hypersurface Calabi-Yau threefolds

In previous work, we have commenced the task of unpacking the $473,800,776$ reflexive polyhedra by Kreuzer and Skarke into a database of Calabi-Yau threefolds (see http://www.rossealtman.com). In this paper, following a pedagogical introduction, we present a new algorithm to isolate Swiss cheese solutions characterized by "holes," or small 4-cycles, descending from the toric divisors inherent to the original four dimensional reflexive polyhedra. Implementing these methods, we...

We argue that the existence of genus one fibrations with multisections of high degree on certain Calabi-Yau threefolds implies the existence of pairs of such varieties that are not birational, but are derived equivalent. It also (likely) implies the existence of non-birational counterexamples to the Torelli problem for Calabi-Yau threefolds.

The Hodge numbers of generic elliptically fibered Calabi-Yau threefolds over toric base surfaces fill out the "shield" structure previously identified by Kreuzer and Skarke. The connectivity structure of these spaces and bounds on the Hodge numbers are illuminated by considerations from F-theory and the minimal model program. In particular, there is a rigorous bound on the Hodge number h_{21} <= 491 for any elliptically fibered Calabi-Yau threefold. The threefolds with the la...

The diffeomorphism class of simply-connected smooth Calabi-Yau threefolds with torsion-free cohomology is determined via certain basic topological invariants: the Hodge numbers, the triple intersection form, and the second Chern class. In the present paper, we shed some light on this classification by placing bounds on the number of diffeomorphism classes present in the set of smooth Calabi-Yau threefolds constructed from the Kreuzer-Skarke list of reflexive polytopes up to P...

We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories....

In this work we systematically enumerate genus one fibrations in the class of 7,890 Calabi-Yau manifolds defined as complete intersections in products of projective spaces, the so-called CICY threefolds. This survey is independent of the description of the manifolds and improves upon past approaches that probed only a particular algebraic form of the threefolds (i.e. searches for "obvious" genus one fibrations as in [1,2]). We also study K3-fibrations and nested fibration str...

Using an inclusion of one reflexive polytope into another is a well-known strategy for connecting the moduli spaces of two Calabi-Yau families. In this paper we look at the question of when an inclusion of reflexive polytopes determines a torically-defined extremal transition between smooth Calabi-Yau hypersurface families. We show this is always possible for reflexive polytopes in dimensions two and three. However, in dimension four and higher, obstructions can occur. This l...

Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi-Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of...

After a brief introduction into the use of Calabi--Yau varieties in string dualities, and the role of toric geometry in that context, we review the classification of toric Calabi-Yau hypersurfaces and present some results on complete intersections. While no proof of the existence of a finite bound on the Hodge numbers is known, all new data stay inside the familiar range $h_{11}+h_{12}\le 502$.

We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface $B$ that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface $\tilde{B}$ that is related through toric geometry to the line bundle $-6K_B$. The Kreuzer-Skarke database in...