On the Classification of Reflexive Polyhedra

We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedr...

We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase.

It has been shown by Batyrev and Borisov that nef partitions of reflexive polyhedra can be used to construct mirror pairs of complete intersection Calabi--Yau manifolds in toric ambient spaces. We construct a number of such spaces and compute their cohomological data. We also discuss the relation of our results to complete intersections in weighted projective spaces and try to recover them as special cases of the toric construction. As compared to hypersurfaces, codimension t...

We present an inductive algebraic approach to the systematic construction and classification of generalized Calabi-Yau (CY) manifolds in different numbers of complex dimensions, based on Batyrev's formulation of CY manifolds as toric varieties in weighted complex projective spaces associated with reflexive polyhedra. We show how the allowed weight vectors in lower dimensions may be extended to higher dimensions, emphasizing the roles of projection and intersection in their du...

We describe the C program mori.x. It is part of PALP, a package for analyzing lattice polytopes. Its main purpose is the construction and analysis of three--dimensional smooth Calabi--Yau hypersurfaces in toric varieties. The ambient toric varieties are given in terms of fans over the facets of reflexive lattice polytopes. The program performs crepant star triangulations of reflexive polytopes and determines the Mori cones of the resulting toric varieties. Furthermore, it com...

We extend the construction of Calabi-Yau manifolds to hypersurfaces in non-Fano toric varieties, requiring the use of certain Laurent defining polynomials, and explore the phases of the corresponding gauged linear sigma models. The associated non-reflexive and non-convex polytopes provide a generalization of Batyrev's original work, allowing us to construct novel pairs of mirror models. We showcase our proposal for this generalization by examining Calabi-Yau hypersurfaces in ...

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric...

Generalizing the notions of reflexive polytopes and nef-partitions of Batyrev and Borisov, we propose a mirror symmetry construction for Calabi-Yau complete intersections in Fano toric varieties.

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi--Yau 4-folds. We find 914,164 weights with degree $d\le150$ whose maximal Newton polyhedra are reflexive and 525,572 weights with degree $d\le4000$ that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrev's formulas (de...

We present a universal normal algebra suitable for constructing and classifying Calabi-Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a `dual' construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi-Yau spaces ...