An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts

TL;DR

This paper explores the rich structure of Calabi-Yau threefolds with K3 fibrations, revealing how reflexive polytopes and their subdivisions explain patterns in their Hodge numbers and mirror symmetry.

Contribution

It introduces a novel geometric construction using reflexive polytopes with interchangeable parts to explain the abundance of K3 fibrations in Calabi-Yau threefolds.

Findings

Patterns in the Hodge plot correspond to webs of elliptic-K3 fibrations.

Reflexive polytopes can be decomposed and recombined along slices to generate new fibrations.

Additivity of Hodge numbers under polytope subdivision explains observed structures.

Abstract

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 fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.