An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts

TL;DR

This paper investigates the structure of the Hodge plot for Calabi–Yau threefolds arising as toric hypersurfaces and shows that the intricate patterns reflect nested webs of elliptic–K3 fibrations whose mirrors are also K3-fibered. By exploiting reflexive polytopes that admit a K3 slice, the authors develop a framework where a top and bottom partition a 4D polytope, and compatible halves glue to form a reflexive polytope; they prove an additivity relation for the Hodge numbers $(h^{1,1},h^{1,2})$ under suitable conditions. They explicitly construct large catalogs of elliptic–K3 fibrations for self-dual and non-self-dual K3 polyhedra, notably $E_8\times\{1\}$ and $E_7\times SU(2)$ families, counting hundreds of thousands of 4-polytopes and thousands of distinct Hodge-pair combinations, while showing how Hodge data transfer via translation vectors recovers the observed grid-like structure. The results provide a principled, polytope-based explanation for the nested, fractal-like webs in the Hodge plots and open avenues to discover additional webs via systematic polytopal classifications and dualities.

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.

Figures (15)

• Figure 1: The Hodge plot for the list or reflexive 4-polytopes. The Euler number $\chi = 2\left( h^{1,1}-h^{1,2} \right)$ is plotted against the height $y=h^{1,1}+h^{1,2}$. The oblique axes correspond to $h^{1,1}=0$ and $h^{1,2}=0$.
• Figure 2: The structure in red, on the left, contains points which have mirror images when reflected about the axis $\chi = -480$. The red arrows show that these points can be mapped into points corresponding to manifolds with positive Euler number by a change in Hodge numbers $\Delta(h^{1,1},h^{1,2} )=(240,-240)$, corresponding to $\Delta(\chi, y)=(960,0)$.
• Figure 3: The blue points can be translated into other points of the plot by a change in Hodge numbers $\Delta(h^{1,1},h^{1,2} )=(1,-29)$. The red arrows illustrate this action on two pairs of points. The extra grey arrow corresponds to $h^{1,2} = 30$.
• Figure 4: The points in black are the only ones that have neither a left nor a right descendant.
• Figure 5: The 465 pairs of Hodge numbers that result from combining the 1263 possible $E_8{\times}\{1\}$ tops with the maximal bottom. Note that this structure is contained in, though is not identical to, the structure shown on the left in Fig. \ref{['YStructure']}. Exchanging the maximal bottom for the minimal bottom shifts the entire structure by the vector $\Delta(\chi,y)=(960,0)$, corresponding to the red arrows.