Searching for K3 Fibrations

TL;DR

The paper addresses identifying K3 fibrations in Calabi–Yau hypersurfaces within toric varieties defined by single weight systems. It develops two complementary toric-geometric methods and applies them to a large catalog of 184,026 spaces, yielding 124,701 K3 fibrations (and 167,406 total fibrations when multiple types are counted). It extends the analysis to elliptic fibrations and K3 surfaces and significantly expands the Calabi–Yau Hodge-number catalog, revealing over threefold more spectra than previously known and highlighting a lack of mirror symmetry in this single-weight setting. The approach relies on reflexive polyhedra, projection criteria, and maximal Newton polyhedra to enable scalable fibrations detection, with implications for heterotic–Type II duality studies and broader Calabi–Yau landscape explorations in toric geometry.

Abstract

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 obtaining more than three times as many as were previously known.

Figures (4)

• Figure 1: A dual pair of reflexive polygons
• Figure 2: The dual polytope $\Delta^*$ of Example 1
• Figure 3: The dual polytope $\Delta^*_3$ of Example 2
• Figure 4: $b_{11}+b_{21}\; vs\; \chi_E \;$ for single weight systems