On the Hodge structure of elliptically fibered Calabi-Yau threefolds

TL;DR

The paper investigates the Hodge structure of generic elliptically fibered Calabi–Yau threefolds over toric bases, revealing that their (h_{11},h_{21}) pairs populate the familiar shield region identified by Kreuzer–Skarke and establishing a rigorous bound h_{21} ≤ 491. By exploiting base geometry through clusters and F-theory anomaly constraints, it computes Hodge numbers from base data and shows the 61{,}539 toric bases form a connected network via blowups, with h_{11} increasing and h_{21} decreasing along blowups unless gauge content expands. It also demonstrates that tuning Weierstrass moduli over these bases yields many additional CYs with larger h_{11} and smaller h_{21}, including explicit SU(N) families on F_m and P^2, many of which appear in Kreuzer–Skarke, and discusses non-toric bases and extensions to non-elliptic fibrations and Calabi–Yau fourfolds. The results suggest that the shield boundary may be universal across a broad class of Calabi–Yau manifolds and provide a framework for systematic exploration of tunings, mirror relationships, and higher-dimensional analogs in F-theory and birational geometry.

Abstract

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 largest known Hodge numbers are associated with a sequence of blow-ups of toric bases beginning with the Hirzebruch surface F_{12} and ending with the toric base for the F-theory model with largest known gauge group.

Figures (4)

• Figure 1: The 30,108 distinct Hodge numbers of the 473.8 million Calabi-Yau threefolds identified by Kreuzer and Skarke. Data from ks-data.
• Figure 2: The 7524 distinct Hodge number pairs for generic elliptically fibered Calabi-Yau threefolds over toric bases (dark/blue data points). Plot axes are Hodge numbers $h_{11}, h_{21}$. Kreuzer-Skarke Hodge pairs are shown in background in light gray for comparison.
• Figure 3: The "tip" of the shield region containing known Calabi-Yau threefold Hodge numbers with large $h_{21}$. Large (blue) dots represent generic threefolds over toric bases, small (gray) dots are from Kreuzer-Skarke list. Dots connected with a solid line represent the line of toric bases connected by blow-up transitions closest to the shield boundary. Dotted line and mid-size (red) dot represents a threefold realized by tuning a Weierstrass model over the toric base corresponding to Hodge numbers (13, 433). (Many other Hodge pairs from the Kreuzer-Skarke list can be realized in a similar fashion by tuning Weierstrass moduli, though only one example is depicted.)
• Figure 4: The region around the central point of the "shield" region of known allowed Hodge numbers for Calabi-Yau threefolds. Large (blue) dots represent generic threefolds over toric bases, small (gray) dots are from Kreuzer-Skarke list. Dots connected with a solid line represent the line of toric bases connected by blow-up transitions closest to the shield boundary.