Calabi-Yau threefolds with large h^{2, 1}

TL;DR

This paper systematically classifies elliptically fibered Calabi-Yau threefolds with large h^{2,1} values, identifying all known and some new examples with h^{2,1} >= 350, and discusses the challenges in fully enumerating such geometries.

Contribution

It provides a complete classification of EFS Calabi-Yau threefolds with h^{2,1} >= 350, including new examples, and analyzes the limitations of current enumeration methods.

Findings

Identified all EFS Calabi-Yau threefolds with h^{2,1} >= 350.

Discovered three new Calabi-Yau threefolds with large Hodge numbers.

Established bounds suggesting no higher Hodge number EFS Calabi-Yau threefolds exist.

Abstract

We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section ("EFS") and have a large Hodge number h^{2, 1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have h^{2, 1} >= 350 by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefolds with h^{2, 1} >= 350, as well as three apparently new Calabi-Yau threefolds. We speculate that there are no other Calabi-Yau threefolds (elliptically fibered or not) with Hodge numbers that exceed this bound. We summarize the theoretical and practical obstacles to a complete enumeration of all possible EFS Calabi-Yau threefolds and fourfolds, including those with small Hodge numbers, using this approach.