Classifying Calabi-Yau threefolds using infinite distance limits

TL;DR

This paper introduces a new classification method for Calabi-Yau threefolds based on their infinite volume limits and associated degeneration patterns, providing a novel invariant and graphical tools for analysis.

Contribution

It develops a systematic classification of Calabi-Yau threefolds using infinite distance limits and degeneration patterns, offering new invariants and graphical representations.

Findings

Classifies Calabi-Yau threefolds via infinite volume limits and mixed Hodge structures.

Introduces Hasse diagrams to analyze decompactification limits and fibrations.

Applies the method to hypersurfaces in toric spaces and complete intersections.

Abstract

We present a novel way to classify Calabi-Yau threefolds by systematically studying their infinite volume limits. Each such limit is at infinite distance in Kahler moduli space and can be classified by an associated limiting mixed Hodge structure. We then argue that the such structures are labeled by a finite number of degeneration types that combine into a characteristic degeneration pattern associated to the underlying Calabi-Yau threefold. These patterns provide a new invariant way to present crucial information encoded in the intersection numbers of Calabi-Yau threefolds. For each pattern, we also introduce a Hasse diagram with vertices representing each, possibly multi-parameter, decompactification limit and explain how to read off properties of the Calabi-Yau manifold from this graphical representation. In particular, we show how it can be used to count elliptic, K3, and nested fibrations and determine relations of elliptic fibrations under birational equivalence. We exemplify this for hypersurfaces in toric ambient spaces as well as for complete intersections in products of projective spaces.