Classifying Calabi–Yau Threefolds Using Infinite Distance Limits

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 Kähler moduli space and can be classified by an associated limiting mixed Hodge structure. We then argue that 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.