Finiteness of Calabi–Yau topological types in any complex dimension

Establish whether, for every complex dimension n, the set of distinct topological types of Calabi–Yau n-folds is finite, thereby determining a finiteness classification for Calabi–Yau manifolds across all dimensions.

Background

The paper studies distinguishing topological data of Complete Intersection Calabi–Yau (CICY) threefolds using machine learning, focusing on invariants such as triple intersection numbers and the second Chern class. In Kähler geometry, the topological type of a compact Kähler threefold is determined by Hodge numbers, triple intersection numbers, and the first Pontrjagin class (equivalently related to Chern classes for Calabi–Yau threefolds), and for simply connected Calabi–Yau threefolds this reduces to a finite list of integer invariants.

Within this context, the authors highlight a broader conjectural question attributed to Shing-Tung Yau concerning the global classification landscape: whether the number of possible topological types of Calabi–Yau n-folds is finite in any complex dimension. Resolving this conjecture would provide a foundational understanding of the scope of Calabi–Yau topology beyond the specific threefold case studied in the paper.

References

It is a standing conjecture of one of the authors (STY), that in any complex dimension n, the possible topological types of a Calabi-Yau n-fold is finite.

— Distinguishing Calabi-Yau Topology using Machine Learning  (2408.05076 - He et al., 2024) in Section 1, Introduction

A famous folklore conjecture of Yau asserts that there are only finitely many diffeomorphism types of Kähler Calabi-Yau threefolds (cf. e.g. ). In fact this is conjectured to hold in all dimensions.

— Topology of low-dimensional generalized Ricci solitons and string backgrounds  (2608.25829 - Streets, 26 Aug 2026) in Introduction, Remark item 4