Yau’s finiteness conjecture for topological types of Calabi–Yau manifolds

Prove that in every complex dimension, the number of distinct topological types of smooth, compact Calabi–Yau manifolds is finite, where topological type is characterized by invariants such as Betti numbers, Hodge numbers, and intersection numbers.

Background

The paper contrasts the classification behaviors across the trichotomy of complex manifolds with positive curvature (Fano), zero curvature (Calabi–Yau), and negative curvature (general type). Fano varieties have a finite number of topological types in each dimension, while varieties of general type are known to be infinite and unclassifiable.

Within this framework, Yau’s conjecture asserts finiteness for Calabi–Yau manifolds in every dimension. The conjecture is verified in complex dimensions one and two—there is only the 2-torus in dimension one, and only the 4-torus and K3 surface in dimension two. In complex dimension three and above, despite large numbers of explicit constructions, finiteness is still not established. This conjecture is central to both mathematical classification efforts and to understanding the structure of the string theory landscape.