2025/04/24 by Joshua Enwright, Enwright, Joshua, Jennifer Li +4
Computer Science · Mathematics · #14E30 (Primary) 14M25 (Secondary) #Advanced Algebra and Logic #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #FOS: Mathematics #Rings, Modules, and Algebras #math.AG #msc:14E30 #msc:14M25
paper · pdf · doi:10.48550/arxiv.2504.17369
29 pages v2: Improved presentation and fixed minor mistakes/typos
openalex publication_date 2025/04/24 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28 · arxiv created 2026/08/03 · arxiv updated 2026/08/04
The complexity of a Calabi-Yau pair (X,B) is an invariant that relates the dimension of X, the rank of the group of divisors, and the coefficients of B. If the complexity is less than one, then X is a toric variety. We prove that if the complexity is less than two, then X is a Fano type variety. Furthermore, if the complexity is less than 3/2, then X admits a Calabi-Yau structure of complexity one and index at most two, and it admits a finite cover Y → X of degree at most 2, where Y is a cluster type variety. In particular, if the complexity is one and the index is one, (X,B) is cluster type. Finally, we establish a connection with the theory of T-varieties. We prove that a variety of T-complexity one admits a similar finite cover from a cluster type variety.