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arXiv · 2504.17369

Complexity one varieties are cluster type

Joshua Enwright·Jennifer Li·José Ignacio Yáñez
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Abstract

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 $Y \to X$ 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.

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Joshua Enwright, Jennifer Li, José Ignacio Yáñez. 2026-08-03. Complexity one varieties are cluster type. https://arxiv.org/abs/2504.17369

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