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Log Calabi-Yau pairs of birational complexity zero

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arxiv 2404.05878 v1 pith:YRPD67V6 submitted 2024-04-08 math.AG

Joshua Enwright , Fernando Figueroa , Joaquín Moraga This is my paper · ORCID
classification math.AG
keywords pairsopenbirationalcalabi-yaucomplexitysubvarietieszeroalgebraic
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abstract

In this article, we study the geometry of log Calabi-Yau pairs $(X,B)$ of index one and birational complexity zero. Firstly, we propose a conjecture that characterizes such pairs $(X,B)$ in terms of their dual complex and the rationality of their log canonical places. Secondly, we show that for these pairs the open set $X\setminus B$ is divisorially covered by open affine subvarieties which are isomorphic to open subvarieties of algebraic tori. We introduce and study invariants that measure the geometry and the number of these open subvarieties of algebraic tori. Thirdly, we study boundedness properties of log Calabi-Yau pairs of index one and birational complexity zero. For instance, in dimension $2$ we prove that such pairs are affinely bounded.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Bott-Chern complexity of K\"ahler pairs

    math.AG 2025-05 accept novelty 7.0 of 10

    The Bott-Chern complexity of a non-projective Calabi-Yau Kähler pair is at least 3, and this bound is attained by singular non-projective K3 surfaces with Picard rank zero.

  2. Complexity one varieties are cluster type

    math.AG 2025-04 conditional novelty 7.0 of 10

    A log Calabi-Yau pair of index one and complexity at most one is a cluster type variety, and varieties of absolute complexity one admit a finite cluster type cover of degree at most two.

  3. Calabi-Yau pairs of complexity two

    math.AG 2024-12 conditional novelty 7.0 of 10

    A Calabi-Yau pair of complexity two is cluster type exactly when its standard model over a toric base is nodal, component-compatible, and volume-large; this classifies all rank-one Gorenstein del Pezzo surfaces.

  4. Geometry of tropical mutation surfaces with a single mutation

    math.AG 2025-10 conditional novelty 6.0 of 10

    A projective log Calabi–Yau surface with reduced boundary, ample boundary support and a G_m-action is exactly a tropical mutation surface with a single shear, and its complexity equals the number of distinct roots of f.

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