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arXiv · 2506.14118

Examples of complete Calabi--Yau metrics on affine smoothings of irregular toric Calabi--Yau cones

Ronan J. Conlon·Tran-Trung Nghiem
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Abstract

We present infinitely many new examples of affine Calabi--Yau manifolds of Euclidean volume growth and quadratic curvature decay, whose tangent cones at infinity are irregular and have smooth links. In the process, we demonstrate (and provide the relevant computer code) how to explicitly compute the Reeb field, the Minkowski summand cone, and its maximal decompositions for a given toric Calabi--Yau cone with smooth link from the data of its toric diagram. In complex dimension three, the component corresponding to a maximal Minkowski decomposition into lattice summands is a smoothing component if and only if every summand is either a primitive lattice segment or a unimodular lattice triangle. Furthermore, we propose an effective strategy to generate smoothable Calabi--Yau cones from a given non-smoothable one by taking Minkowski sums of certain toric diagrams, and provide an example to illustrate the method.

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Ronan J. Conlon, Tran-Trung Nghiem. 2026-09-17. Examples of complete Calabi--Yau metrics on affine smoothings of irregular toric Calabi--Yau cones. https://arxiv.org/abs/2506.14118

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