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

TL;DR

This paper constructs new examples of affine Calabi--Yau manifolds with irregular tangent cones at infinity, providing explicit computational methods and strategies for generating smoothable Calabi--Yau cones from toric data.

Contribution

It introduces explicit examples of affine Calabi--Yau manifolds with irregular tangent cones and develops computational tools and strategies for smoothing toric Calabi--Yau cones.

Findings

Examples of affine Calabi--Yau manifolds with irregular tangent cones.

Explicit computation methods for Reeb fields and Minkowski decompositions.

A strategy to generate smoothable Calabi--Yau cones from non-smoothable ones.

Abstract

We present 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 and all Minkowski decompositions of a given toric Calabi--Yau cone with smooth link from the data of its toric polytope. Minkowski decompositions of this polytope into lattice segments and/or triangles give rise to smoothings of the given cone. 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.