An effective way of constructing asymptotically conical Calabi--Yau manifolds

Abstract: I will present an effective strategy to construct non-compact Calabi--Yau manifolds via the purely combinatorial method of summing two polygons. From toric geometry, giving a three-dimensional toric Calabi--Yau cone is equivalent to specifying a special type of lattice polygon, called a toric diagram. The affine smoothing of such a cone, if exists, has a Calabi--Yau metric, and the metric is moreover asymptotic to the toric Calabi--Yau cone in the sense that will be explained during the talk. On the other hand, Altmann's theory establishes a correspondence between  smoothing families of a toric Calabi--Yau cone and maximal Minkowski decompositions of the toric diagram into lattice summands. Namely, the toric Calabi--Yau is smoothable if and only if its toric diagram has a Minkowski decomposition into sums of lattice segments and/or triangles. I will explain how the process of adding a lattice segment to the toric diagram of a non-smoothable toric Calabi--Yau cone can give rise to a smoothable one, hence to new asymptotically conical Calabi--Yau metrics. The talk will be mainly example-based and technical terms will be explained. This is joint work with Ronan Conlon.