Algebraic Geometry Seminar

The complexity of a Calabi-Yau pair (X,B) is an invariant that relates the dimension of X, the Picard rank of X, and the coefficients of B. It was proven by Brown, McKernan, Svaldi and Zong that the complexity of a Calabi-Yau pair is nonnegative, and a variety X admits a Calabi-Yau pair of complexity 0 if and only if X is toric. In this talk we will discuss the geometry of Calabi-Yau pairs of index one and complexity one or two. Both descriptions are done in terms of cluster type varieties, a generalization of toric varieties. In the case of complexity one, we prove that (X,B) is of cluster type. In the case of complexity two, we give a criterion to decide whether the pair is cluster type or not. This is joint work with Joshua Enwright, Jennifer Li and Joaquin Moraga.