Updates on Calabi-Yau manifolds from pairs of non-compact Calabi-Yau manifolds

Abstract

We discuss developments in the construction of Calabi--Yau manifolds by smoothing normal crossing unions of quasi-Fano manifolds, following the introduction of this method to the physics community in 2010. We describe constructions of Calabi--Yau threefolds with unbounded second Betti numbers and with very small Hodge numbers, as well as the first non-K\"ahler examples in dimensions greater than three. The role of Landau--Ginzburg models in mirror constructions is also explained.