Finiteness, Boundedness, and Constructibility for Calabi-Yau Varieties

Grothendieck’s foundational work shows that many geometric properties of fibers in a projective family, for example smoothness, define constructible subsets of the base. However, one natural property is absent from this list: given a fiber, is the locus of points whose fibers are isomorphic to it constructible? Surprisingly, this question is closely related to the Minimal Model Program and the Kawamata-Morrison cone conjecture, a longstanding central problem in birational geometry with connections to mathematical physics. In this talk, I will explain the relationship between the cone conjecture and this constructibility problem and show how it leads to finiteness results for Calabi-Yau varieties within a fixed birational class.