The well-known Bogomolov-Tian-Todorov theorem says that the deformations of Calabi-Yau manifolds are unobstructed. The unobstructedness of deformations continues to hold Calabi-Yau varieties with ordinary nodal singularities (Kawamata, Ran, Tian), but surprisingly the smoothability of such varieties is subject to topological constrains. These obstructions to the existence of smoothings are linear in dimension 3 (Friedman), and non-linear in higher dimensions (Rollenske-Thomas).

In this talk, I will give vast generalizations to both the unobstructedness of deformations for mildly singular Calabi-Yau varieties, and to the constraints on the existence of smoothings for certain classes of singular Calabi-Yau varieties. Additionally, I will establish the proper context for these results: the Hodge theory of degenerations with prescribed singularities (specifically higher rational/higher Du Bois and liminal singularities).

This is joint work with Robert Friedman.