In this talk, I will discuss the newly-emerged concepts of higher Du Bois and higher rational singularities (due to Mustata-Popa, M. Saito, Friedman-Laza, and their collaborators). These are generalizations of the standard notions of Du Bois and rational singularities respectively. They should be understood as local conditions on the singularities that guarantee good behavior for the cohomology of families of projective complex algebraic varieties. Namely, higher Du Bois singularities guarantee the preservation of Hodge numbers in a certain range. Higher rational singularities are also higher Du Bois, and give additionally some partial Hodge symmetries.
I will then discuss applications of these concepts to the deformation theory of singular Calabi-Yau n-folds. Specifically, I will discuss a generalization of a theorem of Kawamata, Ran, and Tian on the unobstructedness of nodal Calabi-Yau varieties. In a different direction, I will discuss generalizations of the smoothability results of Friedman, Namikawa, Namikawa-Steenbrink from dimension 3 to higher dimensions.

I will close by relating all these results to some conjectures of Odaka on the global geometry of the moduli space of Calabi-Yau’s.

This is joint work with Robert Friedman. Parts of it are also joint with Matt Kerr and Morihiko Saito.